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**[Пономарев И. Н.\*]{}\
*[\*Московский физико-технический институт]{}\
*[E-mail: <[email protected]>]{}****
title: Реализация быстродействующего табличного элемента управления для отображения записей базы данных
---
Введение. Постановка задачи {#введение.-постановка-задачи .unnumbered}
---------------------------
Классическая работа с полосой прокрутки таблицы базы данных предполагает поддержку двух ключевых операций: отображение записей, соответствующих положению бегунка полосы прокрутки, и переход к записи, заданной по комбинации ключевых полей.
Из-за большого числа записей их полная загрузка в оперативную память бывает недопустима, а перекладывание указанных операций на СУБД ведёт к серьёзным проблемам производительности. Операции типа select count(\*) и select ... offset ... являются медленными, т. к. предполагают перебор записей таблицы. В результате часто при работе с таблицами происходит либо отказ от использования таблицы (использование постраничного отображения), либо пользователю тем или иным способом предоставляется лишь иллюзия того, что с помощью бегунка полосы прокрутки можно быстро перейти к любой из записей.
Однако в некоторых случаях пользователю требуется именно классическая работа с бегунком, т. е. возможность прямо перейти с помощью полосы прокрутки к любой части таблицы. Алгоритм, обеспечивающий такую работу грида при сохранении быстродействия, мы рассматриваем в настоящей работе.
Ограничивающими условиями для нас будут следующие: 1) набор данных, отображаемых в таком гриде, может быть отсортирован не произвольным образом, а только лишь по индексированному набору полей, 2) если в составе индекса содержится строковое поле, алгоритм должен быть <<обучен>> правилам сопоставления базы данных (collation rules) для всех символов, которые могут встретиться в данном строковом поле.
<<Быстрые>> и <<медленные>> запросы к СУБД {#быстрые-и-медленные-запросы-к-субд .unnumbered}
------------------------------------------------------------------
Как уже говорилось, быстрыми запросами к СУБД являются запросы, не использующие переборы записей, т. е. такие, в которых получение записей производится поиском по индексу. При условии, что по полю `key` построен индекс, быстрыми являются следующие запросы:
1. Найти первую и последнюю запись в наборе данныx (запрос A):
`select ... order by key [desc] limit 1`
2. Найти $h$ первых записей с ключом, большим или равным данному значению $K$ (запрос B):
`select ... order by key where key >= K limit h`
Эти запросы можно обобщить на случай сортировки по набору из нескольких полей `order by key1, key2, key3...` при условии, что на этих полях построен составной индекс. Условие $k {\geqslant}K$ должно быть обобщено на логическое условие сравнения нескольких значений в лексикографическом порядке:
$$\label{eq:lex1}
\begin{split}
k_1 > K_1 \vee (k_1 = K_1 \wedge k_2 {\geqslant}K_2) \\
\vee (k_1 = K_1 \wedge k_2 = K_2 \wedge k_3 {\geqslant}K_3) \vee \ldots
\end{split}$$
На практике тонкость заключается в том, что запрос с выражением вида `where k1 > K1 or (k1 = K1 and k2 >= K2)` может не выполняться за быстрое время: например, выполнение такого запроса в СУБД PostgreSQL с анализом плана выполнения показывает, что система не в состоянии использовать имеющийся индекс и использует вместо этого сканирование таблицы (table scan). Причиной тому выражение `or` на верхнем уровне логического условия, которое плохо поддаётся оптимизации. Проблема решается заменой условия `where` на логически эквивалентную формуле формулу
$$\label{eq:lex2}
\begin{split}
k_1 {\geqslant}K_1 \wedge (k_1 > K_1 \vee \\(k_2 {\geqslant}K_2 \wedge (k_2 > K_2 \vee \ldots)))
\end{split}$$
При таком ограничивающем условии СУБД распознаёт возможность применения составного индекса.
Не являются быстрыми следующие запросы:
1. Подсчитать общее количество записей в наборе данных (запрос C):
`select count(*) ...`
2. Подсчитать число записей, предшествующих записи с ключом, большим или равным данному (запрос D):
`select count(*) ... where key < K`
В общем случае системе понадобится перечислить все попадающие в фильтр записи для их подсчёта, поэтому на эти запросы затрачивается время, пропорциональное общему количеству записей.
Основная идея нашего подхода заключается в том, чтобы в процедурах, требующих быстрого отклика для пользователя, использовать только быстрые запросы к СУБД.
Зависимость ключа и номера записи {#зависимость-ключа-и-номера-записи .unnumbered}
---------------------------------
Для начала допустим, что интересующий нас набор данных отсортирован по одному единственному целочисленному полю (далее это ограничение будет снято).
Сопоставим каждому положению бегунка полосы прокрутки целочисленное значение $\lambda$, обозначающее количество записей <<сверху>> от границы отображаемого окна, или, иначе говоря, число <<пропущенных перед началом вывода>> записей. Таким образом, при $\lambda = 0$ отображаются записи с самого начала, при $\lambda = 1$ записи отображаются, начиная со второй и т. д. Если в окно умещается $h$ записей, а всего в наборе $N$ записей, то значения полосы прокрутки должны изменяться в пределах от $0$ до $N-h$ включительно.
Теперь рассмотрим функцию $f(k) = \lambda$, сопоставляющую каждому ключу $k$ результат выполнения запроса D, т. е. число записей, предшествующих записи с ключом $k$. Обратная ей функция $f^{-1}(\lambda)$ обладает тем свойством, что подстановка её результата в качестве параметра в быстро выполняющийся запрос B возвратит набор строк, в котором записи будут начинаться с $\lambda + 1$-й по счёту. Таким образом, если есть возможность за малое время вычислять функцию $f$, то задача перехода к записи, имеющей ключ $k$, сводится к вызову запроса B, а также вычислению $\lambda = f(k)$ и выставлению полосы прокрутки в положение $\lambda$. Если есть возможность за малое время вычислять $f^{-1}$, то отображение записей, соответствующих заданному положению полосы прокрутки, сводится к вычислению $k = f(\lambda)$ и вызову запроса B с параметром $k$.
Функция $f$ целиком определяется данными в таблице, поэтому, чтобы точно восстановить взаимозависимость значений ключа и номера записи, необходимо и достаточно прочесть из базы все записи *через одну*: для корректного срабатывания запроса B в промежуточных точках можно считать, что $f^{-1}(2n+1) = f^{-1}(2n)+1$, $n = 0, 1\ldots$). На практике без чтения большого количества данных можно обойтись: как мы покажем, значения $f$ могут быть достаточно точно приближены при помощи кусочной интерполяции по относительно малому числу промежуточных точек.
Допустим, что при помощи запросов A и D нам стали известны минимальное и максимальное значения ключа $k_{\min}$ и $k_{\max}$, а также значение $\lambda_{\max}$ (равное, естественно, числу записей в таблице минус единица). Тот факт, что запрос D — <<медленный>>, не играет существенной роли, что будет видно из дальнейшего. Помимо значений в крайних точках $f(k_{\min}) = 0$ и $f(k_{\max}) = \lambda_{\max}$, про функцию $f$ нам известны следующие факты:
1. $f(k)$ монотонно растёт,
2. при увеличении $k$ на единицу, $\lambda$ увеличивается на единицу или не увеличивается, поэтому график функции $f$, кроме точки $(k_{\min}, 0)$, целиком лежит в параллелограмме $(k_{\min} + 1, 1)$, $(k_{\min}+\lambda_{\max}, \lambda_{\max})$, $(k_{\max}, \lambda_{\max})$, $(k_{\max} - \lambda_{\max} + 1, 1)$,
3. общее число возможных функций $f$ равно числу способов распределения $\lambda-1$ записей по $k_{\max} - k_{\min} - 1$ значениям ключа (позиции первой и последней записи фиксированы), т. е. $$\label{eq:totalfuncnum}F = \binom{k_{\max} - k_{\min} - 1}{\lambda_{\max} - 1}$$
4. число возможных функций $f$, которые в точке $k$ принимают значение $\lambda$, определяется произведением количества комбинаций записей с ключом, меньшим $k$, и большим или равным $k$:
$$\label{eq:funcnum}F_{k,\lambda}=\binom{k - k_{\min} - 1}{\lambda - 1}\binom{k_{\max} - (k - k_{\min})}{ \lambda_{\max} - \lambda}$$
Если каждый из возможных вариантов считать равновероятным, то вероятность того, что для заданного значения $k$ имеется ровно $\lambda$ записей с ключом, строго меньшим $k$, задаётся отношением $F_{k,\lambda}/F$, являющимся гипергеометрическим распределением вероятностей.
В случае $k=k_{\min}$ всегда $\lambda = 0$. На отрезке $k=(k_{\min} + 1)\ldots k_{\max}$ среднее значение $\lambda$ определяется формулой (см. напр. [@F2011]): $$\label{eq:meanlambda}
\overline{\lambda} = \frac{(\lambda_{\max} - 1)(k - k_{\min} - 1)}{k_{\max} - k_{\min} - 1} + 1$$
Дисперсия значения $\lambda$, по [@F2011], имеет форму перевернутой параболы с нулями на краях отрезка $(k_{\min} + 1)\ldots k_{\max}$ и максимумом посередине: $$\label{eq:variance}
\begin{split}
D_\lambda = \frac{(\lambda_{\max} - 1)(k - k_{\min} - 1)(k_{\max} - k)}{(k_{\max} - k_{\min} - 1)^2}\\
\times \frac{(k_{\max} - k_{\min} - \lambda_{\max})}{(k_{\max} - k_{\min} - 2)}
\end{split}$$
На рис. \[fig:combinations\] показаны границы возможных значений функции, её среднее значение для всех комбинаций, а также границы среднеквадратичного отклонения при $k_{\max} - k_{\min} = 60$, $\lambda_{\max} = 6$.
(0,7) – (0,0) – (70,0); at (70,0) [$k$]{}; at (0,7) [$\lambda$]{}; (0, 0) – (6, 6) – (60, 6) – (55, 1) – (1, 1); (1, 1) – (60, 6); (1,1) – (10,1) – (11,1.04) – (13,1.15) – (15,1.27) – (19,1.53) – (21,1.67) – (23,1.82) – (27,2.13) node\[below right\] [$\overline{\lambda} - \sqrt{D_\lambda}$]{} – (29,2.3) – (31,2.46) – (35,2.82) – (37,3) – (39,3.19) – (43,3.58) – (45,3.79) – (47,4) – (49,4.23) – (50,4.34) – (51,4.46) – (53,4.71) – (55,4.98) – (57,5.27) – (59,5.64) – (60,6) – (1, 1) – (2,1.36) – (4,1.73) – (6,2.02) – (8,2.29) – (10,2.54) – (11,2.66) – (13,2.89) – (15,3.1) – (19,3.52) – (21,3.72) – (23,3.91) – (27,4.27) – (29,4.45) – (31,4.62) node\[above left\] [$\overline{\lambda} + \sqrt{D_\lambda}$]{} – (35,4.95) – (37,5.1) – (39,5.25) – (43,5.54) – (45,5.67) – (47,5.79) – (49,5.91) – (50,5.96) – (51,6) – (60,6); (0,0) circle \[radius=1pt\]; (1,1) circle \[radius=1pt\]; (60,6) circle \[radius=1pt\];
(0,1) circle \[radius=1pt\]; (0,6) circle \[radius=1pt\]; (1,0) circle \[radius=1pt\]; (55,0) circle \[radius=1pt\]; (60,0) circle \[radius=1pt\];
(20, 4)node\[left\] [$\overline{\lambda}$]{} to\[out=0,in=140\] (36.5, 4); (60, 6) – (60,0)node\[below right\][$k_{\max}$]{}; (6, 6) – (0,6)node\[left\][$\lambda_{\max}$]{}; (1, 1) – (0,1)node\[left\][$1$]{}; (1, 1) – (1,0)node\[below right\][$k_{\min} + 1$]{}; (55, 1) – (55,0)node\[below left\][$k_{\max} - \lambda_{\max} + 1$]{};
Обратная к функция, в соответствии с [@F2011], даёт несмещенную оценку с минимальной дисперсией для значения $k$:
$$\label{eq:meank}
k = \frac{(\lambda - 1)(k_{\max} - k_{\min} - 1)}{\lambda_{\max} - 1} +k_{\min} + 1$$
Формулы и мы примем в качестве оценки $f$ и $f^{-1}$ для $k=(k_{\min} + 1)\ldots k_{\max}$, а в точке $k_{\min}$ заведомо $f = 0$.
Если в дальнейшем для какого-то $k'$, $k_{\min} < k' < k_{\max}$ мы узнаём новое точное значение $0 < \lambda' < \lambda_{\max}$, мы можем добавить пару $(k', \lambda')$ к интерполяционной таблице и получить уточнённый расчёт значения $k(\lambda)$ для решения задачи прокрутки, а применяя обратную интерполяцию, вычислять уточнённое значение $\lambda(k)$ при решении задачи позиционирования. Разумеется, все эти операции можно реализовать так, чтобы они работали за время, логарифмическое по количеству точек в интерполяционной таблице.
Использование кусочно-линейной интерполяции для поиска записей в таблице лежит в основе алгоритма интерполяционного поиска, исследованного, например, в [@P1978]. В частности, там приводится оценка сверху для средней ошибки значения $\lambda$ (менее чем $\frac{1}{2}\sqrt{\lambda_{\max}}$), которую легко вывести из .
Обобщение на практически встречающиеся случаи {#обобщение-на-практически-встречающиеся-случаи .unnumbered}
---------------------------------------------
На практике дело не ограничивается единственным целочисленным полем для сортировки набора данных. Во-первых, тип данных может быть другим (строка, дата,…) Во-вторых, сортируемых полей может быть несколько. Это затруднение устраняется, если мы умеем вычислять
1. функцию-нумератор $g(K_1,\ldots K_n)=\kappa$, переводящую набор значений полей произвольных типов в натуральное число,
2. обратную ей функцию $g^{-1}(\kappa)=(K_1,\ldots K_n)$, переводящую натуральное число обратно в набор значений полей, $g^{-1}g(K_1,\ldots K_n)=(K_1,\ldots K_n)$.
Функция-нумератор должна обладать тем свойством, что если набор $(K_1,\ldots K_n)$ меньше набора $(K'_1,\ldots K'_n)$ в лексикографическом смысле (см. формулы и ), то должно быть $$g(K_1,\ldots K_n) < g(K'_1,\ldots K'_n).$$
Для представления значений $\kappa$ не подходят стандартные 32- и 64-битовые целочисленные типы: так, чтобы перенумеровать одни лишь всевозможные 10-байтовые строки, уже не хватит 64-битового (8-байтового) целого. В своей реализации мы использовали класс java.math.BigInteger из стандартной библиотеки Java, способный представлять целые числа произвольной величины. При этом объём оперативной памяти, занимаемой значением $\kappa$, примерно равен объёму, занимаемому набором значений $K_1,\ldots K_n$.
Говоря языком математики, биекция $g$ должна устанавливать изоморфизм порядка между множеством возможных значений наборов полей и множеством натуральных чисел. При наличии обратимой функции-нумератора $g$ и обратимой функции-интерполятора $f$,
- **прокрутка** грида сводится к вычислению значений ключевых полей $(K_1,\ldots K_n)=g^{-1}(f^{-1}(\lambda))$, где $\lambda$ – положение вертикальной полосы прокрутки, после чего быстрый запрос к БД находит $h$ первых записей, больших или равных $(K_1,\ldots K_n)$,
- **позиционирование** сводится к считыванию $h$ первых записей из БД по заранее известным значениям $(K_1,\ldots K_n)$, и к вычислению положения бегунка полосы прокрутки $\lambda = f(g(K_1,\ldots K_n))$.
Общая схема взаимодействия процедур {#общая-схема-взаимодействия-процедур .unnumbered}
-----------------------------------
Общая схема взаимодействия процедур системы показана на рис. \[fig:scheme\]. Cплошной стрелкой показана последовательность выполнения процедур, пунктирной стрелкой — асинхронный вызов в отдельном потоке выполнения.
; (s5) – (s4); (s4) – (s3); (s3) – (s2); (s2) – (s1); (s1) – (s0); (p0) – (p1); (p1) – (p2); (p2) – (p3); (p3) – (p4); (p4) – (p5); (r1) – (r2); (r2) – (r3); (r3) – (r4); (r4) – (r5);
(s1)\[dashed\] edge \[bend left = 30\] (r1); (p1)\[dashed\] – (r1);
Допустим, что пользователь изменил положение бегунка вертикальной полосы прокрутки (см. левый нижний угол диаграммы рис. \[fig:scheme\]).
Интерполятор вычисляет значение номера комбинации значений ключевых полей ($\kappa=f^{-1}(\lambda)$) с типом BigInteger. На основе этого значения нумератор восстанавливает комбинацию ключевых полей $(K_1,\ldots K_n)=g^{-1}(\kappa)$. Важно понимать, что на данном этапе в полях $K_1,\ldots K_n$ не обязательно будут находиться значения, действительно присутствующие в базе данных: там будут лишь приближения. В строковых полях, скорее всего, будет бессмысленный набор символов. Тем не менее, вывод из базы данных $h$ строк с ключами, большими или равными набору $K_1,\ldots K_n$, окажется приблизительно верным результатом для данного положения полосы прокрутки.
Если пользователь отпустил полосу прокрутки, асинхронно (в отдельном потоке выполнения) запускается запрос к БД, определяющий порядковый номер записи, а значит, и точное положение полосы прокрутки, которое соответствует тому, что отображено пользователю. Когда запрос будет завершён, на основе полученных данных будет пополнена интерполяционная таблица. Кроме того, если на экране пользователя к тому моменту ничего не изменится, бегунок полосы прокрутки <<отскочит>> на новое, уточнённое положение.
При переходе к записи последовательность вызовов процедур происходит в обратную сторону. Т. к. значения ключевых полей уже известны, для пользователя сразу извлекаются данные из базы. Нумератор вычисляет $\kappa = g(K_1,\ldots K_n)$, и затем интерполятор определяет приблизительное положение полосы прокрутки как $\lambda = f(\kappa)$. Параллельно, в асинхронном потоке выполнения, выполняется уточняющий запрос, по результатам которого в интерполяционную таблицу добавляется новая точка. Если на экране пользователя к тому моменту ничего не изменится, бегунок полосы прокрутки <<отскочит>> на новое, уточнённое положение.
Реализация интерполятора {#реализация-интерполятора .unnumbered}
------------------------
Объект-интерполятор должен хранить в себе промежуточные точки монотонно растущей функции между множеством 32-битных целых чисел (номеров записей в таблице) и множеством объектов типа BigInteger (порядковых номеров комбинаций значений ключевых полей).
Сразу же после инициализации грида необходимо в отдельном потоке выполнения запросить общее количество записей в таблице, чтобы получить корректное значение $\lambda_{\max}$. До того момента, как это значение будет получено при помощи выполняющегося в параллельном потоке запроса, можно использовать некоторое значение по умолчанию (например, 1000) – это не повлияет на корректность работы.
Интерполятор должен уметь за быстрое по количеству интерполяционных точек время вычислять значение как в одну, так и в другую сторону. Однако заметим, что чаще требуется вычислять значение порядкового номера комбинации по номеру записи: такие вычисления производятся много раз за секунду в процессе прокрутки грида пользователем. Поэтому за основу реализации модуля интерполятора удобно взять словарь на основе бинарного дерева, ключами которого являются номера записей, а значениями – порядковые номера комбинаций (класс TreeMap<Integer, BigInteger> в языке Java).
Ясно, что по заданному номеру $\lambda$ такой словарь за логарифмическое время находит две точки ($\underline{\lambda} {\leqslant}\lambda {\leqslant}\overline{\lambda}$), между которыми строит интерполяцию по формуле . Но тот факт, что функция растёт монотонно, позволяет за быстрое время производить и обратное вычисление. В самом деле: если дан номер комбинации $\kappa$, $\kappa_{\min} {\leqslant}\kappa {\leqslant}\kappa_{\max}$, поиск кусочного сегмента, в котором лежит $\kappa$, можно произвести в словаре методом дихотомии. Отыскав нужный сегмент, мы производим обратную интерполяцию по формуле и находим номер $\lambda$, соответствующий $\kappa$.
При пополнении словаря интерполяционными точками необходимо следить за тем, чтобы интерполируемая функция оставалась монотонной. Так как другие пользователи могут удалять и добавлять записи в просматриваемую таблицу, актуальность известных словарю интерполяционных точек может утратиться, а вновь добавляемая точка может нарушить монотонность. Поэтому метод добавления новой интерполяционной точки должен проверять, что <<точке слева>> от только что добавленной соответствует меньшее, а <<точке справа>> – большее значение. Если оказывается, что это не так, следует исходить из предположения, что последняя добавленная точка соответствует актуальному положению вещей, а некоторые из старых точек утратили свою актуальность. По отношению к вновь добавленной точке следует удалять все точки слева, содержащие большее значение, и все точки справа, содержащие меньшее значение (см. рис. \[fig:interp\]).
(0,8) – (0,0) – (8,0); at (8,0) [$\lambda$]{}; at (0,8) [$\kappa$]{}; (1,1) circle \[radius=1pt\]; (3,4) circle \[radius=1pt\]; (5,6) circle \[radius=1pt\]; (7,7) circle \[radius=1pt\]; (7,0) circle \[radius=1pt\]; (0,0) – (1,1) – (3,4) – (5,6) – (7,7); (7, 7) – (7,0); at (7,0) [$\lambda_{\max}$]{}; (10,8) – (10,0) – (18,0); at (18,0) [$\lambda$]{}; at (10,8) [$\kappa$]{}; (11,1) circle \[radius=1pt\]; (14,3) circle \[radius=1pt\]; (15,6) circle \[radius=1pt\]; (17,7) circle \[radius=1pt\]; (10,0) – (11,1) – (14,3) – (15,6) – (17,7); (11, 1) – (13,4) – (14,3); (13,4) circle \[radius=1.5pt\]; (17, 7) – (17,0); (17,0) circle \[radius=1pt\]; at (17,0) [$\lambda_{\max}$]{};
Также интерполятор должен содержать в себе механизм, в целях экономии памяти защищающий словарь от переполнения излишними точками, и отбрасывающий наименее существенные из них.
Нумераторы для числовых типов данных {#нумераторы-для-числовых-типов-данных .unnumbered}
------------------------------------
Назовём мощностью машинного типа данных количество различных значений, которые представимы при помощи этого типа.
Мощность типа BIT равна 2, нумерация его значений тривиальна: $\mathrm{false} \leftrightarrow 0$, $\mathrm{true} \leftrightarrow 1$.
Мощность типа INT (32-битовое целое со знаком) равна $2^{32}$. INT-значение есть число между $-2147483648$ и $2147483647$. Таким образом, нумератор для типа INT есть просто $g(k) = k + 2147483648$ (конечно, выполнять сложение следует, уже приведя $k$ к типу BigInteger). Обратная функция $g^{-1}(\kappa) = \kappa - 2147483648$.
Подобным же образом можно построить нумератор и для 64-битовых целых чисел со знаком.
Числа с типом DOUBLE (двойной точности с плавающей точкой), представленные в формате IEEE 754, обладают тем свойством, что их можно (за несущественными исключениями вроде NaN и $\pm0$) сравнивать как целые 64-битовые числа со знаком. В языке Java получить для значения с типом double его 64-битовый образ в формате IEEE 754 можно с помощью метода Double.doubleToLongBits.
Наконец, значения DATETIME, определяющие момент времени с точностью до миллисекунды, также могут быть сведены к 64-битовому целому числу со знаком, задающему так называемое <<UNIX-время>>, т. е. количество миллисекунд от полуночи 1 января 1970 года. В языке Java это делается при помощи метода Date.getTime.
Методы реализации нумераторов для строковых (VARCHAR(m)) типов и составных ключей рассмотрены далее.
Нумератор для составных ключей {#нумератор-для-составных-ключей .unnumbered}
------------------------------
Пусть типы данных составного ключа имеют мощности $N_1,\ldots, N_n$. Тогда общее количество возможных комбинаций значений ключевых полей равно $N_1N_2\ldots N_n$. Если вычислены функции нумераторов для значения каждого из полей, $\kappa_i$ – порядковый номер значения $i$-го поля, то функция нумератора составного ключа может быть представлена как
$$\label{eq:comp1}
\begin{split}
g(K_1,\ldots K_n) = \kappa_n + N_n\kappa_{n-1} \\
+ N_nN_{n-1}\kappa_{n-2} + \ldots.
\end{split}$$
Значение $\kappa_1$ имеет наибольший вес, $\kappa_n$ – наименьший. Также легко проверить, что $g(N_1 - 1, N_2-1,\ldots N_{n-1}-1) = N_1N_2\ldots N_n - 1$.
Вычисление $g$ напрямую по формуле требует $(n-1)n/2$ операций умножения. Сократить количество умножений до $n-1$ при том же количестве сложений можно, воспользовавшись аналогом схемы Горнера:
$$\label{eq:comp2}
\begin{split}
g(K_1,\ldots K_n) = ((\ldots(\kappa_1N_2 + \kappa_2)\ldots)N_{n-1} \\
+ \kappa_{n-1})N_n + \kappa_n.
\end{split}$$
Вычислить обратную функцию $g^{-1}$, получив из значения $g$ массив значений $\kappa_i$, можно по следующему простому алгоритму:
$i \leftarrow n$
Нумератор для строк (простой лексикографический порядок) {#нумератор-для-строк-простой-лексикографический-порядок .unnumbered}
--------------------------------------------------------
Сперва заметим, что если дан алфавит из $a$ символов, то общее количество строк длины не более $m$ в этом алфавите равно
$$\label{eq:str1}
1 + a + a^2 + \ldots + a^{m} = \frac{a^{m+1}-1}{a-1}.$$
Здесь единица соответствует пустой строке, $a$ – количество строк из одного символа, $a^2$ – количество строк из двух символов и т. д., а в итоге получается сумма геометрической прогрессии.
Представим теперь произвольную строку $c$ длины $l {\leqslant}m$ как массив $(c_0, c_1,\ldots c_{l-1})$, где $c_i$ – номер $i$-го символа строки в алфавите (считая с нуля), позиции символов в строке тоже считаем с нуля. Тогда строка $c$ в простом лексикографическом порядке будет иметь номер $$\label{eq:str2}
\begin{split}
g(c) = l + \frac{a^m - 1}{a - 1}c_0 + \frac{a^{m -1} - 1}{a - 1}c_1 + \ldots \\
+ \frac{a^{m - l + 1} - 1}{a - 1}c_{l-1}.
\end{split}$$
Докажем формулу индукцией по $m$ и $l$. Для иллюстрации примем, что алфавит состоит всего из двух букв: a и b.
Если $m = 0$, то единственный вариант – это пустая строка с номером 0.
Если $m = 1$, то пустая строка будет иметь номер 0, а каждая односимвольная будет иметь номер $1 + c_0$. Единица прибавляется потому, что при сравнении строк меньше любой односимвольной строки будет пустая строка: для двухсимвольного алфавита $0 \leftrightarrow \textnormal{`'}$, $1 \leftrightarrow \textnormal{`a'}$, $2 \leftrightarrow \textnormal{`b'}$.
Если $l {\leqslant}1$, $m {\geqslant}1$, то по-прежнему $0 \leftrightarrow \textnormal{`'}$, $1 \leftrightarrow \textnormal{`a'}$. Но теперь между строками ‘a’ и ‘b’ в пространстве лексикографически отсортированных строк находятся все возможные строки вида <<‘a’ плюс любая строка длиной не более $m - 1$>>: a, aa, aaa…, aab…:
$$\overbrace{a\underbrace{\square\square\square \ldots \square}_{{\leqslant}m-1}}^{{\leqslant}m}.$$
Количество таких строк равно, по , $(a^m-1)/(a-1)$, и окончательно для односимвольных строк
$$g(c) = 1 + \frac{a^m - 1}{a - 1}c_0.$$
Пусть к строке добавляется ещё один символ, при этом по-прежнему $l{\leqslant}m$. В качестве последнего слагаемого к добавляется номер этого символа с соответствующим весом (равным числу строк длиной $m - l$). К первому слагаемому в для каждого дополнительного символа добавляется единица, за счёт того, что строка, полученная отбрасыванием последнего символа, будет в лексикографическом порядке меньше любой из строк длиной $l$. Формула доказана.
Для оптимизации вычисления $g$ по формуле и для вычисления обратной функции необходимо заранее заготовить массив коэффициентов
$$\label{eq:qi}
q_i = \frac{a^{m - i} - 1}{a - 1}, i = 0,\ldots m-1.$$
Разумеется, пользоваться формулой напрямую при заготовке значений $q_i$ не нужно: сэкономить на арифметических операциях можно, заметив, что все $q_i$ являются частичными суммами геометрической прогрессии, которую можно вычислять <<на ходу>> при заполнении массива $q_i$.
Алгоритм для вычисления обратной функции $g^{-1}$, как и в случае нумератора составного ключа, является вариацией алгоритма преобразования числа в систему счисления с произвольным основанием. Необходимо только на каждом шаге перед получением очередного символа вычитать единицу, помня о первом слагаемом в формуле :
$i \leftarrow 0$
Нумератор для строк с учётом правил сопоставлений {#нумератор-для-строк-с-учётом-правил-сопоставлений .unnumbered}
-------------------------------------------------
Порядок, в котором база данных сортирует строковые значения, в действительности отличается от простого лексикографического и использует так называемые правила сопоставления (collation rules, [@U2015]).
Базой данных при сравнении строк с учётом этих правил каждый символ рассматривается в трёх аспектах: собственно символ, его регистр (case) и вариант написания (accent). Например, русская буква <<е>> в большинстве случаев рассматривается как имеющая два варианта написания, каждый из которых имеет два регистра: е, Е; ё, Ё. Это позволяет при сортировке, нечувствительной к варианту написания (accent insensitive), не различать <<е>> и <<ё>>, а при сортировке, нечувствительной к регистру (case insensitive), не различать строчные и заглавные буквы. При этом, что считать отдельной буквой, а что – вариантом написания другой буквы, зависит от культурных традиций и может различаться даже в языках, использующих один и тот же алфавит.
Общий алгоритм сравнения строк с учётом правил сопоставления следующий:
1. Строки сравниваются посимвольно без учёта регистров и вариантов. Если обнаружено различие, возвращается результат (<<больше>> или <<меньше>>).
2. Если сортировка accent sensitive, сравниваются номера вариантов каждого из символов. Если обнаружено различие, возвращается результат.
3. Если сортировка case sensitive, сравниваются регистры каждого из символов. Если обнаружено различие, возвращается результат.
4. Если выход из алгоритма не произошёл до сих пор – строки равны.
Таким образом, нам, во-первых, необходимо модифицировать алгоритм работы нумератора для строк таким образом, чтобы он учитывал правила сопоставления, а во-вторых, необходимо уметь задавать различные правила сопоставления, <<обучая>> грид работе с той или иной базой данных.
Первая из этих задач относительно проста с учётом уже полученных результатов. Всякое строковое значение необходимо рассматривать не как одномерный массив символов $c_i$, а как массив трёхкомпонентных значений $c_{ij}$, $0 {\leqslant}j {\leqslant}2$. Здесь $c_{i0}$ – номер $i$-го символа в алфавите, $c_{i1}$ – его же вариант написания, $c_{i2}$ – его же регистр. Тогда работу со строковым полем можно производить аналогично работе с составным ключом, состоящим из трёх полей.
Если известны $a_0$ – количество символов в алфавите, $a_1$ – максимальное число вариантов написания и $a_2$ – максимальное число регистров (в известных нам языках $a_2 = 2$), то
$$\label{eq:coll1}
g = (k_0 a_1^m + k_1) a_2^m + k_2,$$
где $k_0$ – вычисленное по первым компонентам строки значение формулы ($a = a_0$),
$$\begin{array}{l}
k_1 = a_1^{m-1} c_{01} + a_1^{m -2 } c_{11} + \ldots,\\
k_2 = a_2^{m-1} c_{02} + a_2^{m -2 } c_{12} + \ldots
\end{array}$$
Обратную функцию легко построить, используя уже вышеизложенные принципы: сперва необходимо разделить $g$ на три компоненты $k_1$, $k_2$ и $k_3$, затем получить массив трёхкомпонентных значений, на основании которого восстанавливается исходная строка.
В стандартной библиотеке Java имеются абстрактный класс java.text.Collator и его реализация java.text.RuleBasedCollator. Назначением этих классов является сравнение строк с учётом разнообразных правил сопоставлений. Доступна обширная библиотека готовых правил. К сожалению, эти классы не пригодны для использования с какой-либо иной, чем сравнение строк, целью: вся информация о правилах сопоставлений инкапсулирована, и её невозможно получить, штатным образом используя системную библиотеку.
Поэтому для решения нашей задачи понадобилось создать интерпретатор правил сопоставлений самостоятельно.
Эту задачу облегчило изучение класса RuleBasedCollator. Главной заимствованной идеей стал язык определения правил сортировки, формальное описание которого приведено в документации [@J2016]. Понять принцип работы этого языка проще всего, рассмотрев пример правила:
`<г,Г<д,Д<е,Е;ё,Ё<ж,Ж<з,З<и,И;й,Й<к,К<л,Л`
Следующие знаки являются служебными в языке правил сопоставления:
1. < – разделение символов,
2. ; – разделение вариантов написания,
3. , – разделение регистров.
При помощи выражений, подобных вышеприведённому, можно определить правила, соответствующие различным сопоставлениям различных баз данных. Т. к. язык правил сопоставлений достаточно примитивен, для его разбора достаточно алгоритма, работающего как детерминированный конечный автомат. В итоге по заданному выражению правил мы получаем экземпляр класса, способный
1. по заданным правилам получить значения $a = a_0$ (количество символов) для вычисления по формуле , а также $a_1$ (максимальное число вариантов) и $a_2$ (максимальное число регистров) для вычисления по формуле и соответствующих обратных функций,
2. по заданному символу определить три его компоненты (номер символа в алфавите, номер варианта, номер регистра),
3. по заданной тройке компонентов определить символ.
Это позволяет завершить реализацию нумератора для строковых значений.
Практическая реализация {#практическая-реализация .unnumbered}
-----------------------
Грид по приведённым здесь принципам был реализован на языке Java с использованием PostgreSQL в качестве СУБД. В качестве нагрузочного тестового набора данных использовалась база данных КЛАДР [@K2016], содержащая $1075429$ названий улиц населённых пунктов России, сортировка производилась по различным полям и их комбинациям. Тест продемонстрировал работоспособность изложенных здесь принципов. При постоянном перемещении бегунка вертикальной полосы прокрутки для пользователя создаётся полная иллюзия прокручивания всех записей в реальном времени, через малое время после окончания прокручивания (когда срабатывает уточняющий запрос и в интерполяционную таблицу добавляется ещё одна точка) позиция бегунка полосы прокрутки уточняется, перескакивая на небольшое расстояние. Позиционирование позволяет моментально отобразить нужные записи и сразу же приблизительно выставить бегунок полосы прокрутки, через малое время его позиция также уточняется. По мере работы с гридом и накопления интерполяционных точек, <<отскоки>> становятся всё менее и менее заметными.
Прокрутка на малый шаг {#прокрутка-на-малый-шаг .unnumbered}
----------------------
Важным нюансом при практической реализации явилась необходимость отдельной обработки *передвижения бегунка прокрутки на малый шаг*.
Прокрутка на одну строку вверх или вниз происходит при щелчке мышью на стрелки <<вверх>> и <<вниз>> вертикальной полосы прокрутки. При щелчке на свободное поле полосы прокрутки сверху или снизу от бегунка происходит прокрутка на фиксированное (малое) количество строк. В этих случаях пользователь ожидает сдвига всех видимых на экране строчек на фиксированное число позиций. Интерполятор, не набравший достаточно интерполяционных точек, может повести себя непредсказуемо, отбросив отображаемую пользователю картину слишком далеко назад или вперёд, и после уточнения позиция полосы прокрутки не будет соответствовать тому, что хотел пользователь.
В этом случае, однако, использование интерполятора и не оправдано. Если известен предыдущий набор значений ключевых полей, то получить одну предыдущую (или одну следующую) запись можно быстрым запросом к базе данных (см. (\[eq:lex1\]) и (\[eq:lex2\])). После извлечения этих данных, помимо отображения их пользователю, можно пополнить интерполяционную таблицу ещё одной точкой, *не прибегая к запросу на подсчёт записей*, т. к. полученная запись имеет номер, отличающийся от предыдущей на известное значение.
Начальное заполнение интерполяционной таблицы {#начальное-заполнение-интерполяционной-таблицы .unnumbered}
---------------------------------------------
Другим важным нюансом при практической реализации явилась необходимость заполнять интерполяционную таблицу данными до того, как пользователь начинает прокрутку грида.
Нет ничего удивительного в том, что номера комбинаций $\kappa$ на основе данных в реальной таблице распределяются на числовой прямой очень неравномерно. Поэтому, при недостаточном количестве точек в интерполяционной таблице, пользователь, сместив бегунок полосы прокрутки на некоторое расстояние, может получить после уточнения позиции сильный <<отскок>> вперёд или назад. В итоге реальная позиция просматриваемых данных переместится или намного дальше, или, наоборот, намного ближе, чем хотел пользователь.
Испытания показывают, что погрешность на 20-25% от длины полосы прокрутки при позиционировании является психологически допустимой, но не более того. Поэтому после отображения грида пользователю желательно обеспечить, чтобы максимальная длина <<отскока>> составляла не более, чем 20-25% длины полосы прокрутки даже в самом начале работы, когда статистика в интерполяционной таблице ещё не накоплена.
Сделать это эффективным образом можно в параллельном потоке выполнения, запускаемом после отображения грида. В этом потоке выполняется серия уточняющих запросов, по результатам которых пополняется интерполяционная таблица. Значение комбинации ключей для уточняющего запроса всякий раз выбирается как лежащее посередине самого большого зазора в значениях порядковых номеров записей интерполяционной таблицы. Процесс выполняется до тех пор, пока ширина максимального зазора не уменьшится до желаемого размера, либо до достижения ограничения на количество итераций.
[99]{} Statistical Distributions. Wiley, 2011. P. 117. Interpolation Search – A Log Log N Search / Communications of the ACM, 21, 7 (1978). Pp. 550-554. Unicode Technical Standard \#10. Unicode Collation Algorithm. <http://www.unicode.org/reports/tr10/> RuleBasedCollator (Java Platform SE 8). 2016. <https://docs.oracle.com/javase/8/docs/api/java/text/RuleBasedCollator.html> Классификатор адресов России (КЛАДР). 2016. <http://www.gnivc.ru/inf_provision/classifiers_reference/kladr/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A new categorical framework is provided for dealing with multiple arguments in a programming language with effects, for example in a language with imperative features. Like related frameworks (Monads, Arrows, Freyd categories), we distinguish two kinds of functions. In addition, we also distinguish two kinds of equations. Then, we are able to define a kind of product, that generalizes the usual categorical product. This yields a powerful tool for deriving many results about languages with effects.'
author:
- |
Jean-Guillaume Dumas\
LJK, University of Grenoble, France. [[`J`[email protected]]{}]{}\
Dominique Duval\
LJK, University of Grenoble, France. [[`D`[email protected]]{}]{}\
Jean-Claude Reynaud\
Malhiver, 38640 Claix, France. [[`J`[email protected]]{}]{}
bibliography:
- 'prod.bib'
date: 'July 4., 2007'
title: Sequential products in effect categories
---
\[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Assumption]{}
\[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Example]{} \[thm\][Examples]{} \[thm\][Definition]{} \[thm\][Convention]{} \[thm\][Conjecture]{} \[thm\][Problem]{} \[thm\][Open Problem]{} \[thm\][Algorithm]{} \[thm\][Observation]{} \[thm\][Question]{}
Introduction
============
The aim of this paper is to provide a new categorical framework dealing with multiple arguments in a programming language with effects, for example in a language with imperative features. In our *cartesian effect categories*, as in other related frameworks (Monads, Arrows, Freyd categories), two kinds of functions are distinguished. The new feature here is that two kinds of equations are also distinguished. Then, we define a kind of product, that is mapped to the usual categorical product when the distinctions (between functions and between equations) are forgotten. In addition, we prove that cartesian effect categories determine Arrows.
A well-established framework for dealing with computational effects is the notion of *strong monads*, that is used in Haskell [@Moggi91; @Wadler93]. Monads have been generalized on the categorical side to *Freyd categories* [@PowerRobinson97] and on the functional programming side to *Arrows* [@Hughes00]. The claims that Arrows generalize Monads and that Arrows are Freyd categories are made precise in [@HeunenJacobs06]. In all these frameworks, effect-free functions are distinguished among all functions, generalizing the distinction of *values* among all *computations* in [@Moggi91]. In this paper, as in [@BentonHyland03; @HeunenJacobs06], effect-free functions are called *pure* functions; however, the symbols ${\mathbf{C}}$ and ${\mathbf{V}}$, that are used for the category of all functions and for the subcategory of pure functions, respectively, are reminiscent of Moggi’s terminology.
In all these frameworks, one major issue is about the order of evaluation of the arguments of multivariate operations. When there is no effect, the order does not matter, and the notion of product in a cartesian category provides a relevant framework. So, the category ${\mathbf{V}}$ is cartesian, and *products of pure funtions* are defined by the usual characteristic property of products. But, when effects do occur, the order of evaluation of the arguments becomes fundamental, which cannot be dealt with the categorical product. So, the category ${\mathbf{C}}$ is not cartesian, and products of functions do not make sense, in general. However, some kind of *sequential product of computations* should make sense, in order to evaluate the arguments in a given order. This is usually defined, by composition, from some kinds of products of a computation with an identity. This is performed by the *strength* of the monad [@Moggi91], by the *symmetric premonoidal category* of the Freyd category [@PowerRobinson97], and by the *first* operator of Arrows [@Hughes00].
In this paper, the framework of *cartesian effect categories* is introduced. We still distinguish two kinds of functions: pure functions among arbitrary functions, that form two categories ${\mathbf{V}}$ and ${\mathbf{C}}$, with ${\mathbf{V}}$ a subcategory of ${\mathbf{C}}$, and ${\mathbf{V}}$ cartesian. Let us say that the functions are *decorated*, either as pure or as arbitrary. The new feature that is introduced in this paper is that we also distinguish two kinds of equations: strong equations and *semi-equations*, respectively denoted ${\equiv}$ and ${\lesssim}$, so that equations also are *decorated*. Strong equations can be seen, essentially, as equalities between computations, while semi-equations are much weaker, and can be seen as a kind of approximation relation. Moreover, as suggested by the symbols ${\equiv}$ and ${\lesssim}$, the strong equations form an equivalence relation, while the semi-equations form a preorder relation. Then, we define the *semi-product* of two functions when at least one is pure, by a characteristic property that is a decorated version of the characteristic property of the usual product. Since all identities are values, we get the semi-product of any function with an identity, that is used for building sequential products of functions.
Cartesian effect categories give rise to Arrows, in the sense of [@Hughes00], and they provide a deduction system: it is possible to decorate many proofs on cartesian categories in order to get proofs on cartesian effect categories.
As for terminology, our *graphs* are directed multi-graphs, made of *points* (or vertices, or objects) and *functions* (or edges, arrows, morphisms). We use *weak* categories rather than categories, i.e., we use a congruence $\equiv$ rather than the equality, however this “syntactic” choice is not fundamental here. As for notations, we often omit the subscripts in the diagrams and in the proofs.
Cartesian weak categories are reminded in section \[sec:weak\], then cartesian effect categories are defined in section \[sec:effect\]; they are compared with Arrows in section \[sec:related\], and examples are presented in section \[sec:exam\]. In appendix \[app:proof\] are given the proofs of some properties of cartesian weak categories, that are well-known, followed by their decorated versions, that yield proofs of properties of cartesian effect categories.
Cartesian weak categories {#sec:weak}
=========================
Weak categories are reminded in this section, with their notion of product. Except for the minor fact that equality is weakened as a congruence, all this section is very well known. Some detailed proofs are given in appendix \[app:proof\], with their decorated versions.
Weak categories {#subsec:weak-cat}
---------------
A weak category is like a category, except that the equations (for unitarity and associativity) hold only “up to congruence”.
\[defi:weak-cat\] A *weak category* is a graph where:
- for each point $X$ there is a loop ${\mathrm{id}}_X:X\to X$ called the *identity* of $X$,
- for each consecutive functions $f:X\to Y$, $g:Y\to Z$, there is a function $g\circ f:X\to Z$ called the *composition* of $f$ and $g$,
- and there is a relation $\equiv$ between parallel functions (each $f_1\equiv f_2$ is called an *equation*), such that:
- $\equiv$ is a *congruence*, i.e., it is an equivalence relation and for each $f:X\to Y$, $g_1,g_2:Y\to Z$, $h:Z\to W$, if $g_1\equiv g_2$ then $g_1\circ f\equiv g_2\circ f$ (*substitution*) and $h\circ g_1 \equiv h\circ g_2$ (*replacement*),
- for each $f:X\to Y$, the *unitarity equations* hold: $f\circ{\mathrm{id}}_X\equiv f$ and ${\mathrm{id}}_Y\circ f\equiv f$,
- and for each $f:X\to Y$, $g:Y\to Z$, $h:Z\to W$, the *associativity equation* holds: $h\circ (g \circ f) \equiv (h\circ g) \circ f$.
So, a weak category is a special kind of a bicategory, and a category is a weak category where the congruence is the equality.
Products {#subsec:weak-prod}
--------
In a weak category, a *weak product*, or simply a *product*, is defined as a product “up to congruence”. We focus on nullary products (i.e., terminal points) and binary products; it is well-know that products of any arity can be recovered from those.
\[defi:weak-term\] A *(weak) terminal point* is a point $U$ (for “*Unit*”) such that for every point $X$ there is a function ${{\langle \, \rangle}}_X:X\to U$, unique up to congruence.
\[defi:weak-prod\] A *binary cone* is made of two functions with the same source ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$. A *binary (weak) product* is a binary cone ${Y_1{\stackrel{q_1}{\longleftarrow}}Y_1\times Y_2{\stackrel{q_2}{\longrightarrow}}Y_2}$ such that for every binary cone with the same base ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$ there is a function ${\langle f_1,f_2 \rangle}:X\to Y_1\times Y_2$, called the *pair* of $f_1$ and $f_2$, unique up to congruence, such that: $$q_1\circ {\langle f_1,f_2 \rangle}\equiv f_1 \;\mbox{ and }\;
q_2\circ {\langle f_1,f_2 \rangle}\equiv f_2 \;.$$
As usual, all terminal points are isomorphic, and the fact of using $U$ for denoting a terminal point corresponds to the choice of one terminal point. Similarly, all products on a given base are isomorphic (in a suitable sense), and the notations correspond to the choice of one product for each base.
\[defi:weak-ccat\] A *cartesian weak category* is a weak category with a chosen terminal point and chosen binary products.
Products of functions {#subsec:weak-prodfn}
---------------------
\[defi:weak-arr-prod\] In a cartesian weak category, the *(weak) binary product* of two functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ is the function: $$f_1\times f_2 = {\langle f_1\circ p_1,f_2\circ p_2 \rangle}:
X_1\times X_2 \to Y_1\times Y_2 \;.$$
So, the binary product of functions is characterized, up to congruence, by the equations: $$q_1 \circ (f_1\times f_2) \equiv f_1 \circ p_1 \;\mbox{ and }\;
q_2 \circ (f_1\times f_2) \equiv f_2 \circ p_2 \;.$$ The defining equations of a pair and a product can be illustrated as follows: $$\xymatrix@C=5pc{
& Y_1 \\
X \ar[ru]^{f_1} \ar[rd]_{f_2} \ar[r]^{{\langle f_1,f_2 \rangle}} &
Y_1 \times Y_2 \ar[u]_{q_1} \ar[d]^{q_2}
\ar@{}[ld]|(.3){\equiv} \ar@{}[lu]|(.3){\equiv} \\
& Y_2 \\
}
\qquad\qquad
\xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \\
X_1\times X_2 \ar[u]^{p_1} \ar[d]_{p_2} \ar[r]^{f_1\times f_2}
\ar@{}[rd]|{\equiv} \ar@{}[ru]|{\equiv } &
Y_1 \times Y_2 \ar[u]_{q_1} \ar[d]^{q_2} \\
X_2 \ar[r]^{f_2} & Y_2 \\
}$$ So, the products are defined from the pairs (note that we use the same symbols $f_1,f_2$ for the general case $f_i:X_i\to Y_i$ and for the special case $f_i:X\to Y_i$). The other way round, the pairs can be recovered from the products and the *diagonals*, i.e., the pairs ${\langle {\mathrm{id}},{\mathrm{id}}\rangle}$; indeed, it is easy to prove that for each cone ${X_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{f_2}{\longrightarrow}}X_2}$ $${\langle f_1,f_2 \rangle} \equiv (f_1\times f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$
In the following, we consider products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$, ${Y_1{\stackrel{q_1}{\longleftarrow}}Y_1\times Y_2{\stackrel{q_2}{\longrightarrow}}Y_2}$ and ${Z_1{\stackrel{r_1}{\longleftarrow}}Z_1\times Z_2{\stackrel{r_2}{\longrightarrow}}Z_2}$.
\[prop:weak-equiv\] For each $f_1\equiv f'_1:X_1\to Y_1$ and $f_2\equiv f'_2:X_2\to Y_2$
1. if $X_1=X_2$ $${\langle f_1,f_2 \rangle} \equiv {\langle f'_1,f'_2 \rangle} \;,$$
2. in all cases $$f_1 \times f_2 \equiv f'_1 \times f'_2 \;.$$
\[prop:weak-comp\] For each $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$, $g_1:Y_1\to Z_1$, $g_2:Y_2\to Z_2$
1. if $X_1=X_2$ and $Y_1=Y_2$ and $f_1=f_2(=f)$ $${\langle g_1,g_2 \rangle} \circ f \equiv {\langle g_1\circ f,g_2\circ f \rangle} \;,$$
2. if $X_1=X_2$ $$(g_1 \times g_2) \circ {\langle f_1,f_2 \rangle} \equiv
{\langle g_1\circ f_1,g_2\circ f_2 \rangle} \;,$$
3. in all cases $$(g_1 \times g_2) \circ (f_1 \times f_2) \equiv
(g_1\circ f_1) \times (g_2\circ f_2) \;.$$
Let us consider the products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$ and ${X_2{\stackrel{p'_2}{\longleftarrow}}X_2\times X_1{\stackrel{p'_1}{\longrightarrow}}X_1}$. The *swap* function is the isomorphism: $$\gamma_{(X_1,X_2)} = {\langle p'_1,p'_2 \rangle}_{p_1,p_2} = {\langle p'_1,p'_2 \rangle}:
X_2\times X_1\to X_1\times X_2\;,$$ characterized by: $$p_1\circ \gamma_{(X_1,X_2)}\equiv p'_1
\;\mbox{ and }\; p_2\circ \gamma_{(X_1,X_2)}\equiv p'_2 \;.$$
\[prop:weak-swap\] For each $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$, let $\gamma_Y=\gamma_{(Y_1,Y_2)}$ and $\gamma_X=\gamma_{(X_1,X_2)}$, then:
1. if $X_1=X_2$ $$\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv {\langle f_1,f_2 \rangle} \;,$$
2. in all cases $$\gamma_Y \circ (f_2\times f_1) \circ \gamma_X^{-1} \equiv
f_1 \times f_2 \;.$$
Let us consider the products ${X_1{\stackrel{p_1}{\longleftarrow}}X_1\times X_2{\stackrel{p_2}{\longrightarrow}}X_2}$, ${X_1\times X_2{\stackrel{p_{1,2}}{\longleftarrow}}(X_1\times X_2)\times X_3{\stackrel{p_3}{\longrightarrow}}X_3}$, ${X_2{\stackrel{p'_2}{\longleftarrow}}X_2\times X_3{\stackrel{p'_3}{\longrightarrow}}X_3}$ and ${X_1{\stackrel{p'_1}{\longleftarrow}}X_1\times (X_2\times X_3){\stackrel{p'_{2,3}}{\longrightarrow}}X_2\times X_3}$. The *associativity* function is the isomorphism: $$\alpha_{(X_1,X_2,X_3)} =
{\langle {\langle p'_1,p'_2\circ p'_{2,3} \rangle}_{p_1,p_2},
p'_3\circ p'_{2,3} \rangle}_{p_{1,2},p_3}:
X_1\times (X_2\times X_3) \to (X_1\times X_2)\times X_3\;,$$ characterized by: $$p_1\circ p_{1,2}\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_1
\,,\;
p_2\circ p_{1,2}\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_2\circ p'_{2,3}
\;\mbox{ and }\;
p_3\circ \alpha_{(X_1,X_2,X_3)} \equiv p'_3\circ p'_{2,3} \;.$$
\[prop:weak-assoc\] For each $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and $f_3:X_3\to Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then:
1. if $X_1=X_2=X_3$ $$\alpha_Y \circ {\langle f_1,{\langle f_2,f_3 \rangle} \rangle}
\equiv {\langle {\langle f_1,f_2 \rangle},f_3 \rangle} \;,$$
2. in all cases $$\alpha_Y \circ (f_1\times (f_2\times f_3)) \equiv
((f_1 \times f_2)\times f_3) \circ \alpha_X \;.$$
In the definition of the binary product $f_1\times f_2$, both $f_1$ and $f_2$ play symmetric rôles. This symmetry can be broken: “first $f_1$ then $f_2$” corresponds to $({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})$, using the intermediate product $Y_1\times X_2$, while “first $f_2$ then $f_1$” corresponds to $(f_1\times{\mathrm{id}}_{Y_2}) \circ ({\mathrm{id}}_{X_1}\times f_2)$, using the intermediate product $X_1\times Y_2$. These are called the *(left and right) sequential products* of $f_1$ and $f_2$. The three versions of the binary product of functions coincide, up to congruence; this is a kind of *parallelism* property, meaning that both $f_1$ and $f_2$ can be computed either simultaneously, or one after the other, in any order:
\[prop:weak-seq\] For each $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ $$f_1\times f_2 \equiv ({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})
\equiv (f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2) \;.$$
Cartesian effect categories {#sec:effect}
===========================
Sections \[subsec:effect-cat\] to \[subsec:effect-semifn\] form a decorated version of section \[sec:weak\]. Roughly speaking, a kind of structure is *decorated* when there is some classification of its ingredients. Here, the classification involves two kinds of functions and two kinds of equations. Effect categories are defined in section \[subsec:effect-cat\] as decorated weak categories. In section \[subsec:effect-semi\], semi-products are defined as decorated weak products, then cartesian effect category as decorated cartesian weak categories. Decorated propositions are stated here, and the corresponding decorated proofs are given in appendix \[app:proof\]. Then, in sections \[subsec:effect-seq\] and \[subsec:effect-proj\], the sequential product of functions is defined by composing semi-products, and some of its properties are derived.
Effect categories {#subsec:effect-cat}
-----------------
A *(weak) subcategory* ${\mathbf{V}}$ of a weak category ${\mathbf{C}}$ is a subcategory of ${\mathbf{C}}$ such that each equation of ${\mathbf{V}}$ is an equation of ${\mathbf{C}}$. It is a *wide (weak) subcategory* when ${\mathbf{V}}$ and ${\mathbf{C}}$ have the same points, and each equation of ${\mathbf{C}}$ between functions in ${\mathbf{V}}$ is an equation in ${\mathbf{V}}$. Then only one symbol ${\equiv}$ can be used, for both ${\mathbf{V}}$ and ${\mathbf{C}}$.
\[defi:effect-cat\] Let ${\mathbf{V}}$ be a weak category. An *effect category extending* ${\mathbf{V}}$ is a weak category ${\mathbf{C}}$, such that ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$, together with a relation ${\lesssim}$ between parallel functions in ${\mathbf{C}}$ such that:
- the relation ${\lesssim}$ is weaker than ${\equiv}$ for $f_1,f_2$ in ${\mathbf{C}}$, $\,f_1{\equiv}f_2 \Rightarrow f_1{\lesssim}f_2$;
- ${\lesssim}$ is transitive;
- ${\lesssim}$ and ${\equiv}$ coincide on ${\mathbf{V}}$ for $v_1,v_2$ in ${\mathbf{V}}$, $\,v_1{\equiv}v_2 \iff v_1{\lesssim}v_2$;
- ${\lesssim}$ satisfies the substitution property:\
if $f:X\to Y$ and $g_1{\lesssim}g_2:Y\to Z$ then $g_1\circ f{\lesssim}g_2\circ f:X\to Z$;
- ${\lesssim}$ satisfies the replacement property *with respect to ${\mathbf{V}}$*:\
if $g_1{\lesssim}g_2:Y\to Z$ and $v:Z\to W$ in ${\mathbf{V}}$ then $v\circ g_1 {\lesssim}v\circ g_2:Y\to W$.
The first property implies that ${\lesssim}$ is reflexive, and when ${\equiv}$ is the equality it means precisely that ${\lesssim}$ is reflexive. Since ${\lesssim}$ is transitive and weaker than ${\equiv}$, if either $f_1{\equiv}f_2{\lesssim}f_3$ or $f_1{\lesssim}f_2{\equiv}f_3$, then $f_1{\lesssim}f_3$; this is called the *compatibility* of ${\lesssim}$ with ${\equiv}$. An effect category is *strict* when ${\equiv}$ is the equality. In this paper, there is no major difference between effect categories and strict effect categories.
A *pure function* is a function in ${\mathbf{V}}$. The symbol ${\rightsquigarrow}$ is used for pure functions, and $\to$ for all functions. It follows from definition \[defi:effect-cat\] that all the identities of ${\mathbf{C}}$ are pure, the composition of pure functions is pure, and more precisely a composition of functions is pure if and only if all the composing functions are pure. It should be noted that there can be equations $f{\equiv}v$ between a non-pure function and a pure one; then the function $f$ is proved effect-free, without being pure. This “syntactic” choice could be argued; note that this situation disappears when the congruence ${\equiv}$ is the equality. The relation ${\lesssim}$ is called the *semi-congruence* of the effect category, and each $f_1{\lesssim}f_2$ is called a *semi-equation*. The semi-congruence generally is not a congruence, for two reasons: it may not be symmetric, and it may not satisfy the replacement property for all functions.
Examples of strict effect categories are given in section \[sec:exam\]. For dealing with partiality in section \[subsec:exam-partial\], the semi-congruence ${\lesssim}$ coincides with the usual ordering of partial functions, it is not symmetric but it satisfies the replacement property for all partial functions. On the other hand, in section \[subsec:exam-state\], the semi-congruence ${\lesssim}$ means that two functions in an imperative language have the same result but may act differently on the state, it is an equivalence relation that does not satisfy the replacement property for non-pure functions.
Clearly, if the decorations are forgotten, i.e., if both the distinction between pure functions and arbitrary functions and the distinction between the congruence and the semi-congruence are forgotten, then an effect category is just a weak category.
A cartesian effect category, as defined below, is an effect category where ${\mathbf{V}}$ is cartesian and where this cartesian structure on ${\mathbf{V}}$ has some kind of generalization to ${\mathbf{C}}$, that does *not*, in general, turn ${\mathbf{C}}$ into a cartesian weak category.
Semi-products {#subsec:effect-semi}
-------------
Now, let us assume that ${\mathbf{C}}$ is an effect category extending ${\mathbf{V}}$, and that ${\mathbf{V}}$ is cartesian. We define nullary and binary *semi-products* in ${\mathbf{C}}$, for building pairs of functions when at least one of them is pure.
\[defi:effect-term\] A *semi-terminal point* in ${\mathbf{C}}$ is a terminal point $U$ in ${\mathbf{V}}$ such that every function $g:X\to U$ satisfies $g{\lesssim}{{\langle \, \rangle}}_X$.
\[defi:effect-prod\] A *binary semi-product* in ${\mathbf{C}}$ is a binary product ${Y_1{\stackrel{q_1}{\leftsquigarrow}}Y_1\times Y_2{\stackrel{q_2}{\rightsquigarrow}}Y_2}$ in ${\mathbf{V}}$ such that:
- for every binary cone with the same base ${Y_1{\stackrel{f_1}{\longleftarrow}}X{\stackrel{v_2}{\rightsquigarrow}}Y_2}$ and with $v_2$ pure, there is a function ${\langle f_1,v_2 \rangle}_{q_1,q_2}={\langle f_1,v_2 \rangle}:X\to Y_1\times Y_2$, unique up to ${\equiv}$, such that $$q_1\circ {\langle f_1,v_2 \rangle} {\equiv}f_1 \;\mbox{ and }\;
q_2\circ {\langle f_1,v_2 \rangle} {\lesssim}v_2 \;,$$
- and for every binary cone with the same base ${Y_1{\stackrel{v_1}{\leftsquigarrow}}X{\stackrel{f_2}{\longrightarrow}}Y_2}$ and with $v_1$ pure, there is a function ${\langle v_1,f_2 \rangle}_{q_1,q_2}={\langle v_1,f_2 \rangle}:X\to Y_1\times Y_2$, unique up to ${\equiv}$, such that $$q_1\circ {\langle v_1,f_2 \rangle} {\lesssim}v_1 \;\mbox{ and }\;
q_2\circ {\langle v_1,f_2 \rangle} {\equiv}f_2 \;.$$
The defining (semi-)equations of a binary semi-product can be illustrated as follows: $$\xymatrix@C=5pc{
& Y_1 \\
X \ar@{~>}[ru]^{v_1} \ar@{~>}[rd]_{v_2} \ar@{~>}[r]^{{\langle v_1,v_2 \rangle}} &
Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2}
\ar@{}[ld]|(.3){{\equiv}} \ar@{}[lu]|(.3){{\equiv}}\\
& Y_2 \\
}
\qquad
\xymatrix@C=5pc{
& Y_1 \\
X \ar[ru]^{f_1} \ar@{~>}[rd]_{v_2} \ar[r]^{{\langle f_1,v_2 \rangle}\;\;} &
Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2}
\ar@{}[ld]|(.3){{\gtrsim}} \ar@{}[lu]|(.3){{\equiv}}\\
& Y_2 \\
}
\qquad
\xymatrix@C=5pc{
& Y_1 \\
X \ar@{~>}[ru]^{v_1} \ar[rd]_{f_2} \ar[r]^{{\langle v_1,f_2 \rangle}\;\;} &
Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2}
\ar@{}[ld]|(.3){{\equiv}} \ar@{}[lu]|(.3){{\gtrsim}}\\
& Y_2 \\
}$$ Clearly, if the decorations are forgotten, then semi-products are just products.
The notation is not ambiguous. Indeed, if ${Y_1{\stackrel{v_1}{\leftsquigarrow}}X{\stackrel{v_2}{\rightsquigarrow}}Y_2}$ is a binary cone in ${\mathbf{V}}$, then the three definitions of the pair ${\langle v_1,v_2 \rangle}$ above coincide, up to congruence: let $t$ denote any one of the three pairs, then $t$ is characterized, up to congruence, by $q_1\circ t {\equiv}v_1$ and $q_2\circ t {\equiv}v_2$, because ${\equiv}$ and ${\lesssim}$ coincide on pure functions.
\[defi:effect-ccat\] A *cartesian effect category extending* a cartesian weak category ${\mathbf{V}}$ is an effect category extending ${\mathbf{V}}$ such that each terminal point of ${\mathbf{V}}$ is a semi-terminal point of ${\mathbf{C}}$ and each binary product of ${\mathbf{V}}$ is a binary semi-product of ${\mathbf{C}}$.
Semi-products of functions {#subsec:effect-semifn}
--------------------------
\[defi:effect-arr-prod\] In a cartesian effect category, the *binary semi-product* $f_1\times v_2$ of a function $f_1:X_1\to Y_1$ and a pure function $v_2:X_2{\rightsquigarrow}Y_2$ is the function: $$f_1\times v_2={\langle f_1\circ p_1,v_2\circ p_2 \rangle} :
X_1\times X_2 \to Y_1\times Y_2$$
It follows that $f_1\times v_2$ is characterized, up to ${\equiv}$, by: $$q_1 \circ (f_1\times v_2) {\equiv}f_1 \circ p_1 \;\mbox{ and }\;
q_2 \circ (f_1\times v_2) {\lesssim}v_2 \circ p_2$$ $$\xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \\
X_1\times X_2 \ar@{~>}[u]^{p_1} \ar@{~>}[d]_{p_2} \ar[r]^{f_1\times v_2}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{{\equiv}} &
Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \\
X_2 \ar@{~>}[r]^{v_2} & Y_2 \\
}$$ The *binary semi-product* $v_1\times f_2:X_1\times X_2\to Y_1\times Y_2$ of a pure function $v_1:X_1{\rightsquigarrow}Y_1$ and a function $f_2:X_2\to Y_2$ is defined in the symmetric way, and it is characterized, up to ${\equiv}$, by the symmetric property.
The notation is not ambiguous, because so is the notation for pairs; if $v_1$ and $v_2$ are pure functions, then the three definitions of $v_1\times v_2$ coincide, up to congruence.
Propositions about products in cartesian weak categories are called *basic* propositions. It happens that each basic proposition in section \[sec:weak\] has a *decorated* version, about semi-products of the form $f_1\times v_2$ in cartesian effect categories, that is stated below. The symmetric decorated version also holds, for semi-products of the form $v_1\times f_2$. Each function in the basic proposition is replaced either by a function or by a pure function, and each equation is replaced either by an equation (${\equiv}$) or by a semi-equation (${\lesssim}$ or ${\gtrsim}$).
In addition, in appendix \[app:proof\], the proofs of the decorated propositions are *decorated* versions of the *basic* proofs. It happens that no semi-equation appears in the decorated propositions below, but they are used in the proofs. Indeed, a major ingredient in the basic proofs is that a function ${\langle f_1,f_2 \rangle}$ or $f_1\times f_2$ is characterized, up to $\equiv$, by its projections, both up to $\equiv$. The decorated version of this property is that a function ${\langle f_1,f_2 \rangle}$ or $f_1\times f_2$, where $f_1$ or $f_2$ is pure, is characterized, up to $\equiv$, by its projections, one up to $\equiv$ *and the other one up to ${\lesssim}$*. It should be noted that even when some decorated version of a basic proposition is valid, usually not all the basic proofs can be decorated. In addition, when equations are decorated as semi-equations, some care is required when the symmetry and replacement properties are used.
\[prop:effect-equiv\] For each congruent functions $f_1{\equiv}f'_1:X\to Y_1$ and pure functions $v_2{\equiv}v'_2:X{\rightsquigarrow}Y_2$
1. if $X_1=X_2$ $${\langle f_1,v_2 \rangle} {\equiv}{\langle f'_1,v'_2 \rangle} \;.$$
2. in all cases $$f_1 \times v_2 {\equiv}f'_1 \times v'_2 \;.$$
\[prop:effect-comp\] For each functions $f_1:X_1\to Y_1$, $g_1:Y_1\to Z_1$ and pure functions $v_2:X_2{\rightsquigarrow}Y_2$, $w_2:Y_2{\rightsquigarrow}Z_2$
1. if $X_1=X_2$ and $Y_1=Y_2$ and $f_1=v_2(=v)$ $${\langle g_1,w_2 \rangle} \circ f {\equiv}{\langle g_1\circ v,w_2\circ v \rangle} \;,$$
2. if $X_1=X_2$ $$(g_1 \times w_2) \circ {\langle f_1,v_2 \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle} \;,$$
3. in all cases $$(g_1 \times w_2) \circ (f_1 \times v_2) {\equiv}(g_1\circ f_1) \times (w_2\circ v_2) \;.$$
The swap and associativity functions are defined in the same way as in section \[sec:weak\]; they are products of projections, so that they are pure functions. It follows that the swap and associativity functions are characterized by the same equations as in section \[sec:weak\], and that they are still isomorphisms.
\[prop:effect-swap\] For each function $f_1:X\to Y_1$ and pure function $v_2:X{\rightsquigarrow}Y_2$, let $\gamma_Y=\gamma_{(Y_1,Y_2)}$ and $\gamma_X=\gamma_{(X_1,X_2)}$, then:
1. if $X_1=X_2$ $$\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}{\langle f_1,v_2 \rangle} \;,$$
2. in all cases $$\gamma_Y \circ (v_2\times f_1) \circ \gamma_X^{-1} {\equiv}f_1 \times v_2 \;.$$
\[prop:effect-assoc\] For each function $f_1:X_1\to Y_1$ and pure functions $v_2:X_2{\rightsquigarrow}Y_2$, $v_3:X_3{\rightsquigarrow}Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then:
1. if $X_1=X_2=X_3$ $$\alpha_Y \circ {\langle f_1,{\langle v_2,v_3 \rangle} \rangle}
{\equiv}{\langle {\langle f_1,v_2 \rangle},v_3 \rangle} \;,$$
2. in all cases: $$\alpha_Y \circ (f_1\times (v_2\times v_3)) {\equiv}((f_1 \times v_2)\times v_3) \circ \alpha_X \;.$$
The sequential product of a function $f_1:X_1\to Y_1$ and a pure function $v_2:X_2{\rightsquigarrow}Y_2$ can be defined as in section \[sec:weak\], using the intermediate products ${Y_1{\stackrel{s_1}{\leftsquigarrow}}Y_1\times X_2{\stackrel{s_2}{\rightsquigarrow}}X_2}$ and ${X_1{\stackrel{t_1}{\leftsquigarrow}}X_1\times Y_2{\stackrel{t_2}{\rightsquigarrow}}Y_2}$. It does coincide with the semi-product of $f_1$ and $v_2$, up to congruence:
\[prop:effect-seq\] For each function $f_1:X_1\to Y_1$ and pure function $v_2:X_2{\rightsquigarrow}Y_2$ $$f_1\times v_2 {\equiv}({\mathrm{id}}_{Y_1}\times v_2) \circ (f_1\times{\mathrm{id}}_{X_2})
{\equiv}(f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times v_2) \;.$$
Sequential products of functions {#subsec:effect-seq}
--------------------------------
It has been stated in proposition \[prop:weak-seq\] that, in a cartesian weak category, the binary product of functions coincide with both sequential products, up to congruence: $$f_1\times f_2 \equiv ({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})
\equiv (f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2) \;.$$ In a cartesian effect category, when $f_1$ and $f_2$ are any functions, the product $f_1\times f_2$ is not defined. But $({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2})$ and $(f_1\times{\mathrm{id}}_{X_2}) \circ ({\mathrm{id}}_{Y_1}\times f_2)$ make sense, thanks to semi-products, because identities are pure. They are called the sequential products of $f_1$ and $f_2$, and they do not coincide up to congruence, in general: parallelism is not satisfied.
\[defi:effect-seq-prod\] The *left binary sequential product* of two functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ is the function: $$f_1\ltimes f_2 =
({\mathrm{id}}_{Y_1}\times f_2) \circ (f_1\times{\mathrm{id}}_{X_2}) : X_1\times X_2\to Y_1\times Y_2 \;.$$
So, the left binary sequential product is obtained from: $$\xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\
X_1\times X_2 \ar@{~>}[u]^{p_1} \ar@{~>}[d]_{p_2}
\ar[r]^{f_1\times {\mathrm{id}}}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{{\equiv}} &
Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2}
\ar[r]^{{\mathrm{id}}\times f_2}
\ar@{}[rd]|{{\equiv}} \ar@{}[ru]|{{\gtrsim}} &
Y_1 \times Y_2 \ar@{~>}[u]^{q_1} \ar@{~>}[d]_{q_2} \\
X_2 \ar@{~>}[r]^{{\mathrm{id}}} & X_2 \ar[r]^{f_2} & Y_2 \\
}$$ The left sequential product extends the semi-product:
\[prop:effect-seq-lproduct\] For each function $f_1$ and pure function $v_2$, $ f_1\ltimes v_2 {\equiv}f_1\times v_2 $.
>From proposition \[prop:effect-comp\], $f_1\ltimes v_2 = ({\mathrm{id}}\times v_2) \circ (f_1\times{\mathrm{id}})
{\equiv}({\mathrm{id}}\circ f_1) \times (v_2\circ{\mathrm{id}})
{\equiv}f_1 \times v_2$.
Note that the diagonal ${\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}$ is a pair of pure functions. So, by analogy with the property ${\langle f_1,f_2 \rangle} \equiv (f_1\times f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}$ in weak categories:
\[defi:effect-seq-pair\] The *left sequential pair* of two functions $f_1:X\to Y_1$ and $f_2:X\to Y_2$ is: $${\langle f_1,f_2 \rangle}_l = (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$
The left sequential pairs do not satisfy the usual equations for pairs, as in definition \[defi:weak-prod\]. However, they satisfy some weaker properties, as stated in corollary \[cor:seq-pair\].
The *right binary sequential product* of $f_1$ and $f_2$ is defined in the symmetric way; it is the function: $$f_1\rtimes f_2 =
(f_1\times{\mathrm{id}}_{Y_2}) \circ ({\mathrm{id}}_{X_1}\times f_2) :
X_1\times X_2\to Y_1\times Y_2 \;.$$ It does also extend the product of a pure function and a function: for each pure function $v_1$, $ v_1\rtimes f_2 {\equiv}v_1\times f_2 $. The *right sequential pair* of $f_1:X\to Y_1$ and $f_2:X\to Y_2$ is: $${\langle f_1,f_2 \rangle}_r = (f_1\rtimes f_2) \circ {\langle {\mathrm{id}}_X,{\mathrm{id}}_X \rangle}\;.$$
Here are some properties of the sequential products that are easily deduced from the properties of semi-products in \[subsec:effect-semi\]. The symmetric properties also hold.
\[prop:seq-equiv\] For each congruent functions $f_1{\equiv}f'_1:X_1\to Y_1$ and $f_2{\equiv}f'_2:X_2\to Y_2$ $$f_1 \ltimes f_2 {\equiv}f'_1 \ltimes f'_2\;.$$
Clear, from \[prop:effect-equiv\].
\[prop:seq-comp\] For each functions $f_1:X_1\to Y_1$, $g_1:Y_1\to Z_1$, $g_2:Y_2\to Z_2$ and pure function $v_2:X_2{\rightsquigarrow}Y_2$ $$(g_1 \ltimes g_2) \circ (f_1 \times v_2) {\equiv}(g_1\circ f_1) \ltimes (g_2\circ v_2) \;.$$
$$\xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \ar@{~>}[r]^{{\mathrm{id}}} & Z_1 \\
X_1 \times X_2 \ar@{~>}[u] \ar@{~>}[d] \ar[r]^{f_1\times v_2}
\ar@{}[rd]|{{\equiv}} \ar@{}[ru]|{{\equiv}} &
Y_1\times Y_2 \ar@{~>}[u] \ar@{~>}[d]
\ar[r]^{g_1\times {\mathrm{id}}}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{{\equiv}} &
Z_1 \times Y_2 \ar@{~>}[u] \ar@{~>}[d]
\ar[r]^{{\mathrm{id}}\times g_2}
\ar@{}[rd]|{{\equiv}} \ar@{}[ru]|{{\gtrsim}} &
Z_1 \times Z_2 \ar@{~>}[u] \ar@{~>}[d] \\
X_2 \ar@{~>}[r]^{v_2} & Y_2 \ar@{~>}[r]^{{\mathrm{id}}} & Y_2 \ar[r]^{g_2} & Z_2 \\
}$$
>From several applications of proposition \[prop:effect-comp\] and its symmetric version:\
$({\mathrm{id}}\times g_2) \circ (g_1\times{\mathrm{id}}) \circ (f_1 \times v_2)
{\equiv}({\mathrm{id}}\times g_2) \circ ((g_1\circ f_1) \times v_2)
{\equiv}({\mathrm{id}}\times g_2) \circ ({\mathrm{id}}\times v_2)
\circ ((g_1\circ f_1)\times{\mathrm{id}})
{\equiv}({\mathrm{id}}\times (g_2\circ v_2)) \circ ((g_1\circ f_1)\times{\mathrm{id}})$.
\[prop:seq-swap\] For each functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$, the left and right sequential products are related by swaps: $$\gamma_Y \circ (f_2\rtimes f_1) \circ \gamma_X^{-1} {\equiv}f_1 \ltimes f_2 \;.$$
>From proposition \[prop:effect-swap\] and its symmetric version:\
$ \gamma \circ ({\mathrm{id}}\times f_2) \circ (f_1\times{\mathrm{id}})
{\equiv}(f_2\times{\mathrm{id}}) \circ \gamma \circ (f_1\times{\mathrm{id}})
{\equiv}(f_2\times{\mathrm{id}}) \circ ({\mathrm{id}}\times f_1) \circ \gamma$.
\[prop:seq-assoc\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and $f_3:X_3\to Y_3$, let $\alpha_Y=\alpha_{(Y_1,Y_2,Y_3)}$ and $\alpha_X=\alpha_{(X_1,X_2,X_3)}$, then: : $$\alpha_Y \circ (f_1\ltimes (f_2\ltimes f_3)) {\equiv}((f_1 \ltimes f_2)\ltimes f_3) \circ \alpha_X \;.$$
>From proposition \[prop:effect-assoc\].
Projections of sequential products {#subsec:effect-proj}
----------------------------------
Let us come back to a weak category, as in section \[sec:weak\]. The binary product of functions is characterized, up to congruence, by the equations: $$q_1 \circ (f_1\times f_2) \equiv f_1 \circ p_1 \;\mbox{ and }\;
q_2 \circ (f_1\times f_2) \equiv f_2 \circ p_2 \;,$$ so that for all constant functions $x_1:U\to X_1$ and $x_2:U\to X_2$ $$q_1 \circ (f_1\times f_2) \circ {\langle x_1,x_2 \rangle} \equiv f_1 \circ x_1
\;\mbox{ and }\;
q_2 \circ (f_1\times f_2) \circ {\langle x_1,x_2 \rangle} \equiv f_2 \circ x_2 \;.$$ In a cartesian effect category, it is proved in theorem \[thm:seq-prod\] that $f_1\ltimes f_2$, when applied to a pair of constant pure functions ${\langle x_1,x_2 \rangle}$, returns on the $Y_1$ side a function that is semi-congruent to $f_1(x_1)$, and on the $Y_2$ side a function that is congruent to $f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1\circ x_1$, which means “first $f_1(x_1)$, then forget the result, then $f_2(x_2)$”. More precise statements are given in propositions \[prop:seq-val\] and \[prop:seq-com\]. Proofs are presented in the same formalized way as in appendix \[app:proof\].
As above, we consider the semi-terminal point $U$ and semi-products\
${X_1{\stackrel{p_1}{\leftsquigarrow}}X_1\times X_2{\stackrel{p_2}{\rightsquigarrow}}X_2}$, ${Y_1{\stackrel{q_1}{\leftsquigarrow}}Y_1\times Y_2{\stackrel{q_2}{\rightsquigarrow}}Y_2}$ and ${Y_1{\stackrel{s_1}{\leftsquigarrow}}Y_1\times X_2{\stackrel{s_2}{\rightsquigarrow}}X_2}$.
\[prop:seq-val\] For each functions $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ $$q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1 : X_1\times X_2\to Y_1\;.$$
$$\xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\
X_1\times X_2 \ar@{~>}[u]^{p_1}
\ar[r]^{f_1\times {\mathrm{id}}}
\ar@{}[ru]|{{\equiv}} \ar@/_4ex/[rr]_{f_1\ltimes f_2}^{=} &
Y_1 \times X_2 \ar@{~>}[u]^{s_1}
\ar[r]^{{\mathrm{id}}\times f_2}
\ar@{}[ru]|{{\gtrsim}} &
Y_1 \times Y_2 \ar@{~>}[u]^{q_1} \\
}$$
\
------- ------------------------------------------------------------------------- ----------------------------------------
$(a)$ $q_1\circ ({\mathrm{id}}\times f_2) {\lesssim}s_1$
$(b)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}s_1\circ (f_1\times{\mathrm{id}})$ $(a)$, ${\mathit{subst}}_{{\lesssim}}$
$(c)$ $s_1\circ (f_1\times{\mathrm{id}}) {\equiv}f_1\circ p_1$
$(d)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1$ $(b)$, $(c)$, ${\mathit{comp}}$
------- ------------------------------------------------------------------------- ----------------------------------------
\
\[lem:seq-terminal\] For each function $f_1:X_1\to Y_1$ and pure function $x_2:U{\rightsquigarrow}X_2$ $${\langle {\mathrm{id}}_{Y_1},x_2\circ{{\langle \, \rangle}}_{Y_1} \rangle}\circ f_1 {\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}_{X_1} \rangle}
: X_1 \to Y_1\times X_2 \;.$$
Both handsides can be illustrated as follows: $$\xymatrix@C=5pc{
& Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\
X_1 \ar[r]^{f_1} & Y_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}}
\ar@{~>}[r]^{{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}}
\ar@{}[rd]|{{\equiv}} \ar@{}[ru]|{{\equiv}} &
Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2} \\
& U \ar@{~>}[r]^{x_2} & X_2 \\
} \qquad \xymatrix@C=5pc{
X_1 \ar[r]^{f_1} & Y_1 \\
X_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}}
\ar[r]^{{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{{\equiv}} &
Y_1 \times X_2 \ar@{~>}[u]^{s_1} \ar@{~>}[d]_{s_2} \\
U \ar@{~>}[r]^{x_2} & X_2 \\
}$$
\
--------- -------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------
$(a_1)$ $s_1\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}{\mathrm{id}}$
$(b_1)$ $s_1\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}f_1$ $(a_1)$, ${\mathit{subst}}_{{\equiv}}$
$(a_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}x_2\circ{{\langle \, \rangle}}$
$(b_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}x_2\circ{{\langle \, \rangle}}\circ f_1$ $(a_2)$, ${\mathit{subst}}_{{\equiv}}$
$(c_2)$ ${{\langle \, \rangle}}\circ f_1 {\lesssim}{{\langle \, \rangle}}$ semi-terminality of $U$
$(d_2)$ $x_2\circ{{\langle \, \rangle}}\circ f_1{\lesssim}x_2\circ{{\langle \, \rangle}}$ $(c_2)$, ${\mathit{repl}}_{{\lesssim}}$ ($x_2$ is pure)
$(e_2)$ $s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\lesssim}x_2\circ{{\langle \, \rangle}}$ $(b_2)$, $(d_2)$, ${\mathit{trans}}_{{\lesssim}}$
$(f)$ ${\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 {\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}$ $(b_1)$, $(e_2)$
--------- -------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------
\
\[prop:seq-com\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and pure function $x_2:U{\rightsquigarrow}X_2$ $$q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle}
{\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 : X_1\to Y_2 \;.$$
Both handsides can be illustrated as follows: $$\xymatrix@C=3pc{
X_1 \ar@{~>}[r]^{{\mathrm{id}}} & X_1 & & \\
X_1 \ar@{=}[u] \ar@{~>}[d]_{{{\langle \, \rangle}}}
\ar@{~>}[r]^{{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\;} &
X_1 \times X_2 \ar@/^4ex/[rr]^{f_1\ltimes f_2}_{=}
\ar@{~>}[u]_{p_1} \ar@{~>}[d]^{p_2} \ar[r]^{f_1\times{\mathrm{id}}}
\ar@{}[ld]|{{\equiv}} \ar@{}[lu]|{{\equiv}} \ar@{}[rd]|{{\gtrsim}} &
Y_1 \times X_2 \ar@{~>}[d]^{s_2} \ar[r]^{{\mathrm{id}}\times f_2}
\ar@{}[rd]|{{\equiv}} &
Y_1 \times Y_2 \ar@{~>}[d]^{q_2} \\
U \ar@{~>}[r]^{x_2} & X_2 \ar@{~>}[r]^{{\mathrm{id}}} & X_2 \ar[r]^{f_2} & Y_2 \\
} \quad
\xymatrix@C=2pc{
\mbox{ } \\
X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[d]_{{{\langle \, \rangle}}} \\
& U \ar@{~>}[r]^{x_2} & X_2 \ar[r]^{f_2} & Y_2 \\
}$$
\
------- ---------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------
$(a)$ $q_2\circ ({\mathrm{id}}\times f_2) {\equiv}f_2\circ s_2$
$(b)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(a)$, ${\mathit{subst}}_{{\equiv}}$
{\equiv}f_2\circ s_2\circ (f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle}$
$(c)$ $(f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} prop. \[prop:effect-comp\]
{\equiv}{\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle}$
$(d)$ ${\langle f_1,x_2\circ{{\langle \, \rangle}}\rangle} lemma \[lem:seq-terminal\]
{\equiv}{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$
$(e)$ $(f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(c)$, $(d)$, ${\mathit{trans}}_{{\equiv}}$
{\equiv}{\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$
$(f)$ $f_2\circ s_2\circ (f_1\times{\mathrm{id}}) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(e)$, ${\mathit{repl}}_{{\equiv}}$
{\equiv}f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$
$(g)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(b)$, $(f)$, ${\mathit{trans}}_{{\equiv}}$
{\equiv}f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1$
$(h)$ $p_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle} {\equiv}x_2\circ {{\langle \, \rangle}}$
$(i)$ $f_2\circ s_2\circ {\langle {\mathrm{id}},x_2\circ{{\langle \, \rangle}}\rangle}\circ f_1 $(h)$, ${\mathit{subst}}_{{\equiv}}$, ${\mathit{repl}}_{{\equiv}}$
{\equiv}f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1 $
$(j)$ $q_2\circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}},x_2\circ {{\langle \, \rangle}}\rangle} $(g)$, $(i)$, ${\mathit{trans}}_{{\equiv}}$
{\equiv}f_2\circ x_2\circ {{\langle \, \rangle}}\circ f_1$
------- ---------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------
\
\[thm:seq-prod\] For each functions $f_1:X_1\to Y_1$, $f_2:X_2\to Y_2$ and pure functions $x_1:U{\rightsquigarrow}X_1$ and $x_2:U{\rightsquigarrow}X_2$, the function $(f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle}$ satisfies: $$q_1 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle}
{\lesssim}f_1 \circ x_1 \;\mbox{ and }\;
q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle}
{\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 \circ x_1 \;.$$
$$\xymatrix@C=3pc{
U \ar@{=}[d] \ar@{~>}[rr]^{x_1} && X_1 \ar[rrr]^{f_1} &&& Y_1 \\
U \ar@{=}[d] \ar@{~>}[rr]^{{\langle x_1,x_2 \rangle}} &&
X_1\times X_2 \ar[rrr]^{f_1\ltimes f_2}
\ar@{}[ru]|{{\gtrsim}} \ar@{}[rd]|{{\equiv}} &&&
Y_1 \times Y_2 \ar@{~>}[u]_{q_1} \ar@{~>}[d]^{q_2} \\
U \ar@{~>}[r]^{x_1} & X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{{\langle \, \rangle}}}
& U \ar@{~>}[r]^{x_2} & X_2 \ar[r]^{f_2} & Y_2 \\
}$$
\
--------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------
$(a_1)$ $q_1\circ (f_1\ltimes f_2) {\lesssim}f_1\circ p_1$ prop. \[prop:seq-val\]
$(b_1)$ $q_1\circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} $(a_1)$, ${\mathit{subst}}_{{\lesssim}}$
{\lesssim}f_1\circ p_1 \circ {\langle x_1,x_2 \rangle}$
$(c_1)$ $p_1 \circ {\langle x_1,x_2 \rangle}{\equiv}x_1$ (on values)
$(d_1)$ $f_1\circ p_1 \circ {\langle x_1,x_2 \rangle}{\equiv}f_1\circ x_1$ $(c_1)$, ${\mathit{repl}}_{{\equiv}}$
$(e_1)$ $q_1 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} $(b_1)$, $(d_1)$, ${\mathit{comp}}$
{\lesssim}f_1 \circ x_1$
$(a_2)$ ${\langle x_1,x_2 \rangle} {\equiv}{\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} \circ x_1$ (on values)
$(b_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\equiv}q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} \circ x_1$ $(a_2)$, ${\mathit{repl}}_{{\equiv}}$
$(c_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} prop. \[prop:seq-com\]
{\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1$
$(d_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle {\mathrm{id}}_{X_1},x_2\circ {{\langle \, \rangle}}_{X_1} \rangle} $(c_2)$, ${\mathit{subst}}_{{\equiv}}$
\circ x_1
{\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1\circ x_1$
$(e_2)$ $q_2 \circ (f_1\ltimes f_2) \circ {\langle x_1,x_2 \rangle} {\equiv}f_2 \circ x_2 \circ {{\langle \, \rangle}}_{Y_1} \circ f_1\circ x_1$ $(b_2)$, $(d_2)$, ${\mathit{trans}}_{{\equiv}}$
--------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------
\
The corresponding properties of left sequential pairs easily follow.
\[cor:seq-pair\] For each functions $f_1:X\to Y_1$, $f_2:X\to Y_2$ and pure function $x:U{\rightsquigarrow}X$ $$q_1\circ {\langle f_1,f_2 \rangle}_l {\lesssim}f_1 \,\mbox{, hence }\,
q_1\circ {\langle f_1,f_2 \rangle}_l\circ x {\lesssim}f_1\circ x
\,\mbox{, and }\, q_2\circ {\langle f_1,f_2 \rangle}_l\circ x {\equiv}f_2 \circ x \circ {{\langle \, \rangle}}_{Y_1} \circ f_1 \circ x \;.$$
Effect categories and Arrows {#sec:related}
============================
Starting from [@Moggi91; @Wadler93], *monads* are used in Haskell for dealing with computational effects. A *Monad type* in Haskell is a unary type constructor that corresponds to a *strong monad*, in the categorical sense. Monads have been generalized on the categorical side to *Freyd categories* [@PowerRobinson97] and on the functional programming side to *Arrows* [@Hughes00]. A precise statement of the facts that Arrows generalize Monads and that Arrows are Freyd categories can be found in [@HeunenJacobs06], where each of the three notions is seen as a monoid in a relevant category. Now we prove that cartesian effect categories determine Arrows. In section \[sec:exam\] our approach is compared with the Monads approach, for two fundamental examples. In this section, all effect categories are strict: the congruence ${\equiv}$ is the equality.
Arrows {#subsec:related-arrows}
------
According to [@Paterson01], Arrows in Haskell are defined as follows.
\[defi:related-arr\] An *Arrow* is a binary type constructor class ${\mathtt{A}}$ of the form:
999 = 999 = 999 `class Arrow {\mathtt{A}} where`\
${\mathtt{arr}}:: (X\to Y)\to {\mathtt{A}}\;X\;Y$\
$({>\!\!>\!\!>}):: {\mathtt{A}}\;X\;Y \to {\mathtt{A}}\;Y\;Z \to {\mathtt{A}}\;X\;Z$\
${\mathtt{first}}:: {\mathtt{A}}\;X\;Y \to {\mathtt{A}}\;(X,Z)\;(Y,Z)$\
satisfying the following equations:
----- ------------------------------------------------------------------------------------------------ --- -------------------------------------------------------------------------------
(1) ${\mathtt{arr}}\; {\mathrm{id}}{>\!\!>\!\!>}f $ = $ f$
(2) $f {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathrm{id}}$ = $ f$
(3) $(f {>\!\!>\!\!>}g) {>\!\!>\!\!>}h $ = $ f {>\!\!>\!\!>}(g {>\!\!>\!\!>}h)$
(4) ${\mathtt{arr}}\;(w.v) $ = $ {\mathtt{arr}}\; v {>\!\!>\!\!>}{\mathtt{arr}}\; w$
(5) ${\mathtt{first}}\; ({\mathtt{arr}}\; v) $ = $ {\mathtt{arr}}\; (v\times{\mathrm{id}})$
(6) ${\mathtt{first}}\;(f {>\!\!>\!\!>}g) $ = $ {\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{first}}\; g$
(7) ${\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{arr}}\; ({\mathrm{id}}\times v) $ = $ {\mathtt{arr}}\; ({\mathrm{id}}\times v) {>\!\!>\!\!>}{\mathtt{first}}\; f$
(8) ${\mathtt{first}}\; f {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathtt{fst}}$ = $ {\mathtt{arr}}\; {\mathtt{fst}}{>\!\!>\!\!>}f$
(9) $\;\;{\mathtt{first}}\; ({\mathtt{first}}\; f) {>\!\!>\!\!>}{\mathtt{arr}}\; {\mathtt{assoc}}$ = $ {\mathtt{arr}}\; {\mathtt{assoc}}{>\!\!>\!\!>}{\mathtt{first}}\; f$
----- ------------------------------------------------------------------------------------------------ --- -------------------------------------------------------------------------------
where the functions $(\times)$, ${\mathtt{fst}}$ and ${\mathtt{assoc}}$ are defined as: $$\begin{array}{lll}
(\times) :: & (X\to X')\to(Y\to Y')\to (X,Y)\to (X',Y') &
(f\times g)(x,y)=(f\;x,g\;y) \\
{\mathtt{fst}}:: & (X,Y)\to X &
{\mathtt{fst}}(x,y)=x \\
({\mathtt{assoc}}) :: & ((X,Y),Z)\to (X,(Y,Z)) &
{\mathtt{assoc}}((x,y),z) = (x,(y,z)) \\
\end{array}$$
Cartesian effect categories determine Arrows {#subsec:related-cec-arr}
--------------------------------------------
Let ${\mathbf{V}}_H$ denote the category of Haskell types and ordinary functions, so that the Haskell notation $\mathtt{(X\to Y)}$ represents ${\mathbf{V}}_H(X,Y)$, made of the Haskell ordinary functions from $X$ to $Y$. An arrow ${\mathtt{A}}$ contructs a type ${\mathtt{A}}\;X\;Y$ for all types $X$ and $Y$. We slightly modify the definition of Arrows by allowing $\mathtt{(X\to Y)}$ to represent ${\mathbf{V}}(X,Y)$ for any cartesian category ${\mathbf{V}}$ and by requiring that ${\mathtt{A}}\;X\;Y$ is a set rather than a type. In addition, we use categorical notations instead of Haskell syntax.
So, from now on, for any cartesian category ${\mathbf{V}}$, an *Arrow $A$ on ${\mathbf{V}}$* associates to each points $X$, $Y$ of ${\mathbf{V}}$ a set $A(X,Y)$, together with three operations:
999 = 999 = 999 ${\mathtt{arr}}: {\mathbf{V}}(X,Y)\to A(X,Y)$\
${>\!\!>\!\!>}: A(X,Y) \to A(Y,Z) \to A(X,Z)$\
${\mathtt{first}}: A(X,Y) \to A(X\times Z,Y\times Z)$\
that satisfy the equations (1)-(9).
Basically, the correspondence between a cartesian effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$ and an Arrow $A$ on ${\mathbf{V}}$ identifies ${\mathbf{C}}(X,Y)$ with $A(X,Y)$ for all types $X$ and $Y$. More precisely:
\[thm:related-arr\] Every cartesian effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$ gives rise to an Arrow $A$ on ${\mathbf{V}}$, according to the following table:
The first and second line in the table say that $A(X,Y)$ is made of the functions from $X$ to $Y$ in ${\mathbf{C}}$ and that ${\mathtt{arr}}$ is the convertion from pure functions to arbitrary functions. The third and fourth lines say that ${>\!\!>\!\!>}$ is the (reverse) composition of functions and that ${\mathtt{first}}$ is the semi-product with the identity. Let us check that $A$ is an Arrow; the following table translates each property (1)-(9) in terms of cartesian effect categories (where $\rho_X:X\times U \to X$ is the projection), and gives the argument for its proof.
----- ----------------------------------------------------------------- --- --------------------------------------------------------- --------------------------------------------------
(1) $f\circ {\mathrm{id}}$ = $f$ unitarity in ${\mathbf{C}}$
(2) $ {\mathrm{id}}\circ f $ = $f$ unitarity in ${\mathbf{C}}$
(3) $h\circ (g\circ f) $ = $ (h\circ g)\circ f $ associativity in ${\mathbf{C}}$
(4) $w\circ v$ in ${\mathbf{V}}$ = $w\circ v$ in ${\mathbf{C}}$ ${\mathbf{V}}\subseteq{\mathbf{C}}$ is a functor
(5) $v\times{\mathrm{id}}$ in ${\mathbf{V}}$ = $ v\times{\mathrm{id}}$ in ${\mathbf{C}}$ non-ambiguity of “$\times$”
(6) $(g\circ f)\times{\mathrm{id}}$ = $ (g\times{\mathrm{id}})\circ (f\times{\mathrm{id}})$ proposition \[prop:effect-comp\]
(7) $({\mathrm{id}}\times v) \circ (f\times{\mathrm{id}}) $ = $ (f\times{\mathrm{id}}) \circ ({\mathrm{id}}\times v)$ proposition \[prop:effect-comp\]
(8) $\rho\circ (f\times{\mathrm{id}}_U) $ = $ f\circ \rho$ definition \[defi:effect-arr-prod\]
(9) $\alpha^{-1}\circ ((f\times{\mathrm{id}})\times{\mathrm{id}}) $ = $ (f\times{\mathrm{id}})\circ\alpha^{-1}$ proposition \[prop:effect-assoc\]
----- ----------------------------------------------------------------- --- --------------------------------------------------------- --------------------------------------------------
The translation of the Arrow combinators follows easily, using ${\langle f,g \rangle}_l=(f\ltimes g) \circ {\langle {\mathrm{id}},{\mathrm{id}}\rangle}$ as in section \[subsec:effect-seq\]:
--------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------
$ (id \times f) = \gamma \circ (f \times id) \circ \gamma $ ${\mathtt{second}}\;f = {\mathtt{arr}}\;{\mathtt{swap}}{>\!\!>\!\!>}{\mathtt{first}}\;f {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{swap}}$
$f \ltimes g = ({\mathrm{id}}\times g) \circ (f\times{\mathrm{id}})$ $f {*\!\!*\!\!*}g = {\mathtt{first}}\;f {>\!\!>\!\!>}{\mathtt{second}}\;g $
${\langle f,g \rangle}_l = (f\ltimes g) \circ {\langle {\mathrm{id}},{\mathrm{id}}\rangle}$ $f {\&\!\!\&\!\!\&}g = {\mathtt{arr}}(\lambda b \rightarrow (b,b)) {>\!\!>\!\!>}(f {*\!\!*\!\!*}g) $
--------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------
For instance, in [@Hughes00], the author states that ${\&\!\!\&\!\!\&}$ is not a categorical product since in general $(f {\&\!\!\&\!\!\&}g) {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{fst}}$ is different from $f$. We can state this more precisely in the effect category, where $(f {\&\!\!\&\!\!\&}g) {>\!\!>\!\!>}{\mathtt{arr}}\;{\mathtt{fst}}$ corresponds to $q_1 \circ {\langle f,g \rangle}_l $. Indeed, according to corollary \[cor:seq-pair\]: $$q_1 \circ {\langle f,g \rangle}_l {\lesssim}f \;.$$
Examples {#sec:exam}
========
Here are presented some examples of strict cartesian effect categories. Several versions are given, some of them rely on monads.
Partiality {#subsec:exam-partial}
----------
Let ${\mathbf{V}}={\mathbf{Set}}$ be the category of sets and maps, and ${\mathbf{C}}={\mathbf{Part}}$ the category of sets and partial maps, so that ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. Let ${\lesssim}$ denote the usual ordering on partial maps: $f{\lesssim}g$ if and only if ${\mathcal{D}}(f)\subseteq{\mathcal{D}}(g)$ (where ${\mathcal{D}}$ denotes the domain of definition) and $f(x)=g(x)$ for all $x\in{\mathcal{D}}(f)$. The restriction of ${\lesssim}$ to ${\mathbf{V}}$ is the equality of total maps. Clearly ${\lesssim}$ is not symmetric, but it satisfies all the other properties of a congruence, in particular the replacement property with respect to all maps. So, ${\lesssim}$ is a semi-congruence (which satisfies replacement), that makes ${\mathbf{C}}$ a strict effect category extending ${\mathbf{V}}$. *Warning:* usually the notations are $v:X\to Y$ for a total map and $f:X{\rightharpoonup}Y$ for a partial map, but here we use respectively $v:X{\rightsquigarrow}Y$ (total) and $f:X\to Y$ (partial).
Let us define the pair ${\langle f,v \rangle}$ of a partial map $f:X\to Y_1$ and a total map $v:X{\rightsquigarrow}Y_2$ as the partial map ${\langle f,v \rangle}:X\to Y_1\times Y_2$ with the same domain of definition as $f$ and such that ${\langle f,v \rangle}(x)={\langle f(x),v(x) \rangle}$ for all $x\in{\mathcal{D}}(f)$. It is easy to check that we get a cartesian effect category. For illustrating the semi-product $f\times v$, there are two cases: either $f(x_1)$ is defined, or not, in which case we note $f(x_1)=\bot$. We use the traditional notation $\xymatrix@=1.5pc{ x \ar@{|->}[r]^{f} & y \\ }$ when $y=f(x)$ and its analog $\xymatrix@=1.5pc{ x \ar@{|~>}[r]^{v} & y \\ }$ when $y=v(x)$ and $v$ is pure. $$\xymatrix@C=5pc{
x_1 \ar@{|->}[r]^{f} & y_1 \\
{\langle x_1,x_2 \rangle} \ar@{|~>}[u] \ar@{|~>}[d] \ar@{|->}[r]^{f\times v}
\ar@{}[rd]|{=} \ar@{}[ru]|{=} &
{\langle y_1,y_2 \rangle} \ar@{|~>}[u] \ar@{|~>}[d] \\
x_2 \ar@{|~>}[r]^{v} & y_2 \\
} \qquad \mbox{ or }\qquad
\xymatrix@C=5pc{
x_1 \ar@{|->}[r]^{f} & \bot \\
{\langle x_1,x_2 \rangle} \ar@{|~>}[u] \ar@{|~>}[d] \ar@{|->}[r]^{f\times v}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{=} &
\bot \ar@{|~>}[u] \ar@{|~>}[d] \\
x_2 \ar@{|~>}[r]^{v} & y_2\ne \bot \\
}$$
It can be noted that, in the previous example, ${\mathbf{C}}$ is a 2-category, with a 2-cell from $f$ to $g$ if and only if $f{\lesssim}g$. More generally, let ${\mathbf{C}}$ be a 2-category and ${\mathbf{V}}$ a sub-2-category where the unique 2-cells are the identities. Then by defining $f{\lesssim}g$ whenever there is a 2-cell from $f$ to $g$, we get a strict effect category. In such effect categories, the replacement property holds with respect to all functions in ${\mathbf{C}}$, but the semi-congruence is usually not symmetric.
Let us come back to the partiality example, from the slightly different point of view of the *Maybe monad*. First, let us present this point of view in a naive way, without monads. Let $U=\{\bot\}$ be a singleton, let “$+$” denote the disjoint union of sets, and for each set $X$ let $GX=X+U$ and let $\eta_X:X\to GX$ be the inclusion. Each partial map $f$ from $X$ to $Y$ can be extended as a total map $Gf$ from $X$ to $GY$, such that $Gf(x)=f(x)$ for $x\in{\mathcal{D}}(f)$ and $Gf(x)=\bot$ otherwise. This defines a bijection between the partial maps from $X$ to $Y$ and the total maps from $X$ to $GY$. Let ${\mathbf{C}}$ be the category such that its points are the sets, and a function $X\to Y$ in ${\mathbf{C}}$ is a function $X\to GY$ in ${\mathbf{Set}}$; we say that $X\to Y$ in ${\mathbf{C}}$ *stands for* $X\to GY$ in ${\mathbf{Set}}$. Let $J:{\mathbf{Set}}\to{\mathbf{C}}$ be the functor that is the identity on points and associates to each map $v_0:X\to Y$ the map $\eta_Y\circ v_0$. Let ${\mathbf{V}}=J({\mathbf{Set}})$. Then ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. For all $f,g:X\to Y$ in ${\mathbf{C}}$, that stand for $f,g:X\to GY$ in ${\mathbf{Set}}$, let: $$f{\lesssim}g \iff \forall x\in X\; (f(x)\ne\bot\Rightarrow (g(x)\ne\bot \wedge g(x)=f(x))\;.$$ This yields a strict effect category ${\mathbf{C}}$ extending ${\mathbf{V}}$, with the semi-congruence ${\lesssim}$, and as above the replacement property holds with respect to all functions in ${\mathbf{C}}$ but ${\lesssim}$ is not symmetric. Let $f:X\to Y_1$ in ${\mathbf{C}}$ and $v:X\to Y_2$ in ${\mathbf{V}}$, they stand respectively for $f:X\to GY_1$ and $v=\eta_{Y_2}\circ v_0$ with $v_0:X\to Y_2$. Then, in ${\mathbf{Set}}$, the pair ${\langle f,v_0 \rangle}:X\to GY_1 \times Y_2$ can be composed with: $$t:GY_1\times Y_2=(Y_1+U)\times Y_2\to (Y_1\times Y_2)+U =G(Y_1\times Y_2)\;,$$ that maps ${\langle y_1,y_2 \rangle}$ to itself and ${\langle \bot,y_2 \rangle}$ to $\bot$. Now, let ${\langle f,v \rangle}:X\to Y_1\times Y_2$ in ${\mathbf{C}}$ stand for ${\langle f,v \rangle}=t\circ {\langle f,v_0 \rangle}:X\to G(Y_1\times Y_2)$ in ${\mathbf{Set}}$. Then ${\langle f,v \rangle}$ is a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category. The diagrams for illustrating the semi-product $f\times v$ are the same as above.
This point of view can also be presented using the the *Maybe monad* for managing failures, as follows. We have defined a functor $G:{\mathbf{Part}}\to{\mathbf{Set}}$, that is a right adjoint to the inclusion functor $I:{\mathbf{Set}}\subseteq{\mathbf{Part}}$. The corresponding monad has endofunctor $M=GI$ on ${\mathbf{Set}}$, the category ${\mathbf{C}}$ is the Kleisli category of $M$, and $J:{\mathbf{Set}}\to{\mathbf{C}}$ is the canonical functor associated to the monad. In addition, this monad $M$ is *strong*, and $t$ is the $(Y_1,Y_2)$ component of the *strength* of $M$. But the definition of the semi-congruence ${\lesssim}$, as above, is not part of the usual framework of monads.
State {#subsec:exam-state}
-----
Let ${\mathbf{V}}_0$ be a cartesian category, with a distinguished point $S$ for “the type of states”; for all $X$, let $\pi_X:S\times X\to X$ denotes the projection. Let ${\mathbf{C}}$ be the category with the same points as ${\mathbf{V}}_0$ and with a function $f:X\to Y$ for each function $f:S\times X\to S\times Y$ in ${\mathbf{V}}_0$; we say that $f:X\to Y$ in ${\mathbf{C}}$ *stands for* $f:S\times X\to S\times Y$ in ${\mathbf{V}}_0$. Let $J:{\mathbf{V}}_0\to{\mathbf{C}}$ be the identity-on-points functor which maps each $v_0:X\to Y$ in ${\mathbf{V}}_0$ to the function $J(v_0):X\to Y$ in ${\mathbf{C}}$ that stands for ${\mathrm{id}}_S\times v_0:S\times X\to S\times Y$ in ${\mathbf{V}}_0$. Let ${\mathbf{V}}=J({\mathbf{V}}_0)$, it is a wide subcategory of ${\mathbf{C}}$. For all $f,g:X\to Y$ in ${\mathbf{C}}$, let: $$f{\lesssim}g \iff \pi_Y\circ g = \pi_Y\circ f \;.$$ We get a strict effect category, where the semi-congruence ${\lesssim}$ is symmetric, but does not satisfy the replacement property with respect to all functions in ${\mathbf{C}}$. The semi-product of $f:X\to Y_1$ and $v:X{\rightsquigarrow}Y_2$ is defined as follows. Since $f:S\times X\to S\times Y_1$ in ${\mathbf{V}}_0$ and $v={\mathrm{id}}_S\times v_0$ for some $v_0:X\to Y$ in ${\mathbf{V}}_0$, the pair ${\langle f,v_0\circ\pi_X \rangle}:S\times X\to (S\times Y_1)\times Y_2$ exists in ${\mathbf{V}}_0$. By composing it with the isomorphism $(S\times Y_1)\times Y_2 \to S\times (Y_1\times Y_2)$ we get ${\langle f,v \rangle}:S\times X\to S\times (Y_1\times Y_2)$ in ${\mathbf{V}}_0$, i.e., ${\langle f,v \rangle}:X\to Y_1\times Y_2$ in ${\mathbf{C}}$. It is easy to check that this defines a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category, where the characteristic property of the semi-product $f\times v$ can be illustrated as follows: $$\xymatrix@C=5pc{
(s,x_1) \ar@{|->}[r]^{f} & (s',y_1) \\
(s,x_1,x_2) \ar@{|~>}[u] \ar@{|~>}[d] \ar@{|->}[r]^{f\times v}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{=} &
(s',y_1,y_2) \ar@{|~>}[u] \ar@{|~>}[d] \\
(s,x_2) \ar@{|~>}[r]^{v} & (s,y_2)\ne(s',y_2)
\ar@{|~>}[r]^{\quad\pi_Y} & y_2 \\
}$$
The example above can be curried, thus recovering the *State monad*. A motivation for the introduction of Freyd categories in [@PowerRobinson97] is the possibility of dealing with state in a linear way, as above, rather than in the exponential way provided by the State monad. Now ${\mathbf{V}}_0$ is still a cartesian category with a distinguished point $S$, the “type of states”, and in addition ${\mathbf{V}}_0$ has exponentials $(S\times X)^S$ for each $X$. Then the endofunctor $M(X)=(S\times X)^S$ defines the State monad on ${\mathbf{V}}_0$, with composition defined as usual. It is well-known that $M$ is a strong monad, with strength $t_{Y_1,Y_2}=(S\times Y_1)^S\times Y_2 \to (S\times Y_1\times Y_2)^S$ obtained from ${\mathrm{app}}_{S\times Y_1}\times{\mathrm{id}}_{Y_2}:
S\times (S\times Y_1)^S\times Y_2 \to S\times Y_1\times Y_2$, where “${\mathrm{app}}$” denotes the application function. Hence, from $f:X\to M(Y_1)$ and $v_0:X\to Y_2$ in ${\mathbf{V}}_0$, we can build ${\langle f,v \rangle}=t_{Y_1,Y_2}\circ{\langle f,v_0 \rangle}:X\to M(Y_1\times Y_2)$. Let ${\mathbf{C}}$ be the Kleisli category of the monad $M$, let $J:{\mathbf{V}}_0\to{\mathbf{C}}$ be the canonical functor associated to the monad, and let ${\mathbf{V}}=J({\mathbf{V}}_0)$, then ${\mathbf{V}}$ is a wide subcategory of ${\mathbf{C}}$. A function $f:X\to Y$ in ${\mathbf{C}}$ stands for a function $f:X\to (S\times Y)^S$ in ${\mathbf{V}}_0$. Now, in addition to the usual framework of monads, for all $f,g:X\to Y$ in ${\mathbf{C}}$, i.e., $f,g:X\to (S\times Y)^S$ in ${\mathbf{V}}_0$, let: $$f{\lesssim}g \iff {\pi_Y}^S \circ g = {\pi_Y}^S \circ f \;,$$ where ${\pi_Y}^S:(S\times Y)^S\to Y^S$ associates to each map $m:S\to S\times Y$ the map $\pi_Y\times m:S\to Y$. The relation ${\lesssim}$ defines a semi-conguence on ${\mathbf{C}}$, and ${\langle f,v \rangle}$ is a semi-product, so that ${\mathbf{C}}$ is a cartesian effect category. The characteristic property of the semi-product $f\times v$ can be illustrated as follows: $$\xymatrix@C=5pc{
x_1 \ar@{|->}[r]^{f} & (s\mapsto(s',y_1)) \\
(x_1,x_2) \ar@{|~>}[u] \ar@{|~>}[d] \ar@{|->}[r]^{f\times v}
\ar@{}[rd]|{{\gtrsim}} \ar@{}[ru]|{=} &
(s\mapsto(s',y_1,y_2)) \ar@{|~>}[u] \ar@{|~>}[d] \\
x_2 \ar@{|~>}[r]^{v} & (s\mapsto(s,y_2))\ne(s\mapsto(s',y_2))
\ar@{|~>}[r]^{\qquad\pi_Y^{S}} & (s\mapsto y_2) \\
}$$
Conclusion
==========
We have presented a new categorical framework, called a *cartesian effect category*, for dealing with the issue of multiple arguments in programming languages with computational effects. The major new feature in cartesian effect categories is the introduction of a *semi-congruence*, which allows to define *semi-products* and to prove their properties by decorating the usual definitions, properties and proofs about products in a category. Forthcoming work should study the nesting of several effects.
In order to deal with other issues related to effects, we believe that the idea of *decorations* in logic can be more widely used. This is the case for dealing with exceptions [@B-DuvRey05] (note that a previous attempt to define decorated products can be found in [@F-DuvRey04]). The framework of decorations might be used for generalizing this work in the direction of closed Freyd categories [@PowerThielecke99]. or traced premonoidal categories [@BentonHyland03]. Moreover, with one additional level of abstraction, decorations can be obtained from *morphisms between logics*, in the context of *diagrammatic logics* [@F-DuvLai02; @A-Duv03].
Proofs in cartesian effect categories {#app:proof}
=====================================
Here are proofs for some results in section \[subsec:weak-prod\], called *basic* proofs, followed by their *decorated* versions for the corresponding results in section \[subsec:effect-semi\]. All basic proofs are straightforward. All proofs are presented in a formalized way: each property is preceded by its label and followed by its proof. For the basic proofs, the properties of the congruence are denoted ${\mathit{trans}}$, ${\mathit{sym}}$, ${\mathit{subst}}$, ${\mathit{repl}}$, for respectively transitivity, symmetry, substitution, replacement. For the decorated proofs, the properties of the congruence and the semi-congruence are still denoted ${\mathit{trans}}$, ${\mathit{sym}}$, ${\mathit{subst}}$, ${\mathit{repl}}$, with subscript either ${\equiv}$ or ${\lesssim}$. It should be reminded that ${\mathit{sym}}_{{\lesssim}}$ does *not* hold, and that ${\mathit{repl}}_{{\lesssim}}$ is allowed *only* with respect to a pure function: if $g_1{\lesssim}g_2:Y\to Z$ and $v:Z{\rightsquigarrow}W$ then $v\circ g_1 {\lesssim}v\circ g_2:Y\to W$. In addition, ${\mathit{comp}}$ means compatibiblity of ${\lesssim}$ with ${\equiv}$, which means that if either $f_1{\equiv}f_2{\lesssim}f_3$ or $f_1{\lesssim}f_2{\equiv}f_3$ then $f_1{\lesssim}f_3$. In decorated proofs, “like *basic*” means that this part of the proof is exactly the same as in the basic proof. Proofs of propositions \[prop:weak-assoc\], \[prop:effect-assoc\](associativity) and \[prop:weak-seq\], \[prop:effect-seq\] (parallelism) are left to the reader.
\
----------- ---------------------------------------------------------------- --------------------------------------
1.$\quad$ When $X_1=X_2$
$(a_1)$ $q_1\circ {\langle f_1,f_2 \rangle} \equiv f_1$
$(b_1)$ $f_1 \equiv f'_1$
$(c_1)$ $q_1\circ {\langle f_1,f_2 \rangle} \equiv f'_1$ $(a_1)$, $(b_1)$, ${\mathit{trans}}$
$(c_2)$ $q_2\circ {\langle f_1,f_2 \rangle} \equiv f'_2$ like $(c_1)$
$(d)$ ${\langle f_1,f_2 \rangle} \equiv {\langle f'_1,f'_2 \rangle}$ $(c_1)$, $(c_2)$
2.$\quad$ In all cases
$(e_1)$ $f_1\equiv f'_1$
$(f_1)$ $f_1\circ p_1\equiv f'_1\circ p_1$ $(e_1)$, ${\mathit{subst}}$
$(f_2)$ $f_2\circ p_2\equiv f'_2\circ p_2$ like $(f_1)$
$(g)$ ${\langle f_1\circ p_1,f_2\circ p_2 \rangle} \equiv $(f_1)$, $(f_2)$, $(1)$
{\langle f'_1\circ p_1,f'_2\circ p_2 \rangle}$
----------- ---------------------------------------------------------------- --------------------------------------
\
\
----------- ----------------------------------------------------------------------------------------------------- -------------------------------------
1.$\quad$ When $X_1=X_2$
$(c_1)$ $q_1\circ {\langle f_1,f_2 \rangle} {\equiv}f'_1$ like *basic*
$(a_2)$ $q_2\circ {\langle f_1,f_2 \rangle} {\lesssim}f_2$
$(b_2)$ $f_2 {\equiv}f'_2$
$(c_2)$ $q_2\circ {\langle f_1,f_2 \rangle} {\lesssim}f'_2$ $(a_2)$, $(b_2)$, ${\mathit{comp}}$
$(d)$ ${\langle f_1,f_2 \rangle} {\equiv}{\langle f'_1,f'_2 \rangle}$ $(c_1)$, $(c_2)$
2.$\quad$ In all cases
$(g)$ ${\langle f_1\circ p_1,f_2\circ p_2 \rangle} {\equiv}{\langle f'_1\circ p_1,f'_2\circ p_2 \rangle}$ like *basic*
----------- ----------------------------------------------------------------------------------------------------- -------------------------------------
\
\
The three left handsides can be illustrated as follows: $$\xymatrix@C=3pc@R=1.5pc{
& & Z_1 \\
X \ar[r]^{f} & Y \ar[ru]^{g_1} \ar[rd]_{g_2} \ar[r] &
\bullet \ar[u] \ar[d]
\ar@{}[ld]|(.3){\equiv} \ar@{}[lu]|(.3){\equiv} \\
& & Z_2 \\
} \quad
\xymatrix@C=3pc@R=1.5pc{
& Y_1 \ar[r]^{g_1} & Z_1 \\
X \ar[ru]^{f_1} \ar[rd]_{f_2} \ar[r] &
\bullet \ar[u] \ar[d] \ar[r]
\ar@{}[ld]|(.3){\equiv} \ar@{}[lu]|(.3){\equiv} &
\bullet \ar[u] \ar[d]
\ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} \\
& Y_2 \ar[r]_{g_2} & Z_2 \\
} \quad
\xymatrix@C=3pc@R=1.5pc{
X_1\ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \\
\bullet \ar[u] \ar[d] \ar[r] &
\bullet \ar[u] \ar[d] \ar[r]
\ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} &
\bullet \ar[u] \ar[d]
\ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} \\
X_2 \ar[r]_{f_2} & Y_2 \ar[r]_{g_2} & Z_2 \\
}$$
----------- ------------------------------------------------------------------------------------ ---------------------------------------------
1.$\quad$ When $f_1=f_2(=f)$
$(a_1)$ $r_1\circ {\langle g_1,g_2 \rangle} \equiv g_1$
$(b_1)$ $r_1\circ{\langle g_1,g_2 \rangle}\circ f\equiv g_1\circ f $ $(a_1)$, ${\mathit{subst}}$
$(b_2)$ $r_2\circ {\langle g_1,g_2 \rangle} \circ f \equiv g_2 \circ f $ like $(b_1)$
$(c)$ ${\langle g_1,g_2 \rangle} \circ f \equiv {\langle g_1\circ f,g_2\circ f \rangle}$ $(b_1)$, $(b_2)$
2.$\quad$ When $X_1=X_2$
$(d)$ $(g_1\times g_2) \circ {\langle f_1,f_2 \rangle} $(1)$
\equiv {\langle g_1\circ q_1 \circ {\langle f_1,f_2 \rangle},
g_2\circ q_2 \circ {\langle f_1,f_2 \rangle} \rangle}$
$(e_1)$ $q_1 \circ {\langle f_1,f_2 \rangle} \equiv f_1$
$(f_1)$ $g_1\circ q_1\circ{\langle f_1,f_2 \rangle} \equiv g_1\circ f_1$ ${\mathit{repl}}$
$(f_2)$ $g_2\circ q_2 \circ {\langle f_1,f_2 \rangle}\equiv g_2\circ f_2$ like $(f_1)$
$(g)$ ${\langle g_1\circ q_1 \circ {\langle f_1,f_2 \rangle}, $(f_1)$, $(f_2)$, prop. \[prop:weak-equiv\]
g_2\circ q_2 \circ {\langle f_1,f_2 \rangle} \rangle} \equiv
{\langle g_1\circ f_1,g_2\circ f_2 \rangle}$
$(h)$ $(g_1\times g_2) \circ {\langle f_1,f_2 \rangle} \equiv $(d)$, $(g)$, ${\mathit{trans}}$
{\langle g_1\circ f_1,g_2\circ f_2 \rangle}$
3.$\quad$ In all cases
$(k)$ $ (g_1 \times g_2) \circ {\langle f_1\circ p_1,f_2\circ p_2 \rangle} \equiv $(2)$
{\langle g_1\circ f_1\circ p_1,g_2\circ f_2\circ p_2 \rangle}$
----------- ------------------------------------------------------------------------------------ ---------------------------------------------
\
\
The three left handsides can be illustrated as follows: $$\xymatrix@C=3pc@R=1.5pc{
& & Z_1 \\
X \ar@{~>}[r]^{v} & Y \ar[ru]^{g_1} \ar@{~>}[rd]_{w_2} \ar[r] &
\bullet \ar@{~>}[u] \ar@{~>}[d]
\ar@{}[ld]|(.3){{\gtrsim}} \ar@{}[lu]|(.3){{\equiv}} \\
& & Z_2 \\
} \quad
\xymatrix@C=3pc@R=1.5pc{
& Y_1 \ar[r]^{g_1} & Z_1 \\
X \ar[ru]^{f_1} \ar@{~>}[rd]_{v_2} \ar[r] &
\bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r]
\ar@{}[ld]|(.3){{\gtrsim}} \ar@{}[lu]|(.3){{\equiv}} &
\bullet \ar@{~>}[u] \ar@{~>}[d]
\ar@{}[ld]|{{\gtrsim}} \ar@{}[lu]|{{\equiv}} \\
& Y_2 \ar@{~>}[r]_{w_2} & Z_2 \\
} \quad
\xymatrix@C=3pc@R=1.5pc{
X_1\ar[r]^{f_1} & Y_1 \ar[r]^{g_1} & Z_1 \\
\bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r] &
\bullet \ar@{~>}[u] \ar@{~>}[d] \ar[r]
\ar@{}[ld]|{{\gtrsim}} \ar@{}[lu]|{{\equiv}} &
\bullet \ar@{~>}[u] \ar@{~>}[d]
\ar@{}[ld]|{{\gtrsim}} \ar@{}[lu]|{{\equiv}} \\
X_2 \ar@{~>}[r]_{v_2} & Y_2 \ar@{~>}[r]_{w_2} & Z_2 \\
}$$
----------- --------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------
1.$\quad$ When $f_1=v_2(=v)$
$(b_1)$ $r_1\circ{\langle g_1,w_2 \rangle}\circ v{\equiv}g_1\circ v $ like *basic*
$(a_2)$ $r_2\circ {\langle g_1,w_2 \rangle} {\lesssim}w_2 $
$(b_2)$ $r_2\circ{\langle g_1,w_2 \rangle}\circ v{\lesssim}w_2\circ v $ $(a_1)$, ${\mathit{subst}}_{{\lesssim}}$
$(c)$ ${\langle g_1,w_2 \rangle} \circ v {\equiv}{\langle g_1\circ v,w_2\circ v \rangle}$ $(b_1)$, $(b_2)$
2.$\quad$ When $X_1=X_2$
$(d)$ $(g_1\times w_2) \circ {\langle f_1,v_2 \rangle} $(1)$
{\equiv}{\langle g_1\circ q_1 \circ {\langle f_1,v_2 \rangle},
w_2\circ q_2 \circ {\langle f_1,v_2 \rangle} \rangle}$
$(f_1)$ $g_1\circ q_1\circ{\langle f_1,v_2 \rangle} {\equiv}g_1\circ f_1$ like *basic*
$(e_2)$ $q_2 \circ {\langle f_1,v_2 \rangle} {\lesssim}v_2$
$(f_2)$ $w_2\circ q_2\circ{\langle f_1,v_2 \rangle} {\lesssim}w_2\circ v_2$ ${\mathit{repl}}_{{\lesssim}}$ ($w_2$ is pure)
$(g)$ ${\langle g_1\circ q_1 \circ {\langle f_1,v_2 \rangle}, $(f_1)$, $(f_2)$, prop. \[prop:weak-equiv\]
w_2\circ q_2 \circ {\langle f_1,v_2 \rangle} \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle}$
$(h)$ $(g_1\times w_2) \circ {\langle f_1,v_2 \rangle} {\equiv}{\langle g_1\circ f_1,w_2\circ v_2 \rangle}$ $(d)$, $(g)$, ${\mathit{trans}}_{{\equiv}}$
3.$\quad$ In all cases
$(k)$ $ (g_1 \times w_2) \circ {\langle f_1\circ p_1,v_2\circ p_2 \rangle} {\equiv}{\langle g_1\circ f_1\circ p_1,w_2\circ v_2\circ p_2 \rangle}$ $(2)$
----------- --------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------
\
\
The two left handsides can be illustrated as follows: $$\xymatrix@C=2pc@R=1.5pc{
& Y_1 \ar[r]^{{\mathrm{id}}} & Y_1 \\
X \ar[ur]^{f_1} \ar[dr]_{f_2} \ar[r] &
Y_2\times Y_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]|(.3){\equiv} \ar@{}[lu]|(.3){\equiv} &
Y_1\times Y_2 \ar[u]\ar[d] \ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} \\
& Y_2 \ar[r]_{{\mathrm{id}}} & Y_2 \\
} \quad \xymatrix@C=2pc@R=1.5pc{
X_1 \ar[r]^{{\mathrm{id}}} & X_1 \ar[r]^{f_1} & Y_1 \ar[r]^{{\mathrm{id}}} & Y_1 \\
X_1\times X_2 \ar[u]\ar[d] \ar[r] &
X_2\times X_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} &
Y_2\times Y_1 \ar[u]\ar[d] \ar[r] \ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} &
Y_1\times Y_2 \ar[u]\ar[d] \ar@{}[ld]|{\equiv} \ar@{}[lu]|{\equiv} \\
X_2 \ar[r]_{{\mathrm{id}}} & X_2 \ar[r]_{f_2} & Y_2 \ar[r]_{{\mathrm{id}}} & Y_2 \\
}$$
----------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------
1.$\quad$ When $X_1=X_2$
$(a_1)$ $q_1\circ\gamma_Y \equiv q'_1$
$(b_1)$ $q_1\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} $(a_1)$, ${\mathit{subst}}$
\equiv q'_1 \circ {\langle f_2,f_1 \rangle} $
$(c_1)$ $q'_1 \circ {\langle f_2,f_1 \rangle} \equiv f_1$
$(d_1)$ $q_1\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv f_1$ $(b_1)$, $(c_1)$, ${\mathit{trans}}$
$(d_2)$ $q_2\circ\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv f_2 $ like $(d_1)$
$(e)$ $\gamma_Y \circ {\langle f_2,f_1 \rangle} \equiv {\langle f_1,f_2 \rangle}$ $(d_1)$, $(d_2)$
2.$\quad$ In all cases
$(f)$ ${\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} prop. \[prop:weak-comp\], ${\mathit{sym}}$
\equiv {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $
$(g)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} ${\mathit{repl}}$
\equiv \gamma_Y \circ {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $
$(h)$ $\gamma_Y \circ {\langle f_2\circ p'_2\circ \gamma_X^{-1}, f_1\circ p'_1\circ \gamma_X^{-1} \rangle} $(1)$
\equiv {\langle f_1\circ p'_1\circ \gamma_X^{-1},f_2\circ p'_2\circ \gamma_X^{-1} \rangle}$
$(i_1)$ $p'_1\circ \gamma_X^{-1} \equiv p_1$
$(j_1)$ $f_1\circ p'_1\circ \gamma_X^{-1} \equiv f_1\circ p_1$ $(i_1)$, ${\mathit{repl}}$
$(j_2)$ $f_2\circ p'_2\circ \gamma_X^{-1} \equiv f_2\circ p_2$ like $(j_1)$
$(k)$ $ {\langle f_1\circ p'_1\circ \gamma_X^{-1},f_2\circ p'_2\circ \gamma_X^{-1} \rangle} $(j_1)$, $(j_2)$, prop. \[prop:weak-equiv\]
\equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$
$(l)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} $(g)$, $(h)$, $(k)$, ${\mathit{trans}}$
\equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$
----------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------
\
\
The two left handsides can be illustrated as follows: $$\xymatrix@C=2pc@R=1.5pc{
& Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\
X \ar[ur]^{f_1} \ar@{~>}[dr]_{v_2} \ar[r] &
Y_2\times Y_1 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r]
\ar@{}[ld]|(.3){{\gtrsim}} \ar@{}[lu]|(.3){{\equiv}} &
Y_1\times Y_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{}[ld]|{{\equiv}} \ar@{}[lu]|{{\equiv}} \\
& Y_2 \ar@{~>}[r]_{{\mathrm{id}}} & Y_2 \\
} \quad \xymatrix@C=2pc@R=1.5pc{
X_1 \ar@{~>}[r]^{{\mathrm{id}}} & X_1 \ar[r]^{f_1} & Y_1 \ar@{~>}[r]^{{\mathrm{id}}} & Y_1 \\
X_1\times X_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r] &
X_2\times X_1 \ar@{~>}[u]\ar@{~>}[d] \ar[r] \ar@{}[ld]|{{\equiv}} \ar@{}[lu]|{{\equiv}} &
Y_2\times Y_1 \ar@{~>}[u]\ar@{~>}[d] \ar@{~>}[r] \ar@{}[ld]|{{\gtrsim}} \ar@{}[lu]|{{\equiv}} &
Y_1\times Y_2 \ar@{~>}[u]\ar@{~>}[d] \ar@{}[ld]|{{\equiv}} \ar@{}[lu]|{{\equiv}} \\
X_2 \ar@{~>}[r]_{{\mathrm{id}}} & X_2 \ar@{~>}[r]_{v_2} & Y_2 \ar@{~>}[r]_{{\mathrm{id}}} & Y_2 \\
}$$
----------- ------------------------------------------------------------------------------------ ----------------------------------------
1.$\quad$ When $X_1=X_2$
$(d_1)$ $q_1\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}f_1$ like *basic*
$(a_2)$ $q_2\circ\gamma_Y {\equiv}q'_2$
$(b_2)$ $q_2\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} $(a_2)$, ${\mathit{subst}}_{{\equiv}}$
{\equiv}q'_2 \circ {\langle v_2,f_1 \rangle} $
$(c_2)$ $q'_2 \circ {\langle v_2,f_1 \rangle} {\lesssim}v_2$
$(d_2)$ $q_2\circ\gamma_Y \circ {\langle v_2,f_1 \rangle} {\lesssim}v_2$ $(b_2)$, $(c_2)$, ${\mathit{comp}}$
$(e)$ $\gamma_Y \circ {\langle v_2,f_1 \rangle} {\equiv}{\langle f_1,v_2 \rangle}$ $(d_1)$, $(d_2)$
2.$\quad$ In all cases
$(l)$ $\gamma_Y \circ {\langle f_2\circ p'_2, f_1\circ p'_1 \rangle} \circ \gamma_X^{-1} like *basic*
\equiv {\langle f_1\circ p_1,f_2\circ p_2 \rangle}$
----------- ------------------------------------------------------------------------------------ ----------------------------------------
\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In hard interactions between external particles incident on a heat-bath, we show that large logarithms are generated when a radiated or absorbed gauge boson is collinear with the initial fermion momentum. These logarithms can be absorbed into process independent splitting/absorption probabilities. Unlike the zero-temperature case, however, they depend explicitly on the temperature and the scale of the interaction.'
---
[**Collinear Divergences at One-loop Order\
for External Particles in a Heat-bath.**]{}
[Saumen Datta[Theory Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, Email— [email protected]]{}, Sourendu Gupta[Theory Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, Email— [email protected]]{}, V. Ravindran[Theory Group, Tata Institute of Fundamental Research, Navrangpura, Ahmedabad 380009, India, Email— [email protected]]{}.\
]{}
In recent years the problem of calculating scattering cross sections or rates for external particles travelling in a thermal medium has become phenomenologically interesting [@pheno]. In a previous paper we have shown that to one-loop order there are no infrared divergences when two external particles collide in a heatbath kept at a temperature $T$ [@old]. We were able to resum the one-loop result and calculate the distribution of the mismatch in initial and final momenta due to soft radiation. The result was finite and contained some large logarithms of the type which usually arise from collinear emissions. In this paper we study this collinear behaviour in detail. We find that divergences exist, and are signalled by the familiar “large terms” of the order $\log(Q^2/m^2)$ ($Q^2$ is the scale of the process and $m$ is a regulating mass). We show that the collinear part gives rise to a certain universal splitting/absorption probability which explicitly depends on $Q$ and $T$.
There has been recent interest in collinear singularities associated with soft particles [@many], because they might spoil the hard thermal loop resummation [@htl]. Since we study a hard particle in a heat-bath, we have nothing further to say about this.
Consider the simplest scattering process in which the collinear singularities make their appearance. This is the inclusive cross section for the scattering of a charged electron with a space-like photon, $\gamma^*$, in a QED heat-bath, which we consider in a perturbation theory of the gauge coupling. The initial electron momentum is denoted by $p$, the $\gamma^*$ momentum by $q$ and the final electron momentum by $p'$. The 4-velocity of the heat-bath in any frame, $u$ ($u^2=1$), is a new vector in the problem [@wel82]. Compared to the $T=0$ case, there are extra scalars $p\cdot u$ and $q\cdot u$ which have to be taken into account. For the rest, we use the standard notation $$q^2=-Q^2\qquad{\rm and}\qquad p\cdot q={Q^2\over2x},
\label{int:kine}$$ and work in the limit $Q^2\gg m^2$, where $p^2=p'^2=m^2$. We also take $Q^2\gg T^2$ where $T$ is the temperature of the heat-bath.
The inclusive cross section can be written as a perturbation expansion in the form $$\sigma=\alpha\left(\sigma_0+\alpha\sigma_1+\alpha^2\sigma_2+\cdots\right),
\label{int:cross}$$ where $\alpha$ is the gauge coupling. At the lowest order in perturbation theory, one has only the interaction between $e$ and $\gamma^*$, and hence $\sigma_0$ is independent of $T$. At higher orders, interactions with thermal photons have to be taken into account, and $\sigma_i$ ($i>0$) may change from its $T=0$ value.
It is useful to write the cross section in the form $$\sigma\;=\;{\rho^{\mu\nu} W_{\mu\nu}\over4\sqrt{m^2Q^2+(p\cdot q)^2}},
\qquad{\rm where}\qquad
\rho_{\mu\nu}=\sum_\lambda \epsilon^{*\lambda}_\mu(q)
\epsilon^\lambda_\nu(q).
\label{int:sfdef}$$ $\epsilon^\lambda_\mu(q)$ is the polarisation vector of the off-shell photon in the polarisation state $\lambda$. The tensor $\rho$ is a density matrix for the polarisation states of $\gamma^*$ and is symmetric in its indices. The quantity to be computed in perturbation theory is the symmetric part of the rank-2 tensor $W_{\mu\nu}$, which is the vacuum expectation value of the product of the electromagnetic current coupling to $\gamma^*$.
The computation is simplified by decomposing the tensor $W_{\mu\nu}$ into scalar functions multiplying all symmetric tensors which can be built out of the vectors in the problem. Furthermore, the gauge invariance of the current implies that only those tensors orthogonal to $q$ are relevant. There are four such tensors— $$\begin{array}{rl}
T^1_{\mu\nu}\;=\;g_{\mu\nu}
+{\displaystyle 1\over\displaystyle Q^2}q_\mu q_\nu\quad&\quad
T^2_{\mu\nu}\;=\;P_\mu P_\nu\\
T^3_{\mu\nu}\;=\;U_\mu U_\nu\qquad\quad\;\quad&\quad
T^4_{\mu\nu}\;=\;U_\mu P_\nu+U_\nu P_\mu.\\
\end{array}\label{apten:basis}$$ We have used a shorthand notation for the components of $p$ and $u$ orthogonal to $q$— $$P_\mu\;=\;p_\mu+{\displaystyle p\cdot q\over \displaystyle Q^2}q_\mu,
\quad{\rm and}\quad
U_\mu\;=\;u_\mu+{\displaystyle u\cdot q\over \displaystyle Q^2}q_\mu.
\label{apten:ortho}$$
As a result, $$W_{\mu\nu}=\sum_{i=1}^4 W_i(x,Q^2,p\cdot u,q\cdot u) T^i_{\mu\nu},
\label{apten:sfdef}$$ and hence there are four “structure functions” in this problem. At $T=0$ only the two structure functions $W_1$ and $W_2$ appear. Even for $T>0$, at the leading order of perturbation theory $W_3=W_4=0$, since $\sigma_0$ does not contain any terms involving $u$. Hence the Callan-Gross relation [@cg] is also valid to this order, with corrections generated at higher orders, through the usual vacuum ($T=0$) processes, as well as by additional interactions with real thermal gauge bosons.
9truecm
Reviews of the real-time thermal field theory techniques we use can be found in [@realtime]. Using these we can generate the processes contributing to $\sigma_1$ and the rules for their evaluation. The relevant diagrams with real thermal photon emission and absorption are shown in Figure (\[fg:feynr\]). For one photon emission (absorption) the two-body phase space measure can be taken to be $$d\Gamma_\pm\;=\;
{1\over(2\pi)^4}d^4k 2\pi\delta^\pm(k^2)2\pi\delta^+((p+q-k)^2)
B(k\cdot u),
\label{apsud:measure}$$ where $k$ is the four-momentum of the thermal photon, $\delta^\pm(x^2)=
\delta(x^2)\theta(\pm x_0)$ and $B(x)$ is the Bose distribution $1/[\exp(|x|/T)-1]$. The positive sign in eq. (\[apsud:measure\]) corresponds to the emission process and negative to absorption.
Note that for the diagrams in Figure (\[fg:feynr\]), the vector $u$ appears only in the measure. This leads to a singularity as $k\cdot u\to0$. However, this is not a collinear singularity but the previously analysed soft singularity [@old]. Since $u^2=1$, a boost to the rest frame of $u$ can always be done. This gives us the correct interpretation of the divergence. The only collinear singularities then arise from the matrix elements. These can be identified as divergences in the limit $k\cdot p\to0$ when $m\to0$, and have the same origin as those occurring at $T=0$.
We choose to work in a planar gauge [@ddt], which is a ghost-free gauge specified by the gauge fixing part of the Lagrangian— $${\cal L}_{gf}\;=\; -{1\over2 v^2}(v_\mu A^\mu)\partial^2(v_\nu A^\nu).
\label{pert:gauge}$$ The vector $$v_\mu\;=\;C_1 p'+ C_2 p
\label{pert:gaugedef}$$ defines the gauge choice. The coefficients $C_1$ and $C_2$ are chosen such that $v^2\ne0$ and the sum over polarisations of the real photons becomes $$d_{\alpha\beta}\;\equiv\;
\sum_\lambda\epsilon_\alpha^\lambda(k)\epsilon_\beta^{\lambda *}(k)
\;=\;-g_{\alpha\beta}+{\displaystyle k_\alpha v_\beta
+k_\beta v_\alpha\over k\cdot v},
\label{pert:completeness}$$ where $\epsilon_\mu^\lambda(k)$ is the polarisation vector of a photon with polarisation $\lambda$ and momentum $k$.
It can be verified that in this gauge all collinear singularities come from the squares of the diagrams with the real photon attached to the initial fermion leg. For the first emission diagram, we obtain $$\begin{array}{rl}
|M|^2\;=&\;-2e^4{\displaystyle1\over\displaystyle(p-k)^2}\\&{\rm Tr}
\left[\gamma_\nu p'\!\!\!/ \gamma_\mu \left\{
p\!\!/\left({\displaystyle2p\cdot v\over\displaystyle k\cdot v}-1\right)
+k\!\!/\left(1-{\displaystyle p\cdot v\over\displaystyle k\cdot v}\right)
+v\!\!/{\displaystyle k\cdot p\over\displaystyle k\cdot v}\right\}\right].
\end{array}\label{coll:eq2}$$
We are interested in extracting the leading terms in the collinear limit, $p\cdot k\to0$. The most transparent way of doing this is to use the Sudakov parametrisation [@ddt]— $$k\;=\;(1-\rho)p+\beta(q+xp)+k_{\scriptscriptstyle T},
\qquad{\rm where}\qquad
p\cdot k_{\scriptscriptstyle T}=q\cdot k_{\scriptscriptstyle T}=0.
\label{coll:sud}$$ It is clear that this is a Lorentz invariant decomposition. The integration variables are changed to $\rho$, $\beta$ and the two independent components of $k_{\scriptscriptstyle T}$. The Jacobian is simply $${d^4k\over d\rho d\beta d^2k_{\scriptscriptstyle T}}\;=\;{Q^2\over2x}.
\label{apsud:jacob}$$ The variables $\rho$ and $\beta$ are fixed by the $\delta$-function constraints in the measure $d\Gamma_+$. In the collinear limit, the solution which leads to $p\cdot k\to0$ as $k_{\scriptscriptstyle T}^2\to0$ is $$\rho\;=\;x+{\cal O}\left(k_{\scriptscriptstyle T}^2\right),\qquad\beta\;=\;
{\displaystyle xk_{\scriptscriptstyle T}^2\over\displaystyle Q^2(1-x)}
+{\cal O}\left(k_{\scriptscriptstyle T}^4\right).
\label{apsud:coll}$$ Requiring $\rho$ and $\beta$ to be real, in the limit $m\to0$ we find $$0\;\le\;k_{\scriptscriptstyle T}^2\;\le\;{Q^2\over4x}(1-x).
\label{apsud:limits}$$ The $\theta$-functions place no further restrictions, and may be dropped to give the phase space measure $$d\Gamma_+\;=\;
{\displaystyle xd\rho d\beta d^2k_{\scriptscriptstyle T}
\over\displaystyle 2Q^2(1-x)(2\pi)^2}
\delta\left(\beta-{xz\over1-\rho}\right)\delta(\rho-x).
\label{apsud:final}$$
Absorption is handled by making the change of variables $k_\mu\to-k_\mu$, and then writing a Sudakov parametrisation as before. $d\Gamma_-$ differs from $d\Gamma_+$ only in the change $$\delta(\rho-x)\;\longrightarrow\;\delta(\rho-2+x).
\label{apsud:finalm}$$
In the collinear limit, the denominator on the right of eq. (\[coll:eq2\]) becomes zero, and hence some care is required in taking this limit. We retain the fermion mass, $m$, as a regulator in the denominator and write $${1\over(p-k)^2}\;=\;{1\over m^2(1-2\rho)-Q^2\beta/x}.
\label{coll:den}$$ The most singular terms in the collinear limit can then be found by evaluating the trace in eq. (\[coll:eq2\]) for $m=0$ and neglecting terms in $k_{\scriptscriptstyle T}^2$ (and hence in $\beta$). After some straightforward manipulations, we find that the most singular contribution to the tensor $W_{\mu\nu}$ is $$\begin{array}{rl}
-8e^4&\left[2xT^2_{\mu\nu}
-{\displaystyle Q^2\over\displaystyle2x}T^1_{\mu\nu}\right]\\
&\;\;\int d\Gamma_+\,B[(1-\rho)p\cdot u]
\left({\displaystyle1+\rho^2\over\displaystyle1-\rho}\right)
{\displaystyle1\over\displaystyle m^2(1-2\rho)-Q^2\beta/x}
\end{array}\label{coll:eq3}$$ The contribution from the corresponding absorption diagram, obtained by changing $k\to-k$ in the matrix element and using the phase space measure $d\Gamma_-$, is simply obtained by setting $\rho$ to $2-\rho$ is the above expression.
In the chosen gauge, the squares of the other two diagrams do not have singular denominators and hence can be neglected in the collinear limit. The cross terms do have a singular denominator. However, the trace contains $$\stackrel{\displaystyle{\rm Lt}}{\scriptscriptstyle k\to(1-\rho)p}
d_{\alpha\beta}(k)p^\alpha\;=\;0.$$ Hence this term can also be neglected.
Two facts about eq. (\[coll:eq3\]) and its analogue for photon absorption are worth pointing out. First, both terms contain only the tensors $T^1$ and $T^2$. Hence there are no collinear contributions to the thermal structure functions $W_3$ and $W_4$ at this order. Second, the integrand has a probability interpretation. The part $(1+\rho^2)/(1-\rho)$ can be interpreted as the probability that an incoming electron radiates a photon carrying a fraction $1-\rho$ of the initial momentum. The Bose distribution factor is the probability that the radiated photon is indistinguishable from a thermal photon. The integrand has support on $0\le\rho\le1$, which is consistent with this interpretation. Similarly, the absorption process can also be interpreted as the product of two probabilities. The Bose distribution is the probability of finding a photon in the heat-bath carrying a fraction $\rho-1$ of the incoming electron’s momentum and the factor $(1+(2-\rho)^2)/(1-(2-\rho))$ is the probability of absorption. This integral has support on $1\le\rho\le2$.
The integrals can be performed completely. The thermal leading log part is $$W_{\mu\nu}\;=\;4\pi\alpha^2 P(x,Q/T)
\log\left({\displaystyle Q^2\over \displaystyle m^2}\right)
{\displaystyle x\over\displaystyle Q^2}
\left[2xT^2_{\mu\nu}
-{\displaystyle Q^2\over\displaystyle2x}T^1_{\mu\nu}\right].
\label{coll:result}$$ We have introduced the finite temperature part of the “splitting function” $$P(x,Q/T)\;=\;{2\over\exp[(1-x)p\cdot u/T]-1}\,
\left({\displaystyle 1+x^2\over\displaystyle1-x}\right).
\label{coll:splitting}$$ This can be given an interpretation as the probability of an external electron splitting off a thermal photon. Unlike the case at $T=0$, this factor has an explicit dependence on $Q/T$. Note also that $P(x,Q/T)$ has a singularity as $x\to1$. This is the region of phase space where the thermal photon is soft. We have shown in [@old] that the cross section is finite in this limit provided virtual corrections are taken into account. Using this result, we can simply write down a regulated version of eq. (\[coll:splitting\]) as the splitting probability for the external electron.
At $T=0$ the leading soft divergence in the real diagrams is logarithmic, and shows up in the splitting functions as a divergence of the form $1/(1-x)$, in the limit $x\to1$. It is cured by taking into account the virtual diagrams. The regulated form of the splitting functions is then given by the familiar prescription $$\int dx P_+(x) f(x)\;=\;\int dx P(x)\left[f(x)-f(1)\right],
\qquad\qquad(T=0),
\label{appl:zero}$$ where $f(x)$ is a test function. Note that the first moment of any distribution vanishes when convoluted with the splitting function so regularised.
For $T>0$ the leading soft divergence is quadratic. This is signalled by a divergence of the form $1/(1-x)^2$ ($x\to1$) in the splitting functions. Its cancellation against virtual contributions has been shown in [@old]. The sub-leading logarithmic divergence has also been shown to cancel against virtual corrections [@indu]. Consequently, we just write down an appropriately regularised version of the $T>0$ part of the splitting function— $$\int dx P_+(x) f(x)\;=\;\int dx P(x)
\left[f(x)-f(1)-(x-1) f'(1) \right].
\label{appl:def}$$ Note that the first two moments of $P_+$ vanish with this regularisation. The vanishing second moment implies that finite temperature effects do not change the expectation value of the parton’s momentum. This is expected [@old] and is a consequence of detailed balance.
The universality of this additional finite temperature term in the splitting functions derived here can be easily checked. The calculation for Fermion pair-production is very similiar to the computation presented in this paper, and yields precisely the same collinear term derived here.
In QED, since the electron mass is non-zero, our results are complete. However, in a real experiment we shall have to deal with a QCD heatbath. Our main results, eq. (\[coll:result\]), along with eqs. (\[coll:splitting\]) and (\[appl:def\]), can be carried over to this case with the simple replacement $\alpha^2\to C_{\scriptscriptstyle F}\alpha\alpha_{\scriptscriptstyle S}$. The crucial change is that $m=0$ for quarks and hence the results are singular.
At $T=0$, these collinear singularities are handled by factoring them into universal quark distributions inside hadrons. A similiar procedure will have to be developed for external hadrons or jets impinging on a plasma. In order to complete this program, a suitable definition of the QCD running coupling at finite temperature [@baier] must be provided. At $T=0$, this is sufficient information to sum the one-loop iterated ladder diagrams into the DGLAP equations [@dglap].
For $T>0$ the situation is more complicated. This is clear from the fact that the analogue of the splitting function contains the dimensionless variable $Q/T$ in addition to $x$. At finite temperature and arbitrary scale $Q^2$, we are forced to consider two scales in the renormalisation group [@twoscale]. In various domains these simplify. For example, when $Q\gg T$, one expects to be able to use a single scale. In this limit, the regularised version of eq (\[coll:splitting\]) vanishes, and the evolution in $Q^2$ is the same as at $T=0$. However, the parton distributions at each $T$ must then be seperately measured. Only by keeping two scales can information at $T=0$ be evolved to $T>0$. This work is left to the future.
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| {
"pile_set_name": "ArXiv"
} |
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1.6cm
[**Correlations in flavor-changing $\mib{tqZ}$ couplings**]{} 0.18cm [**constrained via experimental data**]{}
Zenrō HIOKI$^{\:1),\:}$[^1] Kazumasa OHKUMA$^{\:2),\:}$[^2] andAkira UEJIMA$^{\:2),\:}$[^3]
*$1)$ Institute of Theoretical Physics, University of Tokushima*
*Tokushima 770-8502, Japan*
0.2cm
*$2)$ Department of Information and Computer Engineering,*
*Okayama University of Science*
*Okayama 700-0005, Japan*
ABSTRACT
plus 0.1pt minus 0.1pt Possible non-standard $tqZ$ couplings, where $q=c$ or $u$, originated from general flavor-changing-neutral-current interactions are studied model-independently using the effective Lagrangian consisting of several $SU(3)\times SU(2) \times U(1)$ invariant dimension-6 operators. After the electroweak symmetry breaking, these operators are recombined to form four kinds of independent terms whose coefficients are complex in general. The Lagrangian could therefore include up to eight independent coupling parameters. Through searches for experimentally allowed regions of these parameters, it is found that some correlations exist among the signs and sizes of those couplings. 1.5cm
PACS: 12.38.Qk, 12.60.-i, 14.65.Ha
Searches for Flavor-Changing Neutral Current (FCNC) are a quite attractive mission at future collider experiments: The existence of physics beyond the standard model is strongly indicated if new phenomena originated from FCNC are observed, because the event probability of such phenomena is too tiny to detect within the standard-model framework [@Glashow:1970gm]. In exploring such rare processes, the top quark is expected to play an especially important role, since it decays without being affected by non-perturbative effects thanks to its short lifetime [@Bigi:1980az; @Bigi:1986jk] in contrast to the other heavy quarks. We therefore studied top-quark FCNC processes model-independently in our latest paper [@Hioki:2018asl] and derived constraints on $tqZ$ ($q=u/c$) couplings, which induce FCNC interactions, using the effective Lagrangian.[^4] We however did not deal with any correlation among those coupling constants there, so we focus on this issue in this short letter and study how these couplings are related with each other. In our analysis, we use the following effective Lagrangian to describe the general $tqZ$ interactions [@Buchmuller:1985jz; @Arzt:1994gp; @AguilarSaavedra:2008zc; @Grzadkowski:2010es] : $$\begin{aligned}
{1}\label{eq:efflag_decay}
&{\cal L}_{tqZ} = -\frac{g}{2 \cos \theta_W}
\Bigl[\,\bar{\psi}_q(x)\gamma^\mu(f_1^L P_L + f_1^R P_R)\psi_t(x)Z_\mu(x) \Bigr.
\nonumber\\
&\phantom{========}
+\bar{\psi}_q(x)\frac{\sigma^{\mu\nu}}{M_Z}(f_2^L P_L + f_2^R P_R)
\psi_t(x)\partial_\mu Z_\nu(x) \,\Bigr],\end{aligned}$$ where $g$ and $\theta_W$ are the $SU(2)$ coupling constant and the weak mixing angle, $P_{L/R}\equiv(1\mp\gamma_5)/2$, $f_{1/2}^{L/R}$ stand for the non-standard couplings parameterizing contributions from relevant $SU(3) \times SU(2) \times U(1)$ gauge invariant dimension-6 effective operators [@Hioki:2018asl; @AguilarSaavedra:2008zc]. We treat these coupling parameters as complex numbers independent of each other in order to perform analyses as model-independently as possible. Thus, the resultant $tqZ$ couplings are expressed by up to eight independent parameters.
Using the above Lagrangian, we can derive the theoretical partial decay width as an eight-variable function ${\mit \Gamma}^{\rm th}_{tqZ}({f_ {1/2}^{L/R}})$. On the other hand, the experimental partial decay width ${\mit \Gamma}^{\rm exp}_{tqZ}$ is obtained by the product of the branching ratio ${\rm Br}(t \to q Z)$ and the top-quark total decay width ${\mit\Gamma}_t$ : ${\mit \Gamma}^{\rm exp}_{tqZ}={\rm Br}(t \to q Z)\times {\mit\Gamma}_t$. Then, allowed regions for $f_ {1/2}^{L/R}$ are obtained by varying their real and imaginary parts at the same time and searching for the parameter space that satisfies $ {\mit \Gamma}^{\rm th}_{tqZ}({f_ {1/2}^{L/R}}) < {\mit \Gamma}^{\rm exp}_{tqZ}$.
Let us briefly show the result for the $tcZ$ couplings as an example (see Ref. [@Hioki:2018asl] for more detailed results): their allowed regions at 95% confidence level for ${\mit\Gamma}_t=1.322$ GeV [@Gao:2012ja] [^5] and ${\rm Br}(t \to c Z)<2.3 \times 10^{-4}$ [@ATLAS:2017beb] (thus, ${\mit \Gamma}^{\rm exp}_{tqZ}=3.0\times 10^{-4}$) are derived as $$\left|{\rm Re/Im}(f_{1}^{L/R})\right|\leq 3.4\times 10^{-2},~~
\left|{\rm Re/Im}(f_{2}^{L/R})\right|\leq 2.8\times 10^{-2}.$$ Here, we should comment on the meaning of the [*allowed*]{} regions. This means that if we give one parameter a value outside its allowed range, we can no longer reproduce the current experimental data no matter how we vary the other parameters.
Now, using these results and carrying out similar computations, we investigate if there is a certain relationship among the couplings by setting one of them to its [*maximum*]{} value and varying all the others. When ${\rm Re}(f_1^L)$ is taken as such a fixed parameter, the allowed regions of the remaining couplings are derived as Table \[tab:one\_fix\_sample1\]. From this table, we can see a relation between ${\rm Re}(f_1^L)$ and ${\rm Re} (f_2^R)$: the sign of ${\rm Re} (f_2^R)$ is opposite to that of ${\rm Re}(f_1^L)$ and the size of ${\rm Re} (f_2^R)$ is the same order as ${\rm Re}(f_1^L)$. Furthermore, in Table \[tab:one\_fix\_sample2\], we show the almost same one as Table \[tab:one\_fix\_sample1\] but in the case that ${\rm Re}(f_2^R)$ is set to its [*minimum*]{} value. Then, we find a quite similar correlation between ${\rm Re}(f_1^L)$ and ${\rm Re}(f_2^R)$ again.
There it must be also very meaningful to point out the following fact: While we are now observing the relation between ${\rm Re}(f_1^L)$ and ${\rm Re}(f_2^R)$ assuming that they take the maximal/minimal values, the other parameters can also have some allowed space though its size is one order of magnitude smaller. In fact, even if those other parameters would have no allowed area, the allowed space for the correlated pair does not change drastically.
On the other hand, if we assume as an extreme case that only one non-standard coupling exists in the $tqZ$ interactions, the allowed region of that coupling becomes rather small : In the case that only ${\rm Re}(f_{1}^{L})$ or ${\rm Re}(f_{2}^{R})$ exists in the $tcZ$ couplings, we can get the corresponding allowed region as $\left|{\rm Re}(f_{1}^{L})\right|\leq 1.9\times 10^{-2}$ or $\left|{\rm Re}(f_{2}^{R})\right|\leq 1.5\times 10^{-2}$.
-- ------------- -------------------------------- -------------------------------- --------------------------------
Re($f_1^L$) Im($f_1^L)$ Re($f_1^R$) Im($f_1^R$)
$-4.0\times 10^{-3}$ $-4.0\times 10^{-3}$ $-4.0\times 10^{-3}$
[(Fixed)]{} $\phantom{-}4.0\times 10^{-3}$ $\phantom{-}4.0\times 10^{-3}$ $\phantom{-}4.0\times 10^{-3}$
-- ------------- -------------------------------- -------------------------------- --------------------------------
: Allowed minimum and maximum values of the $tcZ$-coupling parameters for ${\mit\Gamma}_{tcZ} = 3.0 \times 10^{-4}$ in the case that Re($f_2^R$) is fixed to its minimum value $-2.8\times 10^{-2}$.[]{data-label="tab:one_fix_sample2"}
\
-- -------------------------------- -------------------------------- ---------------------- --------------------------------
Re($f_2^L$) Im($f_2^L)$ Re($f_2^R$) Im($f_2^R$)
$-3.0\times 10^{-3}$ $-3.0\times 10^{-3}$ $-2.5\times 10^{-2}$ $-3.0\times 10^{-3}$
$\phantom{-}3.0\times 10^{-3}$ $\phantom{-}3.0\times 10^{-3}$ $-2.2\times 10^{-2}$ $\phantom{-}3.0\times 10^{-3}$
-- -------------------------------- -------------------------------- ---------------------- --------------------------------
: Allowed minimum and maximum values of the $tcZ$-coupling parameters for ${\mit\Gamma}_{tcZ} = 3.0 \times 10^{-4}$ in the case that Re($f_2^R$) is fixed to its minimum value $-2.8\times 10^{-2}$.[]{data-label="tab:one_fix_sample2"}
-- -------------------------------- -------------------------------- -------------------------------- --------------------------------
Re($f_1^L$) Im($f_1^L)$ Re($f_1^R$) Im($f_1^R$)
$\phantom{-}2.6\times 10^{-2}$ $-5.0\times 10^{-3}$ $-5.0\times 10^{-3}$ $-5.0\times 10^{-3}$
$\phantom{-}3.1\times 10^{-2}$ $\phantom{-}5.0\times 10^{-3}$ $\phantom{-}5.0\times 10^{-3}$ $\phantom{-}5.0\times 10^{-3}$
-- -------------------------------- -------------------------------- -------------------------------- --------------------------------
: Allowed minimum and maximum values of the $tcZ$-coupling parameters for ${\mit\Gamma}_{tcZ} = 3.0 \times 10^{-4}$ in the case that Re($f_2^R$) is fixed to its minimum value $-2.8\times 10^{-2}$.[]{data-label="tab:one_fix_sample2"}
\
-- -------------------------------- -------------------------------- ------------- --------------------------------
Re($f_2^L$) Im($f_2^L)$ Re($f_2^R$) Im($f_2^R$)
$-4.0\times 10^{-3}$ $-4.0\times 10^{-3}$ $-4.0\times 10^{-3}$
$\phantom{-}4.0\times 10^{-3}$ $\phantom{-}4.0\times 10^{-3}$ [(Fixed)]{} $\phantom{-}4.0\times 10^{-3}$
-- -------------------------------- -------------------------------- ------------- --------------------------------
: Allowed minimum and maximum values of the $tcZ$-coupling parameters for ${\mit\Gamma}_{tcZ} = 3.0 \times 10^{-4}$ in the case that Re($f_2^R$) is fixed to its minimum value $-2.8\times 10^{-2}$.[]{data-label="tab:one_fix_sample2"}
We then performed the same analyses for all the remaining parameters. The results are summarized as follows:
- There hold relational expressions ${\rm Re/Im}(f_{1/2}^{L/R})=-{C\,\rm Re/Im}(f_{2/1}^{R/L})$, where the maximum ranges of $f_1^{L/R}$ and $f_2^{L/R} $ are given with $0.93 \lesssim C \lesssim 1.1$ and $0.65 \lesssim C \lesssim 0.73$ respectively by substituting the maximum or minimum values of $f_2^{R/L}$ and $f_1^{R/L}$ in the right-hand side of the expressions.
- Three or more non-standard couplings cannot take large values (within the allowed regions) at the same time.
- The allowed region could be roughly twice larger when two non-standard coupling constants exist than in the case that only one non-standard coupling constant exists.
We also confirmed that these results are the same as those of similar analyses performed for $t \to u Z$. That is, what we found here is common to both the $tcZ$ couplings and the $tuZ$ couplings.
Finally, let us note to what extent our results depend on the upper limit of the branching ratio of $t \to q Z$. We used here the present data ${\rm Br}(t \to c Z)
<2.3 \times 10^{-4}$ [@ATLAS:2017beb], but it is expected to be improved by half at High-Luminosity Large Hadron Collider [@CMS-PAS-FTR-13-016]. Therefore we performed the same computations assuming that reduced upper bound. Of course, the allowed parameter ranges are thereby narrowed, which we already showed in the previous work [@Hioki:2018asl], but interestingly enough we found that the coupling-parameter correlations and related results are little affected and still hold even there. In conclusion, we have studied here possible non-standard $tqZ$ couplings and correlations among them in the framework of the effective Lagrangian. This Lagrangian can incorporate up to eight independent coupling parameters describing FCNC interactions. The allowed regions of these couplings satisfying the current experimental limits get larger when several numbers of couplings exist than when only one coupling exists. It was found that the allowed region becomes the largest when there are relations as ${\rm Re/Im}(f_{1/2}^{L/R})=-C\,{\rm Re/Im}(f_{2/1}^{R/L})$ with $C \simeq 0.65 \sim 0.73$ ($C \simeq 0.93 \sim 1.1$) in the case that the maximum or minimum value of $f_1^{R/L}$ ($f_2^{R/L}$) is substituted in the right-hand side. Tables \[tab:one\_fix\_sample1\] and \[tab:one\_fix\_sample2\] also tell us that the non-standard couplings except for the two correlated ones can take non-zero values as well though their sizes are one order of magnitude smaller than those of the correlated ones. However, even if they had no allowed regions, the current experimental limits could be realized by the existence of correlated two couplings alone.
Our results are from the current experimental data, but it is quite interesting to note that the parameter correlations and related results are little affected even if the experimental limits on ${\rm Br}(t \to c Z)$ are improved, e.g., by half at future facilities. Since this analysis was performed in a very general framework and does not depend on any special assumptions, the results pointed out here will be useful information for constructing a specific model inducing rather strong FCNC interactions. This work was partly supported by the Grant-in-Aid for Scientific Research (C) Grant Number 17K05426 from the Japan Society for the Promotion of Science. plus 0.1pt minus 0.1pt
[99]{} S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D [**2**]{} (1970) 1285. I.I.Y. Bigi and H. Krasemann, Z. Phys. C [**7**]{} (1981) 127. I.I.Y. Bigi, Y.L. Dokshitzer, V.A. Khoze, J.H. Kuhn and P.M. Zerwas, Phys. Lett. B [**181**]{} (1986) 157. Z. Hioki, K. Ohkuma and A. Uejima, Mod. Phys. Lett. A, to appear (arXiv:1809.01389 \[hep-ph\]). W. Buchmuller and D. Wyler, Nucl. Phys. B [**268**]{} (1986) 621. C. Arzt, M.B. Einhorn and J. Wudka, Nucl. Phys. B [**433**]{} (1995) 41 (hep-ph/9405214). J.A. Aguilar-Saavedra, Nucl. Phys. B [**812**]{} (2009) 181 (arXiv:0811.3842 \[hep-ph\]). B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, JHEP [**1010**]{} (2010) 085 (arXiv:1008.4884 \[hep-ph\]). J. Gao, C.S. Li and H.X. Zhu, Phys. Rev. Lett. [**110**]{} (2013) 042001 (arXiv:1210.2808 \[hep-ph\]). The ATLAS collaboration , ATLAS-CONF-2017-070. CMS Collaboration, CMS-PAS-FTR-13-016.
[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
[^3]: E-mail address: [email protected]
[^4]: We have given a detailed list of preceding works by other authors in [@Hioki:2018asl].
[^5]: The direct measurement of the total decay width of the top quark is consistent with the prediction by the standard model, but the measured one still has a large uncertainty. Therefore, we use the standard-model value here instead of the experimental value (see Ref.[@Hioki:2018asl] for related discussions). Since we are focusing on the rare decays, this replacement does not bring any significant problem into our analysis.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present and discuss the results of a Monte-Carlo simulation of the phase transition in pure compact U(1) lattice gauge theory with Wilson action on a hypercubic lattice with periodic boundary conditions. The statistics are large enough to make a thorough analysis of the size dependence of the gap. In particular we find a non-zero latent heat in the infinite volume limit. We also find that the critical exponents $\nu$ and $\alpha$ are consistent with the hyperscaling relation but confirm that the critical behavior is different from a conventional first-order transition.'
address: |
Centre de Physique Théorique, Ecole Polytechnique,\
Centre National de la Recherche Scientifique, UPR A0014,\
91128 Palaiseau Cedex, France
author:
- Claude Roiesnel
title: |
PC-561-09-97
Finite size effects at phase transition in compact U(1) gauge theory [^1]
---
INTRODUCTION
============
The interest about the nature of the phase transition in compact 4D U(1) lattice gauge theory has been revived by the recent development of two new line of results. On the one hand, Kerler, Rebbi and Weber [@KRW] studied the critical properties of the model by adding to the standard Wilson action a coupling $\lambda$ controlling the density of monopoles. They concluded to the existence of a non-gaussian second-order critical point in the $(\beta, \lambda)$ plane. Damm and Kerler [@KD] are also investigating whether the critical exponent of this transition is universal or changes with $\lambda$.
On the other hand, Jersàk, Lang and Neuhaus [@JLN] studied the compact U(1) gauge theory on lattices with sphere-like topology with a Wilson action extended by a coupling $\gamma$ of charge 2 $$S=\beta \sum_{P} \cos \Theta_P + \gamma \sum_{P} \cos 2\Theta_P$$$$ They found no gap on these lattices for $\gamma \leq 0$. They also made a thorough finite-size size scaling analysis of their data and concluded to the existence, for $\gamma \leq 0$, of a second-order transition with a non-gaussian continuum limit.
Finally there is an investigation [@COX] of the scaling behaviour of gauge-ball masses and of the static potential, which seems to confirm the second-order nature of the transition also on lattices with periodic boundary conditions at $\gamma = -0.2$ and $\gamma =
-0.5$.
However we must note that the critical exponents of these two approaches are different. Moreover there is always an apparent contradiction between the simulations on lattices with spherical topology and on lattices with periodic boundary conditions since one observes a gap on the latter even when $\gamma < 0$. One can fairly state that some confusion about the nature of the transition still persists.
Therefore it is useful to reconsider the simulation of compact pure 4D U(1) lattice gauge theory with the standard Wilson action ($\gamma = 0$) and periodic boundary conditions provided that such a simulation fulfills two goals not met previously [@JNZ; @FSS; @GNC]:
- give an estimation of the infinite volume limit of the gap which is observed on finite-size lattices with periodic boundary conditions.
- make a careful modern finite-size scaling analysis of the bulk critical behavior (as has been done on lattices with sphere-like topology [@JLN]).
In order to estimate the infinite volume limit of the gap, one needs extrapolation formulas which require at least 3 parameters. Also the asymptotic scaling formulas depend in general upon 3 parameters. Determining these parameters from the measurement of one observable only requires at least 6 to 7 data points. One needs more data points if one wants to have some chance of estimating the subleading correction terms.
Therefore to be as systematic as possible the simulation has been done on 9 lattice sizes from $L=4$ to $L=16$. One could argue that these linear sizes are too small to reach the asymptotic scaling regime. If this turns out to be indeed the case, then these smaller lattice sizes are anyhow needed to determine the corrections to scaling which will be required even when data on larger lattices become available.
SIMULATION 1
============
Since accurate data were lacking for many of the above lattice sizes, a first simulation has been done to determine their pseudo-critical coupling to about 1 part in $10^{-4}$. In order to reach this accuracy a scanning of the pseudo-critical regions was done with a step $\Delta\beta=10^{-4}$ and $10^{5}$ iterations at each coupling constant $\beta$.
We used standard histogramming techniques to locate the double-peak structure found in these simulations which is usually characteristic of a first-order transition. A pseudo-critical coupling was defined as the coupling for which peaks have equal statistical weight. The latent heat was defined as the gap between the two peaks at this pseudo-critical coupling.
The details of this simulation are described in [@ROI] and we only give here a short summary of the results.
A three-paramater fit to the gap $\Delta e(L)$ of the form $$\label{fss}
\Delta e(L) = \Delta e(\infty) + a L^{-b}$$ gives $\Delta e(\infty) = 0.014(5)$ and $b = 1.03(16)$. We note that the exponent $b$ is not consistent with the first-order prediction (at least in Potts models) $b=D=4 \cite{BK,BGB}$. Fixing $b$ to 1 reproduces the data quite well but with an infinite volume limit of the latent heat which is definitely different from zero. However it should be stressed that it is very difficult to constrain the functional form of a three-parameter fit to the gap data. Indeed an exponential fit $$\label{exp}
\Delta e(L) = \Delta e(\infty) + a \exp(-b L)$$ reproduces the data as well with $\Delta e(\infty) = 0.0278(15)$.
An asymptotic finite-size scaling analysis of the pseudo-critical couplings $$\label{bc}
\beta_{c}(L)=\beta_{c}(\infty) + a L^{-\frac{1}{\nu}}$$ yields the results $b_{c}(\infty)=1.01132(10)$ and $\nu=0.326(8)$.
We have also checked the scaling of the maxima of the specific heat for lattice sizes in the range $L=4-12$. A two-parameter ansatz $$C_{V,max}(L) = a L^{\frac{2}{\nu}-D}$$ gives the independent determination $\nu=0.330(2)$ but with a rather high $\chi^{2} \approx 3$ which hints at the need of correction terms to the asymptotic formula.
All these results taken together confirm the rather paradoxical nature of the U(1) phase transition. The critical exponents are completely consistent with a second-order phase transition but with an index $\nu
\approx 0.33$ which is different from the value, $\nu \approx 0.36$, quoted in [@JLN; @COX]. This discrepancy raises the suspicion about universality at different values of $\gamma$. On the other hand any reasonable fitting ansatz to the gap data yields a non-zero value of the latent heat in the infinite volume limit. But again the approach to this limit is different from the asymptotic formula expected within the description of first-order transitions in the double gaussian approximation [@BGB]. This disagreement might mean that the asymptotic regime is not yet reached with lattice sizes up to $L=16$.
SIMULATION 2
============
Going to larger lattices is impracticable with local algorithms since already we could not overcome the hysteresis on the $16^4$ lattice with $10^5$ iterations. However it is possible to attack the problem indirectly by increasing the statistics on the smaller lattices so as to make a full finite size analysis including corrections to scaling. The comparison of the finite size scaling of several cumulants can unravel the systematic errors in the critical exponents induced by the corrections to scaling. Combined fits can reveal whether the critical exponents vary when excluding the smallest lattice sizes.
Therefore we have made a second simulation at 3 to 5 coupling constants selected in the pseudo-critical interval determined in simulation 1 at each lattice size, except $L=16$. $10^6$ iterations have been done at each coupling constant. These $10^6$ iterations were divided in two independent runs, $5\times 10^5$ sweeps each, respectively from a hot start and a cold start, using different random generators.
The data analysis, which is not yet completed, makes an extensive use of the reweighting technique [@FS]. All independent runs on the same lattice size are used as independent samples at the same $\beta$. The total amount of statistics that we get is quite comparable to many of the Monte-Carlo simulations of 3D spin models.
We are making a finite-size scaling analysis of 3 cumulants: the specific heat per plaquette $C_{v}$, the Binder Cumulant $U_{4}$ and the second cumulant $U_{2}$. We are also adding an analysis of their derivatives $dC_{v}/d\beta,
dU_{4}/d\beta, dU_{2}/d\beta$. These 6 cumulants are algebraically independent and the position of their extrema defines a pseudo-critical coupling if located in the scaling region. The value of each pseudo-critical coupling is determined independently for every run by minimizing the corresponding reweighted histogram with respect to $\beta$. Finally we take the statistical average over all runs at each lattice size. The caveat of the method, and the limiting factor of its applicability, is to ensure that the Ferrenberg-Swendsen technique remains valid throughout the minimization process.
CONCLUSION
==========
Preliminary results of the second simulation confirm the non-zero value of the latent heat in the infinite volume limit. Fits for $\Delta e(\infty)$ with Eq. \[fss\] and Eq. \[exp\] are completely consistent between both simulations. However the value of the parameter $b$ in Eq. \[fss\] increases to $\approx 1.40$ and the corresponding $\chi^2$ is much larger than for the exponential fit which becomes highly favored.
Preliminary results from independent asymptotic finite size scaling fits to the various definitions of the pseudo-critical couplings show deviations $\approx 2\%$ among the values of the critical exponent $\nu$. Even if these deviations are much larger than the statistical errors, it will be very difficult to extract the corrections to scaling.
[99]{} W. Kerler, C. Rebbi and A. Weber, Phys. Lett. [**B392**]{} (1997) 438. G. Dammm and W. Kerler, HUB-EP-97/63 (1997), hep-lat/9709061. J. Jersák, C.B Lang and T. Neuhaus, Phys. Rev. Lett. [**77**]{} (1996) 1933; Phys. Rev. [**D54**]{} (1996) 6909. J. Cox, W. Franzki, J. Jersàk, C.B. Lang, T. Neuhaus, A. Seyfried, P. W. Stephenson, Nucl. Phys. [**B499**]{} (1997) 371. J. Jersák, T. Neuhaus and P.M. Zerwas, Phys. Lett. [**133B**]{} (1983) 103. H.G. Evertz, J. Jersák, T. Neuhaus and P.M. Zerwas, Nucl. Phys. [**B251**]{} \[FS13\] (1985) 279; J. Jersák, T. Neuhaus and P.M. Zerwas, Nucl. Phys. [**B251**]{} \[FS13\] (1985) 299. R. Gupta, M.A. Novotny and R. Cordery, Phys. Lett. [**B172**]{} (1986) 86. C. Roiesnel, Phys. Lett. [**B405**]{} (1997) 126. C. Borgs and R. Kotecký, J. Stat. Phys. [**61**]{} (1990) 79. A. Billoire, T. Neuhaus and B. Berg, Nucl. Phys. [**B413**]{} (1994) 795. A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. [**61**]{}, (1988) 2635; [**63**]{} (1989) 1658; [**63**]{} (1989) 1195.
[^1]: Contribution to Lattice ’97, International Symposium, Edinburgh, UK, 1997.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Because of a logarithmic enhancement from soft, collinear magnetic gluons, in dense quark matter the gap for a color superconducting condensate with spin zero depends upon the QCD coupling constant $g$ not as $\exp(-1/g^2)$, like in BCS theory, but as $\exp(-1/g)$. In weak coupling, the ratio of the transition temperature to the spin-zero gap at zero temperature is the same as in BCS theory. We classify the gaps with spin one, and find that they are of the same order in $g$ as the spin-zero gap.'
address: |
a) Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA\
b) RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
author:
- 'Robert D. Pisarski$^{a}$ and Dirk H. Rischke$^{b}$'
title: Gaps and Critical Temperature for Color Superconductivity
---
In cold, dense quark matter, the attractive interaction between quarks of different colors generates color superconductivity [@bl; @general; @schafer0; @prlett; @prscalar; @son; @schuster; @schafer; @hong; @hongetal]. In this Letter we discuss in what aspects color superconductivity differs from the classic model of Bardeen, Cooper, and Schrieffer (BCS) [@BCS], and in which aspects it resembles it.
One way in which color superconductivity differs from BCS theory is the dependence of the condensate on the coupling constant. In theories with short-ranged interactions, such as BCS theory, the gap depends upon the coupling constant $g$ as the exponential of $1/g^2$. We argued previously, though, that static magnetic interactions are [*not*]{} screened to any finite order in $g$ [@prlett; @son]. The scattering of quarks near the Fermi surface is then logarithmically enhanced by the emission of collinear, nearly static magnetic gluons, and this changes the gap from an exponential in $1/g^2$ to one in $1/g$.
The explicit value of the gap in weak coupling was first computed by Son [@son]. Using an elegant renormalization group argument, he found that there is an instability at a scale $\phi_0 \sim b_0 \, \mu \, g^{-5}\, \exp(-c_0/g)$, where $\mu$ is the quark-chemical potential, $c_0 = 3 \pi^2/\sqrt{2}$, and $b_0$ is a pure number. To explicitly compute the magnitude of the spin-zero gap at zero temperature, $\phi_0$, it is necessary to solve a gap equation. This was initiated by Son [@son]. In this Letter we first extend Son’s analysis to estimate the constant $b_0$. To the order in $g$ at which we work, all of our results are manifestly gauge invariant. Details will be presented elsewhere [@wip].
Next, we solve the gap equation at non-zero temperature, $T$, and show that the critical temperature for the onset of color superconductivity, $T_c$, divided by $\phi_0$ is [*equal*]{} to the value in BCS theory [@BCS], $T_c/\phi_0 \simeq 0.567 + O(g)$.
Finally, we classify the gaps for massless fermions with total spin $J=1$. There are two types, longitudinal and transverse to the direction of momentum of the quarks in the condensate, $\phi_1^{\parallel}$ and $\bbox{\phi}_1^{\perp}$, respectively. In agreement with Son [@son], we find that all spin-one gaps are of the same order as the spin-zero gap: $\phi_1/\phi_0$ is a pure number of order one.
Our results are of practical importance. Bailin and Love assumed in their original analysis [@bl] that static magnetic interactions are screened, so that the gaps are BCS-like, and thus tiny, $\phi_0 \sim 10^{-3} \mu$. Our results are only valid perturbatively, but if we extrapolate to strong coupling, we find that because the constant $b_0$ is huge, the gaps can become quite large: as seen in fig. 1, $\phi_0$ peaks at $\phi_0 \sim 0.26 \, \mu$, with a big $T_c \sim 0.15 \, \mu$. At AGS energies, heavy-ion collisions can probe the region of $\mu \sim 600$ MeV and $T \sim 100$ MeV. Therefore, by triggering on collisions in which [*cool*]{}, dense nuclear matter is formed, it may be possible to observe color superconductivity.
That $J=1$ gaps are not exponentially suppressed is important for quark stars. At very high densities, the chemical potential of up, down, and strange quarks are nearly equal, so the $J=0$ color-flavor locked condensate is surely favored [@general]. At intermediate densities, however, because of the large strange quark mass, and the requirement of charge neutrality, these chemical potentials will differ. This suppresses the formation of $J=0$ gaps, which are predominantly flavor off-diagonal. The $J=1$ gaps, however, can form between quarks of the same flavor, and will be significant.
We follow the notations and conventions of our previous work [@prlett; @prscalar]. For massless quarks there are four types of spin-zero condensates [@bl; @prlett; @prscalar]: right-handed condensates $\phi^\pm_{r,\pm}$, and left-handed condensates $\phi^\pm_{\ell,\mp}$. The superscript refers to particles or antiparticles, while the subscript denotes helicity. In perturbation theory, QCD is manifestly chirally symmetric, so that the gap equations for $\phi^\pm_r$ and $\phi^\pm_\ell$ are [*identical*]{} order by order in $g^2$. Although the magnitude of the gaps for $\phi_{r}$ and $\phi_{\ell}$ must then be equal, because they are complex numbers, they differ by an arbitrary phase. This phase represents the spontaneous breaking of parity by a spin-zero, color superconducting gap in an instanton-free regime [@prlett; @prconf].
Without loss of generality, then, we can consider only the right-handed gaps, denoted as $\phi^+$ and $\phi^-$, and take them to be real and positive. Suppressing chiral projectors, and the color and flavor indices [@color], the gap function is $$\Phi(Q) = \phi^+(Q) \; \Lambda^+({\bf q})
+ \phi^-(Q) \; \Lambda^-({\bf q}) \; . \;
\label{e3}$$ The condensate is formed from a quark, with four-momentum $Q=(q^0,{\bf q})$, and a charge conjugate antiquark. $\Lambda^\pm ({\bf q}) \equiv (1 \pm \gamma_0 \bbox{\gamma}
\cdot \hat{\bf q})/2$ are projectors for energy; ${\bf q} = q \, \hat{\bf q}$, $\hat{\bf q}^2 = 1$.
Including the gap, from (15) of [@prscalar] the quark propagator is $$G(Q) = \left[
\frac{\Lambda^{+}({\bf q})}{q_0^2 -
\epsilon^+_q\,^2}
+\, \frac{ \Lambda^{-}({\bf q})}{q_0^2-
\epsilon^-_q\,^2}
\right]
(\gamma \cdot Q - \mu \gamma_0) ,
\label{e4}$$ where $\epsilon^\pm_q$ is the energy of the quark relative to the Fermi surface: $$\epsilon^\pm_q \equiv \sqrt{( q \mp \mu)^2 +
\phi^\pm(Q)^2} \,\, .
\label{e5}$$ The poles with $\mp \epsilon^+_q$ represent quasiparticles and their holes, those with $\mp \epsilon^-_q$ quasi-antiparticles and their holes [@prscalar]. At the Fermi surface, $q = \mu$, it takes very little energy to excite a quasiparticle, $\epsilon^+_q = -\phi^+$, and a lot to excite a quasi-antiparticle, $\epsilon^-_q \approx -2 \mu$.
At one-loop order, from (A35) of [@prscalar] the equation for the gap function $\Phi(K)$ is $$\Phi(K) = \frac{2 g^2}{3}
%\int \frac{d^4 Q}{(2 \pi)^4}
\frac{T}{V}\sum_Q
\Delta_{\mu \nu}(K\!\!-\!\!Q)
\gamma^\mu G_0^-(Q) \Phi(Q) G(Q) \gamma^\nu .
\label{e6}$$ Here $G_0^-(Q) = 1/(\gamma \cdot Q - \mu \gamma_0)$ is the bare propagator for charge-conjugate quarks [@color]. To evaluate the Matsubara sum over $q^0$ we use spectral representations [@lebellac].
In the gap equation, the gluon propagator $\Delta^{\mu \nu}$ includes the effects of “hard dense loops” (HDL) [@lebellac]. The basic parameter of the HDL Lagrangian is the gluon “mass”, $m_g$; for $N_c$ colors and $N_f$ flavors of massless quarks, $$m_g^2 \; = \; N_f \, \frac{g^2 \mu^2}{6 \pi^2} \; + \;
\left( N_c + \frac{N_f}{2} \right)
\frac{g^2 T^2}{9} \; .
\label{e2}$$ For the time being we take strict Coulomb gauge for the HDL propagator. HDL corrections can be neglected for the quark propagator and the quark-gluon vertex, as the quark lines are hard, $q \sim \mu$.
We solve the gap equation by including the effects of the superconducting state in the simplest possible way for the quark, Eq. (\[e4\]), and not at all for the gluon. This is reasonable in weak coupling, because the scale of the condensate, $\phi_0 \sim \mu \, \exp(-c_0/g)$, is much smaller than either $\mu$ or $m_g \sim g \mu$ [@gap].
As in strong coupling BCS theory [@BCS], $\Phi(K)$ has an imaginary part, but for small $g$ this can be neglected in QCD [@imag]. Consequently, the only values of the gap functions $\phi^\pm(Q)$ which enter into the gap equation are those on either the quasiparticle mass shell, $\phi^+(\pm \epsilon^+_q,q)$, or the quasi-antiparticle mass shell, $\phi^-(\pm \epsilon^-_q,q)$.
Gap equations for $\phi^\pm(\epsilon^\pm_k,k)$ are derived from (\[e6\]) via projection with $\Lambda^\pm({\bf k})$. As is typical in models of superconductivity [@BCS], the dominant terms arise from the quasiparticle poles. These correspond physically to scattering of quarks near the Fermi surface. As this involves little energy transfer between the quarks, it suffices to use the nearly static limit of the gluon propagator.
With these approximations, denoting $\epsilon_k^+ = \epsilon_k$, the gap equation for $\phi(k) \equiv
\phi^+(\epsilon_k,k)$ becomes [@wip] $$\phi(k) = \frac{g^2}{36\pi^2} \int_{\mu-\delta}^{\mu +\delta}
\frac{{\rm d}q}{\epsilon_q} \; \frac{1}{2}
\, \ln \left( \frac{b^2 \mu^2}{\epsilon_q^2 -\epsilon_k^2} \right)
\tanh \left(\frac{\epsilon_q}{2T} \right) \phi(q) \,\, ,
\label{e7}$$ $$b = \frac{b_0}{g^5} = b_{\rm t}^2 \; b_{\rm l}^3 \; b_0'
= 256 \; \pi^4 \left(\frac{2}{g^2 N_f}\right)^{5/2} b_0'\;,
\label{e8}$$ where $ b_{\rm t} = 4 \sqrt{2}\, \mu/(\sqrt{3 \pi} \,m_g)$, and $b_{\rm l} = 2 \, \mu/(\sqrt{3} \, m_g)$. The logarithm $\sim \ln[1/(\epsilon_q^2 - \epsilon_k^2)]$ arises from the cut term in the spectral density of a nearly static transverse gluon [@prlett; @son]. In the gap equation, there are also terms $\sim \ln(1/g)$ which arise from the non-static transverse gluons and from static longitudinal gluons; these produce the constants $b_{\rm t}$ and $b_{\rm l}$, respectively. In addition, there are terms $\sim 1$ in the gap equation which contribute to the constant $b_0'$; we did not compute these terms. In deriving (\[e7\]) we assume that $\epsilon_k, \epsilon_q < \mu$, so we introduce a cut-off $\delta$ on the $q$-integration; the final result is independent of $\delta$.
At $T=0$, an approximate solution of (\[e7\]) is [@wip] $$\begin{aligned}
\label{e10}
\phi(k) & = & \phi_0 \, \sin (\bar{g}\, y_k) \,\, , \\
\bar{g} \equiv \frac{g}{3\sqrt{2}\pi} & , & \;\;\;
y_k \equiv \ln \left( \frac{2b\mu}{|k-\mu|+\epsilon_k} \right)\;,\nonumber\end{aligned}$$ where $\phi_0$ denotes the value of the condensate at the Fermi surface, $k = \mu$. (This is similar, but not identical to the solution of [@son; @schafer; @hongetal].) As $y_\mu = \ln(2b\mu/\phi_0)$, Eq. (\[e10\]) requires $\bar{g}\, y_\mu = \pi/2$, [*i.e.*]{}, $$\label{e1}
\phi_0 = 2\, b \mu \, \exp\left(-\frac{\pi}{2 \bar{g}}\right)\;.$$ Our results for $c_0$ and the prefactor $1/g^5$ are in agreement with Son [@son]. The constants $b_{\rm t}$ and $b_{\rm l}$ are the same found in an independent analysis by Schäfer and Wilczek [@schafer]; see also Hong [*et al.*]{} [@hongetal].
In BCS-like theories with zero-ranged interactions, such as Nambu–Jona-Lasinio (NJL) models [@general], all particle pairs around the Fermi surface contribute with [*equal*]{} weight to build up the BCS-logarithm, so that the gap function is constant: $\sim g^2 \int
{\rm d}q / \epsilon_q \simeq g^2 \ln(2\delta/\phi_0)$, with solution $\phi_0 \sim 2 \delta \, \exp(-1/g^2)$. In a model where fermions interact with scalar bosons of mass $M_s \sim g \mu$ [@prscalar], scattering of particle pairs through small angles is [*favored*]{}. The collinear singularity is cut off by $M_s \neq 0$, so that logarithmic factors of $\sim \ln (\mu/M_s) \sim \ln(1/g)$ appear in the gap equation.
In QCD, the scattering of quark pairs through small angles is again favored. If the exchanged gluon is electric, the collinear singularity is cut off by the Debye mass, $\sqrt{3}\,
m_g$. This produces $\ln (1/g)$ terms which contribute to the prefactor $1/g^5$ in $b$, Eq. (\[e8\]). If the exchanged gluon is magnetic, the collinear singularity is only cut off by the difference in energies between the incoming and outgoing pairs. In the gap equation (\[e7\]), this generates the logarithmic enhancement factor $\sim \ln[1/(\epsilon_q^2 -
\epsilon_k^2)]$. The dependence of the gap function on $\epsilon_k$ is then not negligible. Quasiparticles with momenta exponentially close to the Fermi surface, $\epsilon_q \sim b \mu \, \exp(-c/g)$, dominate the integral, with a contribution which is enhanced by $ \ln(b\mu/\epsilon_q) \sim c/g$. The gap function $\phi(q)$ is weighted towards these pairs, as $\phi(q)/\phi_0 \sim \sin(\pi c/2c_0) \sim 1$. For quasiparticles which are not exponentially close to the Fermi surface, $\epsilon_q \sim \mu$ and $c\sim g$, the gap function is down by $\phi(q)/\phi_0 \sim g$ [@imag].
The temperature dependence of the condensate can be computed from Eq. (\[e7\]) as follows. We assume that the temperature $T$ is of the order of the gap at zero temperature, $\phi_0$. Let us introduce a dimensionless parameter $\kappa \gg 1$, and divide the integration region into $ \epsilon_q\geq \kappa \phi_0$ and $\epsilon_q < \kappa \phi_0$. Away from the Fermi surface, $\epsilon_q \gg \phi_0$, the Fermi–Dirac distribution becomes a Boltzmann distribution, so $\tanh (\epsilon_q/2T) \simeq 1$, and thermal effects are negligible. Near the Fermi surface, the thermal factor $\tanh (\epsilon_q/2T)$ cuts off any singularity, even at the critical temperature, $T_c$, when $\phi(q) \rightarrow 0$. Then the gap function is the same as (\[e10\]) for $\epsilon_k \gg \kappa \phi_0$, and a constant for $\epsilon_k \ll \kappa \phi_0$. Matching the two regions at $\kappa \phi_0$, and then sending $\kappa \rightarrow \infty$, we derive the condition $$\label{e12}
\int_0^\infty {\rm d}|q-\mu| \left[ \frac{1}{\epsilon_q} \,
\tanh\left(\frac{\epsilon_q}{2T}\right) - \frac{1}{\epsilon_q^0}
\right] = 0\;,$$ where $\epsilon_q = \sqrt{(q-\mu)^2 +\phi^2(T)}$, with $\phi(T)$ the gap at the Fermi surface at a temperature $T$, and $\epsilon_q^0 = \sqrt{(q-\mu)^2 +\phi_0^2}$. This is correct to leading order in $g$. Equation (\[e12\]) implicitly determines the function $\phi(T)/\phi_0$; it is [*identical*]{} to that obtained in BCS theory in weak coupling [@BCS; @wip]. In particular, the ratio of the critical temperature to the zero-temperature gap is the same as in BCS, $T_c/\phi_0 = \zeta/2 +O(g)$, where the constant $\zeta=2\, e^\gamma /\pi\simeq 1.13$. ($\gamma \simeq 0.577$ is the Euler-Mascheroni constant.)
Following the classification of [@prlett; @prscalar], for massless quarks a spin-one condensate has the form $$\sum_{h, e} \left( \bbox{\phi}^{\parallel e}_h(Q)
\cdot \hat{\bf q} + \bbox{\phi}^{\perp e}_h(Q)
\cdot {\bf P}({\bf q}) \cdot \bbox{\gamma}
\right) {\cal P}_h \, \Lambda^{e}({\bf q}) \;,$$ where the sum runs over chiralities, $h= r,\ell$, and energies, $e=\pm$. ${\cal P}_{r,\ell} = (1 \pm \gamma_5)/2$ is the chiral projector, and ${\bf P}({\bf q})={\bf 1} - \hat{\bf q} \hat{\bf q}$ a projector onto the subspace orthogonal to ${\bf q}$. Because a spin-one condensate is a three-vector, there are 12 types of condensates, four longitudinal, $\phi_{1,h}^{\parallel e} \equiv \bbox{\phi}^{\parallel e}_{h} \cdot
\hat{\bf q}$, and eight transverse, $\bbox{\phi}_{1,h}^{\perp e} \equiv
\bbox{\phi}^{\perp e}_h \cdot {\bf P} ({\bf q})$. This classification is equivalent to that of [@bl]. While the spin-zero gaps are symmetric in the simultaneous interchange of color and flavor indices [@bl; @prlett; @prscalar], the longitudinal gaps $\phi_{1,h}^{\parallel e}$ are antisymmetric. The transverse gaps fulfill a more complicated relationship, $(\bbox{\phi}^{\perp \pm}_{r, \ell})^T = - \bbox{\phi}^{\perp \mp}_{\ell,r}$. The spin-zero gaps and the longitudinal spin-one gaps do not mix quarks of different chirality; the transverse spin-one gaps do, and thus break chiral symmetry. The gap equations can be constructed as in the spin-zero case [@wip]. We find that both the longitudinal as well as the transverse gaps fulfill the same gap equation as the spin-zero gaps, with identically the same solution as in (\[e1\]), except that the constant analogous to $b_0'$ (which we do not compute) may differ.
In the static limit, gauge dependent terms in the gluon propagator $\Delta^{\mu \nu}(P)$ are $\sim p^\mu p^\nu/p^2$ [@lebellac]. These terms contribute to the gap equation, but neither to $c_0$, the power of $g$ in the prefactor, $b_{\rm t}$, nor $b_{\rm l}$. They do appear to contribute to the undetermined constant $b_0'$, but we suggest that in the end, $b_0'$ is gauge invariant. There are other effects which contribute to $b_0'$ [@wip]. One-loop diagrams with a [*soft*]{}, transverse HDL gluon propagator renormalize the quark [@wave] and gluon wave functions, and the quark-gluon vertex. Other contributions arise from the influence of the condensate on the gluon propagator [@gap], and the admixture of quasi-antiparticle modes in the quasiparticle gap equation. It is important to calculate $b_0'$, since its numerical value determines exactly [*which*]{} patterns of symmetry breaking are favored.
While the results which we have derived are rigorously valid only in weak coupling, it is interesting to plot $\phi_0/\mu$ as a function of $g$, fig. 1. We take $N_f=2$ (note that $b_0 \sim 1/N_f^{5/2}$). From (\[e8\]), $\phi_0/\mu$ is proportional to $b_0'$; in fig. 1 we set this undetermined constant equal to 1. Equation (\[e1\]) has the form of a semiclassical tunneling probability, including a prefactor from five zero modes. Because of the “zero modes”, the gap function peaks at a value of $\phi_0/\mu \sim 0.26$ when $g \sim 4.2$.
Extending the picture of Schäfer and Wilczek [@schafer0], we view quark matter as a color superconducting “liquid”, and hadronic matter as a color superconducting “vapor”. From [@prlett] there is a first-order phase transition between these liquid and vapor phases at $\mu=\mu_c$ and $T=0$. Then perhaps at several times nuclear matter density, the liquid phase occurs at the maximum of $\phi_0/\mu$, and the vapor phase at larger $g$, providing a qualitative explanation for the smallness of the analogous gaps in hadronic matter [@general].
We conclude by using (\[e6\]) to estimate the validity of perturbation theory. Perturbative calculations break down when $m_g \simeq \mu$ or $T$. For $N_c= 3$ and $N_f = 2$, at $T \neq 0$ and $\mu =0$, $m_g = T$ when the QCD fine structure constant is tiny, $\alpha_s \equiv g^2/4 \pi \sim 0.18$. In contrast, at $\mu \neq 0$ and $T = 0$, $m_g = \mu$ when $\alpha_s$ is much larger, $\alpha_s \sim 2.4$. This suggests to us that while perturbation theory is not a good approximation for hot quark-gluon matter [@temp], it may well be a reasonable guide to understanding dense quark matter, as long as it is cold, $T < 0.3 \mu$.
This work was supported in part by DOE grant DE-AC02-98CH10886. We thank M. Alford, J. Berges, W. Brown, V. Emery, D.K. Hong, S.D.H. Hsu, J.T. Liu, V.N. Muthukumar, K. Rajagopal, H.C. Ren, T. Schäfer, D. Son, and F. Wilczek for enlightening discussions. We especially thank T. Schäfer for discussions on the ratio $T_c/\phi_0$. D.H.R. thanks RIKEN, BNL and the U.S. Department of Energy for providing the facilities essential for the completion of this work.
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The interaction between two quarks contains two pieces, which are symmetric or antisymmetric in the color indices of the fundamental representation. The antisymmetric representation is attractive to lowest order in $g$; for an $SU(N_c)$ gauge theory, the coefficient in (\[e6\]) is $g^2(N_c+1)/(2 N_c)$ [@schuster]. When $N_c = 3$, the antisymmetric representation is the color $\overline{{\bf 3}}$ representation, the symmetric the color ${\bf 6}$. Fermi statistics for a $J=0$ gap imposes constraints which require the number of massless flavors, $N_f \geq 2$ [@prlett]. The form of the quark propagator in (\[e4\]) is only valid when $N_f=2$, and the $(SU(3)_c,SU_{r,\ell}(N_f))$ representation is $(\overline{{\bf 3}}, {\bf 1})$ [@prlett]. When $N_f =3$, the $(\overline{{\bf 3}}, {\overline {\bf 3}})$ representation mixes with the $({\bf 6}, {\bf 6})$ [@prlett]. This mixing only affects the gap equation to higher order $\sim \phi_0^2$, which is negligible in weak coupling.
M. Le Bellac, [*Thermal Field Theory*]{} (Cambridge, Cambridge University Press, 1996).
Due to infrared singular factors, the effective action for the condensate is $\sim |D_\mu \phi|^2/\phi_0^2$, so that the (true) gluon mass from color superconductivity is not $m_{super}
\sim g \phi_0$, as one would naively expect, but much larger, $m_{super} \sim g \mu$. (We thank T. Schäfer for discussions on this point.) To the order at which we work, this is irrelevant for the gap equation, because the dominant momenta are $\gg \phi_0$, and on that scale, corrections from the condensate are small, $\sim g^2 \phi_0/q$ at large $q \gg \phi_0$.
From (\[e7\]) the imaginary part of $\phi(k)$ arises from the cut in the logarithm for $\epsilon_q < \epsilon_k$, $
{\rm Im}\, \phi(k) \sim g^2 \int_{\phi_0}^{\epsilon_k}
{\rm d}\epsilon_q/\epsilon_q\; \phi(q)
\simeq g^2 \, \ln(\epsilon_k/\phi_0) \phi_0 .
$ Taking $\epsilon_k \sim b \mu \, \exp(-c/g)$, momenta exponentially close to the Fermi surface occur when $c \sim 1$. In this region, the imaginary part of the gap function, ${\rm Im}\, \phi(k) \sim g (c_0-c) \phi_0$, is [*down*]{} by $g$ relative to the real part, ${\rm Re}\, \phi(k) \sim \sin(\pi c/2 c_0) \phi_0$. Away from the Fermi surface, $\epsilon_k \sim \mu$, so $c \sim g $, and $\phi(k)$ is strongly damped, with the real and imaginary parts of comparable magnitude, $ {\rm Re}\, \phi(k) \sim {\rm Im}\, \phi(k) \sim g \phi_0$.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
This paper was inspired by four articles: surface cluster algebras studied by Fomin-Shapiro-Thurston [@fst], the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky [@dwz], string modules associated to arcs on unpunctured surfaces by Assem-Br$\ddot{u}$stle-Charbonneau-Plamondon [@acbp] and Quivers with potentials associated to triangulated surfaces, part II: Arc representations by Labardini-Fragoso. [@lf2]. For a surface with marked points ($\Sigma,M$) Labardini-Fragoso associated a quiver with potential $(Q(\tau),S(\tau))$ then for an ideal triangulation of ($\Sigma,M$) and an ideal arc Labardini-Fragoso defined an arc representation of $(Q(\tau),S(\tau))$. This paper focuses on extent the definition of arc representation to a more general context by considering a tagged triangulation and a tagged arc. We associate in an explicit way a representation of the quiver with potential constructed Labardini-Fragoso and prove that the Jacobian relations are met.
\
author:
- |
Salomón Domínguez,\
Facultad de Economía y Negocios,\
Universidad Anáhuac Campus Sur,\
[[email protected]]{}
title: Arc Representations
---
[6]{}
I. Assem, T. Br$\ddot{u}$stle, G. Charbonneau-Jodoin and P. Plamondon . [*Gentle algebras arising from surface triangulations.* ]{} Algebra & Number Theory 4 (2010), No. 2, 201-229.
P. Caldero, F. Chapoton y R. Schiffler. [*Quivers with relations arising from clusters*]{} Transactions of the American Mathematical Society. 358 (2006), No. 3, 1347-1364.
H. Derksen, J. Weyman and A. Zelevinsky. [*Quiver With Potential and Their Representations I: Mutations.*]{} Selecta Mathematica 14 (2008), No. 1, 59-119.
S. Domínguez, C. Geiss. [*A Caldero-Chapoton formula for generalized cluster categories.*]{} Journal of Algebra 399 (2014), 887-893. V. V. Fock, A. B. Goncharov, [*Dual Teichmuler and Lamination Spaces.*]{} Handbook of Teichmuler theory, No. 1, 647-684, IRMA Lectures in Mathematics and Theorical Physics 11, European Mathematical Society, 2007.
S. Fomin, M. Shapiro, D. Thurston. [*Cluster Algebras and Triangulated Surfaces, Part I: Cluster Complexes*]{}. Acta Mathematica 201 (2008), 83-146.
S. Fomin, A. Zelevinsky. [*Cluster Algebras I: Foundations*]{}. Journal of the American Mathematical Society 15 (2002), 497-529.
M. Gekhtman, M. Shapiro, A. Vainstein. [*Cluster Algebras and Weil-Petersson Forms*]{}. Duke Mathematical Journal 127 (2005) 291-311. D. Labardini-Fragoso.[*Quivers with potentials associated to triangulated surfaces*]{}. Proceeding of the London Mathematical Society (2009), No. 98, 797-839.
D. Labardini-Fragoso. [*Quivers with potentials associated to triangulated surfaces, part II: Arc representations.*]{}arXiv:0909.4100
Giovanni Cerulli Irelli and Daniel Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials. Compositio Mathematica (2012),No. 148, 1833-1866. D. Labardini-Fragoso. [*Quiver With Potentials Associated to Triangulated Surfaces, Part IV: Removing Boundary Assumptions.*]{}Selecta Mathematica, No. 22, 145-189.
L. Mosher. [*Tiling the projective foliation space of a punctured surface.*]{} Transactions of the American Mathematical Society. 113 (1987), 229 - 339.
G. Musiker, R. Schiffler and L. Williams. [*Positivity for Cluster From Surfaces.*]{} Advances in Mathematics (2011),No. 227 , 2241 - 2308.
Y. Palu. [*Cluster characters for 2-Calabi-Yau triangulated categories.*]{} Annales De L’Institute Fourier (Grenoble) 58 (6) (2008) 2221 - 2248.
I. Assem, A. Skowronski y D. Simpson (2006). [*Elements of the Representation Theory of Associative Algebras: No. 1: Techniques of Representation Theory.*]{} London Mathematical Society, Student Texts.
| {
"pile_set_name": "ArXiv"
} |
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abstract: |
In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping $$\left\{ \enspace
\begin{aligned}
&{\partial}_t\rho+\opdiv(\rho u)=0,\\
&{\partial}_t(\rho u)+\opdiv\left(\rho u\otimes u+p\,{\textup{\uppercase\expandafter{\romannumeral1}}}_d\right)=-{\alpha}(t)\rho u,\\
&\rho(0,x)=\bar \rho+{\varepsilon}\rho_0(x),\quad u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is ${\alpha}(t)=\frac{\mu}{(1+t)^{\lambda}}$ with ${\lambda}\ge0$ and $\mu>0$, $\bar\rho>0$ is a constant, $\rho_0,u_0 \in C_0^\infty({\mathbb R}^d)$, $(\rho_0,u_0)\not\equiv 0$, $\rho(0,x)>0$, and ${\varepsilon}>0$ is sufficiently small. One can totally divide the range of ${\lambda}\ge0$ and $\mu>0$ into the following four cases:
Case 1: $0\le{\lambda}<1$, $\mu>0$ for $d=2,3$;
Case 2: ${\lambda}=1$, $\mu>3-d$ for $d=2,3$;
Case 3: ${\lambda}=1$, $\mu\le 3-d$ for $d=2$;
Case 4: ${\lambda}>1$, $\mu>0$ for $d=2,3$.
We show that there exists a global $C^{\infty}-$smooth solution $(\rho, u)$ in Case 1, and Case 2 with $\opcurl u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(\rho, u)$ blows up in finite time. Therefore, ${\lambda}=1$ and $\mu=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(\rho, u)$ in $d-$dimensional compressible Euler equations with time-depending damping.
**Keywords.** Compressible Euler equations, damping, time-weighted energy inequality, Klainerman-Sobolev inequality, blowup, hypergeometric function.
**2010 Mathematical Subject Classification.** 35L70, 35L65, 35L67, 76N15.
author:
- |
Fei Hou$^{1, *}$ Huicheng Yin$^{2, }$[^1]\
\[12pt\] [1. Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China]{}\
[2. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China]{}
title: 'On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping'
---
Introduction
============
In this paper, we are concerned with the global existence and blowup of $C^{\infty}-$smooth solution $(\rho, u)$ to the multi-dimensional compressible Euler equations with time-depending damping $$\label{euler-eqn}
\left\{ \enspace
\begin{aligned}
&{\partial}_t\rho+\opdiv(\rho u)=0,\\
&{\partial}_t(\rho u)+\opdiv(\rho u\otimes u+p\,{\textup{\uppercase\expandafter{\romannumeral1}}}_d)=-{\alpha}(t)\rho u,\\
&\rho(0,x)=\bar \rho+{\varepsilon}\rho_0(x),\quad u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $x=(x_1,\cdots,x_d)\in{\mathbb R}^d$, $d=2,3$, $\rho$, $u=(u_1,\cdots,u_d)$, and $p$ stand for the density, velocity and pressure, respectively, ${\textup{\uppercase\expandafter{\romannumeral1}}}_d$ is the $d\times d$ identity matrix, the frictional coefficient is ${\alpha}(t)=\frac{\mu}{(1+t)^{\lambda}}$ with ${\lambda}\ge0$ and $\mu>0$, and $u_0=(u_{1,0},\cdots,u_{d,0})$. The state equation of the gases is described by $p(\rho)=A\rho^{{\gamma}}$, where $A>0$ and ${\gamma}>1$ are constants. In addition, $\bar\rho>0$ is a constant, $\rho_0,u_0\in C_0^\infty({\mathbb R}^d)$, $\supp\rho_0,\supp u_0 \subseteq \{x\colon|x|\le M\}$, $(\rho_0, u_0)\not\equiv 0$, $\rho(0,x)>0$, and ${\varepsilon}>0$ is sufficiently small. For the physical background of , it can be found in [@Da] and the references therein.
For $\mu=0$ in ${\alpha}(t)$, is the standard compressible Euler equation. It is well known that smooth solution $(\rho, u)$ of will generally blow up in finite time. For examples, for a special class of initial data $(\rho(0,x), u(0,x))$, Sideris [@Sideris85] has proved that the smooth solution $(\rho, u)$ of in three space dimensions can develop singularities in finite time, and Rammaha in [@Rammaha89] has proved a blowup result in two space dimensions. For more extensive literature on the blowup results and the blowup mechanism for $(\rho, u)$, see [@Alinhac93; @Alinhac99a; @Alinhac99b; @Chr07; @CM14; @CL15; @DWY16; @Sideris97; @Speck14; @Yin04] and the references therein.
For ${\lambda}=0$ in ${\alpha}(t)$, it has been shown that admits a global smooth solution $(\rho, u)$, moreover, the long-term behavior of the solution $(\rho,u)$ has been established, see [@HL92; @HS96; @KY04; @Nish; @PZ09; @STW03; @TW12; @WY01; @WY07]. In particular, in [@STW03], the authors showed that the vorticity of velocity $u$ decays to zero exponentially in time $t$.
For $\mu>0$ and ${\lambda}>0$ in ${\alpha}(t)$, one naturally asks: does the smooth solution of blow up in finite time or does it exist globally? For the case of $\opcurl u_0\equiv0$, in [@HWY15], we have studied this problem in three space dimensions and proved that for $0\le{\lambda}\le1$ and $\mu>0$ there exists a global smooth solution $(\rho, u)$ of and while for ${\lambda}>1$, in general, the solution will blow up in finite time. In this paper, we will remove the assumption $\opcurl u_0\equiv0$ in [@HWY15] and systematically study this problem both in two and three space dimensions.
Obviously, one can divide ${\lambda}\ge0$, $\mu>0$ into four cases:
[**Case 1**]{}: $0\le{\lambda}<1$, $\mu>0$ for $d=2,3$;
[**Case 2**]{}: ${\lambda}=1$, $\mu>3-d$, for $d=2,3$;
[**Case 3**]{}: ${\lambda}=1$, $\mu\le 3-d$ for $d=2$;
[**Case 4**]{}: ${\lambda}>1$, $\mu>0$ for $d=2,3$.
0.1 true cm At first, we state the global existence results in this paper.
0.1 true cm
\[thm1\] If $0\le{\lambda}<1$ and $\mu>0$, then for small ${\varepsilon}>0$, admits a global $C^\infty-$ smooth solution $(\rho, u)$ which fulfills $\rho>0$ and which is uniformly bounded for $t\ge0$ together with all its derivatives. In addition, the vorticity $\opcurl u$ and its derivatives decay to zero in the rate $e^{-\frac{\mu}{3(1-{\lambda})}[(1+t)^{1-{\lambda}}-1]}$, where $\opcurl u={\partial}_1u_2-{\partial}_2u_1$ for $d=2$, and $\opcurl u=({\partial}_2u_3-{\partial}_3u_2, {\partial}_3u_1-{\partial}_1u_3, {\partial}_1u_2-{\partial}_2u_1)^T$ for $d=3$.
0.1 true cm
\[thm2\] If ${\lambda}=1$, $\mu>3-d$ and $\opcurl u_0\equiv 0$, then for small ${\varepsilon}>0$, admits a global $C^\infty-$ smooth solution $(\rho, u)$ which fulfills $\rho>0$ and which is uniformly bounded for $t\ge0$ together with all its derivatives.
Next we concentrate on Case 3 and Case 4. As in [@Rammaha89], we introduce the two functions $$\begin{aligned}
q_0(l)&\defeq \int_{x_1>l}(x_1-l)^2\left(\rho(0,x)-\bar\rho\right)dx, \\
q_1(l)&\defeq 2\int_{x_1>l}(x_1-l)(\rho u_1)(0,x)\,dx.\end{aligned}$$ Before stating our blowup result for problem , we require to introduce a special hypergeometric function $\Psi(a,b,c;z)$, where the constants $a$ and $b$ satisfy $a+b=1$ and $$ab=\left\{
\begin{aligned}
&\frac{\mu{\lambda}}{2}, && {\lambda}>1, \\
&\frac\mu2(1-\frac\mu2), && {\lambda}=1,
\end{aligned}
\right.$$ $c\in\Bbb R^+$, the variable $z\in\Bbb R$, and $$\Psi(a,b,c;z)={\displaystyle}\sum_{n=0}^{+\infty}{\frac}{(a)_n(b)_n}{n!(c)_n}z^n$$ with $(a)_n=a(a+1)\cdot\cdot\cdot(a+n-1)$ and $(a)_0=1$. It is known from [@EMOT] that $\Psi(a,b,c;z)$ is an analytic function of $z$ for $z\in(-1, 1)$ and $\Psi(a,b,c;0)=\Psi(a+1,b+1,c;0)=1$. In addition, there exists a small constant ${\delta}_0\in(0,1)$ depending on $\mu$ and ${\lambda}$ such that for $-\frac{{\delta}_0}{2}\le z\le 0$, $$\label{Psi-bound}
\frac12 \le \Psi(a,b,1;z), \Psi(a+1,b+1,2;z) \le \frac32.$$
\[thm3\] Suppose $\supp\rho_0,\supp u_0\subseteq \{x\colon|x|\le M\}$ and let $$\begin{aligned}
q_0(l)&>0, \label{q0-positive}\\
q_1(l)&\ge0 \label{q1-positive}\end{aligned}$$ hold for all $l\in ({\tilde}M, M)$, where ${\tilde}M$ is some fixed constant satisfying $0\le {\tilde}M<M$. Moreover, we assume that there exist two constants $M_0$ and $\Lambda$ with $\max\{{\tilde}M, M-{\delta}_0\}\le M_0<M$ and $\Lambda\ge 3ab$ such that $$\label{+condition}
q_1(l) \ge \Lambda q_0(l)$$ holds for all $l\in (M_0, M)$. If ${\lambda}=1$, $\mu\le1$ for $d=2$ or ${\lambda}>1$, $\mu>0$ for $d=2,3$, then there exists an ${\varepsilon}_0>0$ such that, for $0<{\varepsilon}\le{\varepsilon}_0$, the lifespan $T_{\varepsilon}$ of the smooth solution $(\rho, u)$ of is finite.
Our results in Theorem \[thm1\]-\[thm3\] are strongly motivated by considering the 1-D Burgers equation with time-depending damping term $$\label{burgers-eqn}
\left\{ \enspace
\begin{aligned}
&{\partial}_t v+v{\partial}_x v=-\,{\displaystyle}\frac{\mu}{(1+t)^{\lambda}}\,v,\qquad (t,x)\in{\mathbb R}_+\times{\mathbb R},\\
&v(0,x)={\varepsilon}v_0(x),
\end{aligned}
\right.$$ where ${\lambda}\ge0$ and $\mu>0$ are constants, $v_0\in C_0^{\infty}({\mathbb R})$, $v_0\not\equiv 0$, and ${\varepsilon}>0$ is sufficiently small. One may directly obtain that by the method of characteristics $$\left\{ \enspace
\begin{aligned}
&T_{{\varepsilon}}=\infty, & \text{if $0\le{\lambda}<1$ or ${\lambda}=1$, $\mu>1$,}\\
&T_{{\varepsilon}}<\infty, & \text{if ${\lambda}>1$ or ${\lambda}=1$, $0<\mu\le 1$,}
\end{aligned}
\right.$$ where $T_{{\varepsilon}}$ is the lifespan of the smooth solution $v$ of . Especially in the case of $0\le{\lambda}<1$, $v$ exponentially decays to zero with respect to the time $t$. This means that ${\lambda}=1$ and $\mu=1$ appear to be the critical power and critical value respectively, for the global existence of smooth solution $v$ of .
For the three dimensional problem and the case ${\lambda}=0$ in ${\alpha}(t)$, the authors in [@STW03] proved that the fluid vorticity decays to zero exponentially in time, while the solution $(\rho, u)$ does not decay exponentially. In [**Case 1**]{} of $0\le{\lambda}<1$ and $\mu>0$, we have precisely proved that the vorticity $\opcurl u$ decays to zero in the rate $e^{-\frac{\mu}{3(1-{\lambda})}[(1+t)^{1-{\lambda}}-1]}$ in Theorem \[thm1\].
In Theorem \[thm2\], we pose the assumption of $\opcurl u_0\equiv 0$ for [**Case 2**]{}. If not, it seems difficult for us to obtain the uniform control on the vorticity $\opcurl u$ by our method. Namely, so far we do not know whether the assumption of $\opcurl u_0\equiv 0$ can be removed in order to obtain the global existence of $(\rho, u)$ in [**Case 2**]{}.
It is not hard to find a large number of initial data $(\rho,u)(0,x)$ such that - are satisfied. For instance, choosing $\rho_0(x)>0$ and $u_{1,0}(x)=x_1\rho_0(x)\Lambda/\bar\rho$, then we get -.
\[rem1.6\] In [@Sideris85] and , the authors have shown the formation of singularities in multi-dimensional compressible Euler equations (corresponding $\mu=0$ in ) under the assumptions of -. However, in order to prove the blowup result of smooth solution $(\rho, u)$ to problem and overcome the difficulty arisen by the time-depending frictional coefficient ${\frac}{\mu}{(1+t)^{\lambda}}$ with $\mu>0$ and ${\lambda}\ge1$, we pose an extra assumption except -, which leads to the non-negativity lower bound of $P(t,l)$ in so that two ordinary differential blowup inequalities - can be established. One can see more details in $\S 5$.
If the damping term ${\alpha}(t)\rho u$ in is replaced by $({\alpha}_1(t)\rho u_1,\cdots,{\alpha}_d(t)\rho u_d)^T$ with ${\alpha}_i(t)=\frac{\mu_i}{(1+t)^{{\lambda}_i}}$ ($i=1,\cdots,d$), and there exists some $i_0$ $(1\le i_0\le d)$ such that ${\lambda}_{i_0}$ and $\mu_{i_0}$ satisfy [**Case 3**]{} or [**Case 4**]{}. In this case, one can define the new quantities $$\begin{aligned}
q_0(l)&= \int_{x_{i_0}>l}(x_{i_0}-l)^2\left(\rho(0,x)-\bar\rho\right)dx, \\
q_1(l)&= 2\int_{x_{i_0}>l}(x_{i_0}-l)(\rho u_2)(0,x)\,dx\end{aligned}$$ and $$P(t,l)=\int_{x_{i_0}>l}(x_{i_0}-l)^2\left(\rho(t,x)-\bar\rho\right)dx$$ instead of the ones in - and , respectively, we then obtain an analogous result in Theorem \[thm3\] by applying the same procedure in $\S 5$.
Let us indicate the proofs of Theorems \[thm1\]-\[thm3\]. Without loss of generality, from now on we assume that $\bar c=c(\bar\rho)=1$, where $c(\rho)=\sqrt{P'(\rho)}$ is the sound speed. At first, we reformulate problem . Set $$\label{theta-def}
\theta \defeq \frac1{{\gamma}-1}(A{\gamma}\rho^{{\gamma}-1}-1)=\frac1{{\gamma}-1}(c^2(\rho)-1).$$ Then problem can be rewritten as $$\label{euler-reform}
\left\{ \enspace
\begin{aligned}
&{\partial}_t\theta+u\cdot\nabla\theta+(1+({\gamma}-1)\theta)\opdiv u=0, \\
&{\partial}_tu+\frac\mu{(1+t)^{{\lambda}}}u+u\cdot\nabla u+\nabla\theta=0, \\
&\theta(0,x)=\frac{1}{{\gamma}-1}[(1+\frac{{\varepsilon}\rho_0(x)}{\bar\rho})^{{\gamma}-1}-1]
\defeq {\varepsilon}\theta_0(x)+{\varepsilon}^2g(x,{\varepsilon}), \\
& u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $\nabla=({\partial}_1,\cdots,{\partial}_d)=({\partial}_{x_1},\cdots,{\partial}_{x_d})$, $\theta_0(x)=\frac{\rho_0(x)}{\bar\rho}$ and $g(x,{\varepsilon})=({\gamma}-2)\frac{\rho_0^2(x)}{\bar\rho^2}\int_0^1 (1+\frac{\sigma{\varepsilon}\rho_0(x)}{\bar\rho})^{{\gamma}-3}(1-\sigma)\,d\sigma$. Note that $g(x,{\varepsilon})$ is smooth in $(x,{\varepsilon})$ and has compact support in $x$.
To prove Theorem \[thm1\], we introduce such a time-weighted energy $$\label{energy1}
\mathcal{E}_k[\Phi](t) \defeq (1+t)^{\lambda}\sum_{1\le|{\alpha}|+j\le k}
\|{\partial}_t^j\nabla^{\alpha}\Phi(t,\cdot)\|+\|\Phi(t,\cdot)\|,$$ where $k$ is a fixed positive number, and $\|\cdot\|$ stands for the $L_x^2$ norm on ${\mathbb R}^d$, i.e., $$\|\Phi(t,\cdot)\| \defeq \|\Phi(t,x)\|_{L_x^2({\mathbb R}^d)}=
\left(\int_{{\mathbb R}^d}|\Phi(t,x)|^2dx\right)^\frac12.$$ Denote by $$\label{energy+}
\mathcal{E}_k[\Phi_1,\Phi_2](t) \defeq \mathcal{E}_k[\Phi_1](t)+\mathcal{E}_k[\Phi_2](t).$$ For $0\le{\lambda}<1$ and $\mu>0$, one can choose a constant $t_0$ such that $$\label{critical-time}
(1+t_0)^{1-{\lambda}}=\max\,\{\frac2\mu, 1\},$$ so that problem has a local solution $(\theta, u)\in C^{\infty}([0, t_0]\times{\mathbb R}^3)$ by the smallness of ${\varepsilon}>0$ (see the local existence result for the multidimensional hyperbolic systems in [@Ma]). Making use of the vorticity $\opcurl u$ and the conditions of $0\le{\lambda}<1$ and $\mu>0$ in [**Case 1**]{}, and simultaneously taking the delicate analysis on the system , the uniform time-weighted energy estimates for $\mathcal{E}_4[\theta, u](t)$ are obtained. This, together with the continuity argument, yields the proof of Theorem \[thm1\].
Since we have proved Theorem \[thm2\] in [@HWY15] for the [**Case 2**]{} with $\opcurl u_0\equiv 0$ in three space dimensions, we only require to focus on the proof of Theorem \[thm2\] in two space dimensions. For this purpose, we define another energy $$\label{energy2}
E_k[\Phi](t) \defeq (1+t)^\frac12\sum_{0\le|{\alpha}|\le k-1}
\|{\partial}Z^{\alpha}\Phi(t,\cdot)\|+(1+t)^{-\frac12}\|\Phi(t,\cdot)\|,$$ where ${\partial}=({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2})$, $Z=(Z_0, Z_1, \dots, Z_6)=({\partial}, S, R, H)$ with the scaling field $S=t{\partial}_t+x_1{\partial}_1+x_2{\partial}_2$, the rotation field $R=x_1{\partial}_2-x_2{\partial}_1$, the Lorentz fields $H=(H_1, H_2)=(x_1{\partial}_t+t{\partial}_1, x_2{\partial}_t+t{\partial}_2)$ and $Z^{\alpha}=Z_0^{{\alpha}_0}Z_1^{{\alpha}_1}\cdots Z_6^{{\alpha}_6}$. From we may derive a damped wave equation of $\theta$ as follows $$\label{damped-wave}
{\partial}_t^2\theta+\frac\mu{1+t}{\partial}_t\theta-\Delta\theta=Q(\theta,u),$$ where the expression of $Q(\theta,u)$ will be given in below. Thanks to $\opcurl u\equiv0$, we can get the estimates of velocity $u$ from the equations in (see Lemma \[lem-Zvelocity\]). By $\mu>1$ and a rather technical analysis on the damped wave equation , we eventually show in $\S 4$ that $E_5[\theta,u](t) \le \frac12 \,K_3{\varepsilon}$ (see for the definition of $E_5[\theta,u](t)$) holds when $E_5[\theta,u](t) \le K_3{\varepsilon}$ is assumed for some suitably large constant $K_3>0$ and small ${\varepsilon}>0$. Based on this and the continuity argument, the global existence of $(\theta, u)$ and then Theorem \[thm2\] in two space dimensions are established for ${\lambda}=1$, $\mu>1$ and $\opcurl u_0\equiv0$.
To prove the blowup result in Theorem \[thm3\], as in [@Rammaha89; @Sideris85], we shall derive some blowup-type second-order ordinary differential inequalities in $\S 5$. From this and assumptions -, an upper bound of the lifespan $T_{{\varepsilon}}$ is derived by making use of ${\lambda}=1$, $\mu\le 3-d$ or ${\lambda}>1$, and then the proof of Theorem \[thm3\] is completed.
Here we point out that in [@HWY15], for the 3-d [**irrotational**]{} compressible Euler equations, it has been shown that for $0\le{\lambda}\le1$, there exists a global $C^{\infty}-$smooth small amplitude solution $(\rho, u)$, while for ${\lambda}>1$, the smooth solution $(\rho, u)$ generally blows up in finite time. This means that we have extended the global existence and blowup results in [@HWY15] for the 3-D irrotational flows to the 2-D and 3-D full Euler systems.
In the whole paper, we shall use the following convention:
- $C$ will denote a generic positive constant which is independent of $t$ and ${\varepsilon}$.
- $A\ls B$ or $B\gt A$ means $A\le CB$.
- $r=|x|=\sqrt{x_1^2+\cdots+x_d^2}$, $\sigma_-(t,x)\defeq
\sqrt{1+(r-t)^2}$.
- $\|\Phi(t,\cdot)\| \defeq \|\Phi(t,x)\|_{L_x^2({\mathbb R}^d)}$, $|\Phi(t,\cdot)|_\infty \defeq |\Phi(t,x)|_{L_x^\infty}=
{\displaystyle}\sup_{x\in{\mathbb R}^d}|\Phi(t,x)|$.
- $Z$ denotes one of the Klainerman vector fields $\{{\partial}, S, R, H\}$ on ${\mathbb R}_+\times{\mathbb R}^2$, where ${\partial}=({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2})$, $S=t{\partial}_t+x_1{\partial}_1+x_2{\partial}_2$, $R=x_1{\partial}_2-x_2{\partial}_1$ and $H=(H_1, H_2)=
(x_1{\partial}_t+t{\partial}_1, x_2{\partial}_t+t{\partial}_2)$.
- For two vector fields $X$ and $Y$, $[X,Y] \defeq XY-YX$ denotes the Lie bracket.
- Greek letters ${\alpha},\beta,\cdots$ denote multiple indices, i.e., ${\alpha}=({\alpha}_0,\cdots,{\alpha}_m)$, and $|{\alpha}|={\alpha}_0+\cdots+{\alpha}_m$ denotes its length, where ${\alpha}_i$ is some non-negative integer for all $i=0,\cdots,m$.
- For two multiple indices ${\alpha}$ and $\beta$, $\beta\le{\alpha}$ means $\beta_i\le{\alpha}_i$ for all $i=0,\cdots,m$ while $\beta<{\alpha}$ means $\beta\le{\alpha}$ and $\beta_i<{\alpha}_i$ for some $i$.
- For the differential operator $O=(O_0,\cdots,O_m)$, for example, $O=({\partial}_t,
{\partial}_{x_1},\cdots,{\partial}_{x_d})$ in $\S 3$ and $O=
({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2}, S, R, H)$ in $\S 4$, denote $O^{\alpha}\defeq O_0^{{\alpha}_0}\cdots O_m^{{\alpha}_m}$, $O^{\le{\alpha}}
\defeq {\displaystyle}\sum_{0\le\beta\le{\alpha}}O^\beta$, $O^{<{\alpha}} \defeq {\displaystyle}\sum_
{0\le\beta<{\alpha}}O^\beta$ and $O^{\le k} \defeq {\displaystyle}\sum_{0\le|{\alpha}|\le k}
O^{\alpha}$ with $k$ is an integer.
- Leibniz’s rule: $O^{\alpha}(\Phi\Psi)={\displaystyle}\sum_{0\le\beta\le{\alpha}}C_{{\alpha},\beta}
O^\beta\Phi O^{{\alpha}-\beta}\Psi$ will be abbreviated as\
$O^{\alpha}(\Phi\Psi)={\displaystyle}\sum_{0\le\beta\le{\alpha}}O^\beta\Phi O^{{\alpha}-\beta}\Psi$.
- $\Xi$ is the solution of ${\displaystyle}\Xi'(t) = \frac{\mu}{(1+t)^{\lambda}}\,
\Xi(t)$ with $\Xi(0)=1$, i.e., $$\label{Xi-def}
\Xi(t)\defeq \begin{cases} e^{\frac{\mu}{1-{\lambda}}[(1+t)^{1-{\lambda}}-1]}, &
{\lambda}\ge 0,\,{\lambda}\neq1,\\ (1+t)^\mu, & {\lambda}=1. \end{cases}$$
- $c(\bar\rho)=1$ will be assumed throughout (otherwise, introduce $X=x/c(\bar\rho)$ as new space coordinate if necessary).
Some Preliminaries {#section2}
==================
At first, we derive the scalar equation of $\theta$ in . It follows from the first equation in that $$\label{dtdivu1}
{\partial}_t\opdiv u=-\frac1{(1+({\gamma}-1)\theta)} ({\partial}_t^2\theta+u\cdot\nabla{\partial}_t\theta
+{\partial}_tu\cdot\nabla\theta+({\gamma}-1){\partial}_t\theta\opdiv u).$$ Taking divergence on the second equation in yields $$\label{dtdivu2}
\opdiv{\partial}_t u+\frac\mu{(1+t)^{\lambda}}\opdiv u+\Delta\theta+
u\cdot\nabla\opdiv u+\sum_{i,j=1}^d {\partial}_iu_j{\partial}_ju_i=0,$$ where $\Delta={\partial}_1^2+\cdots+{\partial}_d^2$. Substituting into yields the damped wave equation of $\theta$ $$\label{theta-eqn}
{\partial}_t^2\theta+\frac\mu{(1+t)^{\lambda}}{\partial}_t\theta-\Delta\theta=Q(\theta,u),$$ where $$\begin{aligned}
Q(\theta,u) &\defeq Q_1(\theta,u)+Q_2(\theta,u), \label{Q-def}\\
Q_1(\theta,u) &\defeq ({\gamma}-1)\theta\Delta\theta-\frac\mu{(1+t)^{\lambda}}
u\cdot\nabla\theta-2u\cdot\nabla{\partial}_t\theta-\sum_{i,j=1}^d u_iu_j{\partial}_{ij}^2\theta \label{Q1-def},\\
Q_2(\theta,u) &\defeq -\sum_{i,j=1}^d u_i{\partial}_iu_j{\partial}_j\theta-{\partial}_tu\cdot\nabla\theta
+(1+({\gamma}-1)\theta) (\sum_{i,j=1}^d {\partial}_iu_j{\partial}_ju_i+({\gamma}-1)|\opdiv u|^2). \label{Q2-def}\end{aligned}$$ Let $$\label{vorticity-def}
w\defeq \opcurl u=
\begin{cases}
{\partial}_1u_2-{\partial}_2u_1, & d=2, \\
({\partial}_2u_3-{\partial}_3u_2, {\partial}_3u_1-{\partial}_1u_3, {\partial}_1u_2-{\partial}_2u_1)^T, & d=3.
\end{cases}$$ Then the second equation in implies that for $d=2$ $$\label{2dvorticity-eqn}
{\partial}_tw+\frac\mu{(1+t)^{\lambda}}w+u\cdot\nabla w+w\opdiv u=0$$ and for $d=3$ $$\label{3dvorticity-eqn}
{\partial}_tw+\frac\mu{(1+t)^{\lambda}}w+u\cdot\nabla w+w\opdiv u=w\cdot\nabla u.$$ To prove Theorem \[thm1\]-\[thm2\], we require to introduce the following lemma, which is easily shown.
\[lem-divcurl\] Let $U(x)=(U_1(x),\cdots,U_d(x))$ be a vector-valued function with compact support on ${\mathbb R}^d$ ($d=2,3$), then there holds that $$\label{divcurl-estimate}
\|\nabla U\|\le \|\opcurl U\|+\|\opdiv U\|.$$
The following Sobolev type inequality can be found in [@Kl87].
\[lem-Klainerman-ineq\] Let $\Phi(t,x)$ be a function on ${\mathbb R}^{1+2}$, then there exists a constant $C$ such that $$\label{Klainerman-ineq}
(1+t+r)\,\sigma_-(t,x)\,|\Phi(t,x)| \le C\sum_{|{\alpha}|\le2} \|Z^{\alpha}\Phi(t,\cdot)\|^2.$$
In addition, we have
\[lem-weight\] Let $\Phi(t,x)$ be a function on ${\mathbb R}^{1+2}$ and assume $\supp\Phi\subseteq\{(t,x)\colon |x|\le t+M\}$, then there exists a constant $C>0$ such that for $\nu\in (-\infty,1)$ $$\label{weight-pointwise}
|\sigma_-^{\nu-1}(t,\cdot)\Phi(t,\cdot)|_\infty \le C|\sigma_-^{\,\nu}
(t,\cdot)\nabla\Phi(t,\cdot)|_\infty$$ and for $\ell\neq 1$ $$\label{weight-L2}
\|\sigma_-^{-\ell}(t,\cdot)\Phi(t,\cdot)\| \le
C(t+M)^{(1-\ell)_+}\,\|\nabla\Phi(t,\cdot)\|,$$ where $(1-\ell)_+=\max\,\{1-\ell, 0\}$ and $\ell\in{\mathbb R}$.
For the purpose of completeness, we prove - here. In fact, for the proof of , one can also see [@Alinhac93 Lemma 2.2]. By introducing the polar coordinate $(r,\phi)$ such that $x_1=r\cos\phi$ and $x_2=r\sin\phi$, we then have $$\begin{aligned}
\Phi(t,x)&=\Phi(t,r\cos\phi,r\sin\phi)=-\int_r^{t+M} \frac{d}{d\xi}\,
\Phi(t,\xi\cos\phi,\xi\sin\phi) \,d\xi {\nonumber}\\
&=-\int_r^{t+M} (\cos\phi\,{\partial}_1\Phi(t,\xi\cos\phi,\xi\sin\phi)+
\sin\phi\,{\partial}_2\Phi(t,\xi\cos\phi,\xi\sin\phi) ) \,d\xi. \label{2.14}\end{aligned}$$ Together with the mean value theorem, this yields $$\Phi(t,x) \ls |\sigma_-^{\,\nu}(t,\cdot)\nabla\Phi(t,\cdot)|_\infty
\int_r^{t+M} (1+|t-\xi|)^{-\nu} \,d\xi,$$ which immediately derives . On the other hand, applying Cauchy-Schwartz inequality to derives $$|\Phi(t,r\cos\phi,r\sin\phi)|^2 \le \left( \int_r^{t+M}
|\nabla\Phi(t,\xi\cos\phi,\xi\sin\phi)|^2 \,\xi\,d\xi\right)
\left( \int_r^{t+M} \frac1\xi \,d\xi\right),$$ which yields $$\begin{aligned}
\left\|\frac{\Phi(t,\cdot)}{(t+2M-r)^\ell}\right\|^2
&= \int_0^{2\pi}\int_0^{t+M} \frac{|\Phi(t,r\cos\phi,r\sin\phi)|^2}
{(t+2M-r)^{2\ell}} \,r\,drd\phi {\nonumber}\\
&\le \int_0^{2\pi}\int_0^{t+M} \frac{r\int_r^{t+M} \frac1\xi \,d\xi}
{(t+2M-r)^{2\ell}} \,dr \int_r^{t+M} |\nabla\Phi(t,\xi\cos\phi,
\xi\sin\phi)|^2 \,\xi\,d\xi d\phi {\nonumber}\\
&\ls \int_0^{t+M} \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}} \,dr
\,\|\nabla\Phi(t,\cdot)\|^2. \label{2.15}\end{aligned}$$ On the other hand, it follows from direct computation that $$\label{2.16}
\int_0^\frac{t+M}{2}\, \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}} \,dr
\ls \frac1{(t+M)^{2\ell}} \int_0^\frac{t+M}{2}\, r\log\frac{t+M}r \,dr
\ls (t+M)^{2(1-\ell)}$$ and $$\label{2.17}
\int_\frac{t+M}{2}^{t+M}\, \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}}\,dr
=\int_0^\frac{t+M}{2}\, \frac{(t+M-\xi)\log\frac{t+M}{t+M-\xi} }{(M+\xi)^{2\ell}} \,d\xi
\le \int_0^\frac{t+M}{2} \,\frac\xi{(M+\xi)^{2\ell}} \,d\xi,$$ where we have used the fact of $\frac{t+M}{t+M-\xi}=1+\frac\xi{t+M-\xi} \le e^\frac\xi{t+M-\xi}$ in the last inequality.
Substituting - into and taking direct computation yield . Thus, the proof of Lemma \[lem-weight\] is completed.
Proof of Theorem 1.1. {#section3}
=====================
Throughout this section, we will always assume that $\mathcal{E}_4[\theta,u](t) \le K_1{\varepsilon}$ holds, where the definition of $\mathcal{E}_4[\theta,u](t)$ has been given in and . Together with the standard Sobolev embedding theorem, this yields $$\label{3.1}
|(\theta,u)(t,\cdot)|_\infty+(1+t)^{\lambda}|{\partial}{\partial}^{\le1}(\theta,u)(t,\cdot)|_\infty \ls K_1{\varepsilon}.$$ To prove Theorem \[thm1\], we now carry out the following parts.
Estimates of velocity $u$ and vorticity $w$.
--------------------------------------------
The following lemma is an application of Lemma \[lem-divcurl\] and .
\[lem-velocity1\] Under assumption , for all $t>0$, one has $$\label{3.2}
\mathcal{E}_4[u](t)\ls \|u(t,\cdot)\|+(1+t)^{\lambda}\left(\|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}{\partial}^{\le3}\theta(t,\cdot)\|\right),$$ where the definition of $w$ has been given in .
By the equations in , we see that $$\begin{aligned}
\opdiv u &= -\frac{{\partial}_t\theta+u\cdot\nabla\theta}{1+({\gamma}-1)\theta}, \label{3.3} \\
{\partial}_tu &= -\left(u\cdot\nabla u+\frac\mu{(1+t)^{\lambda}}u+\nabla\theta\right). \label{3.4}\end{aligned}$$ Taking $U={\partial}^{\alpha}u$ with $|{\alpha}|\le3$ in , we then arrive at $$\begin{aligned}
\|\nabla{\partial}^{\le3}& u(t,\cdot)\|
\ls \|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}^{\le3}\opdiv u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}_t{\partial}^{\le3}\theta(t,\cdot)\|+ K_1{\varepsilon}\left( \|\nabla{\partial}^{\le3}\theta(t,\cdot)\|+
(1+t)^{-{\lambda}}\|{\partial}^{\le3} u(t,\cdot)\|\right), \label{3.5}\end{aligned}$$ where we have used and in the last inequality. Taking the $L^2$ norm of ${\partial}^{\alpha}$ yields $$\begin{aligned}
\|{\partial}_t{\partial}^{\alpha}u(t,\cdot)\|
&\ls \|\nabla{\partial}^{\alpha}\theta(t,\cdot)\|+(1+K_1{\varepsilon})(1+t)^{-{\lambda}}
\|{\partial}^{\le{\alpha}} u(t,\cdot)\| +K_1{\varepsilon}\|\nabla{\partial}^{\le{\alpha}} u(t,\cdot)\|. \label{3.6}\end{aligned}$$ Rewrite ${\partial}^{\alpha}={\partial}_t^k{\partial}_x^\beta$ with $0\le k+|\beta|\le3$. Summing up from $k=0$ to $k=3$ yields $$\begin{aligned}
\|{\partial}_t{\partial}^{\le3} u(t,\cdot)\|
&\ls \|\nabla{\partial}^{\le3}\theta(t,\cdot)\|+(1+t)^{-{\lambda}}\|u(t,\cdot)\|
+K_1{\varepsilon}\|\nabla{\partial}^{\le3} u(t,\cdot)\|. \label{3.7}\end{aligned}$$ By the smallness of ${\varepsilon}>0$, we immediately derive from and . This completes the proof of Lemma \[lem-velocity1\].
The following lemma shows the estimate of velocity $u$ itself.
\[lem-velocity2\] Let $\mu>0$. Under assumption , for all $t>0$, it holds that $$\label{3.8}
\frac{d}{dt}\|(\theta,u)(t,\cdot)\|^2+\frac\mu{(1+t)^{\lambda}}\|u(t,\cdot)\|^2
\ls (1+t)^{\lambda}|\theta(t,\cdot)|_\infty\|\nabla\theta(t,\cdot)\|^2.$$
Multiplying the second equation in by $u$ derives $$\label{3.9}
\frac12\,{\partial}_t|u|^2+\frac\mu{(1+t)^{\lambda}}|u|^2+u\cdot\nabla\theta=
-\frac12\,u\cdot\nabla|u|^2.$$ From the first equation in , we see that $$\begin{aligned}
u\cdot\nabla\theta &= \opdiv(\theta u)-\theta\opdiv u {\nonumber}\\
&= \opdiv(\theta u)+\theta({\partial}_t\theta+u\cdot\nabla\theta+
({\gamma}-1)\theta\opdiv u) {\nonumber}\\
&= \opdiv(\theta u+({\gamma}-1)\theta^2 u)+\frac12\,{\partial}_t|\theta|^2+
(3-2{\gamma})\theta\,u\cdot\nabla\theta. \label{3.10}\end{aligned}$$ Substituting into and integrating it over ${\mathbb R}^d$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\|(\theta,u)(t,\cdot)\|^2+\frac{2\mu}{(1+t)^{\lambda}}
\|u(t,\cdot)\|^2 {\nonumber}\\
&\ls |\theta(t,\cdot)|_\infty\|u(t,\cdot)\|\,\|\nabla\theta(t,\cdot)\|
+|\nabla u(t,\cdot)|_\infty\|u(t,\cdot)\|^2. \label{3.11}\end{aligned}$$ Substituting into and applying $\mu>0$ and the smallness of ${\varepsilon}$, we derive . This completes the proof of Lemma \[lem-velocity2\].
Next lemma shows the estimates of vorticity $w$ and its derivatives.
\[lem-vorticity\] Let $\mu>0$. Under assumption , for all $t>0$, it holds that $$\label{3.12}
\frac{d}{dt}\|{\partial}^{\le3} w(t,\cdot)\|^2+\frac\mu{(1+t)^{\lambda}}
\|{\partial}^{\le3} w(t,\cdot)\|^2
\ls |{\partial}^{\le1} w(t,\cdot)|_\infty\, \|{\partial}^{\le3} w(t,\cdot)\|\,
\|{\partial}{\partial}^{\le3} u(t,\cdot)\|.$$
It follows from vorticity equation - that for $|{\alpha}|\le3$, $$\begin{aligned}
& {\partial}_t{\partial}^{\alpha}w+\frac{\mu}{(1+t)^{\lambda}}{\partial}^{\alpha}w+u\cdot\nabla{\partial}^{\alpha}w {\nonumber}\\
&= -\sum_{0<\beta\le{\alpha}}\left[{\partial}^\beta\left(\frac\mu{(1+t)^{\lambda}}\right)
{\partial}^{{\alpha}-\beta} w+{\partial}^\beta u\cdot\nabla{\partial}^{{\alpha}-\beta} w\right]
-{\partial}^{\alpha}(w\opdiv u-w\cdot\nabla u),\label{3.13}\end{aligned}$$ here we point out that the last term $w\cdot\nabla u$ in does not appear when $d=2$ . Multiplying by ${\partial}^{\alpha}w$ and integrating it over ${\mathbb R}^d$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\|{\partial}^{\alpha}w(t,\cdot)\|^2+\frac{2\mu}{(1+t)^{\lambda}}
\|{\partial}^{\alpha}w(t,\cdot)\|^2 {\nonumber}\\
&\ls \frac1{(1+t)^{1+{\lambda}}}\|{\partial}^{\alpha}w(t,\cdot)\|\,\|{\partial}^{<{\alpha}}w(t,\cdot)\|+
|{\partial}{\partial}^{\le1} u(t,\cdot)|_\infty \|{\partial}^{\le{\alpha}}w(t,\cdot)\|^2 {\nonumber}\\
&\quad +|{\partial}^{\le1}w(t,\cdot)|_\infty \|{\partial}{\partial}^{\le3} u(t,\cdot)\|\,
\|{\partial}^{\le{\alpha}}w(t,\cdot)\|. \label{3.14}\end{aligned}$$ Note that when ${\alpha}=0$, the firs term $\|{\partial}^{\alpha}w(t,\cdot)\|\,\|{\partial}^{<{\alpha}}w(t,\cdot)\|$ in the right hand side of does not appear. Summing up from $|{\alpha}|=0$ to $|{\alpha}|=3$ and applying , $\mu>0$ and the smallness of ${\varepsilon}$, we then obtain . This completes the proof of Lemma \[lem-vorticity\].
\[rmk3.1\] The proof of Lemma \[lem-velocity2\] and Lemma \[lem-vorticity\] only depends on , $\mu>0$ and the smallness of ${\varepsilon}$.
Estimates of $\theta$ and its derivatives.
------------------------------------------
The next lemma shows the global estimates of $\theta$ and its derivatives for $t>t_0$, where $t_0$ is defined in .
\[lem-thetaglobal\] Let $0\le{\lambda}<1$, $\mu>0$. Under assumption , for all $t>t_0$, it holds that $$\begin{aligned}
&\quad \mathcal{E}_4^2[\theta](t)
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+K_1{\varepsilon}\int_{t_0}^t\left((1+s)^{-{\lambda}}
\|u(s,\cdot)\|^2+(1+s)^{\lambda}\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds. \label{3.15}\end{aligned}$$
Acting ${\partial}^{\alpha}$ with $|{\alpha}|\le3$ on both sides of equation shows $${\partial}_t^2{\partial}^{\alpha}\theta+\frac\mu{(1+t)^{\lambda}}{\partial}_t{\partial}^{\alpha}\theta-\Delta{\partial}^{\alpha}\theta
={\partial}^{\alpha}Q(\theta,u)+\sum_{\beta<{\alpha}} {\partial}^{{\alpha}-\beta}\left(\frac\mu{(1+t)^{\lambda}}
\right) {\partial}_t{\partial}^\beta\theta.$$ Multiplying this by $2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta$ derives $$\begin{aligned}
&{\partial}_t\left[(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu^2}2|{\partial}^{\alpha}\theta|^2-\frac{\mu{\lambda}}{2}(1+t)^{{\lambda}-1}|{\partial}^{\alpha}\theta|^2\right] {\nonumber}\\
& +\left[\mu(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1}\right]|{\partial}{\partial}^{\alpha}\theta|^2-
\opdiv\left[\nabla{\partial}^{\alpha}\theta\left(2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta
+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta\right) \right]{\nonumber}\\
&= \left(2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta\right)
\left({\partial}^{\alpha}Q(\theta,u)+\sum_{\beta<{\alpha}} {\partial}^{{\alpha}-\beta}\left(
\frac\mu{(1+t)^{\lambda}}\right) {\partial}_t{\partial}^\beta\theta\right) {\nonumber}\\
&\qquad +\frac{\mu{\lambda}(1-{\lambda})}{2}(1+t)^{{\lambda}-2}|{\partial}^{\alpha}\theta|^2. \label{3.16}\end{aligned}$$ Thanks to $0\le{\lambda}<1$, $\mu>0$ and the choice of $t_0$ (see ), for all $t>t_0$ one easily knows that for the term in the first square bracket of the second line in , $$\label{3.17}
\mu(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1} \ge \mu(1-{\lambda})(1+t)^{\lambda}.$$ Furthermore, one gets that for the term in the square bracket of the first line in , $$\begin{aligned}
(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu^2}{2}|{\partial}^{\alpha}\theta|^2-\frac{\mu{\lambda}}{2}(1+t)^{{\lambda}-1}|{\partial}^{\alpha}\theta|^2 \\
=(1+t)^{2{\lambda}}\left(\frac{1-{\lambda}}{3-{\lambda}}|{\partial}_t{\partial}^{\alpha}\theta|^2+|\nabla{\partial}^{\alpha}\theta|^2\right)
+\frac{\mu^2(1-{\lambda})}{8}|{\partial}^{\alpha}\theta|^2 \\
+\left((1+t)^{\lambda}\sqrt\frac{2}{3-{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\frac\mu2\sqrt\frac{3-{\lambda}}{2}{\partial}^{\alpha}\theta\right)^2
+\frac{\mu{\lambda}}{4}(\mu-2(1+t)^{{\lambda}-1})|{\partial}^{\alpha}\theta|^2,\end{aligned}$$ which is equivalent to $(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+|{\partial}^{\alpha}\theta|^2$ for $0\le{\lambda}<1$ and $t>t_0$. Consequently, integrating over $[t_0,t]\times{\mathbb R}^d$ gives $$\begin{aligned}
&\quad (1+t)^{2{\lambda}}\|{\partial}{\partial}^{\alpha}\theta(t,\cdot)\|^2+\|{\partial}^{\alpha}\theta(t,\cdot)\|^2
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\alpha}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+\int_{t_0}^t(1-{\lambda})(1+s)^{{\lambda}-2}\|\theta(s,\cdot)\|^2\,ds
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& +\left|\int_{t_0}^t\int_{{\mathbb R}^d} \left({\partial}^{\alpha}Q_1(\theta,u)+{\partial}^{\alpha}Q_2(\theta,u)\right) \left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+
\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right|. \label{3.18}\end{aligned}$$ Next we deal with the last term in the right hand side of . It follows from - and $|{\alpha}|\le3$ that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} {\partial}^{\alpha}Q_2(\theta,u) \left(2(1+s)^{2{\lambda}}
{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}(\theta,u)(s,\cdot)\|^2\,ds {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds
+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.19}\end{aligned}$$ Now we turn our attention to the term ${\partial}^{\alpha}Q_1(\theta,u)$. It is easy to get $$\begin{aligned}
{\partial}^{\alpha}Q_1(\theta,u) \defeq -{\partial}^{\alpha}\left(\frac\mu{(1+t)^{\lambda}}u\cdot\nabla\theta\right)+({\gamma}-1)\theta\Delta{\partial}^{\alpha}\theta {\nonumber}\\
-2u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta-\sum_{i,j=1}^d u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta+Q^{\alpha}_1(\theta,u). \label{3.20}\end{aligned}$$ One easily checks that still holds if ${\partial}^{\alpha}Q_2(\theta,u)$ is replaced by $Q^{\alpha}_1(\theta,u)$. In addition, for ${\alpha}=0$, we see that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \frac\mu{(1+s)^{\lambda}}u\cdot\nabla\theta
\left(2(1+s)^{2{\lambda}}{\partial}_t\theta+\mu(1+s)^{\lambda}\theta\right) \,dxds\right| {\nonumber}\\
&\ls \int_{t_0}^t \left(|\theta(s,\cdot)|_\infty \|u(s,\cdot)\|+(1+s)^{\lambda}|u(s,\cdot)|_\infty\|{\partial}_t\theta(s,\cdot)\|\right) \|\nabla\theta(s,\cdot)\|\,ds {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t \left( (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2+(1+s)^{\lambda}\|{\partial}\theta(s,\cdot)\|^2\right) \,ds, \label{3.21}\end{aligned}$$ where we have used again. If ${\alpha}>0$, by we find that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} {\partial}^{\alpha}\left(\frac\mu{(1+s)^{\lambda}}u\cdot\nabla\theta\right)
\left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.22}\end{aligned}$$ On the other hand, direct computation derives the following identities $$\begin{aligned}
(1+t)^{2{\lambda}}\theta\Delta{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
\opdiv\left[(1+t)^{2{\lambda}}\theta\nabla{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right]
-(1+t)^{2{\lambda}}\nabla\theta\cdot\nabla{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta \\
-\frac12\,{\partial}_t\left[(1+t)^{2{\lambda}}\theta\,|\nabla{\partial}^{\alpha}\theta|^2\right]
+{\lambda}(1+t)^{2{\lambda}-1}\theta\,|\nabla{\partial}^{\alpha}\theta|^2+\frac12(1+t)^{2{\lambda}}{\partial}_t\theta\,|\nabla{\partial}^{\alpha}\theta|^2\end{aligned}$$ and $$(1+t)^{\lambda}\theta\Delta{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=\opdiv\left[(1+t)^{\lambda}\theta\nabla{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]-(1+t)^{\lambda}(\theta\,
|\nabla{\partial}^{\alpha}\theta|^2+\nabla\theta\cdot\nabla{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta).$$ Integrating these two identities over $[t_0,t]\times{\mathbb R}^d$ yields $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \theta\Delta{\partial}^{\alpha}\theta \left(2
(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right)\,dxds\right| {\nonumber}\\
& \ls K_1{\varepsilon}\left(\mathcal{E}_4^2[\theta](t)+\mathcal{E}_4^2[\theta](t_0)\right)
+K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2\,ds, \label{3.23}\end{aligned}$$ where we have used and ${\lambda}\le1$.
Analogously for the remaining items $u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta$ and ${\displaystyle}\sum_{i,j=1}^du_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta$ in ${\partial}^{\alpha}Q_1(\theta,u)$ (see ), direct computations show $$\begin{aligned}
2(1+t)^{2{\lambda}}u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
& \opdiv\left[(1+t)^{2{\lambda}}u\,|{\partial}_t{\partial}^{\alpha}\theta|^2\right]
-(1+t)^{2{\lambda}}\opdiv u\,|{\partial}_t{\partial}^{\alpha}\theta|^2, \\
(1+t)^{\lambda}u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=
& \opdiv\left[(1+t)^{\lambda}u\,{\partial}_t{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]
-(1+t)^{\lambda}{\partial}_t{\partial}^{\alpha}\theta(u\cdot\nabla{\partial}^{\alpha}\theta+\opdiv u\,
{\partial}^{\alpha}\theta)\end{aligned}$$ and $$\begin{aligned}
2(1+t)^{2{\lambda}}u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
&{\partial}_i\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_j{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right]
+{\partial}_j\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right] \\
& -(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta\Big[{\partial}_i(u_iu_j){\partial}_j{\partial}^{\alpha}\theta+
{\partial}_j(u_iu_j){\partial}_i{\partial}^{\alpha}\theta\Big] \\
& -{\partial}_t\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta\right]
+(1+t)^{2{\lambda}}{\partial}_t(u_iu_j){\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta \\
& +2{\lambda}(1+t)^{2{\lambda}-1}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta, \\
(1+t)^{\lambda}u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=
& {\partial}_i\left[(1+t)^{\lambda}u_iu_j{\partial}_j{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]
-(1+t)^{\lambda}{\partial}_j{\partial}^{\alpha}\theta{\partial}_i(u_iu_j{\partial}^{\alpha}\theta).\end{aligned}$$ Then we have that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \left(2u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta
+\sum_{i,j=1}^du_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta\right) \left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta
+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\left(\mathcal{E}_4^2[\theta](t)+\mathcal{E}_4^2[\theta](t_0)\right)
+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds. \label{3.24}\end{aligned}$$ Substituting - into yields $$\begin{aligned}
&\quad (1+t)^{2{\lambda}}\|{\partial}{\partial}^{\alpha}\theta(t,\cdot)\|^2+\|{\partial}^{\alpha}\theta(t,\cdot)\|^2
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\alpha}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+K_1{\varepsilon}\,\mathcal{E}_4^2[\theta](t)
+\int_{t_0}^t(1-{\lambda})(1+s)^{{\lambda}-2}\|\theta(s,\cdot)\|^2\,ds+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds +K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.25}\end{aligned}$$ Summing up from $|{\alpha}|=0$ to $|{\alpha}|=3$ and applying Gronwall’s inequality for ${\lambda}<1$ yield provided that ${\varepsilon}>0$ is small enough. This completes the proof of Lemma \[lem-thetaglobal\].
Proof of Theorem \[thm1\].
--------------------------
First, we assume that $\mathcal{E}_4[\theta,u](t) \le K_1{\varepsilon}$ holds. Multiplying by $(1+t)^{2{\lambda}}$ yields $$\begin{aligned}
&\quad \frac{d}{dt}\|(1+t)^{\lambda}{\partial}^{\le3}w(t,\cdot)\|^2+\left(\mu
(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1}\right)\|{\partial}^{\le3}w(t,\cdot)\|^2 {\nonumber}\\
&\ls K_1{\varepsilon}(1+t)^{\lambda}\|{\partial}^{\le3}w(t,\cdot)\|\,\|{\partial}{\partial}^{\le3} u(t,\cdot)\|, \label{3.26}\end{aligned}$$ where we have used . In view of , the second term on the first line in is bounded below by $(1+t)^{\lambda}\|{\partial}^{\le3}w(t,\cdot)\|^2$. Integrating and over $[t_0,t]\times{\mathbb R}^d$ derives $$\label{3.27}
\|(\theta,u)(t,\cdot)\|^2+\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds
\ls \|(\theta,u)(t_0,\cdot)\|^2+K_1{\varepsilon}\int_{t_0}^t (1+s)^{\lambda}\|\nabla\theta(s,\cdot)\|^2\,ds$$ and $$\begin{aligned}
&\quad \|(1+t)^{\lambda}{\partial}^{\le3}w(t,\cdot)\|^2+\int_{t_0}^t (1+s)^{\lambda}\|{\partial}^{\le3}w(s,\cdot)\|^2\,ds {\nonumber}\\
&\ls \|{\partial}^{\le3}w(t_0,\cdot)\|^2+K_1{\varepsilon}\int_{t_0}^t (1+s)^{\lambda}\|{\partial}{\partial}^{\le3} u(s,\cdot)\|^2\,ds. \label{3.28}\end{aligned}$$ Collecting , - with , we infer $\mathcal{E}_4[\theta,u](t) \le C_1\mathcal{E}_4[\theta,u](t_0)$. It follows from the local existence of the hyperbolic system that $\mathcal{E}_4[\theta,u](t_0) \le C_2{\varepsilon}$. Let $K_1=2C_1C_2$ and choose ${\varepsilon}>0$ sufficiently small. Then, we conclude that $\mathcal{E}_4[\theta,u](t) \le \frac12 K_1{\varepsilon}$, which implies that admits a global solution $(\theta,u)$ for Case 1. It follows from the definition of $\theta$ (i.e. ) that there exists a global solution $(\rho,u)$ to for Case 1.
Next we show $$\label{vorticity-decay}
\|{\partial}^{\le3}w(t,\cdot)\| \ls \Xi(t)^{-\frac13} \,{\varepsilon},$$ where $\Xi(t)=e^{\frac{\mu}{1-{\lambda}}[(1+t)^{1-{\lambda}}-1]}$ has been defined in .
For this purpose, we assume that $\| \Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|\le K_2{\varepsilon}$ holds for sufficiently large constant $K_2>0$ and small ${\varepsilon}>0$. This immediately implies $$\label{3.30}
\Xi(t)^\frac13 |{\partial}^{\le1}w(t,\cdot)|_\infty \ls K_2{\varepsilon}.$$ Multiplying by $ \Xi(t)^\frac23$ yields $$\begin{aligned}
&\frac{d}{dt} \|\Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|^2
+\frac{\mu}{3(1+t)^{\lambda}} \,\|\Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|^2 {\nonumber}\\
&\ls \Xi(t)^\frac23 |{\partial}^{\le1} w(t,\cdot)|_\infty\,
\|{\partial}^{\le3}w(t\cdot)\|\,\|{\partial}{\partial}^{\le3} u(t,\cdot)\|. \label{3.31}\end{aligned}$$ Substituting and into and applying Young’s inequality, we then have $$\begin{aligned}
&\quad \frac{d}{dt} \|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2+
\frac{\mu}{3(1+t)^{\lambda}}\|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2 {\nonumber}\\
&\ls \Xi(t)^\frac23 |{\partial}^{\le1}w(t,\cdot)|_\infty \,\|{\partial}^{\le3}w(t\cdot)\|
\left((1+t)^{-{\lambda}}\|u(t,\cdot)\|+\|{\partial}^{\le3}w(t\cdot)\|
+\|{\partial}{\partial}^{\le3}\theta(t\cdot)\| \right) {\nonumber}\\
&\ls K_2{\varepsilon}\left((1+t)^{-{\lambda}}\|u(t,\cdot)\|^2+\big((1+t)^{-{\lambda}}
+\Xi(t)^{-\frac13}\big)\|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2
+(1+t)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(t,\cdot)\|^2 \right). \label{3.32}\end{aligned}$$ Collecting , , and applying the same argument as in the proof for the global existence of $(\rho, u)$, we infer . This completes the proof of . Thus the proof of Theorem \[thm1\] is completed.
\[rmk3.2\] The proof of Theorem \[thm1\] can be applied to the case of ${\lambda}=1$, $\mu>2$. In this case, Lemma \[lem-velocity1\]-\[lem-vorticity\] still hold and the coefficient of the first term in the second line of is $(\mu-2)(1+t)$, which plays the same role as . Instead of the identity below , we have $$\begin{aligned}
(1+t)^2|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t){\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu(\mu-1)}{2}|{\partial}^{\alpha}\theta|^2 \\
=(1+t)^2\left(\frac{\mu-2}{3\mu-2}|{\partial}_t{\partial}^{\alpha}\theta|^2+|\nabla{\partial}^{\alpha}\theta|^2\right)
+\frac{\mu(\mu-2)}{8}|{\partial}^{\alpha}\theta|^2 \\
+\left((1+t)\sqrt\frac{2\mu}{3\mu-2}{\partial}_t{\partial}^{\alpha}\theta+\sqrt\frac{\mu(3\mu-2)}{8}{\partial}^{\alpha}\theta\right)^2,\end{aligned}$$ which is equivalent to $(1+t)|{\partial}{\partial}^{\alpha}\theta|^2+|{\partial}^{\alpha}\theta|^2$ for $\mu>2$. However, we cannot obtain the exponential decay of the vorticty $w$.
Proof of Theorem 1.2. {#section4}
=====================
Theorem \[thm2\] in three space dimensions has been proved in [@HWY15]. In this section, we fix $d=2$ and assume that $$\label{4.1}
E_5[\theta,u](t) \le K_3{\varepsilon}$$ holds. By the finite propagation speed property of hyperbolic systems, one easily knows that $(\theta, u)$ and their derivatives are supported in $\{(t,x)\colon |x|\le t+M\}$, which implies that for ${\alpha}>0$, $$\label{4.2}
|Z^{\alpha}(\theta,u)(t,x)| \ls (1+t) |{\partial}Z^{<{\alpha}}(\theta,u)(t,x)|.$$ On the other hand, collecting - with assumption derives the following pointwise estimate $$\label{4.3}
|\sigma_-^{-\frac12}(t,\cdot)Z^{\le2}(\theta,u)(t,\cdot)|_\infty+
|\sigma_-^\frac12(t,\cdot){\partial}Z^{\le2}(\theta,u)(t,\cdot)|_\infty
\ls \frac{K_3{\varepsilon}}{1+t}.$$ To prove Theorem \[thm2\] for $d=2$, we shall focus on the following parts.
Estimates of velocity $u$ and its derivatives.
----------------------------------------------
The following lemma is an application of Lemma \[lem-divcurl\] and -.
\[lem-Zvelocity\] Under assumption , for all $t\ge0$, it holds that $$\label{4.4}
E_5[u](t)\ls (1+t)^{-\frac12}\|u(t,\cdot)\|+(1+t)^\frac12\|{\partial}Z^{\le4}\theta(t,\cdot)\|.$$
In view of $\opcurl u_0\equiv0$ and , it is easy to know that $\opcurl u(t,x)\equiv0$ always holds for any $t\ge0$ as long as the smooth solution $(\theta, u)$ of exists. Then, it follows from $\opcurl u\equiv0$ that $$\begin{aligned}
\opcurl Z^{\alpha}u= Z^{\alpha}\opcurl u+\sum_{\beta<{\alpha}}C_{{\alpha},\beta}{\partial}Z^\beta u= \sum_{\beta<{\alpha}}C_{{\alpha},\beta}{\partial}Z^\beta u,\end{aligned}$$ which can be abbreviated as $$\label{4.5}
\opcurl Z^{\alpha}u={\partial}Z^{<{\alpha}}u.$$ Analogously, we have $$\label{4.6}
\opdiv Z^{\alpha}u=Z^{\alpha}\opdiv u+{\partial}Z^{<{\alpha}}u.$$ Taking $U=Z^{\alpha}u$ with $|{\alpha}|\le4$ in and applying - yield $$\begin{aligned}
\|\nabla Z^{\alpha}u(t,\cdot)\|
&\ls \|Z^{\alpha}\opdiv u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+K_3{\varepsilon}(1+t)^{-1}\sum_{0<\beta\le{\alpha}}
\|Z^\beta(\theta,u)(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\|, \label{4.7}\end{aligned}$$ where we have used the first equation in and -. On the other hand, one easily gets $$\begin{aligned}
\|{\partial}_t Z^{\alpha}u(t,\cdot)\|
&\ls \|Z^{\alpha}{\partial}_t u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+K_3{\varepsilon}\|{\partial}Z^{\le{\alpha}}u(t,\cdot)\|
+(1+t)^{-1}\|u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\|. \label{4.8}\end{aligned}$$ Summing up - from $|{\alpha}|=0$ to $|{\alpha}|=4$, then is obtained by the smallness of ${\varepsilon}$. This completes the proof of Lemma \[lem-Zvelocity\].
\[lem-velocity3\] Let $\mu>0$. Under assumption , for all $t\ge0$, it holds that $$\label{4.9}
\frac{d}{dt}\left[(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2\right]
+\frac1{2(1+t)^2}\|(\theta,u)(t,\cdot)\|^2 \le 0.$$
Multiplying the second equation in by $(1+t)^{-1}u$ derives $$\label{4.10}
\frac12\,{\partial}_t\left[(1+t)^{-1}|u|^2\right]+\frac{\mu+\frac12}{(1+t)^2}|u|^2
+(1+t)^{-1}u\cdot\nabla\theta=-\frac12\,(1+t)^{-1}u\cdot\nabla|u|^2.$$ From the first equation in , we see that $$\begin{aligned}
(1+t)^{-1}u\cdot\nabla\theta
&= \opdiv\left[(1+t)^{-1}(\theta u+({\gamma}-1)\theta^2 u)\right]+
\frac12\,{\partial}_t\left[(1+t)^{-1}|\theta|^2\right] {\nonumber}\\
&\quad +\frac1{2(1+t)^2}|\theta|^2+(3-2{\gamma})(1+t)^{-1}\theta\,
u\cdot\nabla\theta, \label{4.11}\end{aligned}$$ which is similar to the expression in . Substituting into and integrating it over ${\mathbb R}^2$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\left[(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2\right]+
\frac{1}{2(1+t)^2}\|(\theta,u)(t,\cdot)\|^2 {\nonumber}\\
&\ls (1+t)^{-1}|\nabla\theta(t,\cdot)|_\infty\|u(t,\cdot)\|\,\|\theta(t,\cdot)\|
+(1+t)^{-1}|\nabla u(t,\cdot)|_\infty\|u(t,\cdot)\|^2. \label{4.12}\end{aligned}$$ Substituting into , then can be obtained from the smallness of ${\varepsilon}$. This completes the proof of Lemma \[lem-velocity3\].
Estimates of $\theta$ and its derivatives.
------------------------------------------
The following lemma shows the estimates of $\theta$.
\[lem-Ztheta\] Let $\mu>1$. Under assumption , for all $t\ge0$, it holds that $$\begin{aligned}
&\quad E_5^2[\theta](t)+\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2
+(1+s)^{-2}\|Z^{\le4}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
& \ls E_5^2[\theta](0)+K_3{\varepsilon}\int_0^t (1+s)^{-2}\|u(s,\cdot)\|^2 \,ds. \label{4.13}\end{aligned}$$
Acting $Z^{\alpha}$ with $|{\alpha}|\le4$ on both sides of equation implies $$\label{4.14}
{\partial}_t^2Z^{\alpha}\theta+\frac\mu{1+t}{\partial}_tZ^{\alpha}\theta-\Delta Z^{\alpha}\theta=Q^{\alpha}_2(\theta,u),$$ where $$\begin{aligned}
Q^{\alpha}_2(\theta,u) &\defeq Z^{\alpha}Q(\theta,u)+Q^{\alpha}_{21}+Q^{\alpha}_{22}+
Q^{\alpha}_{23} {\nonumber}\\
&\defeq Z^{\alpha}Q(\theta,u)+[{\partial}_t^2-\Delta,Z^{\alpha}]\,\theta+
\frac\mu{1+t}[{\partial}_t,Z^{\alpha}]\,\theta
+\sum_{0<\beta\le{\alpha}}Z^\beta\left(\frac\mu{1+t}\right)
Z^{{\alpha}-\beta}{\partial}_t\theta. \label{4.15}\end{aligned}$$ Multiplying by $2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta$ yields $$\begin{aligned}
&{\partial}_t\left[\mu(1+t)|{\partial}Z^{\alpha}\theta|^2+(2\mu-1)Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta+
\frac{\mu(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2\right] {\nonumber}\\
&\quad +(\mu-1)(2\mu-1)|{\partial}_tZ^{\alpha}\theta|^2+(\mu-1)|\nabla Z^{\alpha}\theta|^2+
\frac{\mu(2\mu-1)}{2(1+t)^2}|Z^{\alpha}\theta|^2 {\nonumber}\\
&=\opdiv\left[\nabla Z^{\alpha}\theta \left(2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)
Z^{\alpha}\theta\right) \right]+Q^{\alpha}_2(\theta,u)\left(2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)
Z^{\alpha}\theta\right). \label{4.16}\end{aligned}$$ Thanks to $\mu>1$, we see that for the term in the square bracket of the first line in $$\begin{aligned}
&\quad \mu(1+t)|{\partial}Z^{\alpha}\theta|^2+(2\mu-1)Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta+
\frac{\mu(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2 \\
&=(1+t)\Big(\frac12|{\partial}_tZ^{\alpha}\theta|^2+\mu|\nabla Z^{\alpha}\theta|^2\Big)+
\frac{(\mu-1)(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2+\frac{2\mu-1}{2(1+t)}
\Big( (1+t){\partial}_tZ^{\alpha}\theta+Z^{\alpha}\theta\Big)^2,\end{aligned}$$ which is equivalent to $(1+t)|{\partial}Z^{\alpha}\theta|^2+\frac1{1+t}|Z^{\alpha}\theta|^2$. Integrating over $[0,t]\times{\mathbb R}^2$ yields $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+\left|\int_0^t\int_{{\mathbb R}^2} Q^{\alpha}_2(\theta,u)\Big(2\mu
(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\right)\,dxds\Big|. \label{4.17}\end{aligned}$$ It follows from a direct computation that $$\label{4.18}
Q^{\alpha}_{21}=Z^{<{\alpha}}({\partial}_t^2-\Delta)\theta=Z^{<{\alpha}}Q(\theta,u)-Z^{<{\alpha}}\left(
\frac{\mu}{1+t}{\partial}_t\theta\right) \defeq Z^{<{\alpha}}Q(\theta,u)+Q^{\alpha}_{24}.$$ For ${\alpha}=0$, we find that $Q^{\alpha}_{22}=Q^{\alpha}_{23}=Q^{\alpha}_{24}=0$. For ${\alpha}>0$ and by , we see that $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2}\left(Q^{\alpha}_{22}+Q^{\alpha}_{23}+Q^{\alpha}_{24}\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls \int_0^t\|{\partial}Z^{\le{\alpha}}\theta(s,\cdot)\|\,\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|\,ds. \label{4.19}\end{aligned}$$ Recall the definition of $Q(\theta,u)$ in - as follows $$\begin{aligned}
Q(\theta,u) &=Q_1(\theta,u)+Q_2(\theta,u), \\
Q_1(\theta,u) &=-\frac\mu{1+t}u\cdot\nabla\theta+({\gamma}-1)\theta\Delta\theta
-2u\cdot\nabla{\partial}_t\theta-\sum_{i,j=1,2} u_iu_j{\partial}_{ij}^2\theta, \\
Q_2(\theta,u) &= -\sum_{i,j=1,2}u_i{\partial}_iu_j{\partial}_j\theta-{\partial}_tu\cdot\nabla\theta+(1+({\gamma}-1)\theta)
(\sum_{i,j=1,2}{\partial}_iu_j{\partial}_ju_i+({\gamma}-1)|\opdiv u|^2).\end{aligned}$$ Then it follows from - and $|{\alpha}|\le4$ that $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Z^{\le{\alpha}}Q_2(\theta,u)\Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls \int_0^t \Big(|\theta(s,\cdot)|_\infty+(1+s)|{\partial}Z^{\le2}(\theta,u)
(s,\cdot)|_\infty\Big)\,\|{\partial}Z^{\le4}(\theta,u)(s,\cdot)\|^2 \,ds {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big)\,ds. \label{4.20}\end{aligned}$$ Substituting - into derives $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+\int_0^t\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|^2\,ds
+K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big)\,ds {\nonumber}\\
&\quad +\left|\int_0^t \int_{{\mathbb R}^2} Z^{\le{\alpha}}Q_1(\theta,u)\Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right|. \label{4.21}\end{aligned}$$ Next we focus on the treatment of $Z^{\le{\alpha}}Q_1(\theta,u)$. In view of $|{\alpha}|\le4$, we find that $$Z^{\le{\alpha}}(\theta\Delta\theta)=\theta\Delta Z^{\alpha}\theta+\sum_{1\le|\beta|\le2}
Z^\beta\theta\,{\partial}^2Z^{{\alpha}-\beta}\theta+\sum_{|\beta|\ge3}
Z^\beta\theta\,{\partial}^2Z^{{\alpha}-\beta}\theta,$$ which can be abbreviated as $$Z^{\le{\alpha}}(\theta\Delta\theta)=\theta\Delta Z^{\alpha}\theta+Z^{\le2}\theta
\,{\partial}^2Z^{\le3}\theta+Z^{\le4}\theta\,{\partial}^2Z^{\le1}\theta.$$ From this and the definition of $Q_1(\theta,u)$, we see that $$\begin{aligned}
&\quad Z^{\le{\alpha}}Q_1(\theta,u) {\nonumber}\\
&=-Z^{\le{\alpha}}\left(\frac\mu{1+t}u\cdot\nabla\theta\right)+({\gamma}-1)
\theta\Delta Z^{\alpha}\theta-2u\cdot\nabla{\partial}_tZ^{\alpha}\theta
-\sum_{i,j=1,2}u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta+Q_3(\theta,u),\end{aligned}$$ where $$\begin{aligned}
Q_3(\theta,u) \defeq (Z^{\le2}\theta+Z^{\le2}u){\partial}^2Z^{\le3}\theta+
(Z^{\le4}\theta+Z^{\le4}u){\partial}^2Z^{\le1}\theta {\nonumber}\\
+Z^{\le2}u\,Z^{\le2}u\,{\partial}^2Z^{\le3}\theta+Z^{\le2}u\,Z^{\le4}u\,
{\partial}^2Z^{\le1}\theta.\end{aligned}$$ If ${\alpha}=0$, applying yields $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \frac\mu{1+s}u\cdot\nabla\theta
\Big(2\mu(1+s){\partial}_t\theta+(2\mu-1)\theta\Big) \,dxds\right| {\nonumber}\\
&\ls \int_0^t \Big(|u(s,\cdot)_\infty\|{\partial}\theta(s,\cdot)\|^2+(1+s)^{-1}
|\theta(s,\cdot)|_\infty\|u(s,\cdot)\|\,\|\nabla\theta(s,\cdot)\|\Big) \,ds {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big) \,ds. \label{4.22}\end{aligned}$$ If ${\alpha}>0$, from - we see that $$\begin{aligned}
&\quad \left\|Z^{\le{\alpha}}\left(\frac\mu{1+t}u\cdot\nabla\theta\right) \right\| \\
&\ls (1+t)^{-1} |Z^{\le2}u(t,\cdot)|_\infty\|{\partial}Z^{\le4}\theta(t,\cdot)
\|+(1+t)^{-1} |{\partial}Z^{\le2}\theta(t,\cdot)|_\infty\|Z^{\le4}u(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+K_3{\varepsilon}(1+t)^{-2}\|u(t,\cdot)\|,\end{aligned}$$ which derives $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Z^{\le{\alpha}}\left(\frac\mu{1+s}u\cdot\nabla\theta\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.23}\end{aligned}$$ As in Lemma \[lem-thetaglobal\], direct computation derives the following identities $$\begin{aligned}
(1+t)\theta\Delta Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
& \opdiv\left[(1+t)\theta\nabla Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\right]-(1+t)
\nabla\theta\cdot\nabla Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta \\
& -\frac12\,{\partial}_t\left[(1+t)\theta\,|\nabla Z^{\alpha}\theta|^2\right]+\frac12\,
\theta\,|\nabla Z^{\alpha}\theta|^2+\frac12(1+t){\partial}_t\theta\,|\nabla Z^{\alpha}\theta|^2, \\
\theta\Delta Z^{\alpha}\theta Z^{\alpha}\theta=
&\opdiv\left[\theta\nabla Z^{\alpha}\theta Z^{\alpha}\theta\right]-\theta\,
|\nabla Z^{\alpha}\theta|^2-\nabla\theta\cdot\nabla Z^{\alpha}\theta Z^{\alpha}\theta,\end{aligned}$$ together with -, this yields $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \theta\Delta Z^{\alpha}\theta \Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\Big(E_5^2[\theta](0)+E_5^2[\theta](t)\Big)+K_3{\varepsilon}\int_0^t
\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2 \,ds. \label{4.24}\end{aligned}$$ Analogously, we have $$\begin{aligned}
2(1+t)u\cdot\nabla{\partial}_tZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
& \opdiv\left[(1+t)u\,|{\partial}_tZ^{\alpha}\theta|^2\right]-(1+t)\opdiv u\,|{\partial}_tZ^{\alpha}\theta|^2, \\
u\cdot\nabla{\partial}_tZ^{\alpha}\theta Z^{\alpha}\theta=
& \opdiv\left[u\,{\partial}_tZ^{\alpha}\theta Z^{\alpha}\theta\right]
-{\partial}_tZ^{\alpha}\theta(u\cdot\nabla Z^{\alpha}\theta+\opdiv u\,Z^{\alpha}\theta)\end{aligned}$$ and $$\begin{aligned}
2(1+t)u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
&{\partial}_i\Big[(1+t)u_iu_j{\partial}_jZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\Big]+
{\partial}_j\Big[(1+t)u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\Big] \\
& -(1+t){\partial}_tZ^{\alpha}\theta\Big[{\partial}_i(u_iu_j){\partial}_jZ^{\alpha}\theta+
{\partial}_j(u_iu_j){\partial}_iZ^{\alpha}\theta\Big] \\
& -{\partial}_t\Big[(1+t)u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta\Big]+
u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta \\
& +(1+t){\partial}_t(u_iu_j){\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta, \\
u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta Z^{\alpha}\theta=
& {\partial}_i\left[u_iu_j{\partial}_jZ^{\alpha}\theta Z^{\alpha}\theta\right]-
{\partial}_jZ^{\alpha}\theta{\partial}_i(u_iu_jZ^{\alpha}\theta),\end{aligned}$$ which derives $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \left(2u\cdot\nabla{\partial}_tZ^{\alpha}\theta+
\sum_{i,j=1,2} u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\Big(E_5^2[\theta](0)+E_5^2[\theta](t)\Big)+K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big) \,ds. \label{4.25}\end{aligned}$$ Next we turn our attention to $Q_3(\theta,u)$. Note that $Q_3(\theta,u)=0$ when ${\alpha}=0$. It follows from direct calculation that for any function $\Phi(t,x)$ $$|\sigma_-(t,x){\partial}\Phi(t,x)| \ls |Z\Phi(t,x)|.$$ From this, and -, we see that $$\begin{aligned}
&\quad \|Q_3(\theta,u)\| {\nonumber}\\
&\ls \|Z^{\le2}(\theta,u){\partial}^2Z^{\le3}\theta\|+
\|Z^{\le4}(\theta,u){\partial}^2Z^{\le1}\theta\| \\
&\ls |\sigma_-^{-1}(t,\cdot)Z^{\le2}(\theta,u)(t,\cdot)|_\infty
\|\sigma_-(t,\cdot){\partial}^2Z^{\le3}\theta(t,\cdot)\| \\
&\quad +|\sigma_-^{\frac32}(t,\cdot){\partial}^2Z^{\le1}\theta(t,\cdot)|_\infty
\|\sigma_-^{-\frac32}(t,\cdot)Z^{\le4}(\theta,u)(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+|\sigma_-^{\frac12}(t,\cdot)
{\partial}Z^{\le2}\theta(t,\cdot)|_\infty \|\nabla Z^{\le4}(\theta,u)(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+K_3{\varepsilon}\,
(1+t)^{-2}\|u(t,\cdot)\|,\end{aligned}$$ which implies $$\begin{aligned}
\label{}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Q_3(\theta,u) \Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta
+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.26}\end{aligned}$$ Substituting - into derives $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+K_3{\varepsilon}\,E_5^2[\theta](t)+\int_0^t\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.27}\end{aligned}$$ Summing up from $|{\alpha}|=0$ to $|{\alpha}|=4$ yields . This completes the proof of Lemma \[lem-Ztheta\].
Proof of Theorem \[thm2\].
--------------------------
Integrating over $[0,t]$ yields that $$(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2+\int_0^t (1+s)^{-2}\|(\theta,u)(s,\cdot)\|^2\,ds \ls \|(\theta,u)(0,\cdot)\|^2.$$ Collecting this with and , we conclude that $E_5[\theta,u](t) \le C_3{\varepsilon}$. Let $K_3=2C_3$, and choose ${\varepsilon}>0$ sufficiently small. Then, we infer $E_5[\theta,u](t) \le \frac12 K_3{\varepsilon}$, which implies that admits a global solution for Case 2 with $\opcurl u_0(x)\equiv 0$. Thus we complete the proof of Theorem \[thm2\].
Proof of Theorem \[thm3\]. {#section5}
==========================
In this section, we shall only prove Theorem \[thm3\] for $d=2$ since the corresponding blowup result for Case 4 with $\opcurl u_0(x)\equiv 0$ in three space dimensions has been proved in [@HWY15].
We divide the proof into two parts.
### Part : ${\gamma}=2$. {#part-gamma2. .unnumbered}
Let $(\rho, u)$ be a $C^{\infty}-$smooth solution of . For $l>0$, we define $$\label{5.1}
P(t,l)=\int_{x_1>l}\eta(x,l)\left(\rho(t,x)-\bar\rho\right)dx,$$ where $$\eta(x,l)=(x_1-l)^2.$$ Employing the first equation in and an integration by parts, we see that $$\begin{aligned}
\label{5.2}
{\partial}_tP(t,l)&=\int_{x_1>l}\eta(x,l){\partial}_t\left(\rho(t,x)-\bar\rho\right)dx=
-\,\int_{x_1>l}\eta(x,l)\opdiv(\rho u)(t,x)\,dx {\nonumber}\\
&=\int_{x_1>l}({\partial}_{x_1}\eta)(x,l)(\rho u_1)(t,x)\,dx,\end{aligned}$$ where we have used the facts of $\eta(x,l)=0$ on $x_1=l$ and $u(t,x)=0$ for $|x|\ge t+M$. By differentiating ${\partial}_tP(t,l)$ in again and using the equation of $u_1$ in , we find that $$\begin{gathered}
{\partial}_t^2P(t,l) =\int_{x_1>l}({\partial}_{x_1}\eta)(x,l){\partial}_t(\rho u_1)(t,x)\,dx
=-\sum_{j=1,2}\int_{x_1>l}({\partial}_{x_1}\eta)\,{\partial}_{x_j}(\rho u_1u_j)(t,x)\,dx \\
-\int_{x_1>l}({\partial}_{x_1}\eta)(x,l){\partial}_{x_1}(p(t,x)-\bar p)\,dx-\frac\mu{(1+t)^{\lambda}}
\int_{x_1>l}({\partial}_{x_1}\eta)(x,l)(\rho u_1)(t,x)\,dx,\end{gathered}$$ where $\bar p=p(\bar\rho)$. It follows from the integration by parts that $$\begin{aligned}
\label{5.3}
{\partial}_t^2P(t,l)+ \frac\mu{(1+t)^{\lambda}}\,{\partial}_tP(t,l)&=\sum_{j=1,2}
\int_{x_1>l}({\partial}_{x_1x_j}^2\eta)\rho u_1u_j\,dx+\int_{x_1>l}2(p-\bar p)\,dx {\nonumber}\\
&= \int_{x_1>l}2\rho u_1^2\,dx+\int_{x_1>l}2(p-\bar p)\,dx,\end{aligned}$$ here we have used that ${\partial}_{x_1}\eta(x,l)=0$ on $x_1=l$ and $p(t,x)-\bar p$ vanishes for $|x|\ge t+M$. Note that $${\partial}_l^2\eta(x,l)=\Delta_x\eta(x,l)=2.$$ Then we have $$\label{5.4}
\int_{x_1>l}2(p-\bar p)\,dx=\int_{x_1>l}{\partial}_l^2\eta(x,l)(p(t,x)-\bar p)\,dx=
{\partial}_l^2\int_{x_1>l}\eta(x,l)(p(t,x)-\bar p)\,dx,$$ where we have used the fact that $\eta$ and ${\partial}_l\eta$ vanish on $x_1=l$. Collecting -, we arrive at $$\label{5.5}
{\partial}_t^2P(t,l)-{\partial}_l^2P(t,l)+\frac\mu{(1+t)^{\lambda}}\,{\partial}_tP(t,l)\defeq f(t,l)=\int_{x_1>l}2\rho u_1^2\,dx+G(t,l)\ge G(t,l),$$ where $$\label{5.6}
G(t,l)=\int_{x_1>l}2\left(p-\bar p-(\rho-\bar\rho)\right)dx
={\partial}_l^2\int_{x_1>l}\eta(x,l)\left(p-\bar p-(\rho-\bar\rho)\right)dx
\defeq {\partial}_l^2{\tilde}G(t,l).$$ Due to ${\gamma}=2$ and the sound speed $\bar c=\sqrt{2A\bar\rho}=1$, we have $$\label{5.7}
p-\bar p-(\rho-\bar\rho)=A\left(\rho^2-\bar\rho^2
-2\bar\rho\left(\rho-\bar\rho\right)\right)=A(\rho-\bar\rho)^2.$$ Substituting into gives $$G(t,l),\,{\tilde}G(t,l) \ge 0.$$ For $M_0$ satisfying the condition , let $\Sigma\defeq \{(t,l)\colon t\ge0, t+M_0\le l\le t+M\}$ be the strip domain. By applying Riemann’s representation (see [@CH §5.5 of Chapter 5]) with the assumptions -, we have the following lower bound of the solution $P(t,l)$ to for $(t,l)\in\Sigma$ $$\label{5.8}
P(t,l)\ge \frac14\Xi(t)^{-\frac12}q_0(l-t)+\frac14\int_0^t\int_{l-t+\tau}^{l+t-\tau} \left(\frac{\Xi(\tau)}{\Xi(t)}\right)^\frac12 f(\tau,y)\,dyd\tau.$$ We put the proof of in Appendix. Define the function $$\label{5.9}
F(t)\defeq\int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}P(\tau,l)\,\frac{dl}{\sqrt l} d\tau.$$ From the definition of $\Xi(t)$, i.e., for ${\lambda}=1$, $\mu\le1$ or ${\lambda}>1$, we have $\Xi(t)^{-\frac12}\gt(t+M)^{-\frac12}$ and $\frac{\Xi(\tau)}{\Xi(t)}\gt\frac{\tau+M}{t+M}$. Then, by , we arrive at $$\begin{aligned}
&F''(t)=\int_{t+M_0}^{t+M}P(t,l)\,\frac{dl}{\sqrt l} \gt
(t+M)^{-\frac12}\int_{t+M_0}^{t+M} q_0(l-t)\,\frac{dl}{\sqrt l} {\nonumber}\\
&\quad +\int_{t+M_0}^{t+M}\int_0^t\int_{l-t+\tau}^{l+t-\tau} \left(\frac{\tau+M}{t+M}\right)^\frac12 G(\tau,y)\,dyd\tau
\frac{dl}{\sqrt l} \defeq J_1+J_2. \label{5.10}\end{aligned}$$ From assumption , we see that $$\label{5.11}
J_1\gt \frac{1}{t+M}\int_{t+M_0}^{t+M}q_0(l-t)\,dl=\frac{1}{t+M}\int_{M_0}^M q_0(l)\,dl \gt \frac{{\varepsilon}}{t+M}.$$ To bound $J_2$ from below, we write $$\begin{aligned}
J_2&=\int_0^{t-M_1}\int_{\tau+M_0}^{\tau+M} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{t+M_0}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\quad+\int_{t-M_1}^t\int_{\tau+M_0}^{2t-\tau+M_0} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{t+M_0}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\quad+\int_{t-M_1}^t\int_{2t-\tau+M_0}^{\tau+M} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{y-t+\tau}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\defeq J_{2,1}+J_{2,2}+J_{2,3}, \label{5.12}\end{aligned}$$ where $M_1=\left(M-M_0\right)/2$. For $t<M_1$, $t-M_1$ in the limits of integration will be replaced by $0$. For the integrand in $J_{2,1}$ we have that $$\label{5.13}
\int_{t+M_0}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{y-\tau-M_0}{(t+M)^\frac12}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}.$$ Analogously, for the integrands in $J_{2,2}$ and $J_{2,3}$ we have that $$\label{5.14}
\int_{t+M_0}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}$$ and $$\label{5.15}
\int_{y-t+\tau}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{t-\tau}{(t+M)^\frac12}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}.$$ Substituting - into yields $$\begin{aligned}
J_2 \gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12
\int_{\tau+M_0}^{\tau+M} (y-\tau-M_0)^2{\partial}_y^2{\tilde}G(\tau,y)\,dyd\tau,\end{aligned}$$ where ${\tilde}G(\tau,y)=\int_{x_1>y} (x_1-y)^2 \left(p(\tau,x)-\bar p-(\rho(\tau,x)-\bar\rho)\right)dx$. Note that ${\tilde}G(\tau,y)={\partial}_y{\tilde}G(\tau,y)=0$ for $y=\tau+M$. Then it follows from the integration by parts together with - that $$\begin{aligned}
J_2&\gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M}{\tilde}G(\tau,y)\,dyd\tau {\nonumber}\\
&\gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M} \int_{x_1>y} (x_1-y)^2
\left(\rho(\tau,x)-\bar\rho\right)^2dxdyd\tau {\nonumber}\\
&\defeq \frac{c}{(t+M)^2}\,J_3. \label{5.16}\end{aligned}$$ By applying the Cauchy-Schwartz inequality to $F(t)$ defined by , we arrive at $$\label{5.17}
F^2(t) \le J_3\int_0^t (t-\tau)(\tau+M)^{-\frac12}\int_{\tau+M_0}^{\tau+M}
\int_{{\tilde}\Omega} (x_1-y)^2 \,dx\frac{dy}{y}d\tau\defeq J_3J_4,$$ where ${\tilde}\Omega \defeq \{x\colon x_1>y,~|x|<\tau+M\}$. Note that $$\begin{aligned}
J_4 &\ls \int_0^t (t-\tau)(\tau+M)^{-\frac12}\int_{\tau+M_0}^{\tau+M}\int_y^{\tau+M}
(x_1-y)^2 [(\tau+M)^2-x_1^2]^\frac12 \,dx_1\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}(\tau+M-y)^\frac72\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t\frac{t-\tau}{\tau+M}\,d\tau \ls (t+M)\log(t/M+1). \label{5.18}\end{aligned}$$ Combining - and - gives the following ordinary differential inequalities $$\begin{aligned}
F''(t) &\gt \frac{{\varepsilon}}{t+M}, && t\ge0, \label{5.19} \\
F''(t) &\gt \left[(t+M)^3\log(t/M+1)\right]^{-1} \,F^2(t), && t\ge0. \label{5.20}\end{aligned}$$ Next, we apply - to prove that the lifespan $T_{\varepsilon}$ of smooth solution $F(t)$ is finite for all $0<{\varepsilon}\le{\varepsilon}_0$. The fact that $F(0)=F'(0)=0$, together with , yields $$\begin{aligned}
F'(t) &\gt {\varepsilon}\log(t/M+1), && t\ge0, \label{5.21} \\
F(t) &\gt {\varepsilon}(t+M)\log(t/M+1), && t\ge t_1\defeq Me^2. \label{5.22}\end{aligned}$$ Substituting into derives $$F''(t) \gt {\varepsilon}^2(t+M)^{-1}\log(t/M+1), \qquad t\ge t_1,$$ which leads to the improvement $$\label{5.23}
F(t) \gt {\varepsilon}^2(t+M)\log^2(t/M+1), \qquad t\ge t_2 \defeq Me^3>t_1.$$ Substituting this into yields $$\label{5.24}
F''(t) \gt {\varepsilon}^2(t+M)^{-2}\log(t/M+1)\,F(t), \qquad t\ge t_2.$$ It follows from that $F'(t)\ge0$ for $t\ge0$. Then multiplying by $F'(t)$ and integrating from $t_3$ (which will be chosen later) to $t$ derive $$F'(t)^2 \ge F'(t_3)^2+C_4{\varepsilon}^2\int_{t_3}^t (s+M)^{-2}\log(s/M+1)\,[F(s)^2]'ds.$$ It follows from the integration by parts that $$\begin{gathered}
\label{5.25}
F'(t)^2 \ge
F'(t_3)^2+C_4{\varepsilon}^2 \left((t+M)^{-2}\log(t/M+1)F(t)^2-(t_3+M)^{-2}\log(t_3/M+1)F(t_3)^2\right)\\
-C_4{\varepsilon}^2\int_{t_3}^t \left(\frac{\log(s/M+1)}{(s+M)^2}\right)' F(s)^2\,ds, \quad t\ge t_3,\end{gathered}$$ where $\left(\frac{\log(s/M+1)}{(s+M)^2}\right)'\le0$ for $s\ge t_3\ge t_2$. Since $F''(t)\ge 0$ and $F(0)=0$, the mean value theorem yields $$\label{5.26}
F(t_3)=\int_0^{t_3}F'(s)ds \le t_3F'(t_3).$$ Choose $$\label{5.27}
t_3=Me^\frac1{2C_4{\varepsilon}^2}-M,$$ which satisfies $C_4{\varepsilon}^2\log(t_3/M+1)=\frac12$. Together with -, this yields $$\label{5.28}
F'(t) \ge \sqrt{C_4}{\varepsilon}(t+M)^{-1}\log^\frac12(t/M+1)\,F(t), \quad t\ge t_3.$$ By integrating from $t_3$ to $t$, we arrive at $$\log\frac{F(t)}{F(t_3)} \ge \sqrt{C_4}{\varepsilon}\log^\frac32\left(\frac{t+M}
{t_3+M}\right), \quad t\ge t_3.$$ If $t\ge t_4\defeq Ct_3^2$, then we have $$\log\frac{F(t)}{F(t_3)} \ge 8\log(t+M).$$ Together with for $F(t_3)$, this yields $$\label{5.29}
F(t) \gt {\varepsilon}^2(t+M)^8, \quad t\ge t_4.$$ Substituting this into derives $$F''(t) \gt {\varepsilon}F(t)^\frac32, \quad t\ge t_4.$$ Multiplying this differential inequality by $F'(t)$ and integrating from $t_4$ to $t$ yield $$F'(t)^2 \gt {\varepsilon}\left(F(t)^\frac52-F(t_4)^\frac52\right).$$ On the other hand, $F(t)\ge 0$, $F''(t)\ge 0$, and the mean value theorem imply that, for $t\ge t_4$, $$F(t)=F'(\xi)(t-t_4)+F(t_4) \ge F'(t_4)(t-t_4) \ge F(t_4)\frac{t-t_4}{t_4},$$ where $t_4\le\xi\le t$. For $t\ge t_5\defeq Ct_4$, we have $$F(t)^\frac52-F(t_4)^\frac52 \ge \frac{1}{2}F(t)^\frac52.$$ Thus $$\label{5.30}
F'(t) \gt \sqrt{\varepsilon}F(t)^\frac54, \quad t\ge t_5.$$ If $T_{\varepsilon}>2t_5$, then integrating from $t_5$ to $T_{\varepsilon}$ derives $$F(t_5)^{-\frac14}-F(T_{\varepsilon})^{-\frac14} \gt \sqrt{\varepsilon}T_{\varepsilon}.$$ We see from and $t_5=Ct_3^2$ that $$F(t_5)\gt {\varepsilon}^2e^\frac{C}{{\varepsilon}^2},$$ which together with $F(T_{\varepsilon})>0$ is a contradiction. Thus, $T_{\varepsilon}\le 2t_5=Ct_3^2$. From the choice of $t_3$ in , we see that $T_{\varepsilon}\le e^{C/{\varepsilon}^2}$.
### Part : ${\gamma}>1$ and ${\gamma}\not=2$. {#part-gamma1-and-gammanot2. .unnumbered}
In view of $\bar c=\sqrt{{\gamma}A\bar\rho^{{\gamma}-1}}=1$, instead of we have $$p-\bar p-(\rho-\bar\rho)=
A\left(\rho^{\gamma}-\bar\rho^{\gamma}-{\gamma}\bar\rho^{{\gamma}-1}(\rho-\bar\rho)\right)
\defeq A\psi(\rho,\bar\rho).$$ The convexity of $\rho^{\gamma}$ for ${\gamma}>1$ implies that $\psi(\rho,\bar\rho)$ is positive for $\rho\neq\bar\rho$. Applying Taylor’s theorem, we have $$\psi(\rho,\bar\rho) \ge C_{{\gamma},\bar\rho} \,\Phi_{\gamma}(\rho,\bar\rho),$$ where $C_{{\gamma},\bar\rho}$ is a positive constant and $\Phi_{\gamma}$ is given by $$\Phi_{\gamma}(\rho,\bar\rho)=
\begin{cases}
(\bar\rho-\rho)^{\gamma}, & \rho< \frac12\bar\rho,\\
(\rho-\bar\rho)^2, & \frac12\bar\rho\le\rho\le2\bar\rho,\\
(\rho-\bar\rho)^{\gamma}, & \rho>2\bar\rho.\\
\end{cases}$$ For ${\gamma}>2$, we have that $(\bar\rho-\rho)^{\gamma}=(\bar\rho-\rho)^2(\bar\rho-\rho)^{{\gamma}-2}\ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$ for $2\rho<\bar\rho$ and $(\rho-\bar\rho)^{\gamma}=(\rho-\bar\rho)^2(\rho-\bar\rho)^{{\gamma}-2}\ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$ for $\rho> 2\bar\rho$. Thus, $\Phi_{\gamma}(\rho,\bar\rho) \ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$. In this case, Theorem \[thm3\] can be shown completely analogously to Part .
Next we treat the case $1<{\gamma}<2$. We define $F(t)$ as in $$F(t)\defeq \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M} \int_{x_1>l}
(x_1-l)^2\left(\rho(\tau,x)-\bar\rho\right)\,dx\frac{dl}{\sqrt l}d\tau.$$ Similarly to Part , we have $$\label{5.31}
F''(t)\ge J_1+J_2,$$ where $$\begin{aligned}
J_1 &\gt \frac{{\varepsilon}}{t+M},\\
J_2 &\gt (t+M)^{-2}{\tilde}J_3\end{aligned}$$ and $${\tilde}J_3 =\int_0^t(t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M}
\int_{x_1>y}(x_1-y)^2\,\Phi_{\gamma}(\rho(\tau,x), \bar\rho)\,dxdyd\tau.$$ Denote $\Omega_1=\{(\tau,x)\colon \bar\rho\le\rho(\tau,x)\le2\bar\rho\}$, $\Omega_2=\{(\tau,x)\colon \rho(\tau,x)>2\bar\rho\}$, and $\Omega_3=\{(\tau,x)\colon \rho(\tau,x)<\bar\rho\}$. Divide $F(t)$ into the following three integrals over the domains $\Omega_i$ $(1\le i\le 3)$ $$F(t)=F_1(t)+F_2(t)+F_3(t)\defeq \int_{\Omega_1}\cdots+\int_{\Omega_2}\cdots+\int_{\Omega_3}\cdots.$$ Corresponding to the three parts of $F(t)$, we define ${\tilde}J_3\defeq{\tilde}J_{3,1}+{\tilde}J_{3,2}+{\tilde}J_{3,3}$. In view of $F(t)\ge0$ and $F_3(t)\le0$, we have $$F(t)\le F_1(t)+F_2(t).$$ Applying Hölder’s inequality for the domains $\Omega_1$ and $\Omega_2$, we obtain that $$\begin{aligned}
F(t)&\le {\tilde}J_{3,1}^\frac12\left(\int_0^t(t-\tau)(\tau+M)^{-\frac12}
\int_{\tau+M_0}^{\tau+M}\frac1y\int_{{\tilde}\Omega}(x_1-y)^2\,dxdyd\tau\right)^\frac12 \\
&\quad +{\tilde}J_{3,2}^\frac1{\gamma}\left(\int_0^t(t-\tau)(\tau+M)^{-\frac1{2({\gamma}-1)}}
\int_{\tau+M_0}^{\tau+M}\frac{1}{y^\frac{{\gamma}}{2({\gamma}-1)}}\int_{{\tilde}\Omega}
(x_1-y)^2\,dxdyd\tau\right)^\frac{{\gamma}-1}{\gamma}\\
&\ls {\tilde}J_3^\frac12(t+M)^\frac12\log^\frac12 (t/M+1)
+{\tilde}J_3^\frac1{\gamma}(t+M)^\frac{{\gamma}-1}{\gamma}\\
&=\left({\tilde}J_3(t+M)^{-1}\right)^\frac12(t+M)\log^\frac12 (t/M+1)
+\left({\tilde}J_3(t+M)^{-1}\right)^\frac1{\gamma}(t+M).\end{aligned}$$ In view of $1<{\gamma}<2$, we have ${\displaystyle}\frac1{2{\gamma}}<\frac12<\frac1{\gamma}$. Applying Young’s inequality yields $$F(t) \ls \Big(\big({\tilde}J_3(t+M)^{-1}\big)^\frac1{2{\gamma}}+
\big({\tilde}J_3(t+M)^{-1}\big)^\frac{1}{{\gamma}}\Big)(t+M)\log^\frac12 (t/M+1),
\quad t\ge {\tilde}t_1\defeq Me.$$ Together with the fact that $F(t)\gt {\varepsilon}(t+M)\log(t/M+1)$, this yields $${\tilde}J_3 \gt F(t)^{\gamma}(t+M)^{1-{\gamma}}\log^{-\frac{{\gamma}}{2}}(t/M+1), \quad t\ge {\tilde}t_1.$$ Substituting this into yields $$\begin{aligned}
F''(t) &\gt \frac{\varepsilon}{t+M}, && t\ge0, \label{5.32}\\
F''(t) &\gt F(t)^{\gamma}(t+M)^{-1-{\gamma}}\log^{-\frac{{\gamma}}{2}}(t/M+1), && t\ge {\tilde}t_1. \label{5.33}\end{aligned}$$ Substituting $F(t)\gt {\varepsilon}(t+M)\log(t/M+1)$ into derives $$F''(t) \gt {\varepsilon}^{\gamma}(t+M)^{-1}\log^\frac{{\gamma}}{2}(t/M+1).$$ Integrating this yields $$F(t) \gt {\varepsilon}^{\gamma}(t+M)\log^\frac{{\gamma}+2}{2}(t/M+1).$$ Substituting this into again gives $$F''(t) \gt {\varepsilon}^{{\gamma}^2}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}+1)}{2}(t/M+1)
={\varepsilon}^{{\gamma}^2}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}^2-1)}{2({\gamma}-1)}(t/M+1).$$ Repeating this process $k$ times, we see that $$\label{5.34}
F''(t) \gt {\varepsilon}^{{\gamma}^k}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}^k-1)}{2({\gamma}-1)}(t/M+1),$$ where $k=\left[\log_{\gamma}2\right]$. Solving yields $$F(t) \gt {\varepsilon}^{{\gamma}^k}(t+M)\log^{\frac{{\gamma}({\gamma}^k-1)}{2({\gamma}-1)}+1}(t/M+1), \quad t\ge {\tilde}t_2,$$ where ${\tilde}t_2>0$ is a constant only depending on ${\gamma}$ and $M$. Substituting this into derives $$\label{5.35}
F''(t) \gt F(t){\varepsilon}^{{\gamma}^k({\gamma}-1)}(t+M)^{-2}\log^\frac{{\gamma}^{k+1}-2}{2}(t/M+1), \quad t\ge {\tilde}t_2,$$ where $\frac{{\gamma}^{k+1}-2}{2}>0$ by the choice of $k=\left[\log_{\gamma}2\right]$. Since is analogous to , as in Part , we can choose ${\tilde}t_3\defeq O\Big(e^{C{\varepsilon}^{-\frac{2{\gamma}^k({\gamma}-1)}{{\gamma}^{k+1}-2}}}\Big)$ such that $$F'(t) \gt {\varepsilon}^\frac{{\gamma}^k({\gamma}-1)}2(t+M)^{-1}\log^\frac{{\gamma}^{k+1}-2}{4}(t/M+1)\,F(t), \quad t\ge {\tilde}t_3,$$ which is similar to and yields $$\label{5.36}
F(t) \gt {\varepsilon}^{C_{\gamma}}(t+M)^\frac{2({\gamma}+2)}{{\gamma}-1}, \quad t\ge {\tilde}t_4\defeq C{\tilde}t_3^2,$$ where $C_{{\gamma}}>0$ is a constant depending on ${\gamma}$. Substituting into yields $$\label{5.37}
F''(t) \gt {\varepsilon}^{C_{\gamma}} F(t)^\frac{{\gamma}+1}2, \qquad t\ge {\tilde}t_4.$$ Multiplying by $F'(t)$ and integrating over the variable $t$ as in Part , we have $$F'(t) \gt {\varepsilon}^{C_{\gamma}}F(t)^\frac{{\gamma}+3}4, \quad t\ge {\tilde}t_5\defeq C{\tilde}t_4.$$ Together with ${\gamma}>1$ and the choice of ${\tilde}t_3$, this yields $T_{\varepsilon}<\infty$.
Collecting Part and Part completes the proof of Theorem \[thm3\].
Proof on the lower bound of $P(t,l)$ in $\Sigma\equiv \{(t,l)\colon
t\ge0, t+M_0\le l\le t+M\}$. {#appendix}
===================================================================
We fixed a point $A=(t_A,l_A)\in\Sigma$. In the characteristic coordinates $\xi=1+t-l$ and $\zeta=1+t+l$, can be written as $$\label{A.1}
\mathscr{L}\bar P\defeq {\partial}_{\xi\zeta}^2\bar P+\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}
({\partial}_\xi\bar P+{\partial}_\zeta\bar P)=\frac{\bar f}4,$$ where $\bar P(\xi,\zeta)\defeq P(\frac{\zeta+\xi}2-1,\frac{\zeta-\xi}2)$. The adjoint operator $\mathscr{L}^*$ of $\mathscr{L}$ has the form $$\label{A.2}
\mathscr{L}^*{\mathcal{R}}\defeq {\partial}_{\xi\zeta}^2{\mathcal{R}}-\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}
({\partial}_\xi{\mathcal{R}}+{\partial}_\zeta{\mathcal{R}})+\frac{2^{{\lambda}-1}\mu{\lambda}}{(\xi+\zeta)^{{\lambda}+1}}{\mathcal{R}}.$$ For the point $A=(\xi_A,\zeta_A)$ with $\xi_A+\zeta_A=2(1+t_A)\ge2$, denote $B=(2-\zeta_A,\zeta_A)$, $C=(\xi_A,2-\xi_A)$ and $\mathscr{D}$, the domain surrounded by the triangle $ABC$ (see Figure 1 below).
Let the numbers $a$ and $b$ satisfy $a+b=1$ and $$ab=\left\{
\begin{aligned}
&\frac{\mu{\lambda}}{2}, && {\lambda}>1, \\
&\frac\mu2(1-\frac\mu2), && {\lambda}=1.
\end{aligned}
\right.$$ We define $$\label{A.3}
z\defeq -\frac{(\xi_A-\xi)(\zeta_A-\zeta)}{(\xi_A+\zeta_A)(\xi+\zeta)}$$ and $$\label{A.4}
{\mathcal{R}}(\xi,\zeta;\xi_A,\zeta_A)\defeq \Big[\frac{\Xi(\xi+\zeta-1)}{\Xi
(\xi_A+\zeta_A-1)}\Big]^{2^{{\lambda}-2}} \Psi(a,b,1;z),$$ here the definition of function $\Xi$ is given in and $\Psi$ is the hypergeometric function.
![**$(\xi, \zeta)-$plane**[]{data-label="fig:1"}](graph1.eps){width="9cm" height="6.5cm"}
From this and direct calculation, we infer $$\label{A.5}
\mathscr{L}^*{\mathcal{R}}=[\frac{2^{{\lambda}-2}\mu{\lambda}}{(\xi+\zeta)^{{\lambda}+1}}-
\frac{ab}{(\xi+\zeta)^2}-\frac{4^{{\lambda}-2}\mu^2}{(\xi+\zeta)^{2{\lambda}}}]{\mathcal{R}}.$$ On the other hand, from - we arrive at $${\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}}={\partial}_\zeta({\mathcal{R}}{\partial}_\xi\bar P
+\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}{\mathcal{R}}\bar P)-{\partial}_\xi(\bar P{\partial}_\zeta{\mathcal{R}}-\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}{\mathcal{R}}\bar P).$$ Integrating this over $\mathscr{D}$ yields $$\begin{aligned}
\label{A.6}
\bar P(A)&=\frac12{\mathcal{R}}(C;A)\bar P(C)+\frac12{\mathcal{R}}(B;A)\bar P(B)+{\int\!\!\!\!\!\int}_\mathscr{D}
({\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}})\,d\xi d\zeta {\nonumber}\\
&+\int_{BC}(\frac12{\mathcal{R}}{\partial}_\xi\bar P-\frac12\bar P{\partial}_\xi{\mathcal{R}}+\frac\mu4{\mathcal{R}}\bar P)\,d\xi
+(\frac12\bar P{\partial}_\zeta{\mathcal{R}}-\frac12{\mathcal{R}}{\partial}_\zeta\bar P-\frac\mu4{\mathcal{R}}\bar P)\,d\zeta.\end{aligned}$$ Returning to the variable $(t,l)$ (see Figure 2 below), we find in the second line of that $$\begin{aligned}
\label{A.7}
\int_{BC}\cdots=\int_B^C[\frac14{\mathcal{R}}({\partial}_t-{\partial}_l)P-\frac14P({\partial}_t-{\partial}_l)
{\mathcal{R}}+\frac\mu4{\mathcal{R}}P]\,(-dl) {\nonumber}\\
+[\frac14P({\partial}_t+{\partial}_l){\mathcal{R}}-\frac14{\mathcal{R}}({\partial}_t+{\partial}_l)P-\frac\mu4{\mathcal{R}}P]\,dl {\nonumber}\\
=\int_{l_A-t_A}^{l_A+t_A}\left.[\frac\mu2{\mathcal{R}}P+\frac12{\mathcal{R}}{\partial}_tP
-\frac12P{\partial}_t{\mathcal{R}}]\right|_{t=0}dl {\nonumber}\\
=\int_{l_A-t_A}^{l_A+t_A} \Xi(t_A)^{-\frac12} \Big[\Psi(a,b,1;z|_{t=0})
\Big(\frac\mu4q_0(l)+\frac12q_1(l)\Big) {\nonumber}\\
-\frac{ab}{2}\Psi(a+1,b+1,2;z|_{t=0})q_0(l)z_t|_{t=0}\Big]dl,\end{aligned}$$
![**$(t, l)-$plane**[]{data-label="fig:2"}](graph2.eps){width="8.5cm" height="6.5cm"}
where we have used the formula $\Psi'(a,b,c;z)=\frac{ab}{c}\Psi(a+1,b+1,c+1;z)$ (see page 58 of [@EMOT]). From the definition , we arrive at $$z=-\frac{(t_A-l_A-t+l)(t_A+l_A-t-l)}{4(1+t_A)(1+t)}$$ and $$\label{A.8}
z_t|_{t=0}=\frac{t_A}{2(1+t_A)}-z|_{t=0}.$$ If $(t, l)\in\Sigma\cap\overline{\mathscr{D}}$, we infer $$\label{A.9}
0\ge z \ge -\frac12(M-M_0)\ge -\frac12\delta_0,$$ which implies that holds. This, together with , - and the assumption of $\Lambda \ge 3ab$, yields that the integral in the second line of is non-negative. Next we prove that $P(t,l)\ge0$ for all $(t, l)\in\Sigma$. Define $$\bar t\equiv\inf \{t\colon \exists~l\in(t+M_0,t+M)~s.t.~P(t,l)<0\}.$$ From assumption , we get $\bar t>0$. If $\bar t<+\infty$, we see that there exists $\bar l\in(\bar t+M_0,\bar t+M)$ such that $P(\bar t,\bar l)=0$. Moreover, we have $P(t,l)\ge0$ for $t<\bar t$. Choose $A=(t_A,l_A)=(\bar t,\bar l)$ in . From - and we infer $\mathscr{L}^*{\mathcal{R}}\le0$ for ${\lambda}\ge1$ and $(t,l)\in\Sigma\cap\mathscr{D}$ ($\mathscr{L}^*{\mathcal{R}}\equiv0$ if ${\lambda}=1$). It follows from $f(t,l)\ge0$ in , - and that $$\begin{aligned}
P(\bar t,\bar l)\ge \frac12{\mathcal{R}}(C;A)P(0,\bar l-\bar t)+{\int\!\!\!\!\!\int}_{\Sigma\cap\mathscr{D}}
({\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}})\,d\xi d\zeta \ge \frac14\Xi(\bar t)^{-\frac12}q_0(\bar l-\bar t)>0,\end{aligned}$$ which is a contradiction with $P(\bar t,\bar l)=0$. Consequently, we conclude that $\bar t=+\infty$ and $P(t,l)\ge0$ for all $(t, l)\in\Sigma$. It follows from -, , , $P(t,l)\ge0$ and $\mathscr{L}^*{\mathcal{R}}\le0$ that $$P(t_A,l_A)\ge \frac14\Xi(t_A)^{-\frac12}q_0(l_A-t_A)+\frac14\int_0^{t_A}\int_{l_A-t_A+\tau}^{l_A+t_A-\tau} \left(\frac{\Xi(\tau)}{\Xi(t_A)}\right)^\frac12 f(\tau,y)\,dyd\tau,$$ which is .
Yin Huicheng wishes to express his gratitude to Professor Ingo Witt, University of Göttingen, and Professor Michael Reissig, Technical University Bergakademie Freiberg, for their interests in this problem and some very fruitful discussions in the past.
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[^1]: Fei Hou (`[email protected]`) and Huicheng Yin (`[email protected]`) were supported by the NSFC (No. 11571177) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct a collection of Lorentz violating Yang-Mills theories exhibiting supersymmetry.'
address: |
Division of Natural Science, New College of Florida\
Sarasota, FL 34234, USA\
$^*$E-mail: [email protected]
author:
- 'DON COLLADAY and PATRICK MCDONALD$^*$'
title: LORENTZ VIOLATION AND EXTENDED SUPERSYMMETRY
---
Introduction and background
===========================
Symmetry has played a fundamental role in the construction of field theories purporting to describe fundamental physics. Nowhere is this more true than in the construction of the Minimal Supersymmetric Standard Model where symmetries mixing bosonic and fermionic states lead to tightly constrained theories often exhibiting remarkable properties. This is particularly true for ${\mathcal N}=4$ extended supersymmetric Yang-Mills theories [@BSS], which are known to be finite.
In this note we construct field theories which exhibit ${\mathcal N}=4$ extended supersymmetry and Lorentz violation. To do so we combine ideas of Berger and Kostelecký [@BK] involving supersymmetric scalar theories exhibiting Lorentz violation, and well-known constructions of extended supersymmetric theories involving dimensional reduction (nicely described in the work of Brink, Schwartz and Scherk[@BSS]). We begin by establishing notation in the context of the standard construction.
Consider a gauge theory involving a single fermion $\lambda$ and lagrangian = - F\^2 +|, where $F$ is the field strength, $F^{\mu\nu} =
[D^\mu,D^\nu]/ig$ and $D$ is the covariant derivative, D\^= \^+ig A\^. To simplify notation, we will first consider the abelian case. To implement supersymmetry we introduce a supercharge, $Q,$ satisfying = 0, {Q,|[Q]{}} = 2\^P\_, where $\gamma^\mu$ are the standard Dirac matrices { \^,\^} = 2 g\^, and the energy momentum 4-vector $P_\mu$ generates spacetime translations. The construction of a supercharge is elegantly carried out in the context of [*superspace.*]{} More precisely, introduce four independent anticommuting variables, $\theta,$ and consider the general vector superfield $$\begin{aligned}
V(x,\theta)& = &C(x) + i\bar{\theta}\gamma^5 w(x)
-\frac{i}{2}\bar{\theta}\gamma^5\theta M(x)
-\frac{1}{2}\bar{\theta}\theta N(x)
+\frac{1}{2}\bar{\theta}\gamma^5\gamma^\mu \theta A_\mu \nonumber\\
& & -i\bar{\theta}\gamma^5\theta \bar{\theta}[\lambda +\frac{i}{2} \not
\partial w(x)] + \frac{1}{2}(\bar{\theta}\theta)^2(D(x)
-\frac{1}{2}\partial_\mu\partial^\mu C(x))\end{aligned}$$ and the $Q$ operator Q= -i \_[|]{} -\^\_. Fixing an arbitrary spinor $\alpha,$ standard analysis[@S] of the operator $\delta_Q V = -i\bar{\alpha}QV$ leads to the supersymmetry transformations defining an ${\mathcal N}=1$ supersymmetric theory.
${\mathcal N}=1$ supersymmetry
==============================
Following Berger and Kostelecký[@BK], we introduce Lorentz violation by defining a twisted derivative: $$\label{perturbation}
\tilde{\partial}^\mu = \partial^\mu + k^{\mu \nu} \partial_\nu,$$ where $k^{\mu \nu}$ is a symmetric, traceless, dimensionless tensor parametrizing Lorentz violation. To obtain a gauge invariant theory, we also twist the underlying connection: \^= A\^+ k\^ A\_. These perturbations lead to a general vector superfield $$\begin{aligned}
\tilde{V}(x,\theta)& = &C(x) + i\bar{\theta}\gamma^5 w(x)
-\frac{i}{2}\bar{\theta}\gamma^5\theta M(x)
-\frac{1}{2}\bar{\theta}\theta N(x)
+\frac{1}{2}\bar{\theta}\gamma^5\gamma^\mu \theta \tilde{A}_\mu \nonumber\\
& & -i\bar{\theta}\gamma^5\theta \bar{\theta}[\lambda +\frac{i}{2} \not
\tilde{\partial} w(x)] + \frac{1}{2}(\bar{\theta}\theta)^2(D(x)
-\frac{1}{2}\tilde{\partial}_\mu\tilde{\partial}^\mu C(x))\end{aligned}$$ and perturbed $Q$ operator = -i \_[|]{} -\^\_. Using Wess-Zumino gauge we obtain a lagrangian $$\label{lagrangian1}
\tilde{{\mathcal L}} = \frac{1}{4} \tilde{F}^2 +\frac{i}{2}
\bar{\lambda} \not \tilde{\partial} \lambda +\frac{1}{2} D^2,$$ where $D$ is an auxiliary chiral field and $\tilde{F}$ is the twisted field strength, $\tilde{F}^{\mu \nu} =\tilde{\partial}^\mu
\tilde{A}^\nu -\tilde{\partial}^\nu \tilde{A}^\mu.$ The twisted field strength can be written in terms of the standard SME parameters: $\tilde{F}^2 = F^2 +k_F^{\mu\nu \alpha \beta}F_{\mu\nu}F_{\alpha
\beta}$, where $$\begin{aligned}
k_{F}^{\mu \nu \alpha \beta} & = & 2(2k^{\alpha \mu} +(k^2)^{\alpha
\mu})g^{\beta \nu} +4(k^{\mu \alpha} +(k^2)^{\alpha \mu})k^{\nu
\beta} +(k^2)^{\alpha \mu} (k^2)^{\beta \nu}.
\quad\end{aligned}$$ Direct calculation confirms that the action is invariant under the supersymmetry transformations $$\begin{aligned}
\delta \tilde{A}^\mu & = & -i\bar{\alpha} \gamma^\mu \lambda, \nonumber\\
\delta \lambda & = &
\frac{i}{2}\sigma^{\mu\nu}\tilde{F}_{\mu\nu}\alpha - \gamma^5
D\alpha, \nonumber\\
\delta D & = & \bar{\alpha} \not \partial \gamma^5 \lambda.\end{aligned}$$ This defines an ${\mathcal N}=1$ supersymmetric theory with Lorentz violation.
${\mathcal N}=4$ supersymmetry
==============================
To build an ${\mathcal N}=4$ supersymmetric theory we work in $4+6$-dimensional spacetime. We represent the $32\times 32$ gamma matrices via $\Gamma^\mu = \gamma^\mu \otimes I_8$ where $I_8$ is the $8\times 8$ identity matrix and $\mu = 0, 1, 2, 3, $ and $$\begin{aligned}
\Gamma^4 = \Gamma^{14} +\Gamma^{23}, \hspace{.25in} & \Gamma^6 = \Gamma^{34}
+\Gamma^{12}, &\hspace{.25in} i\Gamma^8 = \Gamma^{24} +\Gamma^{13}, \nonumber\\
\Gamma^5 = \Gamma^{24} -\Gamma^{13},\hspace{.25in} & i\Gamma^7 = \Gamma^{14}
-\Gamma^{23}, &\hspace{.25in} i\Gamma^9 = \Gamma^{34} -\Gamma^{12}, \end{aligned}$$ where \^[ij]{} = \_5 (
[cc]{} 0 & \^[ij]{}\
\_[ij]{} & 0
) and the $4\times 4$ matrices $\rho$ are defined by $$\begin{aligned}
(\rho^{ij})_{kl} & = & \delta_{ik} \delta_{jl} -\delta_{jk}
\delta_{il},\nonumber\\
(\rho_{ij})_{kl} & = & \frac{1}{2} \epsilon_{ijmn}(\rho^{mn})_{kl} =
\epsilon_{ijkl}. \end{aligned}$$ We consider the lagrangian = - \^2 + | , where $\tilde{\not \partial} = \Gamma^\mu (\partial_\mu +
k_{\mu\nu}\partial^\nu) $ is the twisted derivative with $k_{\mu\nu}$ parametrizing $SO(1,9)$ violation and $\tilde{F}$ is the corresponding perturbed field strength.
Imposing both the Weyl and the Majorana condition and compactifying, the fermion $\lambda$ satisfies $$\begin{aligned}
\lambda = \left(\begin{array}{cc}
L\chi \\
R\tilde{\chi}
\end{array} \right) , & & \end{aligned}$$ where $L$ and $R$ denote left and right projection operators, respectively, the spinor $\chi$ transforms as a 4 of $SU(4)$ and the (independent) spinor $\tilde{\chi}$ transforms as a $\bar{4}$ of $SU(4).$
Choosing the Lorentz violating parameters with care leads to Lorentz violating extended supersymmetric theories which are easy to describe. For example, taking the $k_{\mu\nu}$ to vanish in the compactified directions leads to a lagrangian of the form = - \^2 + i | L+ \_\_[ij]{}\^\^[ij]{} , where the complex scalar fields $\phi_{ij}$ transform as a 6 of $SU(4)$ and are given by $$\begin{aligned}
\phi_{i4}& = & \frac{1}{\sqrt{2}}(A_{i+3}+iA_{i+6}),\nonumber\\
\phi^{jk} & = &\frac{1}{2} \epsilon^{jklm}\phi_{lm} = (\phi_{jk})^*. \end{aligned}$$ The associated action is invariant under the supersymmetry transformations $$\begin{aligned}
\delta \tilde{A}^\mu & = & -i(\bar{\alpha}_i \gamma^\mu L\chi^i
-\bar{\chi}_i \gamma^\mu L\alpha^i), \nonumber\\
\delta \phi_{ij} & = & -i\sqrt{2}(\bar{\alpha}_j R\tilde{\chi}_i -
\bar{\alpha}_i R\tilde{\chi}_j
+\epsilon_{ijkl}\bar{\tilde{\alpha}}^k L \chi^l), \nonumber\\
\delta L\chi^i & = &
\frac{i}{2}\sigma^{\mu\nu}\tilde{F}_{\mu\nu}L\alpha^i -
\sqrt{2}\gamma^\mu \tilde{\partial}_\mu \phi^{ij} R\tilde{\alpha}_j, \nonumber\\
\delta R\tilde{\chi}_i & = &
\frac{i}{2}\sigma^{\mu\nu}\tilde{F}_{\mu\nu}R\tilde{\alpha}_i +
\sqrt{2}\gamma^\mu \tilde{\partial}_\mu \phi_{ij} L\alpha^j. \end{aligned}$$
Similarly, choosing the $k_{\mu \nu}$ parameters to vanish in the spacetime directions $\mu = 0, 1, 2, 3,$ we obtain a Lagrangian of the form = - F\^2 + i | L+ \_\_[ij]{}\^\^[ij]{} , where $\tilde{\phi}_{ij} = \phi_{ij} +\Lambda_{ijkl}\phi_{kl}$ with the matrix $\Lambda_{ijkl}$ containing the effect of Lorentz violation in the compactified directions. The associated action is invariant under the supersymmetry transformations
$$\begin{aligned}
\delta A^\mu & = & -i(\bar{\alpha}_i \gamma^\mu L\chi^i
-\bar{\chi}_i \gamma^\mu L\alpha^i), \nonumber \\
\delta \tilde{\phi}_{ij} & = & -i\sqrt{2}(\bar{\alpha}_j R\tilde{\chi}_i -
\bar{\alpha}_i R\tilde{\chi}_j
+\epsilon_{ijkl}\bar{\tilde{\alpha}}^k L \chi^l), \nonumber \\
\delta L\chi^i & = &
\frac{i}{2}\sigma^{\mu\nu}F_{\mu\nu}L\alpha^i -
\sqrt{2}\gamma^\mu \partial_\mu \tilde{\phi}^{ij} R\tilde{\alpha}_j ,\nonumber \\
\delta R\tilde{\chi}_i & = &
\frac{i}{2}\sigma^{\mu\nu}F_{\mu\nu}R\tilde{\alpha}_i +
\sqrt{2}\gamma^\mu \partial_\mu \tilde{\phi}_{ij} L\alpha^j. \end{aligned}$$
Note that if the scalars $\tilde{\phi}_{ij}$ are identified with physical scalars $\phi_{ij}$ we restore $SU(4)$ symmetry and remove any Lorentz violating effects. If, however, the $\phi_{ij}$ couple to other sectors, Lorentz effects may reappear in these sectors.
Extensions and clarifications
=============================
The above results warrant a number of additional comments:
- These constructions can be carried out in the nonabelian case where they yield supersymmetric theories which exhibit Lorentz violation[@cm].
- The same techniques can be applied to obtain an ${\mathcal N}=2$ supersymmetric theory with Lorentz violation. The construction proceeds by working in 4+2-dimensional spacetime and using dimensional reduction[@cm].
- Because these constructions involve changing the structure of the underlying superalgebra,[@cm; @cm2] the no-go results of Nibbelink and Pospelov[@NP] do not apply.
[xx]{}
L. Brink, J. Schwartz and J. Scherk, Nucl. Phys. B [**121**]{}, 77 (1977).
M. Berger and V.A. Kostelecký, Phys. Rev. D. [**65**]{}, 091701 (2002).
M. Srednicki, [*Quantum Field Theory.*]{} Cambridge University Press, Cambridge, 2007.
D. Colladay and P. McDonald, in preparation.
D. Colladay and P. McDonald, J. Math. Phys. [**43**]{}, 3554 (2002).
S. Nibbelink and M. Pospelov, Phys. Rev. Lett. [bf 94]{}, 081601 (2005).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In the paper it is shown that, even without a knowledge of the concrete form of the equations of mathematical physics and field theories, with the help of skew-symmetric differential forms one can see specific features of the equations of mathematical physics, the relation between mathematical physics and field theory, to understand the mechanism of evolutionary processes that develop in material media and lead to emergency of physical structures forming physical fields. This discloses a physical meaning of such concepts like “conservation laws”, “postulates” and “causality” and gives answers to many principal questions of mathematical physics and general field theory.
In present paper, beside the exterior forms, the skew-symmetric differential forms, whose basis (in contrast to the exterior forms) are deforming manifolds, are used. Mathematical apparatus of such differential forms (which were named evolutionary ones) includes nontraditional elements like nonidentical relations and degenerate transformations and this enables one to describe discrete transitions, quantum steps, evolutionary processes, and generation of various structures.
---
[**Analysis of the equations of mathematical physics**]{}
[**and foundations of field theories with the help**]{}
[**of skew-symmetric differential forms**]{}
**L.I. Petrova**
[*Moscow State University, Russia, e-mail: [email protected]*]{}
[**Introduction**]{}
Skew-symmetric differential forms possess unique properties that enable one to carry out a qualitative investigation of the equations of mathematical physics and the foundations of field theories. They can describe a conjugacy of various operators and objects (derivatives, differential equations, and so on).
Such a potentiality of skew-symmetric differential forms is due to the fact that skew-symmetric differential forms, as opposed to differential equations, deal with differentials and differential expressions rather than with derivatives.
In the paper, beside the exterior skew-symmetric differential forms that can describe conjugated objects, the skew-symmetric differential forms, whose basis (in contrast to the exterior forms) are deforming manifolds, are used. Such skew-symmetric differential forms, which were named evolutionary ones, can describe the process of conjugating objects and obtaining conjugated objects - the closed exterior forms. Evolutionary forms are obtained from the equations modelling physical processes, and therefore, they possess evolutionary properties.
The physical meaning of exterior skew-symmetric differential forms is connected with the fact that they correspond to conservation laws. Closed (inexact) exterior forms and relevant dual forms compose conjugated objects (a differential-geometrical structure). Just such conjugated objects, which are invariant ones, correspond to conservation laws. These are conservation laws for physical fields. The physical structures that form physical fields are just such conjugated objects.
The theory of closed exterior forms lies at the basis of field theories (the theories describing physical fields). The invariant properties of exterior forms explicitly or implicitly manifest themselves essentially in all formalisms of field theory, such as the Hamilton formalism, tensor approaches, group methods, quantum mechanics equations, the Yang-Mills theory and others. The gauge transformations (unitary, gradient and so on), the gauge symmetries and the identical relations of field theories are transformations, symmetries and relations of the theory of closed exterior forms.
In the paper it will be shown that the closed exterior forms, whose properties lie at the basis of field theories, are obtained from evolutionary forms related to the equations of mathematical physics. This discloses a relation between mathematical physics and the invariant field theories.
Evolutionary forms, as well as closed exterior forms, reflect the properties of conservation laws. However, they are conservation laws for material media. They are balance (differential) conservation laws. They are conservation laws for energy, linear momentum, angular momentum, and mass.
Below, on the basis of the properties of evolutionary forms it will be shown the noncommutativity of the balance conservation laws and their controlling role in the evolutionary processes developing in material media. In will be shown that such processes lead to origination of physical structures from which physical fields are formatted. During this the origination of physical structures in these processes reveals as an emergency of such formations like waves, vortices, turbulent pulsations, massless particles, and so on.
Such results firstly proves that material media generate physical fields. And secondly, they explain the nature of turbulence and various types of instabilities developed in material media.
Below it will be shown that the parameters of exterior and evolutionary forms allow to introduce a classification of physical fields and interactions.
The methods of investigating concrete material systems and physical fields on the basis of the exterior and evolutionary differential forms are demonstrated by the examples that are presented in Appendices 2.
The existence of evolutionary skew-symmetric differential forms has been established by the author while studying the problems of stability. Mathematical apparatus of evolutionary skew-symmetric differential forms includes nontraditional elements like nonidentical relations and degenerate transformations and this enables one to describe discrete transitions, quantum steps, evolutionary processes, and generation of various structures. Such mathematical apparatus is beyond the scope of existing physical and mathematical formalisms, and this makes difficulties in presentation and perception of this matter.
It is known that the exterior differential form of degree $p$ ($p$-form) can be written as \[1,2\] $$\theta^p=\sum_{i_1\dots i_p}a_{i_1\dots i_p}dx^{i_1}\wedge
dx^{i_2}\wedge\dots \wedge dx^n\quad 0\leq p\leq n\eqno(1)$$ Here $a_{i_1\dots i_p}$ are functions of the variables $x^1$, $x^2$, …, $x^n$, $n$ is the dimension of space, $\wedge$ is the operator of exterior multiplication, $dx^i$, $dx^{i}\wedge dx^{j}$, $dx^{i}\wedge dx^{j}\wedge dx^{k}$, …is the local basis which satisfies the condition of exterior multiplication (the condition of skew-symmetry).
The differential of the exterior form $\theta^p$ is expressed as $$d\theta^p=\sum_{i_1\dots i_p}da_{i_1\dots
i_p}\wedge dx^{i_1}\wedge dx^{i_2}\wedge \dots \wedge dx^{i_p} \eqno(2)$$
The physical sense have the closed exterior forms. \[Below we present only the data on exterior forms which are necessary for further presentation\].
From the closure condition of the exterior form $\theta^p$: $$d\theta^p=0\eqno(3)$$ one can see that the closed exterior form $\theta^p$ is a conserved quantity. This means that this can correspond to a conservation law, namely, to some conservative quantity.
If the form is closed only on pseudostructure, i.e. this form is a closed inexact one, the closure conditions are written as $$d_\pi\theta^p=0\eqno(4)$$ $$d_\pi{}^*\theta^p=0\eqno(5)$$ where ${}^*\theta^p$ is the dual form.
Condition (5), i.e. the closure condition for dual form, specifies the pseudostructure $\pi$. {Cohomology (de Rham cohomology, singular cohomology and so on), sections of cotangent bundles, the surfaces eikonals, potential surfaces, pseudo-Riemannian and pseudo-Euclidean spaces, and others are examples of the psedustructures and manifolds that are formed by pseudostructures.}
From conditions (4) and (5) one can see the following. The dual form (pseudostructure) and closed inexact form (conservative quantity) made up a conjugated conservative object that can also correspond to some conservation law. The conservative object, which corresponds to the conservation law, is a differential-geometrical structure. (Such differential-geometrical structures are examples of G-structures.) The physical structures, which made up physical fields, and corresponding conservation laws are just such structures.
1\. [*Invariance.*]{}
It is known that the closed exact form is a differential of the form of lower degree: $$\theta^p=d\theta^{p-1}\eqno(6)$$ Closed inexact form is also a differential, and yet not a total one but an interior on pseudostructure $$\theta^p_\pi=d_\pi\theta^{p-1}\eqno(7)$$
Since the closed form is a differential (a total one if the form is exact, or an interior one on the pseudostructure if the form is inexact), it is obvious that the closed form turns out to be invariant under all transformations that conserve the differential. The unitary transformations ($0$-form), the tangent and canonical transformations ($1$-form), the gradient transformations ($2$-form) and so on are examples of such transformations. [*These are gauge transformations for spinor, scalar, vector, tensor ($3$-form) fields*]{}. It should be pointed out that just such transformations are used in field theory.
With the invariance of closed forms it is connected the covariance of relevant dual forms.
2\. [*Conjugacy. Duality. Symmetries.*]{}
The closure of exterior differential forms is the result of conjugating the elements of exterior or dual forms. The closure property of the exterior form means that any objects, namely, the elements of exterior form, the components of elements, the elements of the form differential, the exterior and dual forms, the forms of sequential degrees and others, turn out to be conjugated.
With the conjugacy it is connected the duality.
The example of a duality having physical sense: the closed exterior form is a conservative quantity corresponding to conservation law, and the closed form (as the differential) can correspond to a certain potential force. (Below it will be shown in respect to what the closed form manifests itself as a potential force and with what the conservative physical quantity is connected).
The conjugacy is possible if there is one or another type of symmetry.
The gauge symmetries, which are interior symmetries of field theory and with which gauge transformations are connected, are symmetries of closed exterior differential forms. The conservation laws for physical fields are connected with such interior symmetries.
Since the conjugacy is a certain connection between two operators or mathematical objects, it is evident that, to express the conjugacy mathematically, it can be used relations. These are identical relations.
The identical relations express the fact that each closed exterior form is the differential of some exterior form (with the degree less by one). In general form such an identical relation can be written as $$d _{\pi}\varphi=\theta _{\pi}^p\eqno(8)$$
In this relation the form in the right-hand side has to be a [*closed*]{} one.
Identical relations of exterior differential forms are a mathematical expression of various types of conjugacy that leads to closed exterior forms.
Such relations like the Poincare invariant, vector and tensor identical relations, the Cauchi-Riemann conditions, canonical relations, the integral relations by Stokes or Gauss-Ostrogradskii, the thermodynamic relations, the eikonal relations, and so on are examples of identical relations of closed exterior forms that have either the form of relation (8) or its differential or integral analogs.
One can see that identical relations of closed exterior differential forms make itself evident in various branches of physics and mathematics.
Below the mathematical and physical meaning of these relations will be disclosed with the help of evolutionary forms.
The properties of closed exterior differential forms correspond to the conservation laws for physical fields. Therefore, the mathematical principles of the theory of closed exterior differential forms lie at the basis of existing field theories. {The physical fields \[3\] are a special form of the substance, they are carriers of various interactions such as electromagnetic, gravitational, wave, nuclear and other kinds ofinteractions. The conservation laws for physical fields are those that state an existence of conservative physical quantities or objects.}
The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy, and duality, lie at the basis of the group and structural properties of field theory.
The nondegenerate transformations of field theory are transformations of closed exterior forms. As it has been pointed out, the gauge transformations like the unitary, tangent, canonical, gradient and other gauge transformations are such transformations. These are transformations conserving the differential. Applications of nondegenerate transformations to identical relations enables one to find new closed exterior forms and, hence, to find new physical structures.
The gauge, i.e. internal, symmetries of the field theory equations are those of closed exterior forms.
The nondegenerate transformations of exterior differential forms lie at the basis of field theory operators. If, in addition to the exterior differential, we introduce the following operators: (1) $\delta$ for transformations that convert the form of $(p+1)$ degree into the form of $p$ degree, (2) $\delta'$ for cotangent transformations, (3) $\Delta$ for the $d\delta-\delta d$ transformation, (4) $\Delta'$ for the $d\delta'-\delta'd$ transformation, one can write down the operators in the field theory equations in terms of these operators that act on the exterior differential forms. The operator $\delta$ corresponds to Green’s operator, $\delta'$ does to the canonical transformation operator, $\Delta$ does to the d’Alembert operator in 4-dimensional space, and $\Delta'$ corresponds to the Laplace operator.
It can be shown that the equations of existing field theories are those obtained on the basis of the properties of the exterior form theory. The Hamilton formalism is based on the properties of closed exterior form of the first degree and corresponding dual form. The closed exterior differential form $ds=-Hdt+p_j dq_j$ (the Poincare invariant) corresponds to the field equation related to the Hamilton system. The Schrődinger equation in quantum mechanics is an analog to field equation, where the conjugated coordinates are changed by operators. It is evident that the closed exterior form of zero degree (and dual form) correspond to quantum mechanics. Dirac’s [*bra-*]{} and [*cket*]{}- vectors constitute a closed exterior form of zero degree \[4\]. The properties of closed exterior form of the second degree (and dual form) lie at the basis of the electromagnetic field equations. The Maxwell equations may be written as \[5\] $d\theta^2=0$, $d^*\theta^2=0$, where $\theta^2=
\frac{1}{2}F_{\mu\nu}dx^\mu dx^\nu$ (here $F_{\mu\nu}$ is the strength tensor). Closed exterior and dual forms of the third degree correspond to the gravitational field. (However, to the physical field of given type it can be assigned closed forms of less degree. In particular, to the Einstein equation for gravitational field it is assigned the first degree closed form, although it was pointed out that the type of a field with the third degree closed form corresponds to the gravitational field.)
The connection between the field theory equations and gauge transformations used in field theories with closed exterior forms of appropriate degrees shows that there exists a commonness between field theories describing physical fields of different types. This can serve as an approach to constructing the unified field theory. This connection shows that it is possible to introduce a classification of physical fields according to the degree of closed exterior form. Such a classification also exists for interactions (see below).
(But within the framework of only exterior differential forms one cannot understand how this classification is explained. This can be elucidated only by application of evolutionary differential forms.)
And here it should underline that the field theories are based on the properties of closed [*inexact*]{} forms. This is explained by the fact that only inexact exterior forms can correspond to the physical structures that form physical fields. The condition that the closed exterior forms, which constitute the basis of field theory equations, are inexact ones reveals in the fact that essentially all existing field theories include a certain elements of noninvariance. Such elements of noninvariance are, for example, nonzero value of the curvature tensor in Einstein’s theory \[6\], the indeterminacy principle in Heisenberg’s theory, the torsion in the theory by Weyl \[6\], the Lorentz force in electromagnetic theory \[7\], an absence of general integrability of the Schrődinger equations, the Lagrange function in the variational methods, an absence of the identical integrability of the mathematical physics equations and that of identical covariance of the tensor equations, and so on. Only if we assume the elements of noncovariance, we can obtain closed [*inexact*]{} forms that correspond to physical structures.
And yet, the existing field theories are invariant ones because they are provided with additional conditions under which the invariance or covariance requirements have to be satisfied. It is possible to show that these conditions are the closure conditions of exterior or dual forms. Examples of such conditions may be the identity relations: canonical relations in the Schrődinger equations, gauge invariance in electromagnetic theory, commutator relations in the Heisenberg theory, symmetric connectednesses, identity relations by Bianchi in the Einstein theory, cotangent bundles in the Yang-Mills theory, the Hamilton function in the variational methods, the covariance conditions in the tensor methods, etc.
It is known that the equations of existing field theories and the mathematical formalisms of field theories have been obtained on the basis of postulates. One can see that these postulates are obtained from the closure conditions of inexact exterior forms.
Thus one can see that the properties and mathematical apparatus of closed exterior forms made up the basis of existing field theories.
And here it arises the question of how closed inexact exterior forms, which correspond to physical structures and reflect the properties of conservation laws and on whose properties field theories are based, are obtained. This gives the answers to the following questions: (a) how the physical structures, from which physical fields are formatted, originate; (b) what generates physical structures; (c) how the process of generation proceeds, and (d) what is responsible for such processes? That is, this enables one to understand the heart of physical evolutionary processes and their causality. This has to explain both the internal connection between different physical fields and their classification.
Below it will be shown that the closed inexact exterior forms can be obtained from the evolutionary forms.
Skew-symmetric differential forms, that the author named evolutionary ones, differ in their properties from exterior forms. The distinction between exterior and evolutionary skew-symmetric differential forms is connected with the properties of manifolds on which skew-symmetric forms are defined.
It is known that the exterior differential forms are skew-symmetric differential forms whose basis are differentiable manifolds or they can be manifolds with structures of any type \[2,8\]. (Such manifolds have one common property, namely, they locally admit one-to-one mapping into the Euclidean subspaces and into other manifolds or submanifolds of the same dimension \[8\]).
While describing the evolutionary processes in material systems (material media) one is forced to deal with manifolds which do not allow one-to-one mapping described above. Lagrangian manifolds and tangent manifolds of differential equations describing physical processes can be examples of deforming manifolds.
Such manifolds are those constructed of trajectories of the material system elements (particles). These manifolds, which can be called accompanying manifolds, are deforming variable manifolds.
The skew-symmetric differential forms defined on these manifolds are evolutionary ones. The coefficients of these differential forms and the characteristics of corresponding manifolds are interconnected and are varied as functions of evolutionary variables.
A distinction between manifolds on which exterior and evolutionary forms are defined relates to the properties of metric forms of these manifolds.
Below we present some information about the manifolds on which skew-symmetrical differential forms are defined.
Assume that on the manifold one can set the coordinate system with base vectors $\mathbf{e}_\mu$ and define the metric forms of manifold \[6\]: $(\mathbf{e}_\mu\mathbf{e}_\nu)$, $(\mathbf{e}_\mu dx^\mu)$, $(d\mathbf{e}_\mu)$. The metric forms and their commutators define the metric and differential characteristics of the manifold.
If metric forms are closed (the commutators are equal to zero), the metric is defined $g_{\mu\nu}=(\mathbf{e}_\mu\mathbf{e}_\nu)$, and the results of translation over manifold of the point $d\mathbf{M}=(\mathbf{e}_\mu dx^\mu)$ and of the unit frame $d\mathbf{A}=(d\mathbf{e}_\mu)$ prove to be independent of the curve shape (the path of integration).
To describe the manifold differential characteristics and, correspondingly, the metric form commutators, one can use connectednesses \[2,6\]. If the components of metric form can be expressed in terms of connectedness $\Gamma^\rho_{\mu\nu}$ \[6\], the expressions $\Gamma^\rho_{\mu\nu}$, $(\Gamma^\rho_{\mu\nu}-\Gamma^\rho_{\nu\mu})$ and $R^\mu_{\nu\rho\sigma}$ are components of the commutators of the metric forms with zeroth-, first-, and third degrees. (The commutator of the second degree metric form is written down in a more complex manner \[6\], and therefore it is not presented here).
The closed metric forms define the manifold structure. And the commutators of metric forms define the manifold differential characteristics that specify the manifold deformation: bending, torsion, rotation, and twist. (For example, the commutator of the zeroth degree metric form $\Gamma^\rho_{\mu\nu}$ characterizes the bend, that of the first degree form $(\Gamma^\rho_{\mu\nu}-\Gamma^\rho_{\nu\mu})$ characterizes the torsion, the commutator of the third-degree metric form $R^\mu_{\nu\rho\sigma}$ determines the curvature. (For manifolds with closed metric form of first degree the coefficients of connectedness are symmetric ones.)
Is is evident that the manifolds which are metric ones or possess the structure have closed metric forms. It is with such manifolds the exterior differential forms are connected.
If the manifolds are deforming manifolds, this means that their metric form commutators are nonzero. That is, the metric forms of such manifolds turn out to be unclosed.
The evolutionary skew-symmetric differential forms are defined on manifolds with unclosed metric forms.
Specific properties of evolutionary skew-symmetric differential forms and the distinction of evolutionary forms from exterior ones are connected with the properties of commutators of unclosed metric form, which enter into the evolutionary form commutators.
The evolutionary differential form of degree $p$ ($p$-form) is written similarly to exterior differential form. But the evolutionary form differential cannot be written similarly to that presented for exterior differential forms. In the evolutionary form differential there appears an additional term connected with the fact that the basis of evolutionary form changes. For differential forms defined on the manifold with unclosed metric form one has $d(dx^{\alpha_1}\wedge dx^{\alpha_2}\wedge \dots \wedge dx^{\alpha_p})\neq 0$. (For differential forms defined on the manifold with closed metric form one has $d(dx^{\alpha_1}\wedge dx^{\alpha_2}\wedge \dots \wedge dx^{\alpha_p})=0$). For this reason the differential of the evolutionary form $\theta^p$ can be written as $$d\theta^p{=}\!\sum_{\alpha_1\dots\alpha_p}\!da_{\alpha_1\dots\alpha_p}\wedge
dx^{\alpha_1}\wedge dx^{\alpha_2}\dots \wedge
dx^{\alpha_p}{+}\!\sum_{\alpha_1\dots\alpha_p}\!a_{\alpha_1\dots\alpha_p}
d(dx^{\alpha_1}\wedge dx^{\alpha_2}\dots \wedge dx^{\alpha_p})\eqno(9)$$ where the second term is a differential of unclosed metric form of nonzero value.
\[In further presentation the symbol of summing $\sum$ and the symbol of exterior multiplication $\wedge$ will be omitted. Summation over repeated indices will be implied.\]
The second term connected with the differential of the basis can be expressed in terms of the metric form commutator.
For example, let us consider the first-degree form $\theta=a_\alpha dx^\alpha$. The differential of this form can be written as $$d\theta=K_{\alpha\beta}dx^\alpha dx^\beta\eqno(10)$$ where $K_{\alpha\beta}=a_{\beta;\alpha}-a_{\alpha;\beta}$ are the components of commutator of the form $\theta$, and $a_{\beta;\alpha}$, $a_{\alpha;\beta}$ are covariant derivatives. If we express the covariant derivatives in terms of the connectedness (if it is possible), they can be written as $a_{\beta;\alpha}=\partial a_\beta/\partial
x^\alpha+\Gamma^\sigma_{\beta\alpha}a_\sigma$, where the first term results from differentiating the form coefficients, and the second term results from differentiating the basis. We arrive at the following expression for the commutator components of the form $\theta$ $$K_{\alpha\beta}=\left(\frac{\partial a_\beta}{\partial
x^\alpha}-\frac{\partial a_\alpha}{\partial
x^\beta}\right)+(\Gamma^\sigma_{\beta\alpha}-
\Gamma^\sigma_{\alpha\beta})a_\sigma\eqno(11)$$ Here the expressions $(\Gamma^\sigma_{\beta\alpha}-\Gamma^\sigma_{\alpha\beta})$ entered into the second term are just the components of commutator of the first-degree metric form.
If to substitute the expressions (11) for evolutionary form commutator into formula (10), we obtain the following expression for differential of the first degree skew-symmetric form $$d\theta=\left(\frac{\partial a_\beta}{\partial
x^\alpha}-\frac{\partial a_\alpha}{\partial
x^\beta}\right)dx^\alpha dx^\beta+\left((\Gamma^\sigma_{\beta\alpha}-
\Gamma^\sigma_{\alpha\beta})a_\sigma\right)dx^\alpha dx^\beta\eqno(12)$$ The second term in the expression for differential of skew-symmetric form is connected with the differential of the manifold metric form, which is expressed in terms of the metric form commutator.
Thus, the differentials and, correspondingly, the commutators of exterior and evolutionary forms are of different types. In contrast to the exterior form commutator, the evolutionary form commutator includes two terms. These two terms have different nature, namely, one term is connected with the coefficients of evolutionary form itself, and the other term is connected with differential characteristics of manifold. Interaction between terms of the evolutionary form commutator (interactions between coefficients of evolutionary form and its basis) provides the foundation of evolutionary processes that lead to generation of closed inexact exterior forms.
Above it has been shown that the evolutionary form commutator includes the commutator of the manifold metric form which is nonzero. Therefore, the evolutionary form commutator cannot be equal to zero. This means that the evolutionary form differential is nonzero. Hence, the evolutionary form, in contrast to the case of the exterior form, cannot be closed. This leads to that in the mathematical apparatus of evolutionary forms there arise new unconventional elements like nonidentical relations and degenerate transformations. Just such peculiarities allow to describe evolutionary processes.
Nonidentical relations of the evolutionary form theory, as well as identical relations of the theory of closed exterior forms, are relations between the differential and the skew-symmetric form. In the right-hand side of the identical relation of exterior forms (see relation (8)) it stands a closed form, which is a differential as well as the left-hand side. And in the right-hand of the nonidentical relation of evolutionary form it stands the evolutionary form that is not closed and, hence, cannot be a differential like the left-hand side. Such a relation cannot be identical one.
Nonidentical relations are obtained while describing any processes in terms of differential equations. (In Appendix 1 we present an example of a qualitative investigation of differential equations.) The relation of such type is obtained, for example, while analyzing the integrability of the partial differential equation. The equation is integrable if it can be reduced to the form $d\phi=dU$. However it appears that, if the equation is not subject to an additional condition (the integrability condition), the right-hand side turns out to be an unclosed form and it cannot be expressed as a differential.
Nonidentical relations of evolutionary forms are evolutionary relations because they include the evolutionary form. Such nonidentical evolutionary relations appear to be selfvarying ones. A variation of any object of the relation in some process leads to a variation of another object and, in turn, a variation of the latter leads to a variation of the former. Since one of the objects is a noninvariant (i.e. unmeasurable) quantity, the other cannot be compared with the first one, and hence, the process of mutual variation cannot be completed.
The nonidentity of evolutionary relation is connected with a nonclosure of the evolutionary form, that is, it is connected with the fact that the evolutionary form commutator is nonzero. As it has been pointed out, the evolutionary form commutator includes two terms: one term specifies the mutual variations of the evolutionary form coefficients, and the second term (the metric form commutator) specifies the manifold deformation. These terms have a different nature and cannot make the commutator vanish. In the process of selfvariation of the nonidentical evolutionary relation it proceeds an exchange between the terms of the evolutionary form commutator and this is realized according to the evolutionary relation. The evolutionary form commutator describes a quantity that is a moving force of the evolutionary process.
The significance of the evolutionary relation selfvariation consists in the fact that in such a process it can be realized the conditions of degenerate transformation under which the closed inexact exterior form is obtained from the evolutionary form, and from nonidentical relation the identical relation is obtained.
Exterior and evolutionary forms enable one to investigate the integrability of differential equations. This is due to the fact that they make it possible to study the conjugacy of the equations or their derivatives. The type of solutions to differential equations depends on the conjugacy. Solutions are invariant if the equations and their derivatives are conjugated ones. If this is not fulfilled, the solutions prove to be noninvariant, namely, they are functionals rather then functions.
The qualitative analysis of the equations of mathematical physics with the help of differential forms shows that any differential equations describing any processes turn out to be nonintegrable without additional conditions. Additional conditions under which the equations become integrable can be realized only discretely. This points to the fact that the solutions of any differential equations of mathematical physics describing physical processes can be only generalized (discrete) ones (see Appendix 1). These are precisely generalized solutions that describe various structures.
The importance of evolutionary forms consists in the fact that they allow not only to investigate an integrability of the equations and functional properties of the solutions. They also allow to describe the process of realization of invariant solutions in itself and thereby to describe the process of [*origination* ]{} of physical structures and to disclose the mechanism of processes like turbulence, generation of waves and vortices, creation of massless particles, and so on.
Below we will carry out the analysis of the equations of mathematical physics, which describe physical processes in material media. These are precisely material media that generate physical structures making up physical fields.
It will be shown that the closed exterior forms, which correspond to the conservation laws [*for physical fields*]{} and describe physical structures, are obtained from the evolutionary forms that are connected with the equations of conservation laws [*for material media*]{}. The conservation laws for material media are balance (differential) conservation laws. The process of obtaining closed exterior forms from evolutionary ones just describes the process of generating physical structures by material media. These conclusions follow from the analysis of the equations of balance conservation laws with the help of differential forms.
\[Sometimes below it will be used a double notation in subtitles, one in reference to physical meaning and another in reference to mathematical meaning.\]
[*The balance conservation laws are those that establish the balance between the variation of a physical quantity and the corresponding external action. These are the conservation laws for material systems (material media)*]{} \[9\].
The balance conservation laws are the conservation laws for energy, linear momentum, angular momentum, and mass.
The equations of the balance conservation laws are differential (or integral) equations that describe a variation of functions corresponding to physical quantities \[10-13\]. (The specific forms of these equations for thermodynamical and gas dynamical material systems and the systems of charged particles will be presented in the Appendix 2).
But it appears that, even without a knowledge of the concrete form of these equations, with the help of the differential forms one can see specific features of these equations that elucidate the properties of the balance conservation laws. To do so it is necessary to study the conjugacy (consistency) of these equations.
Equations are conjugate if they can be contracted into identical relations for the differential, i.e. for a closed form.
Let us analyze the equations that describe the balance conservation laws for energy and linear momentum.
We introduce two frames of reference: the first is an inertial one (this frame of reference is not connected with the material system), and the second is an accompanying one (this system is connected with the manifold built by the trajectories of the material system elements). The energy equation in the inertial frame of reference can be reduced to the form: $$\frac{D\psi}{Dt}=A$$ where $D/Dt$ is the total derivative with respect to time, $\psi $ is the functional of the state that specifies the material system (every material system has its own functional of the state), $A$ is the quantity that depends on specific features of the system and on external energy actions onto the system. {The action functional, entropy, wave function can be regarded as examples of the functional $\psi $. Thus, the equation for energy presented in terms of the action functional $S$ has a similar form: $DS/Dt\,=\,L$, where $\psi \,=\,S$, $A\,=\,L$ is the Lagrange function. In mechanics of continuous media the equation for energy of an ideal gas can be presented in the form \[13\]: $Ds/Dt\,=\,0$, where $s$ is entropy. In this case $\psi \,=\,s$, $A\,=\,0$. It is worth noting that the examples presented show that the action functional and entropy play the same role.}
In the accompanying frame of reference the total derivative with respect to time is transformed into the derivative along the trajectory. Equation of energy is now written in the form $${{\partial \psi }\over {\partial \xi ^1}}\,=\,A_1 \eqno(13)$$
Here $\psi$ is the functional specifying the state of material system, $\xi^1$ is the coordinate along the trajectory, $A_1$ is the quantity that depends on specific features of material system and on external (with respect to local domain of material system) energy actions onto the system.
In a similar manner, in the accompanying reference system the equation for linear momentum appears to be reduced to the equation of the form $${{\partial \psi}\over {\partial \xi^{\nu }}}\,=\,A_{\nu },\quad \nu \,=\,2,\,...\eqno(14)$$ where $\xi ^{\nu }$ are the coordinates in the direction normal to the trajectory, $A_{\nu }$ are the quantities that depend on the specific features of material system and on external force actions.
Eqs. (13) and (14) can be convoluted into the relation $$d\psi\,=\,A_{\mu }\,d\xi ^{\mu },\quad (\mu\,=\,1,\,\nu )\eqno(15)$$ where $d\psi $ is the differential expression $d\psi\,=\,(\partial \psi /\partial \xi ^{\mu })d\xi ^{\mu }$.
Relation (15) can be written as $$d\psi \,=\,\omega \eqno(16)$$ here $\omega \,=\,A_{\mu }\,d\xi ^{\mu }$ is the skew-symmetrical differential form of the first degree.
Since the balance conservation laws are evolutionary ones, the relation obtained is also an evolutionary relation.
Relation (16) has been obtained from the equation of the balance conservation laws for energy and linear momentum. In this relation the form $\omega $ is that of the first degree. If the equations of the balance conservation laws for angular momentum be added to the equations for energy and linear momentum, this form in the evolutionary relation will be a form of the second degree. And in combination with the equation of the balance conservation law for mass this form will be a form of degree 3.
Thus, in general case the evolutionary relation can be written as $$d\psi \,=\,\omega^p \eqno(17)$$ where the form degree $p$ takes the values $p\,=\,0,1,2,3$.. (The evolutionary relation for $p\,=\,0$ is similar to that in the differential forms, and it was obtained from the interaction of energy and time.)
Let us show that relation obtained from the equation of the balance conservation laws proves to be nonidentical.
To do so we shall analyze relation (16).
In the left-hand side of relation (16) there is a differential that is a closed form. This form is an invariant object. The right-hand side of relation (16) involves the differential form $\omega$ that is not an invariant object because in real processes, as it will be shown below, this form proves to be unclosed. The commutator of this form is nonzero. The components of commutator of the form $\omega \,=\,A_{\mu }d\xi ^{\mu }$ can be written as follows: $$K_{\alpha \beta }\,=\,\left ({{\partial A_{\beta }}\over {\partial \xi ^{\alpha }}}\,-\,
{{\partial A_{\alpha }}\over {\partial \xi ^{\beta }}}\right )$$ (here the term connected with the manifold metric form has not yet been taken into account).
The coefficients $A_{\mu }$ of the form $\omega $ have to be obtained either from the equation of the balance conservation law for energy or from that for linear momentum. This means that in the first case the coefficients depend on the energetic action and in the second case they depend on the force action. In actual processes energetic and force actions have different nature and appear to be inconsistent. The commutator of the form $\omega $ consisted of the derivatives of such coefficients is nonzero. This means that the differential of the form $\omega $ is nonzero as well. Thus, the form $\omega$ proves to be unclosed and cannot be a differential like the left-hand side. This means that relation (16) cannot be an identical one.
In a similar manner one can prove the nonidentity of relation (17).
Hence, without a knowledge of particular expression for the form $\omega$, one can argue that for actual processes the relation obtained from the equations corresponding to the balance conservation laws proves to be nonidentical.
The nonidentity of evolutionary relation means that the balance conservation law equations are inconsistent. And this indicates that the balance conservation laws are noncommutative. (If the balance conservation laws be commutative, the equations would be consistent and the evolutionary relation would be identical).
To what such a noncommutativity of the balance conservation laws leads?
Nonidentical evolutionary relation obtained from the equations of the balance conservation laws involves the functional that specifies the material system state. However, since this relation turns out to be not identical, from this relation one cannot get the differential $d\psi $ that could point out to the equilibrium state of material system. The absence of differential means that the system state is nonequilibrium. That is, due to noncommutativity of the balance conservation laws the material system state turns out to be nonequilibrium under effect of external actions. This points out to the fact that in material system the internal force acts. (External actions onto local domain of material system lead to emergency of internal forces in local domain.)
The action of internal force leads to a distortion of trajectories of material system. The manifold made up by the trajectories (the accompanying manifold) turns out to be a deforming manifold. The differential form $\omega$, as well as the forms $\omega^p$ defined on such manifold, appear to be evolutionary forms. Commutators of these forms will contain an additional term connected with the commutator of unclosed metric form of manifold, which specifies the manifold deformation.
The availability of two terms in the commutator of the form $\omega^p $ and the nonidentity of evolutionary relation lead to that the relation obtained from the balance conservation law equations turns out to be a selfvarying relation.
Selfvariation of nonidentical evolutionary relation points to the fact that the nonequilibrium state of material system turns out to be selfvarying. State of material system changes but remains nonequilibrium during this process.
It is evident that this selfvariation proceeds under the action of internal force whose quantity is described by the commutator of the unclosed evolutionary form $\omega^p $. (If the commutator be zero, the evolutionary relation would be identical, and this would point to the equilibrium state, i.e. the absence of internal forces.) Everything that gives a contribution into the commutator of the form $\omega^p $ leads to emergency of internal force.
What is the result of such a process of selfvarying the nonequilibrium state of material system?
The significance of the evolutionary relation selfvariation consists in the fact that in such a process it can be realized conditions under which the closed exterior form is obtained from the evolutionary form.
These are conditions of degenerate transformation. Since the differential of evolutionary form, which is unclosed, is nonzero, but the differential of closed exterior form equals zero, the transition from evolutionary form to closed exterior form is possible only as a degenerate transformation, namely, a transformation that does not conserve the differential. And this transition is possible exclusively to closed [*inexact*]{} exterior form, i.e. to the external form being closed on pseudostructure. The conditions of degenerate transformation are those of vanishing the commutator (interior one) of the metric form defining the pseudostructure, in other words, the closure conditions for dual form.
As it has been already mentioned, the evolutionary differential form $\omega^p$, involved into nonidentical relation (17) is an unclosed one. The commutator of this form, and hence the differential, are nonzero. That is, $$d\omega^p\ne 0 \eqno(18)$$ If the conditions of degenerate transformation are realized, then from the unclosed evolutionary form one can obtain the differential form closed on pseudostructure. The differential of this form equals zero. That is, it is realized the transition $d\omega^p\ne 0 \to $ (degenerate transformation) $\to d_\pi \omega^p=0$, $d_\pi{}^*\omega^p=0$
The relations obtained $$d_\pi \omega^p=0, d_\pi{}^*\omega^p=0 \eqno(19)$$ are the closure conditions for exterior inexact form. This means that it is realized the exterior form closed on pseudostructure.
The realization of closed (on pseudostructure) inexact exterior form points to emergency of physical structure \[14\].
To the degenerate transformation it must correspond a vanishing of some functional expressions, such as Jacobians, determinants, the Poisson brackets, residues and others. Vanishing these functional expressions is the closure condition for dual form.
The conditions of degenerate transformation that lead to emergency of closed inexact exterior form are connected with any symmetries. Since these conditions are conditions of vanishing the interior differential of the metric form, i.e. vanishing the interior (rather than total) metric form commutator, the conditions of degenerate transformation can be caused by symmetries of coefficients of the metric form commutator (for example, these can be symmetrical connectednesses).
While describing material system the symmetries can be due to degrees of freedom of material system. The translational degrees of freedom, internal degrees of freedom of the system elements, and so on can be examples of such degrees of freedom.
And it should be emphasized once more that [*the degenerate transformation is realized as a transition from the accompanying noninertial coordinate system to the locally inertial system*]{}. The evolutionary form is defined in the noninertial frame of reference (deforming manifold). But the closed exterior form formatted is obtained with respect to the locally-inertial frame of reference (pseudostructure).
The conditions of degenerate transformation (vanishing the dual form commutator) define a pseudostructure. These conditions specify the derivative of implicit function, which defines the direction of pseudostructure.
The speeds of various waves are examples of such derivatives: the speed of light, the speed of sound and of electromagnetic waves (see the Appendix), the speed of creating particles and so on.
It can be shown that the equations for surfaces of potential (of simple layer, double layer), equations for one, two, …eikonals, of the characteristic and of the characteristic surfaces, the residue equations and so on serve as the equations for pseudostructures.
The mechanism of creating the pseudostructures lies at the basis of forming the pseudometric surfaces and their transition into the metric spaces (see below).
On the pseudostructure $\pi$ evolutionary relation (17) converts into the relation $$d_\pi\psi=\omega_\pi^p\eqno(20)$$ which proves to be an identical relation. Indeed, since the form $\omega_\pi^p$ is a closed one, on the pseudostructure this form turns out to be a differential of some differential form. In other words, this form can be written as $\omega_\pi^p=d_\pi\theta$. Relation (20) is now written as $$d_\pi\psi=d_\pi\theta$$ There are differentials in the left-hand and right-hand sides of this relation. This means that the relation is an identical one.
Thus one can see that under degenerate transformation from evolutionary relation the relation identical on pseudostructure is obtained. Here it should be emphasized that in this case the evolutionary relation itself remains to be nonidentical one. The differential, which equals zero, is an interior one. The evolutionary form commutator vanishes only on pseudostructure. The total evolutionary form commutator is nonzero. That is, under degenerate transformation the evolutionary form differential vanishes only on pseudostructure. The total differential of the evolutionary form is nonzero. The evolutionary form remains to be unclosed.
From relation (20) one can obtain a differential which specifies the state of material system (and the state function), and this corresponds to equilibrium state of the system. But identical relation can be realized only on pseudostructure (which is specified by the condition of degenerate transformation). This means that the transition of material system to equilibrium state proceeds only locally (in the local domain of material system). In other words, it is realized the transition of material system from nonequilibrium state to locally equilibrium one. In this case the global state of material system remains to be nonequilibrium.
The transition from nonidentical relation (17) obtained from the balance conservation laws to identical relation (20) means the following. Firstly, an emergency of the closed (on pseudostructure) inexact exterior form (right-hand side of relation (20)) points to an origination of physical structure. And, secondly, an existence of the state differential (left-hand side of relation (20)) points to a transition of material system from nonequilibrium state to the locally-equilibrium state.
Thus one can see that the transition of material system from nonequilibrium state to locally-equilibrium state is accompanied by originating differential-geometrical structures, which are physical structures.
The emergency of physical structures in the evolutionary process reveals in material system as an emergency of certain observable formations, which develop spontaneously. Such formations and their manifestations are fluctuations, turbulent pulsations, waves, vortices, creating massless particles, and others. The intensity of such formations is controlled by a quantity accumulated by the evolutionary form commutator at the instant in time of originating physical structures.
Here the following should be pointed out. Physical structures are generated by local domains of material system. They are elementary physical structures. By combining with one another they can form large-scale structures and physical fields.
The availability of physical structures points out to fulfilment of conservation laws. These are conservation laws for physical fields. The process of generating physical structures (forming physical fields) demonstrates a connection of these conservation laws, which had been named as exact ones, with the balance (differential) conservation laws for material media. The physical structures that correspond to the exact conservation laws are produced by material system in the evolutionary processes, which are based on the interaction of noncommutative balance conservation laws.
[*Noncommutativity of balance conservation laws for material media and their controlling role in evolutionary processes accompanied by emerging physical structures practically have not been taken into account in the explicit form anywhere*]{}. The mathematical apparatus of evolutionary differential forms enables one to take into account and to describe these points \[9\].
The duality of closed exterior forms as conservative quantities and as potential forces points to that an unmeasurable quantity, which is described by the evolutionary form commutator (recall, that all external, with respect to local domain, actions make a contribution into this commutator) and acts as an internal force, is converted into a measurable quantity that acts as a potential force.
Where, from what, and on what the potential force acts?
The potential force is an action of created (quantum) formation onto the local domains of the material system over which it is translated. And if the internal force acts in the interior of the local domain of material system (and it caused that to deform), the potential force acts onto the neighboring domain. The local domain gets rid of its internal force and modifies that into a potential force which acts onto neighboring domains. An unmeasurable quantity, that acts in local domain as an internal force, is transformed into a measurable quantity of the observable formation (and the physical structure as well) that is emitted from the local domain and acts onto neighboring domain as a force equal to this quantity.
If the external actions equal zero (the evolutionary form commutator be equal to zero), then internal and potential forces equal zero.
Thus, one has to distinguish the forces of three types: 1) external forces (the actions being external with respect to local domain), 2) internal forces that originate in local domains of material system due to the fact that the physical quantities of material system changed by external actions turn out to be inconsistent, and 3) the potential forces are forces of the action of the formations (corresponding to physical structures) onto material system.
The potential force, whose value is conditioned by the quantity of the commutator of the evolutionary form $\omega^p$ at the instant of the formation production, acts normally to the pseudostructure, i.e. with respect to the integrating direction, along which the interior differential (the closed form) is formed. The potential forces are described, for example, by jumps of derivatives in the direction normal to characteristics, to potential surfaces and so on. This corresponds to the fact that the evolutionary form commutators along these directions are nonzero.
The duality of closed inexact form as a conservative quantity and as a potential force shows that potential forces are the action of formations corresponding to physical structures onto material system.
Since the closed inexact exterior form corresponding to physical structure was obtained from the evolutionary form, it is evident that the characteristics of physical structure originated has to be connected with those of the evolutionary form and of the manifold on which this form is defined as well as with the conditions of degenerate transformation and with the values of commutators of the evolutionary form and the manifold metric form.
The conditions of degenerate transformation, i.e. symmetries caused by degrees of freedom of material system, determine the equation for pseudostructures.
The closed exterior forms corresponding to physical structures are conservative quantities. These conservative quantities describe certain charges.
The first term of the commutator of evolutionary form determines the value of discrete change (the quantum), which the quantity conserved on the pseudostructure undergoes during transition from one pseudostructure to another. The second term of the evolutionary form commutator specifies a characteristics that fixes the character of the manifold deformation, which took place before physical structure emerged. (Spin is an example of such a characteristics).
Characteristics of physical structures depends in addition on the properties of material system generating these structures.
The closed exterior forms obtained correspond to the state differential for material system. The differentials of entropy, action, potential and others are examples of such differentials.
As it was already mentioned, in material system the created physical structure is revealed as an observable formation. It is evident that the characteristics of the formation, as well as those of created physical structure, are determined by the evolutionary form and its commutator and by the material system characteristics.
It can be shown that the following correspondence between characteristics of the formations emerged and characteristics of evolutionary forms, of the evolutionary form commutators and of material system is established:
1\) an intensity of the formation (a potential force) $\leftrightarrow$ [*the value of the first term in the commutator of evolutionary form*]{} at the instant when the formation is created;
2\) vorticity (an analog of spin) $\leftrightarrow$ [*the second term in the commutator that is connected with the metric form commutator*]{};
3\) an absolute speed of propagation of created formation (the speed in the inertial frame of reference) $\leftrightarrow$ [*additional conditions connected with degrees of freedom of material system*]{};
4\) a speed of the formation propagation relative to material system $\leftrightarrow $ [*additional conditions connected with degrees of freedom of material system plus the velocity of elements of local domain*]{}.
The connection of physics structures with skew-symmetric differential forms allows to introduce a classification of these structures in dependence on parameters that specify skew-symmetric differential forms and enter into nonidentical and identical relation. To determine these parameters one has to consider the problem of integration of nonidentical evolutionary relation.
Since the identical relation obtained from nonidentical evolutionary relation contains only differential and the closed form also is a differential, one can integrate (on pseudostructure) this relation and obtain a relation with the differential forms of less by one degree. From the relation obtained, which will be nonidentical one, under degenerate transformation it can be obtained new identical relation that can be integrated once more.
Thus, from the nonidentical relation, which contains the evolutionary form of degrees $p$, it can be obtained identical relations with closed inexact forms of degrees $k$, where $k$ ranges from $p$ to $0$. That is, evolutionary forms of degree $p$ can generate closed inexact forms of degrees $k=p$, $k=p-1$, …, $k=0$. Under degenerate conditions from closed inexact forms of zero degree it is obtained an exact exterior form of zero degree which the metric structure corresponds to.
In addition to these parameters, another parameter appears, namely, the dimension of space $n$ If the evolutionary relation generates the closed forms of degrees $k=p$, $k=p-1$, …, $k=0$, to them there correspond the pseudostructures of dimensions $(n+1-k)$.
The parameters of evolutionary and exterior forms that follow from the evolutionary forms allow to introduce a classification of physical structures that defines a type of physical structures and, accordingly, of physical fields and interactions.
The type of physical structures (and, accordingly, of physical fields) generated by the evolutionary relation depends on the degree of differential forms $p$ and $k$ and on the dimension of original inertial space $n$. (Here $p$ is the degree of evolutionary form in nonidentical relation that is connected with a number of interacting balance conservation laws, and $k$ is the degree of closed form generated by nonidentical relation. Recall that the interaction of balance conservation laws for energy and linear momentum corresponds to the value $p=1$, with the balance conservation law for angular momentum in addition this corresponds to the value $p=2$, and with the balance conservation law for mass in addition it corresponds to the value $p=3$. The value $p=0$ corresponds to interaction between time and the balance conservation law for energy.)
By introducing a classification by numbers $p$, $k$, $n$ one can understand the internal connection between various physical fields. Since physical fields are the carriers of interactions, such classification enables one to see a connection between interactions.
On the basis of the properties of evolutionary forms that correspond to the conservation laws, one can suppose that such a classification may be presented in the form of the table given below. This table corresponds to elementary particles.
{It should be emphasized the following. Here the concept of “interaction" is used in a twofold meaning: an interaction of the balance conservation laws that relates to material systems, and the physical concept of “interaction" that relates to physical fields and reflects the interactions of physical structures, namely, it is connected with exact conservation laws}.
TABLE
[@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}]{} **interaction&$k\backslash p,n$&**0&**1&**2&**3**********
\
**gravitation&**3&&&&****
-------------
**graviton\
$\Uparrow$\
electron\
proton\
neutron\
photon**
-------------
\
----------------
**electro-\
**magnetic****
----------------
&**2&&&**
-------------
**photon2\
$\Uparrow$\
electron\
proton\
neutrino**
-------------
&**photon3**
\
**weak&**1&&****
--------------
**neutrino1\
$\Uparrow$\
electron\
quanta**
--------------
&**neutrino2&**neutrino3****
\
**strong&**0&****
-------------
**quanta0\
$\Uparrow$\
quarks?**
-------------
&
------------
**quanta1\
\
**
------------
& **quanta2&**quanta3****
\
--------------
**particles\
material\
nucleons?**
--------------
&
-------
exact
forms
-------
&**electron&**proton&**neutron&**deuteron?\
N&&1&2&3&4\
&&time&time+&time+&time+\
&&&1 coord.&2 coord.&3 coord.\
********
In the Table the names of the particles created are given. Numbers placed near particle names correspond to the space dimension. Under the names of particles the sources of interactions are presented. In the next to the last row we present particles with mass (the elements of material system) formed by interactions (the exact forms of zero degree obtained by sequential integrating the evolutionary relations with the evolutionary forms of degree $p$ corresponding to these particles). In the bottom row the dimension of the [*metric*]{} structure created is presented.
From the Table one can see the correspondence between the degree $k$ of the closed forms realized and the type of interactions. Thus, $k=0$ corresponds to strong interaction, $k=1$ corresponds to weak interaction, $k=2$ corresponds to electromagnetic interaction, and $k=3$ corresponds to gravitational interaction. The degree $k$ of the closed forms realized and the number of interacting balance conservation laws determine the type of interactions and the type of particles created. The properties of particles are governed by the space dimension. The last property is connected with the fact that closed forms of equal degrees $k$, but obtained from the evolutionary relations acting in spaces of different dimensions $n$, are distinctive because they are defined on pseudostructures of different dimensions (the dimension of pseudostructure $(n+1-k)$ depends on the dimension of initial space $n$). For this reason the realized physical structures with closed forms of equal degrees $k$ are distinctive in their properties.
The parameters $p$, $k$, $n$ can range from 0 to 3. They determine some completed cycle. The cycle involves four levels, to each of which there correspond their own values of $p$ ($p=0,1,2,3$) and space dimension $n$.
In the Table one cycle of forming physical structures is presented. Each material system has his own completed cycle. This distinguishes one material system from another system. One completed cycle can serve as the beginning of another cycle (the structures formed in the preceding cycle serve as the sources of interactions for beginning a new cycle). This may mean that one material system (medium) proves to be imbedded into the other material system (medium). The sequential cycles reflect the properties of sequentially imbedded material systems. And yet a given level has specific properties that are inherent characteristics of the same level in another cycles. This can be seen, for example, from comparison of the cycle described and the cycle in which to the exact forms there correspond conductors, semiconductors, dielectrics, and neutral elements. The properties of elements of the third level, namely, of neutrons in one cycle and of dielectrics in another, are identical to the properties of so called “magnetic monopole” \[15,16\].
The mechanism of creating the pseudostructures lies at the basis of forming the pseudometric surfaces and their transition into metric spaces \[17\]. (It should be pointed out that the eigenvalues and the coupling constants appear as the conjugacy conditions for exterior or dual forms, the numerical constants are the conjugacy conditions for exact forms.)
It was shown above that the evolutionary relation of degree $p$ can generate (in the presence of degenerate transformations) closed forms of the degrees $p, p-1,.., 0$. While generating closed forms of sequential degrees $k=p, k=p-1,.., k=0$ the pseudostructures of dimensions $(n+1-k)$: $1, ..., n+1$ are obtained. As a result of transition to the exact closed form of zero degree the metric structure of the dimension $n+1$ is obtained. Under influence of external action (and in the presence of degrees of freedom) the material system can transfer the initial inertial space into the space of the dimension $n+1$.
Sections of the cotangent bundles (Yang-Mills fields), cohomologies by de Rham, singular cohomologies, pseudo-Riemannian and pseudo-Euclidean spaces, and others are examples of psedustructures and spaces that are formed by pseudostructures. Euclidean and Riemannian spaces are examples of metric manifolds that are obtained when going to the exact forms.
What can be said about the pseudo-Riemannian manifold and Riemannian space?
The distinctive property of the Riemannian manifold is an availability of the curvature. This means that the metric form commutator of the third degree is nonzero. Hence, the commutator of the evolutionary form of third degree ($p=3$), which involves into itself the metric form commutator, is not equal to zero. That is, the evolutionary form that enters into the evolutionary relation is unclosed, and the relation is nonidentical one.
When realizing pseudostructures of the dimensions $1, 2, 3, 4$ and obtaining the closed inexact forms of the degrees $k=3, k=2,
k=1, k=0$ the pseudo-Riemannian space is formed, and the transition to the exact form of zero degree corresponds to the transition to the Riemannian space.
It is well known that while obtaining the Einstein equations it was suggested that there are fulfilled the conditions \[6,18\]: 1) the Bianchi identity is satisfied, 2) the coefficients of connectedness are symmetric, 3) the condition that the coefficients of connectedness are the Christoffel symbols, and 4) an existence of the transformation under which the coefficients of connectedness vanish. These conditions are the conditions of realization of degenerate transformations for nonidentical relations obtained from the evolutionary nonidentical relation with evolutionary form of the degree $p=3$ and after going to the identical relations. In this case to the Einstein equation the identical relations with forms of the first degree are assigned.
**Conclusion**
Results of the analysis carried out have shown the following.
Invariant and covariant properties of closed exterior and dual forms, which correspond to the conservation laws for physical fields, make up the foundations of field theories. The field theories operators are built on the basis of gauge transformations of closed exterior forms. Properties of closed exterior and dual forms explicitly or implicitly manifest themselves essentially in all formalisms of field theories. The degrees of closed exterior forms establish the classification of physical fields and interactions, and this discloses an internal connection between various physical fields and a common basis of corresponding field theories. This shows that the theory of closed exterior forms can be useful in establishing the unified field theory.
Evolutionary forms, which are obtained from the equations of balance conservation laws for material media, answer the question of how are realized the closed exterior forms that correspond to field theories.
This explains the process of originating physical fields and gives the answer to many questions of field theories.
Firstly, this shows that physical fields are generated by material media. The conservation laws for material media, i.e. the balance conservation laws for energy, linear momentum, angular momentum, and mass, which are noncommutative ones, play a controlling role in these processes. This is precisely the noncommutativity of the balance conservation laws produced by external actions onto material system, which is a moving force of evolutionary processes leading to emergency of physical structures (to which exact conservation laws are assigned). And thus the causality of physical processes and phenomena is explained. Since physical fields are made up by discrete physical structures, this points to a quantum character of field theories.
Secondly, it becomes clear a connection of field theory with the equations of mathematical physics describing physical processes in material media. The postulates, which field theories are built on, are the closure conditions for exterior forms obtained from the evolutionary forms connected with these equations.
The connection between closed exterior forms corresponding to field theories and the evolutionary forms obtained from the equations for material media discloses a meaning of the field theory parameters. They relate to the number ($p$) of interacting noncommutative balance conservation laws and to the degrees ($k$) of closed exterior forms realized. Hence it arises a possibility to classify physical fields and interactions according to the parameters $p$ and $k$.
The results obtained on the basis of the theory of skew-symmetric differential forms do not contradict to any physical theories. And yet the methodical results of this theory enable one to understand internal connections between physical fields, between physical fields and material media, between field theories and the equations of mathematical physics, to understand a mechanism of emergency of physical structures and the causality of physical processes and phenomena.
The theory of skew-symmetric differential forms, which unites the theory of closed exterior forms constituting the basis of field theories and the theory of evolutionary forms generating closed inexact exterior forms, can serve as an approach to the general field theory. Such a theory enables one not only to describe physical fields, but also shows how the physical fields are produced, what generates them, and what is a cause of these processes.
Below in Appendices the example of qualitative investigation of the solutions to differential equations and the analysis of the principles of thermodynamics, the equations for gas dynamic system and the equations of electromagnetic field with the help of skew-symmetric differential forms are presented.
**Acknowledgments**
The author made reports at the seminar of the Institute of General Physics of Russian Academy of Sceince many times. She thanks the head of the seminar Prof. Anry Rukhadze and the participants for useful discussions and comments.
The author also thanks the orgenazers of conferences on gravitation, theory of relativity, nonlinear acoustics, turbulence, the interacions of elementary particles, symmetries, and algebra and geometry for invitations, hospitality and helpful discussions.
I am also thankful to Prof. R.Kiehn for his attention to my works and for multiple stimulating discussions.
**Qualitative investigation of the solutions**
**of differential equations**
The presented method of investigating the solutions to differential equations is not new. Such an approach was developed by Cartan \[19\] in his analysis of the integrability of differential equations. Here this approach is outlined to demonstrate the role of skew-symmetric differential forms.
The basic idea of the qualitative investigation of the solutions to differential equations can be clarified by the example of the first-order partial differential equation.
Let $$F(x^i,\,u,\,p_i)=0,\quad p_i\,=\,\partial u/\partial x^i \eqno(A1.1)$$ be the partial differential equation of the first order. Let us consider the functional relation $$du\,=\,\theta\eqno(A1.2)$$ where $\theta\,=\,p_i\,dx^i$ (the summation over repeated indices is implied). Here $\theta\,=\,p_i\,dx^i$ is the differential form of the first degree.
The specific feature of functional relation (A1.2) is that in the general case this relation turns out to be nonidentical.
The left-hand side of this relation involves a differential, and the right-hand side includes the differential form $\theta\,=\,p_i\,dx^i$. For this relation to be identical, the differential form $\theta\,=\,p_i\,dx^i$ must be a differential as well (like the left-hand side of relation (A1.2)), that is, it has to be a closed exterior differential form. To do this it requires the commutator $K_{ij}=\partial p_j/\partial x^i-\partial p_i/\partial x^j$ of the differential form $\theta $ has to vanish.
In general case, from equation (A1.1) it does not follow (explicitly) that the derivatives $p_i\,=\,\partial u/\partial x^i $, which obey to the equation (and given boundary or initial conditions of the problem), make up a differential. Without any supplementary conditions the commutator of the differential form $\theta $ defined as $K_{ij}=\partial p_j/\partial x^i-\partial p_i/\partial x^j$ is not equal to zero. The form $\theta\,=\,p_i\,dx^i$ occurs to be unclosed and is not a differential like the left-hand side of relation (A1.2). Functional relation (A1.2) appears to be nonidentical: the left-hand side of this relation is a differential, but the right-hand side is not a differential. (The skew-symmetric differential form $\theta\,=\,p_i\,dx^i$, which enters into functional relation (A1.2), is the example of evolutionary skew-symmetric differential forms.)
The nonidentity of functional relation (A1.2) points to a fact that without additional conditions derivatives of the initial equation do not make up a differential. This means that the corresponding solution to the differential equation $u$ will not be a function of $x^i$. It will depend on the commutator of the form $\theta $, that is, it will be a functional.
To obtain the solution that is a function (i.e., derivatives of this solution form a differential), it is necessary to add the closure condition for the form $\theta\,=\,p_idx^i$ and for the dual form (in the present case the functional $F$ plays a role of the form dual to $\theta $): $$\cases {dF(x^i,\,u,\,p_i)\,=\,0\cr
d(p_i\,dx^i)\,=\,0\cr}\eqno(A1.3)$$ If we expand the differentials, we get a set of homogeneous equations with respect to $dx^i$ and $dp_i$ (in the $2n$-dimensional space – initial and tangential): $$\cases {\displaystyle \left ({{\partial F}\over {\partial x^i}}\,+\,
{{\partial F}\over {\partial u}}\,p_i\right )\,dx^i\,+\,
{{\partial F}\over {\partial p_i}}\,dp_i \,=\,0\cr
dp_i\,dx^i\,-\,dx^i\,dp_i\,=\,0\cr} \eqno(A1.4)$$ The solvability conditions for this system (vanishing of the determinant composed of coefficients at $dx^i$, $dp_i$) have the form: $${{dx^i}\over {\partial F/\partial p_i}}\,=\,{{-dp_i}\over
{\partial F/\partial x^i+p_i\partial F/\partial u}} \eqno(A1.5)$$ These conditions determine an integrating direction, namely, a pseudostructure, on which the form $\theta \,=\,p_i\,dx^i$ occurs to be closed one, i.e. it becomes a differential, and from relation (A1.2) the identical relation is produced. If conditions (A1.5), that may be called the integrability conditions, are satisfied, the derivatives constitute a differential $\delta u\,=\,p_idx^i\,=\,du$ (on the pseudostructure), and the solution becomes a function. Just such solutions, namely, functions on pseudostructures formed by the integrating directions, are so-called generalized solutions. The derivatives of the generalized solution constitute the exterior form, which is closed on pseudostructure.
Since the functions that are generalized solutions are defined only on pseudostructures, they have discontinuities in derivatives in the directions being transverse to pseudostructures. The order of derivatives with discontinuities is equal to the exterior form degree. If the form of zero degree is involved in the functional relation, the function itself, being a generalized solution, will have discontinuities.
If we find the characteristics of equation (A1.1), it appears that conditions (A1.5) are equations for characteristics \[20\]. That is, the characteristics are examples of the pseudostructures on which the derivatives of differential equation constitute closed forms and the solutions turn out to be functions (generalized solutions).
Here it is worth noting that coordinates of the equations for characteristics are not identical to independent coordinates of the initial space on which equation (A1.1) is defined. The transition from initial space to characteristic manifold appears to be a [*degenerate*]{} transformation, namely, the determinant of the system of equations (A1.4) becomes zero. The derivatives of equation (A1.1) are transformed from tangent space to cotangent one. The transition from the tangent space, where the commutator of the form $\theta$ is nonzero (the form is unclosed, the derivatives do not form a differential), to the characteristic manifold, namely, the cotangent space, where the commutator becomes equal to zero (a closed exterior form is formed, i.e. the derivatives make up a differential), is the example of degenerate transformation.
The solutions to all differential equations have similar functional properties. And, if the order of differential equation is $k$, the functional relation with $k$-degree form corresponds to this equation. For ordinary differential equations the commutator is produced at the expense of the conjugacy of derivatives of the functions desired and those of the initial data (the dependence of the solution on the initial data is governed by the commutator).
In a similar manner one can also investigate the solutions to a system of partial differential equations and the solutions to ordinary differential equations (for which the nonconjugacy of desired functions and initial conditions is examined).
It can be shown that the solutions to equations of mathematical physics, on which no additional external conditions are imposed, are functionals. The solutions prove to be exact only under realization of additional requirements, namely, the conditions of degenerate transformations like vanishing determinants, Jacobians and so on, that define integral surfaces. Characteristic manifolds, the envelopes of characteristics, singular points, potentials of simple and double layers, residues and others are the examples of such surfaces.
The dependence of the solution on the commutator can lead to instability of the solution. Equations that do not provided with the integrability conditions (the conditions such as, for example, the characteristics, singular points, integrating factors and others) may have unstable solutions. Unstable solutions appear in the case when additional conditions are not realized and no exact solutions (their derivatives form a differential) are formed. Thus, the solutions to the equations of elliptic type can be unstable.
Investigation of nonidentical functional relations lies at the basis of the qualitative theory of differential equations. It is well known that the qualitative theory of differential equations is based on the analysis of unstable solutions and integrability conditions. From the functional relation it follows that the dependence of the solution on the commutator leads to instability, and the closure conditions of the forms constructed by derivatives are integrability conditions. One can see that the problem of unstable solutions and integrability conditions appears, in fact, to be reduced to the question of under what conditions the identical relation for closed form is produced from the nonidentical relation that corresponds to the relevant differential equation (the relation such as (A1.2)), the identical relation for closed form is produced. In other words, whether or not the solutions are functionals? This is, the analysis of the correctness of setting the problems of mathematical physics is reduced to the same question.
Here the following should be emphasized. When the degenerate transformation from the initial nonidentical functional relation is performed, an integrable identical relation is obtained. As the result of integrating, one obtains a relation that contains exterior forms of less by one degree and which once again proves to be (in the general case without additional conditions) nonidentical. By integrating the functional relations obtained sequentially (it is possible only under realization of the degenerate transformations) from the initial functional relation of degree $k$ one can obtain $(k+1)$ functional relations each involving exterior forms of one of degrees: $k, \,k-1, \,...0$. In particular, for the first-order partial differential equation it is also necessary to analyze the functional relation of zero degree.
Thus, application of skew-symmetric differential forms allows one to reveal the functional properties of the solutions to differential equations.
Field theory is based on the conservation laws. The conservation laws are described by the closure conditions of the exterior differential forms. It is evident that the solutions to the equations of field theory describing physical fields can be only generalized solutions, which correspond to closed exterior differential forms. The generalized solutions can have a differential equation, which is subject to the additional conditions.
Let us consider what equations are obtained in this case.
Return to equation (A1.1). Assume that the equation does not explicitly depend on $u$ and is resolved with respect to some variable, for example $t$, that is, it has the form of $${{\partial u}\over {\partial t}}\,+\,E(t,\,x^j,\,p_j)\,=\,0, \quad p_j\,=
\,{{\partial u}\over {\partial x^j}}\eqno(A1.6)$$ Then integrability conditions (A1.5) (the closure conditions of the differential form $\theta =p_idx^i$ and the corresponding dual form) can be written as (in this case $\partial F/\partial p_1=1$) $${{dx^j}\over {dt}}\,=\,{{\partial E}\over {\partial p_j}}, \quad
{{dp_j}\over {dt}}\,=\,-{{\partial E}\over {\partial x^j}}\eqno(A1.7)$$
These are the characteristic relations for equation (A1.6). As it is well known, the canonical relations have just such a form.
As a result we conclude that the canonical relations are the characteristics of equation (A1.6) and the integrability conditions for this equation.
The canonical relations obtained from the closure condition of the differential form $\theta = p_idx^i$ and the corresponding dual form, are the examples of the identical relation of the theory of exterior differential forms.
Equation (A1.6) provided with the supplementary conditions, namely, the canonical relations (A1.7), is called the Hamilton-Jacobi equation \[20\]. In other words, the equation whose derivatives obey the canonical relation is referred to as the Hamilton-Jacobi equation. The derivatives of this equation form the differential, i.e. the closed exterior differential form: $\delta u\,=\,(\partial u/\partial t)\,dt+p_j\,dx^j\,=\,-E\,dt+p_j\,dx^j\,=\,du$.
The equations of field theory belong to this type. $${{\partial s}\over {\partial t}}+H \left(t,\,q_j,\,{{\partial s}\over {\partial q_j}}
\right )\,=\,0,\quad
{{\partial s}\over {\partial q_j}}\,=\,p_j \eqno(A1.8)$$ where $s$ is the field function for the action functional $S\,=\,\int\,L\,dt$. Here $L$ is the Lagrange function, $H$ is the Hamilton function: $H(t,\,q_j,\,p_j)\,=\,p_j\,\dot q_j-L$, $p_j\,=\,\partial L/\partial \dot q_j$. The closed form $ds\,=-\,H\,dt\,+\,p_j\,dq_j$ (the Poincare invariant) corresponds to equation (A1.8).
A peculiarity of the degenerate transformation can be considered by the example of the field equation.
Here the degenerate transformation is a transition from the Lagrange function to the Hamilton function. The equation for the Lagrange function, that is the Euler variational equation, was obtained from the condition $\delta S\,=\,0$, where $S$ is the action functional. In the real case, when forces are nonpotential or couplings are nonholonomic, the quantity $\delta S$ is not a closed form, that is, $d\,\delta S\,\neq \,0$. But the Hamilton function is obtained from the condition $d\,\delta S\,=\,0$ which is the closure condition for the form $\delta S$. The transition from the Lagrange function $L$ to the Hamilton function $H$ (the transition from variables $q_j,\,\dot q_j$ to variables $q_j,\,p_j=\partial L/\partial \dot q_j$) is a transition from the tangent space, where the form is unclosed, to the cotangent space with a closed form. This transition is a degenerate one.
The invariant field theories used only nondegenerate transformations that conserve a differential. There exists a relation between nondegenerate transformations and degenerate transformations. In the case under consideration the degenerate transformation is a transition from the tangent space ($q_j,\,\dot q_j)$) to the cotangent (characteristic) manifold ($q_j,\,p_j$), but the nondegenerate transformation is a transition from one characteristic manifold ($q_j,\,p_j$) to another characteristic manifold ($Q_j,\,P_j$). {The formula of canonical transformation can be written as $p_jdq_j=P_jdQ_j+dW$, where $W$ is the generating function}.
**The analysis of balance conservation laws for**
**thermodynamic and gas dynamic systems and for**
**the system of charged particles**
Thermodynamic systems {#thermodynamic-systems .unnumbered}
---------------------
The thermodynamics is based on the first and second principles of thermodynamics that were introduced as postulates \[21\]. The first principle of thermodynamics, which can be written in the form $$dE\,+\,dw\,=\,\delta Q\eqno(A2.1)$$ follows from the balance conservation laws for energy and linear momentum (but not only from the conservation law for energy). This is analogous to the evolutionary relation for the thermodynamic system. Since $\delta Q$ is not a differential, relation (A2.1) which corresponds to the first principle of thermodynamics, as well as the evolutionary relation, appears to be a nonidentical relation. This points to a noncommutativity of the balance conservation laws (for energy and linear momentum) and to a nonequilibrium state of the thermodynamic system.
If condition of the integrability be satisfied, from the nonidentical evolutionary relation, which corresponds to the first principle of thermodynamics, it follows an identical relation. It is an identical relation that corresponds to the second principle of thermodynamics.
If $dw\,=\,p\,dV$, there is the integrating factor $\theta$ (a quantity which depends only on the characteristics of the system), where $1/\theta\,=\,pV/R$ is called the temperature $T$ \[21\]. In this case the form $(dE\,+\,p\,dV)/T$ turns out to be a differential (interior) of some quantity that referred to as entropy $S$: $$(dE\,+\,p\,dV)/T\,=\,dS \eqno(A2.2)$$
If the integrating factor $\theta=1/T$ has been realized, that is, relation (A2.2) proves to be satisfied, from relation (A2.1), which corresponds to the first principle of thermodynamics, it follows $$dS\,=\,\delta Q/T \eqno(A2.3)$$ This is just the second principle of thermodynamics for reversible processes. This takes place when the heat input is the only action onto the system.
If in addition to the heat input the system experiences a certain mechanical action (for example, an influence of boundaries), we obtain $$dS\, >\,\delta Q/T \eqno (A2.4)$$ that corresponds to the second principle of thermodynamics for irreversible processes.
In the case examined above the differential of entropy (rather than entropy itself) becomes a closed form. $\{$In this case entropy manifests itself as the thermodynamic potential, namely, the function of state. To the pseudostructure there corresponds the state equation that determines the temperature dependence on the thermodynamic variables$\}$.
For entropy to be a closed form itself, one more condition must be realized. Such a condition could be the realization of the integrating direction, an example of that is the speed of sound: $a^2\,=\,\partial p/\partial \rho\,=\,\gamma\,p/\rho$. In this case it is valid the equality $ds\,=\,d(p/\rho ^{\lambda })\,=\,0$ from which it follows that entropy $s\,=\,p/\rho ^{\lambda }\,=\hbox{const}$ is a closed form (of zero degree). $\{$However it does not mean that a state of the gaseous system is identically isoentropic. Entropy is constant only along the integrating direction (for example, on the adiabatic curve or on the front of the sound wave), whereas in the direction normal to the integrating direction the normal derivative of entropy has a break$\}$.
Gas dynamical systems {#gas-dynamical-systems .unnumbered}
---------------------
We take the simplest gas dynamical system, namely, a flow of ideal (inviscous, heat nonconductive) gas \[13\].
Assume that the gas (the element of gas dynamic system) is a thermodynamic system in the state of local equilibrium (whenever the gas dynamic system itself may be in nonequilibrium state), that is, it is satisfied the relation \[21\] $$Tds\,=\,de\,+\,pdV \eqno(A2.5)$$ where $T$, $p$ and $V$ are the temperature, the pressure and the gas volume, $s$ and $e$ are entropy and internal energy per unit volume.
Let us introduce two frames of reference: an inertial one that is not connected with material system and an accompanying frame of reference that is connected with the manifold formed by the trajectories of the material system elements.
The equation of the balance conservation law of energy for ideal gas can be written as \[13\] $${{Dh}\over {Dt}}- {1\over {\rho }}{{Dp}\over {Dt}}\,=\,0 \eqno(A2.6)$$ where $D/Dt$ is the total derivative with respect to time (if to denote the spatial coordinates by $x_i$ and the velocity components by $u_i$, $D/Dt\,=(\,\partial /\partial t+u_i\partial /\partial x_i$). Here $\rho=1/V $ and $h$ are respectively the mass and the entalpy densities of the gas.
Expressing entalpy in terms of internal energy $e$ using the formula $h\,=\,e\,+\,p/\rho $ and using relation (A2.5), the balance conservation law equation (A2.6) can be put to the form $${{Ds}\over {Dt}}\,=\,0 \eqno(A2.7)$$
And respectively, the equation of the balance conservation law for linear momentum can be presented as \[13,22\] $$\hbox {grad} \,s\,=\,(\hbox {grad} \,h_0\,+\,{\bf U}\times \hbox {rot} {\bf U}\,-{\bf F}\,+\,
\partial {\bf U}/\partial t)/T \eqno(A2.8)$$ where ${\bf U}$ is the velocity of the gas particle, $h_0=({\bf U \cdot U})/2+h$, ${\bf F}$ is the mass force. The operator $grad$ in this equation is defined only in the plane normal to the trajectory. \[Here it was tolerated a certain incorrectness. Equations (A2.7), (A2.8) are written in different forms. This is connected with difficulties when deriving these equations themselves. However, this incorrectness will not effect on results of the qualitative analysis of the evolutionary relation obtained from these equations.\]
Since the total derivative with respect to time is that along the trajectory, in the accompanying frame of reference equations (A2.7) and (A2.8) take the form: $${{\partial s}\over {\partial \xi ^1}}\,=\,0 \eqno (A2.9)$$ $${{\partial s}\over {\partial \xi ^{\nu}}}\,=\,A_{\nu },\quad \nu=2, ... \eqno(A2.10)$$ where $\xi ^1$ is the coordinate along the trajectory, $\partial s/\partial \xi ^{\nu }$ is the left-hand side of equation (A2.8), and $A_{\nu }$ is obtained from the right-hand side of relation (A2.8).
{In the common case when gas is nonideal equation (A2.9) can be written in the form $${{\partial s}\over {\partial \xi ^1}} \,=\,A_1 \eqno (A2.11)$$ where $A_1$ is an expression that depends on the energetic actions (transport phenomena: viscous, heat-conductive). In the case of ideal gas $A_1\,=\,0$ and equation (A2.12) transforms into (A2.9). In the case of the viscous heat-conductive gas described by a set of the Navier-Stokes equations, in the inertial frame of reference the expression $A_1$ can be written as \[13\] $$A_1\,=\,{1\over {\rho }}{{\partial }\over {\partial x_i}}
\left (-{{q_i}\over T}\right )\,-\,{{q_i}\over {\rho T}}\,{{\partial T}\over {\partial x_i}}
\,+{{\tau _{ki}}\over {\rho }}\,{{\partial u_i}\over {\partial x_k}} \eqno(A2.12)$$ Here $q_i$ is the heat flux, $\tau _{ki}$ is the viscous stress tensor. In the case of reacting gas extra terms connected with the chemical nonequilibrium are added \[13\].}
Equations (A2.9) and (A2.10) can be convoluted into the relation $$ds\,=\,A_{\mu} d\xi ^{\mu}\eqno(A2.13)$$ where $\,A_{\mu} d\xi ^{\mu}=\omega\,$ is the first degree differential form (here $A_1=0$,$\mu =1,\,\nu $).
Relation (A2.13) is the evolutionary relation for gas dynamic system (in the case of local thermodynamic equilibrium). Here $\psi\,=\,s$. $\{$It worth notice that in the evolutionary relation for thermodynamic system the dependence of entropy on thermodynamic variables is investigated (see relation (A2.5)), whereas in the evolutionary relation for gas dynamic system the entropy dependence on the space-time variables is considered$\}$.
Relation (A2.13) appears to be nonidentical. To make it sure that this is true one must inspect the commutator of the form $\omega $.
Without accounting for terms that are connected with a deformation of the manifold formed by the trajectories the commutator can be written as $$K_{1\nu }\,=\,{{\partial A_{\nu }}\over {\partial \xi ^1}}\,-\,{{\partial A_1}\over
{\partial \xi ^{\nu }}}$$
From the analysis of the expression $A_{\nu }$ and with taking into account that $A_1\,=\,0$ one can see that terms that are related to the multiple connectedness of the flow domain (the second term of equation (A2.8)), the nonpotentiality of the external forces (the third term in (A2.8)) and the nonstationarity of the flow (the forth term in (A2.8)) contribute into the commutator. {The terms connected with transport phenomena and physical and chemical processes will contribute into the commutator (see expression (A2.12)).}
Since the commutator of the form $\omega $ is nonzero, it is evident that the form $\omega$ proves to be unclosed. This means that relation (A2.13) cannot be an identical one.
Nonidentity of the evolutionary relation points to the nonequilibrium state and the development of the gas dynamic instability. Since the nonequilibrium state is produced by internal forces that are described by the commutator of the form $\omega $, it becomes evident that the cause of the gas dynamic instability is something that contributes into the commutator of the form $\omega $.
One can see (see (A2.8)) that the development of instability is caused by not a simply connectedness of the flow domain, nonpotential external (for each local domain of the gas dynamic system) forces, a nonstationarity of the flow.
All these factors lead to emergency of internal forces, that is, to nonequilibrium state and to development of various types of instability. {Transport phenomena and physical and chemical processes also lead to emergency of internal forces and to development of instability.}
And yet for every type of instability one can find the appropriate term giving contribution to the evolutionary form commutator, which is responsible for this type of instability. Thus, there is an unambiguous connection between the type of instability and the terms that contribute to the evolutionary form commutator in the evolutionary relation. {In the general case one has to consider the evolutionary relations that correspond to the balance conservation laws for angular momentum and mass as well.}
As it was shown above, under realization of additional degrees of freedom it can take place the transition from the nonequilibrium state to the locally equilibrium one, and this process is accompanied by emergency of physical structures. The gas dynamic formations that correspond to these physical structures are shocks, shock waves, turbulent pulsations and so on. Additional degrees of freedom are realized as the condition of the degenerate transformation, namely, vanishing of determinants, Jacobians of transformations, etc. These conditions specify the integral surfaces (pseudostructures): the characteristics (the determinant of coefficients at the normal derivatives vanishes), the singular points (Jacobian is equal to zero), the envelopes of characteristics of the Euler equations and so on. Under crossing throughout the integral surfaces the gas dynamic functions or their derivatives undergo the breaks.
Let as analyze which types of instability and what gas dynamic formation can originate under given external action.
1). [*Shock, break of diaphragm and others*]{}. The instability originates because of nonstationarity. The last term in equation (A2.8) gives a contribution into the commutator. In the case of ideal gas whose flow is described by equations of the hyperbolic type the transition to the locally equilibrium state is possible on the characteristics and their envelopes. The corresponding structures are weak shocks and shock waves.
2).[*Flow of ideal (inviscous, heat nonconductive) gas around bodies Action of nonpotential forces*]{}. The instability develops because of the multiple connectedness of the flow domain and a nonpotentiality of the body forces. The contribution into the commutator comes from the second and third terms of the right-hand side of equation (A2.8). Since the gas is ideal one and $\partial s/\partial \xi ^1=A_1=0$, that is, there is no contribution into the each fluid particle, an instability of convective type develops. For $U>a$ ($U$ is the velocity of the gas particle, $a$ is the speed of sound) a set of equations of the balance conservation laws belongs to the hyperbolic type and hence the transition to the locally equilibrium state is possible on the characteristics and on the envelopes of characteristics as well, and weak shocks and shock waves are the structures of the system. If $U<a$ when the equations are of elliptic type, such a transition is possible only at singular points. The structures emerged due to a convection are of the vortex type. Under long acting the large-scale structures can be produced.
3\. [*Boundary layer*]{}. The instability originates due to the multiple connectness of the domain and the transport phenomena (an effect of viscosity and thermal conductivity). Contributions into the commutator produce the second term in the right-hand side of equation (A2.8) and the second and third terms in expression (A2.12). The transition to the locally equilibrium state is allowed at singular points. because in this case $\partial s/\partial \xi^1=A_1\neq 0$, that is, the external exposure acts onto the gas particle separately, the development of instability and the transitions to the locally equilibrium state are allowed only in an individual fluid particle. Hence, the structures emerged behave as pulsations. These are the turbulent pulsations.
{Studying the instability on the basis of the analysis of entropy behavior was carried out in the works by Prigogine and co-authors \[23\]. In that works entropy was considered as the thermodynamic function of state (though its behavior along the trajectory was analyzed). By means of such state function one can trace the development (in gas fluxes) of the hydrodynamic instability only. To investigate the gas dynamic instability it is necessary to consider entropy as the gas dynamic state function, i.e. as a function of the space-time coordinates. Whereas for studying the thermodynamic instability one has to analyze the commutator constructed by the mixed derivatives of entropy with respect to the thermodynamic variables, for studying the gas dynamic instability it is necessary to analyze the commutators constructed by the mixed derivatives of entropy with respect to the space-time coordinates.}
Electromagnetic field {#electromagnetic-field .unnumbered}
---------------------
The system of charged particles is a material medium, which generates electromagnetic field.
If to use the Lorentz force ${\bf F\,= \,\rho (E + [U\times H]}/c)$, the local variation of energy and linear momentum of the charged matter (material system) can be written respectively as \[10\]: $\rho ({\bf U\cdot E})$, $\rho ({\bf E+[U\times H]}/c)$. Here $\rho$ is the charge density, ${\bf U}$ is the velocity of charged matter. These variations of energy and linear momentum are caused by energetic and force actions and are equal to values of these actions. If to denote these actions by $Q^e$, ${\bf Q}^i$, the balance conservation laws can be written as follows: $$\rho \,({\bf U\cdot E})\,=\,Q^e\eqno(A2.14)$$ $$\rho \,({\bf E\,+\,[U\times H]}/c)\,=\, {\bf Q}^i \eqno(A2.15)$$
After eliminating the characteristics of material system (the charged matter) $\rho$ and ${\bf U}$ by using the Maxwell-Lorentz equations \[10\], the left-hand sides of equations (A2.14), (A2.15) can be expressed only in terms of the strengths of electromagnetic field, and then one can write equations (A2.14), (A2.15) as $$c\,\hbox{div} {\bf S}\,=\,-{{\partial}\over {\partial t}}\,I\,+\,Q^e\eqno(A2.16)$$ $${1\over c}\,{{\partial }\over {\partial t}}\,{\bf S}\,=
\,{\bf G}\,+\,{\bf Q^i}\eqno(A2.17)$$ where ${\bf S=[E\times H]}$ is the Pointing vector, $I=(E^2+H^2)/c$, ${\bf G}={\bf E}\,\hbox {div}{\bf E}+\hbox{grad}({\bf E\cdot E})-
({\bf E}\cdot \hbox {grad}){\bf E}+\hbox {grad}({\bf H\cdot H})-({\bf H}\cdot\hbox{grad}){\bf H}$.
Equation (A2.16) is widely used while describing electromagnetic field and calculating energy and the Pointing vector. But equation (A2.17) does not commonly be taken into account. Actually, the Pointing vector ${\bf S}$ must obey two equations that can be convoluted into the [*relation*]{} $$d\bf S=\,\omega ^2\eqno(A2.18)$$ Here $d\bf S$ is the state differential being 2-form and the coefficients of the form $\omega ^2$ (the second degree form) are the right-hand sides of equations (A2.16) and (A2.17). It is just the evolutionary relation for the system of charged particles that generate electromagnetic field.
By analyzing the coefficients of the form $\omega ^2$ (obtained from equations (A2.16) and (A2.17), one can assure oneself that the form commutator is nonzero. This means that from relation (A2.18) the Pointing vector cannot be found. This points to the fact that there is no such a measurable quantity (a potential).
Under what conditions can the Pointing vector be formed as a measurable quantity?
Let us choose the local coordinates $l_k$ in such a way that one direction $l_1$ coincides with the direction of the vector ${\bf S}$. Because this chosen direction coincides with the direction of the vector ${\bf S=[E\times H]}$ and hence is normal to the vectors ${\bf E}$ and ${\bf H}$, one obtains that $\hbox{div} {\bf S}\,=\,\partial s/\partial l_1$, where $S$ is a module of ${\bf S}$. In addition, the projection of the vector ${\bf G}$ on the chosen direction turns out to be equal to $-\partial I/\partial l_1$. As a result, after separating from vector equation (A2.17) its projection on the chosen direction equations (A2.16) and (A2.17) can be written as $${{\partial S}\over {\partial l_1}}\,=\,-{1\over c}{{\partial I}\over {\partial t}}\,+\,
{1\over c}Q^e \eqno(A2.19)$$ $${{\partial S}\over {\partial t}}\,=\,-c\,{{\partial I}\over {\partial l_1}}\,+\,c{\bf Q}'^i\eqno(A2.20)$$ $$0\,=\,-{\bf G}''\,-\,c{\bf Q}''^i$$ Here the prime relates to the direction $l_1$, double primes relate to the other directions. Under the condition $d l_1/d t\,=\,c$ from equations (A2.19) and (A2.20) it is possible to obtain the relation in differential forms $${{\partial S}\over {\partial l_1}}\,dl_1\,+\,{{\partial S}\over {\partial t}}\,dt\,=\,
-\left( {{\partial I}\over {\partial l_1}}\,dl_1\,+\,{{\partial I}\over {\partial t}}\,dt\right )\,+\,
(Q^e\,dt\,+\,{\bf Q}'^i\,dl_1)\eqno(A2.21)$$ Because the expression within the second braces in the right-hand side is not a differential (the energetic and force actions have different nature and cannot be conjugated), one can obtain a closed form only if this term vanishes: $$(Q^e\,dt\,+\,{\bf Q}'^i\,dl_1)\,=\,0\eqno(A2.22)$$ that is possible only discretely (rather than identically).
In this case $dS\,=\,0$, $dI\,=\,0$ and the modulus of the Pointing vector $S$ proves to be a closed form, i.e. a measurable quantity. The integrating direction (the pseudostructure) will be $$-\,{{\partial S/\partial t}\over {\partial S/\partial l_1}}\,=\,{{dl_1}\over {dt}}\,=\,c\eqno(A2.23)$$ The quantity $I$ is the second dual invariant.
Thus, the constant $c$ entered into the Maxwell equations is defined as the integrating direction.
[00]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a protocol for using trapped ions to measure rotations via matter-wave Sagnac interferometry. The trap allows the interferometer to enclose a large area in a compact apparatus through repeated round-trips in a Sagnac geometry. We show how a uniform magnetic field can be used to close the interferometer over a large dynamic range in rotation speed and measurement bandwidth without losing contrast. Since this technique does not require the ions to be confined in the Lamb-Dicke regime, thermal states with many phonons should be sufficient for operation.'
address:
- '$^1$ UCLA Department of Physics and Astronomy, Los Angeles, California 90095, USA'
- '$^2$ California Institute for Quantum Emulation, Santa Barbara, California 93106, USA'
author:
- 'W C Campbell$^{1,2}$ and P Hamilton$^1$'
bibliography:
- 'IonGyro.bib'
title: Rotation sensing with trapped ions
---
Introduction
============
The Sagnac effect can be used to measure the rotational velocity $\Omega$ of a reference frame by observing the phase shift of an interferometer in that frame whose paths enclose an area $A$ perpendicular to any component of $\Omega$ (see, *e.g.* [@BarrettCRP14] for a review). The rotation-induced phase shift is given by $$\Phi=2 \pi\frac{2 E}{h c^2}\mathbf{A}\cdot
\mbox{\boldmath$\Omega$\unboldmath} ,\label{SagnacPhase}$$ where $\mathbf{A}$ is the vector area enclosed by the two paths. $E$ is the total energy of the particles that are interfering, defined using the relativistic energy-momentum relation $$E^2 = \left(mc^2\right)^2 + p^2c^2.$$ For photons, $E=\hbar \omega_{\mathrm{optical}}$, whereas for atoms of rest mass $m$ moving at non-relativistic speeds ($p\ll mc$), $E=mc^2$.
The sensitivity of a gyroscope is the minimum detectable rotation rate within a detection bandwidth $\Delta f$. For a shot-noise-limited interferometer that detects the outcome of individual interference events at a rate $\dot{N}$, the uncertainty in the measured phase after running for a time $t = 1/\Delta f$ will be $\delta \phi \approx
\sqrt{\Delta f/\dot{N}}$. The sensitivity is given by $$\mathcal{S} = \frac{\delta \phi}{\frac{\partial \Phi}{\partial \Omega}\sqrt{\Delta
f}}=\frac{1}{\frac{\partial \Phi}{\partial \Omega}\sqrt{\dot{N}}}$$ where the *scale factor* is given by $$\frac{\partial \Phi}{\partial \Omega} = 2 \pi A \frac{2E}{hc^2} \label{ScaleFactor}$$ and we have assumed an orientation such that $\mathbf{A} \cdot$$\Omega$$= A\Omega$ for algebraic simplicity.
Two primary methods are frequently employed to boost the sensitivity of interferometric gyroscopes. For photons, optical fibers (or ring laser cavities) allow many effective round-trips through the Sagnac interferometer, thereby increasing the effective area $A$ by 2 times the number of round trips ($M$) without increasing the actual area of the apparatus. This, coupled with the large $\dot{N}$ possible in these devices, leads to a state-of-the-art reported sensitivity of $\mathcal{S} = 1.2 \times 10^{-11} \mbox{
rad}/\mbox{s}/\sqrt{\mbox{Hz}}$, achieved by a ring laser with $16
\mbox{ m}^2$ enclosed area [@SchreiberRSI13].
Another approach is to use atoms on ballistic trajectories instead of photons, which increases $E$ by a factor of $mc^2/\hbar
\omega_{\mathrm{optical}} \approx 10^{11}$. The drawbacks of this approach, as compared to an optical gyroscope, are that $\dot{N}$ is smaller, and the free-flight atom trajectories enclose the interferometer area $A$ only once. The latter constraint has meant that increasing $A$ has necessarily involved increasing the physical size of the apparatus, which can be undesirable for some applications. Furthermore, long atom trajectories and large separations make the measurement susceptible to systematics that can produce path-dependent phase shifts, such as magnetic field gradients. Nonetheless, the improvement in $E$ has enabled atom interferometers to demonstrate high rotation rate sensitivity, with demonstrated state-of-the-art short-term sensitivities of $\mathcal{S} = 6 \times
10^{-10} \mbox{ rad}/\mbox{s}/\sqrt{\mbox{Hz}}$ for atomic beams [@GustavsonCQG00] and $\mathcal{S} = 2.4 \times 10^{-7} \mbox{
rad}/\mbox{s}/\sqrt{\mbox{Hz}}$ for laser-cooled atoms [@GauguetPRA09].
Here, we show that trapped atomic ions provide a way to use both methods simultaneously to increase the interferometer sensitivity. While interferometers with enclosed area have been demonstrated with clouds of trapped neutral atoms [@SackettPRA09; @PrentissPRL07], maintaining the coherence across the ensemble needed for a gyroscope has proved difficult. Here we introduce a combination of laser-driven spin-dependent momentum kicks in one direction with ion trap voltage changes along an orthogonal direction that perform interferometry with trapped ions in a Sagnac (as opposed to Mach-Zehnder) configuration. This allows atomic trajectories to repeatedly enclose the same area, thereby accumulating Sagnac phase continuously for a time that is not limited by a ballistic flight trajectory. Since the enclosed area is proportional to the displacement along both directions and only one of these needs to be state-dependent, the interferometer area can be increased with trap voltage alone, circumventing the need to drive more coherent momentum transfer from the laser. The harmonic trapping potential makes the area enclosed independent of the initial ion velocity, eliminating a source of scale factor instability found in free space atom interferometers. These factors, coupled with the extremely long coherence times of trapped ions, gives the trapped ion interferometer the potential to enclose a large effective area in a small apparatus with high stability.
![Trajectories of an ion during interferometer operation in (a) position space, (b) momentum space (c) $x$ phase space and (d) $y$ phase space. The ion’s starting coordinates are indicated by a circle, and the trap center after the $y$-displacement in step (\[Stepiii\]) is indicated by an $\times$. Red and blue curves represent the trajectory for the two spin states. Trajectories for different starting conditions are qualitatively similar to these, with the exception that for a ground-state ion, the trajectories for the two spin states completely overlap.[]{data-label="TrajectoriesFigure"}](figure1a_1d.eps)
Interferometer operation
========================
We begin by introducing the protocol for measuring rotations with a single ion that hosts a qubit with internal states ${{\left| {\uparrow} \right\rangle}}$ and ${{\left| {\downarrow} \right\rangle}}$. As shown in Fig. \[TrajectoriesFigure\], the enclosed area will be in the $x,y$ plane, and the (secular) trap frequencies for the ion in these two directions after the $y$ displacement (see below) will be assumed to be degenerate: $\omega_x =
\omega_y \equiv \omega$. $(x,y,z)$ will be coordinates in real space, while $(X,Y,Z)$ will denote axes of the qubit’s associated Bloch sphere. We will assume that the confinement in the $z$-direction is strong ($\omega_z \gg \omega$) so that the system can be approximated as being 2D. The time-sequence of the trapped ion gyroscope proceeds in the following steps (see also Fig. \[TrajectoriesFigure\] and \[RotatingFrameFigure\]):
1. Prepare the ion in ${{\left| {\downarrow} \right\rangle}}$ and apply a $\pi/2$ pulse of microwaves about $-\mathbf{\hat{Y}}.$\[Stepi\]
2. Apply $N_{\mathrm{k}}$ spin-dependent kicks (SDKs) in the $x$-direction ($\Delta \mathbf{p} = -N_{\mathrm{k}}\hbar \Delta
\mathbf{k}\hat{\sigma}_Z $) to separate the atom in momentum space.\[Stepii\]
3. Apply a step function in electrode voltages to non-adiabatically displace the trap only in the $y$-direction a distance $y_{\mathrm{d}}$.\[Stepiii\]
4. Allow the ion to oscillate in the trap for an integer number ($M$) of round trips $\Delta t = M 2\pi/\omega$.\[Stepiv\]
5. Reverse step (\[Stepiii\]) by non-adiabatically switching the trap voltages back to their original values.\[Stepv\]
6. Reverse step (\[Stepii\]) by applying $N_{\mathrm{k}}$ SDKs in the other direction ($\Delta \mathbf{p} = N_{\mathrm{k}}\hbar \Delta
\mathbf{k}\hat{\sigma}_Z $) to close the interferometer.\[Stepvi\]
7. Apply another $\pi/2$ pulse with microwaves about an axis inclined by $\phi$ in the $X,Y$ plane from the $-\mathbf{\hat{Y}}$ axis of the Bloch sphere, then measure the internal state of the ion in the qubit basis.\[Stepvii\]
We note that steps (\[Stepi\]) and (\[Stepvii\]) are a standard Ramsey sequence, so qubit and microwave oscillator coherence are required for the duration of the protocol. Even for non-clock-state qubits with magnetic sensitivity on the order of a Bohr magneton ($\mu_{\mathrm{B}}$), qubit coherence times of the order $1 \mbox{ s}$ or greater can be achieved [@RusterArXiv]. We will describe the details of the gyroscope protocol assuming the magnetic field on the ion is zero before discussing the magnetic field effects.
Phase-space displacements
=========================
The trapped ion gyroscope relies on two different methods to produce displacements in motional phase space: spin-dependent kicks that transfer photon momenta to the ions in directions that depend upon the ion’s spin (qubit) state, and trap voltage steps that rapidly displace the trap center. Since a displacement operation in phase space necessarily involves a large number of Fock states, both of these operations take place much faster than the resolved-sideband limit ($T
\approx 2 \pi/ \omega$), and can be thought of as driving many different motional transitions at once.
The spin-dependent kicks [@MizrahiPRL13] of steps (\[Stepii\]) and (\[Stepvi\]) act as the beam splitters in the matter-wave interferometer. The speed of the SDK is enabled through the utilization of “ultrafast” mode-locked lasers to transfer $\hbar
\Delta k$ of momentum to the ion (see [@MizrahiPRL13; @MizrahiAPB14] for more detail). Conceptually, the ideal SDK transfers a momentum kick to the ion whose direction is reversed for the two spin states via the operator $$\hat{U}_{\mathrm{SDK}} = \hat{D}_x[\rmi \eta] \hat{\sigma}_+ +
\hat{D}_x[-\rmi \eta] \hat{\sigma}_-\label{SDKOperator}$$ where $\hat{D}_x[s]$ displaces a coherent state in $x$ phase space a distance $s$ (see Fig. \[TrajectoriesFigure\] and \[RotatingFrameFigure\]) and $\eta$ is the Lamb-Dicke factor for the laser-ion interaction in the $x$-direction ($\eta \equiv \Delta k x_0
= \Delta k \sqrt{\hbar/2 m \omega}$). Since step (\[Stepvi\]) drives the same process as (\[Stepii\]) with the direction of the kicks reversed (effectively replacing $\rmi \!\rightarrow -\rmi$ in (\[SDKOperator\])), we have suppressed the laser beat note phase when writing (\[SDKOperator\]) since it plays no role as along as it is stable during a single enactment the interferometer protocol. The qubit raising and lowering operators ($\hat{\sigma}_{\pm}$) flip the spin state of the qubit, so one way that larger displacements (*i.e.* $M$ of them) can be made is by repeating this operation after a delay by half a motional period [@JohnsonPRL15]. For algebraic simplicity, we will assume for our protocol that the number of spin-dependent kicks applied ($N_{\mathrm{k}}$) is even and that an extra half-period of motion is inserted after the last kick to preserve the harmonic oscillation phase of the initial motional state.
Working in the coherent state basis for describing the ion’s motion in $x$ and $y$ (denoted by coherent state parameters $\alpha_x$ and $\alpha_y$), step (\[Stepi\]) results in the state $${{\left| {\psi_{\mathrm{\ref{Stepi}}}} \right\rangle}} = \textstyle \frac{1}{\sqrt{2}} \displaystyle
\left( {{\left| {\downarrow} \right\rangle}} + {{\left| {\uparrow} \right\rangle}} \right) \otimes {{\left| {\alpha_x,\alpha_y} \right\rangle}}.$$ The SDKs in step (\[Stepii\]) induce spin-orbit coupling to produce the entangled state $$\begin{aligned}
\fl {{\left| {\psi_{\mathrm{\ref{Stepii}}}} \right\rangle}} = &\textstyle \frac{1}{\sqrt{2}} \displaystyle
\big( \rme^{\rmi N_{\mathrm{k}} \eta \mathrm{I\!R}(\alpha_x)}
{{\left| {\downarrow} \right\rangle}} \otimes {{\left| {\alpha_x + \rmi N_{\mathrm{k}}\eta} \right\rangle}}
\nonumber \\
& + \rme^{-\rmi N_{\mathrm{k}} \eta \mathrm{I\!R}(\alpha_x)}
{{\left| {\uparrow} \right\rangle}} \otimes {{\left| {\alpha_x - \rmi N_{\mathrm{k}}\eta} \right\rangle}} \big)
\otimes {{\left| {\alpha_y} \right\rangle}}\end{aligned}$$ where $\mathrm{I\!R}(\alpha)$ denotes the real part of the coherent state parameter $\alpha$.
Interferometers based on SDKs have been proposed [@PoyatosPRA96] to measure the Sagnac effect, and have recently been implemented in a 1D non-Sagnac geometry to measure temperature over a wide dynamic range [@JohnsonPRL15]. However, for a Sagnac gyroscope, the second displacement need not be spin-dependent and can therefore be implemented as a simple trap center shift. Electrode voltages can be rapidly changed to displace the trap center, an operation that has been demonstrated to couple to thousands of Fock states to perform a coherent state displacement operation [@AlonsoNatComms16]. Compared to coherent momentum transfer from a laser, voltage-driven motion of this sort can produce a larger displacement, and does so without the detrimental effects of spontaneous emission and differential AC Stark shifts associated with laser-driven gates.
A rapid shift of the trap center by a physical distance $y_{\mathrm{d}}$ along $y$ can be modeled with a displacement operator $$\hat{D}_y\!\big[ \textstyle
-\frac{y_{\mathrm{d}}}{2 y_0} \displaystyle \big] {{\left| {\alpha_y} \right\rangle}} =
\big| \alpha_y - \textstyle
\frac{y_{\mathrm{d}}}{2 y_0} \displaystyle \big\rangle$$ where $y_0 = x_0 \equiv \sqrt{\hbar/2 m \omega}$ and we will refrain from writing phase terms that for this protocol are global.
Rotation-induced phase
======================
The effect of rotation in this system can be described in either the non-rotating frame (the ion’s frame) or the rotating reference frame (the apparatus frame). We choose the former, which means that the rotation manifests itself as change in the direction of the kicks in steps (\[Stepv\]) and (\[Stepvi\]). A constant rotation rate $\Omega$ about the positive $z$-axis of the apparatus will shift the angles of these kicks by $$\theta = \Omega \Delta t = \Omega M \frac{2 \pi}{\omega}.\label{thetaDef}$$ This transforms the displacement operators according to $$\hat{D}^{\prime} = \rme^{-\rmi \theta \hat{J}_z}\hat{D} \,\rme^{\rmi \theta \hat{J}_z}$$ and the state of the ion after step (\[Stepvi\]) is
$$\begin{aligned}
\fl {{\left| {\psi_{\mathrm{\ref{Stepvi}}}} \right\rangle}} = &\textstyle \frac{1}{\sqrt{2}} \displaystyle
\Big( \rme^{\rmi \delta/2}
{{\left| {\downarrow} \right\rangle}} \otimes \big| \alpha_x + \rmi N_{\mathrm{k}}\eta(1 - \cos
\theta) - \textstyle
\frac{y_{\mathrm{d}}}{2 x_0} \displaystyle \sin \theta \big\rangle
\nonumber \\
& \;\; \otimes \big| \alpha_y - \textstyle
\frac{y_{\mathrm{d}}}{2 y_0} \displaystyle(1 - \cos
\theta) - \rmi N_{\mathrm{k}} \eta \sin \theta \big\rangle \nonumber \\
& + \rme^{-\rmi \delta/2}
{{\left| {\uparrow} \right\rangle}} \otimes \big| \alpha_x - \rmi N_{\mathrm{k}}\eta(1 - \cos
\theta) - \textstyle
\frac{y_{\mathrm{d}}}{2 x_0} \displaystyle \sin \theta \big\rangle
\nonumber \\
& \;\; \otimes \big| \alpha_y - \textstyle
\frac{y_{\mathrm{d}}}{2 y_0} \displaystyle(1 - \cos
\theta) + \rmi N_{\mathrm{k}} \eta \sin \theta \big\rangle \Big)\label{psivi}\end{aligned}$$
where the relative phase ($\delta$) is given by $$\begin{aligned}
\delta &=& 2 N_{\mathrm{k}} \eta \Big( \textstyle \frac{y_{\mathrm{d}}}{2
x_0} \displaystyle (1 + \cos \theta )\sin \theta + \textstyle \frac{y_{\mathrm{d}}}{2
y_0} \displaystyle (1 - \cos \theta )\sin \theta \nonumber \\
&& \;\;\;\;\;\;\;+ \mathrm{I\!R}(\alpha_x)( 1 - \cos \theta) -
\mathrm{I\!R}(\alpha_y)\sin \theta \Big).\end{aligned}$$
![Trajectory of an ion in $x$ phase space in the interaction picture with respect to the harmonic oscillation. The ion’s starting coordinates are indicated by a circle, and red and blue curves represent the trajectory for the two spin states. A freely-evolving coherent state in this “rotating frame” (rotating in phase space, as opposed to real space) appears stationary; the trajectories shown are induced by the displacement operators. For small rotations ($\theta \ll 1$), the area enclosed in this phase space is the Sagnac phase (\[SagnacExpanded\]) for an ion that starts at position $y=0$.[]{data-label="RotatingFrameFigure"}](figure2.eps)
(\[psivi\]) shows that this protocol leaves residual entanglement between the spin and motion in both $x$ and $y$. Using ${{\left| {\mu_{i,\mathrm{f}}(\theta)} \right\rangle}}$ to denote the final motional states in $x$ and $y$ for the parts of the wavefunction that are associated with spin state $i \in \{ \uparrow,\downarrow\}$ in (\[psivi\]), the overlap is $$\big\langle \mu_{\downarrow,\mathrm{f}}(\theta) \big|
\mu_{\uparrow,\mathrm{f}}(\theta) \big\rangle =
\rme^{-2(2N_{\mathrm{k}}\eta \sin\frac{\theta}{2})^2} \rme^{-\rmi \delta^{\prime}}$$ where the first term comes from the imperfect state overlap (which is confined entirely to momentum space) and the second is a pure phase term called the *overlap phase* $\delta^{\prime}$: $$\delta^{\prime} \equiv 2 N_{\mathrm{k}}\eta \big(
\mathrm{I\!R}(\alpha_x)(1 - \cos \theta) - \mathrm{I\!R}(\alpha_y)\sin
\theta \big).$$ Residual entanglement between spin and motion will reduce the contrast of the interferometer, and it is the sum of $\delta$ and $\delta^{\prime}$ that contributes the phase shift that is measured using this protocol. However, as we show below, the phase shift that is measured is unaffected by the initial ion temperature, and there is no requirement that this device be operated in the Lamb-Dicke regime.
Readout
=======
In order to measure the rotation-induced phase ($\delta +
\delta^{\prime}$), step (\[Stepvii\]) applies a second Ramsey zone with a controllable phase shift $\phi$, yielding $$\begin{aligned}
\fl {{\left| {\psi_{\mathrm{\ref{Stepvii}}}} \right\rangle}} = &\textstyle \frac{1}{2}
\displaystyle \Big( \rme^{\rmi \delta/2} \left( \rme^{-\rmi \phi}
{{\left| {\uparrow} \right\rangle}} + {{\left| {\downarrow} \right\rangle}} \right) \otimes
{{\left| {\mu_{\downarrow,\mathrm{f}}(\theta)} \right\rangle}} \nonumber \\
& + \rme^{-\rmi
\delta/2} \left( {{\left| {\uparrow} \right\rangle}} - \rme^{\rmi \phi} {{\left| {\downarrow} \right\rangle}}
\right) \otimes {{\left| {\mu_{\uparrow,\mathrm{f}}(\theta)} \right\rangle}}
\Big). \label{psivii}\end{aligned}$$ This step maps the motional phase onto the internal state of the ion, which would then measured using standard fluorescence techniques. The probability of measuring, for instance, spin up $({{\left| {\uparrow} \right\rangle}})$ is given by $$\mathcal{P}(\uparrow, \theta, \phi) = \!\!\int \!\! \mathrm{d}^2
\alpha_x \mathrm{d}^2 \alpha_y\,
P(\alpha_x) P(\alpha_y) \langle \psi_{\mathrm{\ref{Stepvii}}}
{{\left| {\uparrow} \right\rangle \!\!\left\langle {\uparrow} \right|}} \psi_{\mathrm{\ref{Stepvii}}} \rangle\label{MotionTrace}$$ where $$\begin{aligned}
\fl \langle \psi_{\mathrm{\ref{Stepvii}}}
{{\left| {\uparrow} \right\rangle \!\!\left\langle {\uparrow} \right|}} \psi_{\mathrm{\ref{Stepvii}}} \rangle
= \frac{1}{2} + \frac{1}{2} \rme^{-2(2N_{\mathrm{k}}\eta
\sin\frac{\theta}{2})^2} \nonumber \\
\times \cos \left(\!\phi -
\frac{A(\alpha_y)}{\pi x_0^2} \sin \theta - 4 N_{\mathrm{k}}\eta x_0^2
\mathrm{I\!R}(\alpha_x)(1\! - \!\cos \theta )\!
\right)\label{Preadout}\end{aligned}$$ and $P(\alpha_j)$ is the Glauber-Sudarshan $P$-representation describing the (potentially mixed) initial motional state in the coherent state basis. $A(\alpha_y)$ is the classical, geometric area of the ellipse enclosed by the ion trajectories in the $x,y$ plane, which depends upon the initial position in $y$ via $y_i = 2 y_0 \mathrm{I\!R}(\alpha_y)$: $$A(\alpha_y) \equiv \pi \,2 x_0 N_{\mathrm{k}}\eta \, (y_{\mathrm{d}} -
2 y_0\mathrm{I\!R}(\alpha_y)). \label{Area}$$ In \[AreaAppendix\], we show that a semiclassical derivation agrees with this quantum calculation. If we expand the argument of the cosine in (\[Preadout\]) to first order in $\theta$, we see that it simplifies to $\phi - \Phi$, where $$\Phi = \frac{A(\alpha_y)}{\pi x_0^2}\theta = 2 \pi \frac{2 m c^2}{h
c^2} (2 M A(\alpha_y)) \Omega. \label{SagnacExpanded}$$ This is identifiable as the Sagnac phase shift (\[SagnacPhase\]) with an effective area of $A_{\mathrm{eff}} = 2 M A(\alpha_y)$ since the ion encloses the ellipse area $A(\alpha_y)$ twice each period for $M$ periods. (\[SagnacExpanded\]) also provides some insight into the origin of the scale factor of this interferometer: the rotation angle $\theta$ is effectively “amplified” by a gain factor of $A(\alpha_y)/\pi x_0^2$, the ratio of the enclosed area to the area of the ground-state wavefunction. This gain factor is the angular momentum of the ion’s motion divided by $\hbar$, and the interferometer can therefore be thought of as a generalized atomic or nuclear spin gyroscope with a very large effective spin.
Finite temperature
==================
For an ion that is pre-cooled to the motional ground state along $y$ ($\alpha_y=0$), (\[SagnacExpanded\]) gives precisely the desired outcome (\[SagnacPhase\]) for the trapped ion gyroscope. For an ion that is initially in a thermal state with mean phonon occupation numbers $\bar{n}_x = \bar{n}_y\equiv \bar{n}$, (\[MotionTrace\]) can be used to calculate the probability of measuring spin up: $$\begin{aligned}
\mathcal{P}(\uparrow, \theta, \phi)& =& \frac{1}{2} + \frac{1}{2} \rme^{-(4N_{\mathrm{k}}\eta
\sin\frac{\theta}{2})^2(\bar{n} + \frac{1}{2})} \nonumber \\
&&\times \cos \left(\phi - \frac{A(0)}{\pi x_0^2}\sin \theta \right), \label{ThermalStateSpinUp}\end{aligned}$$ which is valid to all orders in $\theta$. The effect of finite temperature is a reduction in the contrast of the interference, but does not produce a phase shift of the signal. However, since the exponent in (\[ThermalStateSpinUp\]) is proportional to $\sin^2(\theta/2)$ and the Sagnac phase shift is proportional to $\sin(\theta)$, the free evolution time ($\Delta t$ in (\[thetaDef\])) can be chosen to satisfy $$\sin^2\left( \frac{\theta}{2} \right) \ll 16 N_{\mathrm{k}}^2 \eta^2
\left( \bar{n} + \frac{1}{2} \right)$$ and the interferometer can be operated at essentially full contrast, even at high temperature. There is therefore no requirement that the ion be cooled to the Lamb-Dicke regime, and as we estimate below, Doppler cooling should be sufficient for full-contrast operation.
Magnetic field effects
======================
Since the ion is moving while it is accumulating rotation-induced phase, a nonzero magnetic field will give rise to a Lorentz force on the moving monopole. For a magnetic field in the $z$-direction, this will cause the ion’s orbit to precess in the $x,y$ plane, which will lead to a false rotation signal. Specifically, the magnetically-induced rotation rate ($\Omega_{\mathrm{m}}$) can be found [@SakuraiPRD80] by equating the Lorentz and Coriolis forces for a uniform, static magnetic field in the $z$-direction ($\mathbf{B}
= B_z \mathbf{\hat{z}}$): $$2 m \Omega_{\mathrm{m}} ( \mathbf{v} \times \mathbf{\hat{z}}) =
e B_z (\mathbf{v} \times \mathbf{\hat{z}}).$$ This rotation rate is half the cyclotron frequency $\omega_{\mathrm{cyc}}= e B/m = 2 \Omega_{\mathrm{m}}$, and the precession angle this will produce is boosted up by the gain factor of $A(0)/\pi x_0^2$ to give a magnetically-induced phase shift of $$\Phi_{\mathrm{m}} \equiv \Delta t\, \Omega_{\mathrm{m}}
\frac{A(0)}{\pi x_0^2} =
\Delta t \frac{2}{\hbar} \left(e \frac{\omega}{2 \pi} \right) A(0) B_z.$$ This phase shift can be interpreted as the dynamical phase from the Zeeman shift of the ion’s motional magnetic moment, $$\mu_{\mathrm{m}} \equiv \frac{1}{2}\frac{\partial \left( \hbar
\frac{\Phi_{\mathrm{m}}}{\Delta t} \right)}{\partial B_z} = I A(0)$$ where $I\equiv e\omega/2 \pi$ is the current from the ion’s motion. This matches the classical expression for the magnetic moment of a current loop of area $A(0)$.
Non-harmonic corrections
========================
The analysis we have presented has thus far assumed a perfectly harmonic potential. We can find the first-order phase correction for small non-harmonic terms of the potential by treating these terms as a perturbation and integrating over the unperturbed trajectories. Assuming the potential remains separable the formulas below hold for each axis. Let us write the general potential as $$V=\frac{1}{2}m\omega^2 x_l^2\left(\frac{x^2}{x_l^2}+C_3
\frac{x^3}{x_l^3}+ C_4 \frac{x^4}{x_l^4}+\cdots\right)$$ where $x_l$ is a length scale for the amplitude of the ion’s motion and the $\{C_i\}$ are dimensionless numbers assumed to be much smaller than one. With a harmonic trajectory $x(t)=x_l \sin(\omega t+\phi)$, only even $i$ terms are non-zero. Integrating over $M$ orbits gives $$\begin{aligned}
\Delta\phi&=\frac{1}{\hbar}\int_0^{2\pi M/\omega}\mathrm{d}t\, \frac{1}{2}m\omega^2 x_l^2\sum_{i\geq 3}{C_i\sin^i(\omega t+\phi)}\\
&=m\omega x_l^2\frac{3\pi M}{8\hbar}C_4+\cdots \end{aligned}$$
Performance
===========
Once the evolution time has been fixed, the sensitivity of the trapped ion gyroscope can be written $$\mathcal{S} = \frac{1}{2 N_{\mathrm{k}} \Delta k \, y_{\mathrm{d}}\sqrt{\Delta t}}.$$
Since this is independent of the trap frequency $\omega$ (we assume $M\! \gg\! 1$ and can therefore be chosen essentially arbitrarily), the trapped ion gyroscope can be operated in a relatively low-frequency trap as compared to typical traps for applications requiring resolved sideband operations. This provides the practical advantage of making the non-adiabatic operations easier to achieve with high fidelity in a fixed time. It also permits the use of a trap whose electrodes are far apart and far from the ion, which will suppress surface-induced heating and patch charge perturbations and improve harmonicity for a fixed absolute length scale.
We also note that the performance of this rotation sensor is independent of the mass of the ion, and depends essentially only on the wavelength of the laser used to drive the SDKs. We will estimate parameters for ${}^{171}\mathrm{Yb}^+$, which was used for the first demonstrations of spin-dependent kicks [@MizrahiPRL13], but estimates for other species will be similar in magnitude.
For the hyperfine clock-state qubit in ${}^{171}\mathrm{Yb}^+$, stimulated Raman transitions can be driven by a tripled vanadate laser at $4 \pi/\Delta k= 355 \mbox{ nm}$ [@CampbellPRL10] with $N_{\mathrm{k}} = 100$ [@MonroePrivateCommunication]. Since this qubit has a demonstrated coherence time exceeding $1000 \mbox{ s}$ [@FiskIEEE97], a free-evolution time of $\Delta t = 1\mbox{ s}$ should be straightforward to achieve. For a (secular) trap frequency of $\omega/2 \pi = 10 \mbox{ kHz}$, a trap displacement of $y_{\mathrm{d}} = 100 \mbox{ }\mu\mbox{m}$ would correspond to $\langle n \rangle \approx 9 \times 10^5$ phonons, where displacements corresponding to $\langle n \rangle \approx 10^4$ have already been demonstrated [@AlonsoNatComms16]. Cooling $\mathrm{Yb}^+$ to the Doppler limit ($T_{\mathrm{D}} = \hbar \gamma/ 2 k_{\mathrm{B}}$) in such a trap will result in an interference contrast of 85% for the (sidereal) rotation rate of the earth $\Omega_{\mathrm{e}} \approx 73
\mbox{ }\mu\mbox{rad}/\mbox{s}$. A trapped ion gyroscope operated with these parameters would have a scale factor of $\partial \Phi /
\partial \Omega = 52 \mbox{ rad/}\Omega_{\mathrm{e}}$ and a sensitivity of $\mathcal{S} = 1.4 \times 10^{-6} \mbox{
rad/s/}\sqrt{\mbox{Hz}}$. Improvements in the numbers of SDKs or the distance of coherent trap displacements would make this competitive with cold atom interferometers that use large numbers of atoms.
For an interferometer using the parameters discussed above, the magnetically-induced rotation rate per unit field is $\Omega_{\mathrm{m}}/ (2 \pi B_z)= 4.5 \mbox{ Hz}/\mbox{G}$. The associated motional magnetic moment is $\mu_{\mathrm{m}} = 1.1
\mu_{\mathrm{B}}$, where $\mu_{\mathrm{B}}$ is the Bohr magneton. Since the magnetic field stabilization required to combat this systematic only needs to be applied to a small volume ($\ll 1 \mbox{
cm}^3$), this magnetic sensitivity resembles the effect of using a Zeeman-sensitive qubit, and many of the technical difficulties associated with this have been overcome in various trapped ion quantum information processing experiments [@RusterArXiv].
In the limit where the magnetically-induced rotation rate is much slower than the (secular) trap frequency ($\Omega_{\mathrm{m}} \ll
\omega$), the ion’s motion is in elliptical orbits of fixed area whose orientation slowly rotates. Too much rotation will reduce the contrast of the interference signal since the kicks in step (\[Stepv\]) and (\[Stepvi\]) will not efficiently close the interferometer loop. However, trapped ions have also been demonstrated as superb magnetic field sensors, and it seems likely that with a periodic measurement of a stationary ion’s Zeeman splitting, a well-controlled field could be applied to cancel this effect.
Likewise, magnetic rotation could be leveraged to cancel the contrast reduction associated with high actual rotation rates (the exponential factor in (\[Preadout\])). In this “closed-loop mode,” the magnetic field needed to cancel the rotation would become the output signal for the interferometer, and low-resolution rotation sensors could be incorporated to feed forward the magnetic field needed to keep the interferometer contrast maximized and on the steepest part of a fringe.
Discussion
==========
As compared to free-flight matter-wave interferometers, the trapped ion device provides many practical advantages. First, the physical size of the interferometer can be compact while still retaining a large effective interferometer area by using multiple orbits. Second, since the ion wavepacket re-combines in space twice per trap period, this interferometer can be interrogated over a wide dynamic range of free-evolution times. Fast rotation rates, which can be problematic in neutral atom systems if the wavepackets don’t re-combine or leave the interferometry region, can be compensated by applying uniform magnetic fields. The operational mode could be to actively stabilize the fringes with an applied field, which becomes the readout signal. There is also no need to keep multiple optical beam paths interferometrically (relatively) stable since the only steps that are sensitive to a laser phase (the SDKs, steps (\[Stepii\]) and (\[Stepvi\])) are driven by the same laser with its beam traversing the same optical path. In addition, by using single ion wavepackets which travel the same average trajectory, only in opposite directions, we eliminate spatially varying systematics. Finally, free-flight interferometers have sensitivities to accelerations and the atomic beam velocities, whereas the scale factor for the ion trap interferometer depends only on the momentum kicks and trap displacement.
Another advantage of using trapped ions instead of neutral atoms for matter-wave interferometry is the potential to leverage the advances in trapped ion quantum information processing to produce sub-shot-noise scaling of the sensitivity with ion number. For example, a collection of $N_{\mathrm{I}}$ ions could be prepared in step (\[Stepi\]) in a GHZ spin state, $${{\left| {\psi} \right\rangle}} = \textstyle \frac{1}{\sqrt{N_{\mathrm{I}}}} \displaystyle
\left({{\left| {\downarrow \downarrow \downarrow \cdots \downarrow} \right\rangle}} +
{{\left| {\uparrow \uparrow \uparrow \cdots \uparrow} \right\rangle}} \right), \label{GHz}$$ and the same protocol could be used as for the single ion to accumulate phase, but with the resolution (and sensitivity) enhanced by a factor of $N_{\mathrm{I}}$. These states (\[GHz\]) have been created for as many as $N_{\mathrm{I}} = 14$ ions [@MonzPRL11], and multiple groups are actively pursuing various ways to scale up the size of entangled trapped ion systems.
We thank Amar Vutha, Chris Monroe, Dana Anderson, and Kale Johnson for helpful discussions. W.C.C. Acknowledges support from the U.S. Army Research Office under award W911NF-15-1-0261 and University of California Office of the President’s Research Catalyst Award No. CA-15-327861. P.H. acknowledges support from the University of California Office of the President’s Research Catalyst Award No. CA-16-377655.
References {#references .unnumbered}
==========
Area formula {#AreaAppendix}
============
We show using semiclassical derivation that the area enclosed by the interferometer is insensitive to the ion’s initial position and momentum in $x$ and initial momentum in $y$. The area for a trajectory enclosed by a periodic trajectory $\mathbf{r}(t)$ is given by the path integral $$\begin{aligned}
\mathbf{A} &=
\frac{1}{2}\oint\mathbf{r}\times\,\mathrm{d}\mathbf{r}=\frac{1}{2}\int_0^T
\mathrm{d}t\,\mathbf{r}(t)\times\mathbf{v}(t)\\
& =\frac{1}{2m}\int_0^T \mathrm{d}t\, \mathbf{J}=\frac{\mathbf{J}\,T}{2m}\end{aligned}$$ where $T \equiv 2 \pi/\omega$ is the period and $\mathbf{J}$ is the angular momentum. A momentum kick, $\Delta\mathbf{p}$, at the start of the trajectory (and taken to be along $x$) changes the angular momentum by $\Delta\mathbf{J}_{\mathrm{SDK}} = \mathbf{r}(0)\times\Delta\mathbf{p}
= \mathbf{r}_\perp(0)\times\Delta\mathbf{p}$, where $\mathbf{r}_\perp(0)$ is the component of the initial displacement perpendicular to the direction of the momentum kick (which we will take to be the $y$-direction). The trap displacement in $y$ then changes the angular momentum by $\Delta
\mathbf{J}_{\mathrm{d}} = -y_{\mathrm{d}} \mathbf{\hat{y}} \times
\mathbf{p}(0)$. We are interested in the areas for two trajectories with initial momentum kicks, $\pm \Delta \mathbf{p} =\pm
N_{\mathrm{k}}\hbar\Delta k \,\mathbf{\hat{x}}$, in opposite directions in $x$. The area enclosed by the interferometer is the difference between these areas, taken over half a motional period (since the interferometer closes at time $T/2$): $$A =
\left|\frac{\Delta\mathbf{J}\,\frac{T}{2}}{2m}\right|= \pi
\frac{\Delta p}{m \omega} (y_{\mathrm{d}} - r_{\perp} (0) ),$$ which agrees with (\[Area\]). We see that the area difference depends only on the initial displacement perpendicular to the SDK direction and the size of the kick. The formula holds for circular, elliptical, or even straight line trajectories and is also independent of the initial momentum of the particle.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings – Noetherian semirings, natural numbers – are proper. Properness matters as the equivalence problem for weighted automata over a semiring which is proper and finitely and effectively presented is decidable. Milius generalised the notion of properness from a semiring to a functor. As a consequence, a semiring is proper if and only if its associated “cubic functor” is proper. Moreover, properness of a functor renders soundness and completeness proofs for axiomatizations of equivalent behaviour.
In this paper we provide a method for proving properness of functors, and instantiate it to cover both the known cases and several novel ones: (1) properness of the semirings of positive rationals and positive reals, via properness of the corresponding cubic functors; and (2) properness of two functors on (positive) convex algebras. The latter functors are important for axiomatizing trace equivalence of probabilistic transition systems. Our proofs rely on results that stretch all the way back to Hilbert and Minkowski.
author:
- Ana Sokolova
- Harald Woracek
title: Proper Semirings and Proper Convex Functors
---
Introduction {#sec-Intro}
============
In this paper we deal with algebraic categories and deterministic weighted automata functors on them. Such categories are the target of generalized determinization [@silva.bonchi.bonsangue.rutten:2010; @silva.sokolova:2011; @jacobs.silva.sokolova:2015] and enable coalgebraic modelling beyond sets. For example, non-deterministic automata, weighted, or probabilistic ones are coalgebraically modelled over the categories of join-semilattices, semimodules for a semiring, and convex sets, respectively. Moreover, expressions for axiomatizing behavior semantics often live in algebraic categories.
In order to prove completeness of such axiomatizations, the common approach [@silva:2010; @bonsangue.milius.silva:2011; @silva.sokolova:2011] is to prove finality of a certain object in a category of coalgebras over an algebraic category. Proofs are significantly simplified if it suffices to verify finality only w.r.t. coalgebras carried by free finitely generated algebras, as those are the coalgebras that result from generalized determinization.
In recent work, Milius [@milius:2017] proposed the notion of a proper functor on an algebraic category that provides a sufficient condition for this purpose. This notion is an extension of the notion of a proper semiring introduced by Esik and Maletti [@esik.maletti:2010]: A semiring is proper if and only if its “cubic" functor is proper. A cubic functor is a functor $\mathbb S \times (-)^A$ where $A$ is a finite alphabet and $\mathbb S$ is a free algebra with a single generator in the algebraic category. Cubic functors model deterministic weighted automata which are models of determinizations of non-deterministic and probabilistic transition systems.
Properness is the property that for any two states that are behaviourally equivalent in coalgebras with free finitely generated carriers, there is a zig-zag of homomorphisms (called a chain of simulations in the original works on weighted automata and proper semirings) that identifies the two states and whose nodes are all carried by free finitely generated algebras.
Even though the notion of properness is relatively new for a semiring and very new for a functor, results on properness of semirings can be found in more distant literature as well. Here is a brief history, to the best of our knowledge:
- The Boolean semiring was proven to be proper in [@bloom.esik:1993].
- Finite commutative ordered semirings were proven to be proper in [@esik.kuich:2001 Theorem 5.1]. Interestingly, the proof provides a zig-zag with at most seven intermediate nodes.
- Any euclidean domain and any skew field were proven proper in [@beal.lombardy.sakarovitch:2005 Theorem 3]. In each case the zig-zag has two intermediate nodes.
- The semiring of natural numbers $\mathbb N$, the Boolean semiring $\mathbb B$, the ring of integers $\mathbb Z$ and any skew field were proven proper in [@beal.lombardy.sakarovitch:2006 Theorem 1]. Here, all zig-zag were spans, i.e., had a single intermediate node with outgoing arrows.
- Noetherian semirings were proven proper in [@esik.maletti:2010 Theorem 4.2], commutative rings also in [@esik.maletti:2010 Corollary 4.4], and finite semirings as well in [@esik.maletti:2010 Corollary 4.5], all with a zig-zag being a span. Moreover, the tropical semiring is not proper, as proven in [@esik.maletti:2010 Theorem 5.4].
Having properness of a semiring, together with the property of the semiring being finitely and effectively presentable, yields decidability of the equivalence problem (decidability of trace equivalence) for weighted automata.
In this paper, motivated by the wish to prove properness of a certain functor ${{\ensuremath{\widehat F}}}$ on convex algebras used for axiomatizing trace semantics of probabilistic systems in [@silva.sokolova:2011], as well as by the open questions stated in [@milius:2017 Example 3.19], we provide a framework for proving properness. We instantiate this framework on known cases like Noetherian semirings and $\mathbb N$ (with a zig-zag that is a span), and further prove new results of properness:
- The semirings $\mathbb Q_+$ and $\mathbb R_+$ of non-negative rationals and reals, respectively, are proper. The shape of the zig-zag is a span as well.
- The functor $[0,1] \times (-)^A$ on is proper, again the zig-zag being a span.
- The functor ${{\ensuremath{\widehat F}}}$ on ${{\text{\tt{PCA}}}}$ is proper. This proof is the most involved, and interestingly, provides the only case where the zig-zag is not a span: it contains three intermediate nodes of which the middle one forms a span.
Our framework requires a proof of so-called *extension* and *reduction lemmas* in each case. While the extension lemma is a generic result that covers all cubic functors of interest, the reduction lemma is in all cases a nontrivial property intrinsic to the algebras under consideration. For the semiring of natural numbers it is a consequence of a result that we trace back to Hilbert; for the case of convex algebra $[0,1]$ the result is due to Minkowski. In the case of ${{\ensuremath{\widehat F}}}$, we use Kakutani’s set-valued fixpoint theorem.
It is an interesting question for future work whether these new properness results may lead to new complete axiomatizations of expressions for certain weighted automata.
The organization of the rest of the paper is as follows. In Section \[sec-ProperFunctors\] we give some basic definitions and introduce the semirings, the categories, and the functors of interest. Section \[sec-PropernessCubic\] provides the general framework as well as proofs of properness of the cubic functors. Section \[sec-SubcubicFunctor\]–Section \[sec-PropernessSubcubic\] lead us to properness of ${{\ensuremath{\widehat F}}}$ on . For space reasons, we present the ideas of proofs and constructions in the main paper and defer all detailed proofs to the appendix.
#### Acknowledgements.
We thank the anonymous reviewers for many valuable comments, in particular for reminding us of a categorical property that shortened the proof of the extension lemma (the proofs of Lemma \[lem:ext-lem-1\] and Lemma \[lem:ext-lem-2\] in Appendix \[app:B\]).
Proper functors {#sec-ProperFunctors}
===============
We start with a brief introduction of the basic notions from algebra and coalgebra needed in the rest of the paper, as well as the important definition of proper functors [@milius:2017]. We refer the interested reader to [@Rut00:tcs; @JR96:eatcs; @jacobs:2017] for more details. We assume basic knowledge of category theory, see e.g. [@maclane:1998] or Appendix \[sec:app-basics\].
Let ${\text{\tt{C}}}$ be a category and $F$ a ${\text{\tt{C}}}$-endofunctor. The category ${{\sf Coalg}({F})}$ of *$F$-coalgebras* is the category having as objects pairs $(X,c)$ where $X$ is an object of ${\text{\tt{C}}}$ and $c$ is a ${\text{\tt{C}}}$-morphism from $X$ to $FX$, and as morphisms $f\colon(X,c)\to(Y,d)$ those ${\text{\tt{C}}}$-morphisms from $X$ to $Y$ that make the diagram on the right commute.
[r]{}[.1]{}
All base categories ${\text{\tt{C}}}$ in this paper will be *algebraic categories*, i.e., categories ${{\text{\tt{Set}}}^{T}}$ of Eilenberg-Moore algebras of a finitary monad [^1] in ${{\text{\tt{Set}}}}$. Hence, all base categories are concrete with forgetful functor that is identity on morphisms.
In such categories behavioural equivalence [@Kurz00:thesis; @Wol00:cmcs; @Staton11] can be defined as follows. Let $(X,c)$ and $(Y,d)$ be $F$-coalgebras and let $x\in X$ and $y\in Y$. Then $x$ and $y$ are *behaviourally equivalent*, and we write $x\sim y$, if there exists an $F$-coalgebra $(Z,e)$ and ${{\sf Coalg}({F})}$-morphisms $f\colon (X,c)\to(Z,e)$, $g\colon(Y,d)\to(Z,e)$, with $f(x)=g(y)$. $$\xymatrix@C=40pt{
(X,c) \ar[r]^f
& (Z,e)
\save[]+<0pt,-13pt>*\txt{$\scriptstyle f(x)=g(y)$}\restore
& (Y,d) \ar[l]_g
}$$ If there exists a final coalgebra in ${{\sf Coalg}({F})}$, and all functors considered in this paper will have this property, then two elements are behaviourally equivalent if and only if they have the same image in the final coalgebra. If we have a *zig-zag diagram* in ${{\sf Coalg}({F})}$ $$\label{zig-zag}
\xymatrix@R=0.7em@C=9pt{
(X,c) \ar[rd]^{f_1} && (Z_2,e_2) \ar[ld]_{f_2} \ar[rd]^{f_3} &&
\cdots \ar[ld]_{f_4} \ar[rd]^{f_{2n-1}} && (Y,d) \ar[ld]_{f_{2n}} &
\\
& (Z_1,e_1)
&& (Z_3,e_1)
&& (Z_{2n-1},e_1)
&
}$$ which relates $x$ with $y$ in the sense that there exist elements $z_{2k}\in Z_{2k}$, $k=1,\ldots,n-1$, with (setting $z_0=x$ and $z_{2n}=y$) $$f_{2k}(z_{2k})=f_{2k-1}(z_{2k-2}),\quad k=1,\ldots, n
,$$ then $x\sim y$.
We now recall the notion of a proper functor, introduced by Milius [@milius:2017] which is central to this paper. It is very helpful for establishing completeness of regular expressions calculi, cf. [@milius:2017 Corollary 3.17].
\[Proper\] Let $T\colon{{\text{\tt{Set}}}}\to{{\text{\tt{Set}}}}$ be a finitary monad with unit $\eta$ and multiplication $\mu$. A ${{\text{\tt{Set}}}^{T}}$-endofunctor $F$ is *proper*, if the following statement holds.
For each pair $(TB_1,c_1)$ and $(TB_2,c_2)$ of $F$-coalgebras with $B_1$ and $B_2$ finite sets, and each two elements $b_1\in B_1$ and $b_2\in B_2$ with $\eta_{B_1}(b_1)\sim\eta_{B_2}(b_2)$, there exists a zig-zag in ${{\sf Coalg}({F})}$ which relates $\eta_{B_1}(b_1)$ with $\eta_{B_2}(b_2)$, and whose nodes $(Z_j,e_j)$ all have free and finitely generated carrier.
This notion generalizes the notion of a proper semiring introduced by Esik and Maletti in [@esik.maletti:2010 Definition 3.2], cf. [@milius:2017 Remark 3.10].
\[fg\] In the definition of properness the condition that intermediate nodes have free *and* finitely generated carrier is necessary for nodes with incoming arrows (the nodes $Z_{2k-1}$ in ). For the intermediate nodes with outgoing arrows ($Z_{2k}$ in ), it is enough to require that their carrier is finitely generated. This follows since every $F$-coalgebra with finitely generated carrier is the image under an $F$-coalgebra morphism of an $F$-coalgebra with free and finitely generated carrier.
Moreover, note that zig-zags which start (or end) with incoming arrows instead of outgoing ones, can also be allowed since a zig-zag of this form can be turned into one of the form by appending identity maps.
Some concrete monads and functors {#some-concrete-monads-and-functors .unnumbered}
---------------------------------
We deal with the following base categories.
- The category ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$ of semimodules over a semiring $\mathbb S$ induced by the monad $T_{\mathbb S}$ of finitely supported maps into $\mathbb S$, see, e.g., [@manes.mulry:2007 Example 4.2.5].
- The category of positively convex algebras induced by the monad of finitely supported subprobability distributions, see, e.g., [@doberkat:2006; @doberkat:2008] and [@pumpluen:1984].
For $n\in\mathbb N$, the free algebra with $n$ generators in ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$ is the direct product $\mathbb S^n$, and in it is the $n$-simplex $\Delta^n=\{(\xi_1,\ldots,\xi_n)\mid \xi_j\geq 0,\sum_{j=1}^n\xi_j\leq 1\}$.
Concerning semimodule-categories, we mainly deal with the semirings $\mathbb N$, $\mathbb Q_+$, and $\mathbb R_+$, and their ring completions $\mathbb Z$, $\mathbb Q$, and $\mathbb R$. For these semirings the categories of $\mathbb S$-semimodules are
- ${{\text{\tt{CMON}}}}$ of commutative monoids for $\mathbb N$,
- ${{\text{\tt{AB}}}}$ of abelian groups for $\mathbb Z$,
- ${{\text{\tt{CONE}}}}$ of convex cones for $\mathbb R_+$,
- ${\text{$\mathbb Q$-{\text{\tt{VEC}}}}}$ and ${\text{$\mathbb R$-{\text{\tt{VEC}}}}}$ of vector spaces over the field of rational and real numbers, respectively, for $\mathbb Q$ and $\mathbb R$.
We consider the following functors, where $A$ is a fixed finite alphabet. Recall that we use the term *cubic functor* for the functor $T1\times(-)^A$ where $T$ is a monad on ${{\text{\tt{Set}}}}$. We chose the name since $T1\times(-)^A$ assigns to objects $X$ a full direct product, i.e., a full cube.
- The *cubic functor* ${{\ensuremath{F_{\,{\mathbb S}}}}}$ on ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$, i.e., the functor acting as $$\begin{aligned}
& {{\ensuremath{F_{\,{\mathbb S}}}}}X=\mathbb S\times X^A\text{ for }X\text{ object of }{\text{$\mathbb S$-{\text{\tt{SMOD}}}}},
\\
& {{\ensuremath{F_{\,{\mathbb S}}}}}f=\operatorname{id}_{\mathbb S}\times(f\circ -)
\text{ for }f\colon X\to Y\text{ morphism of }{\text{$\mathbb S$-{\text{\tt{SMOD}}}}}.
\end{aligned}$$ The underlying ${{\text{\tt{Set}}}}$ functors of cubic functors are also sometimes called deterministic-automata functors, see e.g. [@jacobs.silva.sokolova:2015], as their coalgebras are deterministic weighted automata with output in the semiring.
- The *cubic functor* ${{\ensuremath{F_{\,{[0,1]}}}}}$ on ${{\text{\tt{PCA}}}}$, i.e., the functor ${{\ensuremath{F_{\,{[0,1]}}}}}X=[0,1]\times X^A$ and ${{\ensuremath{F_{\,{[0,1]}}}}}f=\operatorname{id}_{[0,1]}\times(f\circ -)$.
- A *subcubic convex functor* ${{\ensuremath{\widehat F}}}$ on whose action will be introduced in Definition \[Ghat-def\].[^2] The name originates from the fact that ${{\ensuremath{\widehat F}}}X$ is a certain convex subset of ${{\ensuremath{F_{\,{[0,1]}}}}}X$ and that ${{\ensuremath{\widehat F}}}f=({{\ensuremath{F_{\,{[0,1]}}}}}f)|_{{{\ensuremath{\widehat F}}}X}$ for $f\colon X\to Y$.
Cubic functors are liftings of ${{\text{\tt{Set}}}}$-endofunctors, in particular, they preserve surjective algebra homomorphisms. It is easy to see that also the functor ${{\ensuremath{\widehat F}}}$ preserves surjectivity, cf. Lemma \[GhSurj\] (Appendix \[app:D\]) This property is needed to apply the work of Milius, cf. [@milius:2017 Assumptions 3.1].
\[ProperSemiring\] We can now formulate precisely the connection between proper semirings and proper functors mentioned after Definition \[Proper\]. A semiring $\mathbb S$ is proper in the sense of [@esik.maletti:2010], if and only if for every finite input alphabet $A$ the cubic functor ${{\ensuremath{F_{\,{\mathbb S}}}}}$ on ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$ is proper.
We shall interchangeably think of direct products as sets of functions or as sets of tuples. Taking the viewpoint of tuples, the definition of ${{\ensuremath{F_{\,{\mathbb S}}}}}f$ reads as $$({{\ensuremath{F_{\,{\mathbb S}}}}}f)\big((o,(x_a)_{a\in A})\big)=\big(o,(f(x_a))_{a\in A}\big),\quad
o\in\mathbb S,\ x_a\in X\text{ for }a\in A
.$$ A coalgebra structure $c\colon X\to{{\ensuremath{F_{\,{\mathbb S}}}}}X$ writes as $$c(x)=\big({{c}_{\text{\rm o}}}(x),({{c}_{a}}(x))_{a\in A}\big),\quad x\in X
,$$ and we use ${{c}_{\text{\rm o}}}:X\to\mathbb S$ and ${{c}_{a}}:X\to X$ as generic notation for the components of the map $c$. More generally, we define $c_w\colon X\to X$ for any word $w\in A^*$ inductively as ${{c}_{\varepsilon}}=\operatorname{id}_X$ and ${{c}_{wa}}={{c}_{a}}\circ{{c}_{w}},\ w\in A^*,a\in A$.
The map from a coalgebra $(X,c)$ into the final ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebra, the *trace map*, is then given as $\operatorname{tr}_c(x)=\big(({{c}_{\text{\rm o}}}\circ{{c}_{w}})(x)\big)_{w\in A^*}$ for $ x\in X$. Behavioural equivalence for cubic functors is the kernel of the trace map.
Properness of cubic functors {#sec-PropernessCubic}
============================
Our proofs of properness in this section and in Section \[sec-PropernessSubcubic\] below start from the following idea. Let $\mathbb S$ be a semiring, and assume we are given two ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebras which have free finitely generated carrier, say $(\mathbb S^{n_1},c_1)$ and $(\mathbb S^{n_2},c_2)$. Moreover, assume $x_1\in\mathbb S^{n_1}$ and $x_2\in\mathbb S^{n_2}$ are two elements having the same trace. For $j=1,2$, let $d_j\colon \mathbb S^{n_1}\times\mathbb S^{n_2} \to {{\ensuremath{F_{\,{\mathbb S}}}}}(\mathbb S^{n_1}\times\mathbb S^{n_2})$ be given by $$d_j(y_1, y_2) = \Big({{c_j}_{\text{\rm o}}}(y_j),(({{c_1}_{a}}(y_1),{{c_2}_{a}}(y_2)))_{a\in A}\Big).$$ Denoting by $\pi_j\colon \mathbb S^{n_1}\times\mathbb S^{n_2}\to\mathbb S^{n_j}$ the canonical projections, both sides of the following diagram separately commute. $$\xymatrix@C=15pt@R=10pt@M=7pt{
\mathbb S^{n_1} \ar[dd]_{c_1}
&& \mathbb S^{n_1}\times\mathbb S^{n_2} \ar[ll]_{\pi_1} \ar[rr]^{\pi_2}
\ar@/_15pt/[dd]_{d_1} \ar@/^15pt/[dd]^{d_2}
&& \mathbb S^{n_2} \ar[dd]_{c_2}
\\
&& \neq &&
\\
{{\ensuremath{F_{\,{\mathbb S}}}}}\mathbb S^{n_1}
&& {{\ensuremath{F_{\,{\mathbb S}}}}}(\mathbb S^{n_1}\times\mathbb S^{n_2}) \ar[ll]_{{{\ensuremath{F_{\,{\mathbb S}}}}}\pi_1} \ar[rr]^{{{\ensuremath{F_{\,{\mathbb S}}}}}\pi_2}
&& {{\ensuremath{F_{\,{\mathbb S}}}}}\mathbb S^{n_2}
}$$ However, in general the maps $d_1$ and $d_2$ do not coincide.
The next lemma contains a simple observation: there exists a subsemimodule $Z$ of $\mathbb S^{n_1}\times\mathbb S^{n_2}$, such that the restrictions of $d_1$ and $d_2$ to $Z$ coincide and turn $Z$ into an ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebra.
\[Prod\] Let $Z$ be the subsemimodule of $\mathbb S^{n_1}\times\mathbb S^{n_2}$ generated by the pairs $({{c_1}_{w}}(x_1),{{c_2}_{w}}(x_2))$ for $w \in A^*$. Then $d_1|_Z=d_2|_Z$ and $d_j(Z)\subseteq{{\ensuremath{F_{\,{\mathbb S}}}}}(Z)$.
The significance of Lemma \[Prod\] in the present context is that it leads to the diagram (we denote $d=d_j|_Z$) $$\xymatrix@C=15pt@R=35pt@M=7pt{
\mathbb S^{n_1} \ar[d]_{c_1}
&& Z \ar[ll]_{\pi_1} \ar[rr]^{\pi_2} \ar[d]^d
\save[]+<3pt,10pt>*\txt{\begin{rotate}{90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,27pt>*\txt{$\mathbb S^{n_1}\!\!\times\mathbb S^{n_2}$}\restore
&& \mathbb S^{n_2} \ar[d]_{c_2}
\\
{{\ensuremath{F_{\,{\mathbb S}}}}}\mathbb S^{n_1}
&& {{\ensuremath{F_{\,{\mathbb S}}}}}Z \ar[ll]_{{{\ensuremath{F_{\,{\mathbb S}}}}}\pi_1} \ar[rr]^{{{\ensuremath{F_{\,{\mathbb S}}}}}\pi_2}
\save[]+<-3pt,-10pt>*\txt{\begin{rotate}{-90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,-29pt>*\txt{$\mathbb S\!\!\times(\mathbb S^{n_1}\!\!\times\mathbb S^{n_2})^A$}\restore
&& {{\ensuremath{F_{\,{\mathbb S}}}}}\mathbb S^{n_2}
}$$ In other words, it leads to the zig-zag in [[Coalg]{}([${{\ensuremath{F_{\,{\mathbb S}}}}}$]{})]{} $$\label{ZZ}
\xymatrix@C=40pt{
(\mathbb S^{n_1},c_1) & (Z,d) \ar[l]_{\mkern20mu\pi_1} \ar[r]^{\pi_2\mkern20mu}
& (\mathbb S^{n_2},c_2)
}$$ This zig-zag relates $x_1$ with $x_2$ since $(x_1,x_2)\in Z$. If it can be shown that $Z$ is always finitely generated, it will follow that ${{\ensuremath{F_{\,{\mathbb S}}}}}$ is proper.
Let $\mathbb S$ be a Noetherian semiring, i.e., a semiring such that every $\mathbb S$-subsemimodule of some finitely generated $\mathbb S$-semimodule is itself finitely generated. Then $Z$ is, as an $\mathbb S$-subsemimodule of $\mathbb S^{n_1}\times\mathbb S^{n_2}$ finitely generated. Hence, we reobtain the result [@esik.maletti:2010 Theorem 4.2] of Esik and Maletti.
\[Noetherian\] Every Noetherian semiring is proper.
Our first main result is Theorem \[CubProp\] below, where we show properness of the cubic functors ${{\ensuremath{F_{\,{\mathbb S}}}}}$ on , for $\mathbb S$ being one of the semirings $\mathbb N$, $\mathbb Q_+$, $\mathbb R_+$, and of the cubic functor ${{\ensuremath{F_{\,{[0,1]}}}}}$ on . The case of ${{\ensuremath{F_{\,{\mathbb N}}}}}$ is known from [@beal.lombardy.sakarovitch:2006 Theorem 4] [^3], the case of ${{\ensuremath{F_{\,{[0,1]}}}}}$ is stated as an open problem in [@milius:2017 Example 3.19].
\[CubProp\] The cubic functors ${{\ensuremath{F_{\,{\mathbb N}}}}}$, ${{\ensuremath{F_{\,{\mathbb Q_+}}}}}$, ${{\ensuremath{F_{\,{\mathbb R_+}}}}}$, and ${{\ensuremath{F_{\,{[0,1]}}}}}$ are proper.
In fact, for any two coalgebras with free finitely generated carrier and any two elements having the same trace, a zig-zag with free and finitely generated nodes relating those elements can be found, which is a span (has a single intermediate node with outgoing arrows).
The proof proceeds via relating to the Noetherian case. It always follows the same scheme, which we now outline. Observe that the ring completion of each of $\mathbb N$, $\mathbb Q_+$, $\mathbb R_+$, is Noetherian (for the last two it actually is a field), and that $[0,1]$ is the positive part of the unit ball in $\mathbb R$.
*Step 1. The extension lemma:* We use an extension of scalars process to pass from the given category to an associated category E with a Noetherian ring $\mathbb E$. This is a general categorical argument.
To unify notation, we agree that $\mathbb S$ may also take the value $[0,1]$, and that $T_{[0,1]}$ is the monad of finitely supported subprobability distributions giving rise to the category .
$\mathbb S$ $\mathbb N$ $\mathbb Q_+$ $\mathbb R_+$ $[0,1]$
------------- ------------- --------------- --------------- ---------
N () ()
E Z () Q (Q) R (R) R (R)
For the formulation of the extension lemma, recall that the starting category is the Eilenberg-Moore category of the monad ${T_{\mathbb{S}}}$ and the target category E is the Eilenberg-Moore category of ${T_{\mathbb{E}}}$. We write $\eta_{\mathbb S}$ and $\mu_{\mathbb S}$ for the unit and multiplication of ${T_{\mathbb{S}}}$ and analogously for ${T_{\mathbb{E}}}$. We have ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$, via the inclusion monad morphism $\iota\colon {T_{\mathbb{S}}}\Rightarrow {T_{\mathbb{E}}}$ given by $\iota_X(u) = u$, as $\eta_{\mathbb E} = \iota {\mathrel{\circ}}\eta_{\mathbb S}$ and $\mu_{\mathbb E} {\mathrel{\circ}}\iota\iota =\iota{\mathrel{\circ}}\mu_{\mathbb S}$ where $\iota\iota \stackrel{\text{def}}{=}{T_{\mathbb{E}}}\iota{\mathrel{\circ}}\iota \stackrel{\text{nat.}}{=} \iota {\mathrel{\circ}}{T_{\mathbb{S}}}\iota$. Recall that a monad morphism $\iota\colon T_\mathbb S \to T_\mathbb E$ defines a functor $M_\iota\colon {{\text{\tt{Set}}}^{T_\mathbb E}} \to {{\text{\tt{Set}}}^{T_\mathbb S}}$ which maps a $T_\mathbb E$-algebra $(X, \alpha_X)$ to $(X, \iota_X{\mathrel{\circ}}\alpha_X)$ and is identity on morphisms. Obviously, $M_\iota$ commutes with the forgetful functors $U_\mathbb S: {{\text{\tt{Set}}}^{T_\mathbb S}} \to {{\text{\tt{Set}}}}$ and $U_\mathbb E: {{\text{\tt{Set}}}^{T_\mathbb E}} \to {{\text{\tt{Set}}}}$, i.e., $U_\mathbb S {\mathrel{\circ}}M_\iota = U_\mathbb E$.
Let $(X, \alpha_X) \in {{\text{\tt{Set}}}^{{T_{\mathbb{S}}}}}$ and $(Y, \alpha_Y) \in {{\text{\tt{Set}}}^{{T_{\mathbb{E}}}}}$ where ${T_{\mathbb{S}}}$ and ${T_{\mathbb{E}}}$ are monads with ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$ via $\iota\colon {T_{\mathbb{S}}}\Rightarrow {T_{\mathbb{E}}}$. A ${{\text{\tt{Set}}}}$-arrow $h\colon X \to Y$ is a ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$-homomorphism from $(X,\alpha_X)$ to $(Y,\alpha_Y)$ if and only if the following diagram commutes (in ${{\text{\tt{Set}}}}$) $$\xymatrix@R=0.7em{
{T_{\mathbb{S}}}X \ar[rr]^{\iota h} \ar[d]_{\alpha_X}&& {T_{\mathbb{E}}}Y \ar[d]^{\alpha_Y}\\
X \ar[rr]^{h}&& Y
}$$ where $\iota h$ denotes the map $\iota h \stackrel{\text{def}}{=} {T_{\mathbb{E}}}h{\mathrel{\circ}}\iota_X \stackrel{\text{nat.}}{=} \iota_Y {\mathrel{\circ}}{T_{\mathbb{S}}}h$. In other words, a ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$-homomorphism from $(X,\alpha_X)$ to $(Y,\alpha_Y)$ is a morphism in ${{\text{\tt{Set}}}^{T_\mathbb S}}$ from $(X,\alpha_X)$ to $M(Y,\alpha_Y)$.
Now we can formulate the extension lemma.
\[prop:ext-lem\] For every ${{\ensuremath{F_{\,{\mathbb{S}}}}}}$-coalgebra ${T_{\mathbb{S}}}B \stackrel{c}{\to} {{\ensuremath{F_{\,{\mathbb{S}}}}}}({T_{\mathbb{S}}}B)$ with free finitely generated carrier ${T_{\mathbb{S}}}B$ for a finite set $B$, there exists an ${{\ensuremath{F_{\,{\mathbb{E}}}}}}$-coalgebra ${T_{\mathbb{E}}}B \stackrel{\tilde c}{\to} {{\ensuremath{F_{\,{\mathbb{E}}}}}}({T_{\mathbb{E}}}B)$ with free finitely generated carrier ${T_{\mathbb{E}}}B$ such that $$\xymatrix@R=0.7em{
{T_{\mathbb{S}}}B \ar[rr]^{\iota_B} \ar[d]_{c}&& {T_{\mathbb{E}}}B \ar[d]^{\tilde c}\\
{{\ensuremath{F_{\,{\mathbb{S}}}}}}({T_{\mathbb{S}}}B) \ar[rr]^{\iota_1 \times (\iota_B)^A}&& {{\ensuremath{F_{\,{\mathbb{E}}}}}}({T_{\mathbb{E}}}B)
}$$ where the horizontal arrows ($\iota_B$ and $\iota_1 \times \iota_B^A$) are ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$-homomorphisms, and moreover they both amount to inclusion.
*Step 2. The basic diagram:* Let $n_1,n_2\in\mathbb N$, let $B_j$ be the $n_j$-element set consisting of the canonical basis vectors of $\mathbb E^{n_j}$, and set $X_j=T_{\mathbb S}B_j$. Assume we are given ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebras $(X_1,c_1)$ and $(X_2,c_2)$, and elements $x_j\in X_j$ with $\operatorname{tr}_{c_1}x_1=\operatorname{tr}_{c_2}x_2$.
The extension lemma provides ${{\ensuremath{F_{\,{\mathbb E}}}}}$-coalgebras $(\mathbb E^{n_j},\tilde c_j)$ with $\tilde c_j|_{X_j}=c_j$. Clearly, $\operatorname{tr}_{\tilde c_1}x_1=\operatorname{tr}_{\tilde c_2}x_2$. Using the zig-zag diagram in [[Coalg]{}([$F_{\mathbb E}$]{})]{} and appending inclusion maps, we obtain what we call the *basic diagram*. In this diagram all solid arrows are arrows in E, and all dotted arrows are arrows in . The horizontal dotted arrows denote the inclusion maps, and $\pi_j$ are the restrictions to $Z$ of the canonical projections. $$\xymatrix@C=15pt@R=35pt@M=7pt{
X_1 \ar@{.>}[r] \ar@{.>}[d]_{c_1} & \mathbb E^{n_1} \ar[d]_{\tilde c_1}
&& Z \ar[ll]_{\pi_1} \ar[rr]^{\pi_2} \ar[d]^d
\save[]+<3pt,10pt>*\txt{\begin{rotate}{90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,27pt>*\txt{$\mathbb E^{n_1}\!\!\times\mathbb E^{n_2}$}\restore
&& \mathbb E^{n_2} \ar[d]_{\tilde c_2} & X_2 \ar@{.>}[l] \ar@{.>}[d]^{c_2}
\\
{{\ensuremath{F_{\,{\mathbb S}}}}}X_1 \ar@{.>}[r] & {{\ensuremath{F_{\,{\mathbb E}}}}}\mathbb E^{n_1}
&& {{\ensuremath{F_{\,{\mathbb E}}}}}Z \ar[ll]_{{{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1} \ar[rr]^{{{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2}
\save[]+<-3pt,-10pt>*\txt{\begin{rotate}{-90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,-29pt>*\txt{$\mathbb E\!\!\times(\mathbb E^{n_1}\!\!\times\mathbb E^{n_2})^A$}\restore
&& {{\ensuremath{F_{\,{\mathbb E}}}}}\mathbb E^{n_2} & {{\ensuremath{F_{\,{\mathbb S}}}}}X_2 \ar@{.>}[l]
}$$ Commutativity of this diagram yields $d\big(\pi_j^{-1}(X_j)\big)\subseteq({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_j)^{-1}\big({{\ensuremath{F_{\,{\mathbb S}}}}}X_j)$ for $j=1,2$. Now we observe the following properties of cubic functors.
\[CubProperties\] We have ${{\ensuremath{F_{\,{\mathbb E}}}}}X\cap{{\ensuremath{F_{\,{\mathbb S}}}}}Y={{\ensuremath{F_{\,{\mathbb S}}}}}(X\cap Y)$. Moreover, if $Y_j\subseteq X_j$, then $({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_1)\cap({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_2)
={{\ensuremath{F_{\,{\mathbb S}}}}}(Y_1\times Y_2)$.
Using this, yields $$\begin{aligned}
d\big(Z\cap(X_1\times X_2)\big)\subseteq &\,
{{\ensuremath{F_{\,{\mathbb E}}}}}Z\cap({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1)^{-1}\big({{\ensuremath{F_{\,{\mathbb S}}}}}X_1)
\cap({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2)^{-1}\big({{\ensuremath{F_{\,{\mathbb S}}}}}X_2)
\\
= &\, {{\ensuremath{F_{\,{\mathbb E}}}}}Z\cap{{\ensuremath{F_{\,{\mathbb S}}}}}(X_1\times X_2)={{\ensuremath{F_{\,{\mathbb S}}}}}\big(Z\cap(X_1\times X_2)\big)
.
\end{aligned}$$ This shows that $Z\cap(X_1\times X_2)$ becomes an ${{\ensuremath{F_{\,{\mathbb S}}}}}$-coalgebra with the restriction $d|_{Z\cap(X_1\times X_2)}$. Again referring to the basic diagram, we have the following zig-zag in [[Coalg]{}([$F_{\mathbb S}$]{})]{} (to shorten notation, denote the restrictions of $d,\pi_1,\pi_2$ to $Z\cap(X_1\times X_2)$ again as $d,\pi_1,\pi_2$): $$\label{4}
\xymatrix@C=40pt{
(X_1,c_1) & \big(Z\cap(X_1\times X_2),d\big)
\ar[l]_{\pi_1\mkern50mu} \ar[r]^{\mkern60mu\pi_2}
& (X_2,c_2)
}$$ This zig-zag relates $x_1$ with $x_2$ since $(x_1,x_2)\in Z\cap(X_1\times X_2)$.
*Step 3. The reduction lemma:* In view of the zig-zag , the proof of Theorem \[CubProp\] can be completed by showing that $Z\cap(X_1\times X_2)$ is finitely generated as an algebra in . Since $Z$ is a submodule of the finitely generated module $\mathbb E^{n_1}\times\mathbb E^{n_2}$ over the Noetherian ring $\mathbb E$, it is finitely generated as an $\mathbb E$-module. The task thus is to show that being finitely generated is preserved when reducing scalars.
This is done by what we call the *reduction lemma*. Contrasting the extension lemma, the reduction lemma is not a general categorical fact, and requires specific proof in each situation.
\[RedLem\] Let $n_1,n_2\in\mathbb N$, let $B_j$ be the set consisting of the $n_j$ canonical basis vectors of $\mathbb E^{n_j}$, and set $X_j=T_{\mathbb S}B_j$. Moreover, let $Z$ be an $\mathbb E$-submodule of $\mathbb E^{n_1}\times\mathbb E^{n_2}$. Then $Z\cap(X_1\times X_2)$ is finitely generated as an algebra in .
A subcubic convex functor {#sec-SubcubicFunctor}
=========================
Recall the following definition from [@silva.sokolova:2011 p.309].
\[Ghat-def\] We introduce a functor ${{\ensuremath{\widehat F}}}\colon {{\text{\tt{PCA}}}}\to{{\text{\tt{PCA}}}}$.
1. Let $X$ be a . Then $$\begin{aligned}
{{\ensuremath{\widehat F}}}X=\Big\{ (o,\phi)\in[0,1] & \times X^A\mid
\\
& \exists\,n_a\in\mathbb N{{.\kern3pt}}\exists\,p_{a,j}\in[0,1],x_{a,j}\in X\text{ for }j=1,\ldots,n_a,a\in A{{.\kern3pt}}\\
& o+\sum_{a\in A}\sum_{j=1}^{n_a}p_{a,j}\leq 1,\ \phi(a)=\sum_{j=1}^{n_a}p_{a,j}x_{a,j} \Big\}.
\end{aligned}$$
2. Let $X,Y$ be s, and $f\colon X\to Y$ a convex map. Then ${{\ensuremath{\widehat F}}}f\colon {{\ensuremath{\widehat F}}}X\to{{\ensuremath{\widehat F}}}Y$ is the map ${{\ensuremath{\widehat F}}}f=\operatorname{id}_{[0,1]}\times(f\circ -)$.
For every $X$ we have ${{\ensuremath{\widehat F}}}X\subseteq{{\ensuremath{F_{\,{[0,1]}}}}}X$, and for every $f\colon X\to Y$ we have ${{\ensuremath{\widehat F}}}f=({{\ensuremath{F_{\,{[0,1]}}}}}f)|_{{{\ensuremath{\widehat F}}}X}$. For this reason, we think of ${{\ensuremath{\widehat F}}}$ as a *subcubic functor*.
The definition of [[$\widehat F$]{}]{} can be simplified.
\[Ghat-simple\] Let $X$ be a , then $$\begin{aligned}
{{\ensuremath{\widehat F}}}X=\Big\{ (o,f)\in[0,1]\times X^A\mid\ &
\exists\,p_a\in[0,1],x_a\in X\text{ for }a\in A{{.\kern3pt}}\\
& o+\sum_{a\in A}p_a\leq 1,\ f(a)=p_ax_a \Big\}.
\end{aligned}$$
From this representation it is obvious that [[$\widehat F$]{}]{} is monotone in the sense that
- If $X_1\subseteq X_2$, then ${{\ensuremath{\widehat F}}}X_1\subseteq{{\ensuremath{\widehat F}}}X_2$.
- If $f_1\colon X_1\to Y_1,f_2\colon X_2\to Y_2$ with $X_1\subseteq X_2,Y_1\subseteq Y_2$ and $f_2|_{X_1}=f_1$, then ${{\ensuremath{\widehat F}}}f_2|_{{{\ensuremath{\widehat F}}}X_1}={{\ensuremath{\widehat F}}}f_1$.
Note that [[$\widehat F$]{}]{} does not preserve direct products.
For a $X$ whose carrier is a compact subset of a euclidean space, ${{\ensuremath{\widehat F}}}X$ can be described with help of a geometric notion, namely using the Minkowksi functional of $X$. Before we can state this fact, we have to make a brief digression to explain this notion and its properties.
\[Minko\] Let $X\subseteq\mathbb R^n$ be a . The *Minkowski functional* of $X$ is the map ${\ensuremath{\mu_{X}}}\colon \mathbb R^n\to[0,\infty]$ defined as ${\ensuremath{\mu_{X}}}(x)=\inf\{t>0\mid x\in tX\}$, where the infimum of the empty set is understood as $\infty$.
Minkowski functionals, sometimes also called *gauge*, are a central and exhaustively studied notion in convex geometry, see, e.g., [@rudin:1991 p.34] or [@rockafellar:1970 p.28].
We list some basic properties whose proof can be found in the mentioned textbooks.
1. ${\ensuremath{\mu_{X}}}(px)=p{\ensuremath{\mu_{X}}}(x)$ for $x\in\mathbb R^n,p\geq 0$,
2. ${\ensuremath{\mu_{X}}}(x+y)\leq {\ensuremath{\mu_{X}}}(x)+{\ensuremath{\mu_{X}}}(y)$ for $x,y\in\mathbb R^n$,
3. ${\ensuremath{\mu_{X\cap Y}}}(x)=\max\{{\ensuremath{\mu_{X}}}(x),{\ensuremath{\mu_{Y}}}(x)\}$ for $x\in\mathbb R^n$.
4. If $X$ is bounded, then ${\ensuremath{\mu_{X}}}(x)=0$ if and only if $x=0$.
The set $X$ can almost be recovered from ${\ensuremath{\mu_{X}}}$.
5. ${\displaystyle\{x\in\mathbb R^n\mid{\ensuremath{\mu_{X}}}(x)<1\}\subseteq X\subseteq\{x\in\mathbb R^n\mid{\ensuremath{\mu_{X}}}(x)\leq 1\}}$.
6. If $X$ is closed, equality holds in the second inclusion of 5.
7. Let $X,Y$ be closed. Then $X\subseteq Y$ if and only if ${\ensuremath{\mu_{X}}}\geq{\ensuremath{\mu_{Y}}}$.
\[MinkoExa\] As two simple examples, consider the $n$-simplex $\Delta^n\subseteq\mathbb R^n$ and a convex cone $C\subseteq\mathbb R^n$. Then (here $\geq$ denotes the product order on $\mathbb R^n$) $${\ensuremath{\mu_{\Delta^n}}}(x)=
\begin{cases}
\sum_{j=1}^n\xi_j &\hspace*{-3mm},\quad x=(\xi_1,\ldots,\xi_n)\geq 0,
\\
\infty &\hspace*{-3mm},\quad \text{otherwise}.
\end{cases}
\qquad
{\ensuremath{\mu_{C}}}(x)=
\begin{cases}
0 &\hspace*{-3mm},\quad x\in C,
\\
\infty &\hspace*{-3mm},\quad \text{otherwise}.
\end{cases}$$ Observe that $\Delta^n=\{x\in\mathbb R^n\mid {\ensuremath{\mu_{\Delta^n}}}(x)\leq 1\}$.
Another illustrative example is given by general pyramids in a euclidean space. This example will play an important role later on.
\[Pyramid\] For $u\in\mathbb R^n$ consider the set $$X=\big\{x\in\mathbb R^n\mid x\geq 0\text{ and }(x,u)\leq 1\big\}
,$$ where $(\cdot,\cdot)$ denotes the euclidean scalar product on $\mathbb R^n$. The set $X$ is intersection of the cone $\mathbb R_+^n$ with the half-space given by the inequality $(x,u)\leq 1$, hence it is convex and contains $0$. Thus $X$ is a .
Let us first assume that $u$ is strictly positive, i.e., $u\geq 0$ and no component of $u$ equals zero. Then $X$ is a pyramid (in $2$-dimensional space, a triangle).
(0,0) – (0,4-5/12) – (11-12/5,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); (-1,4) – (11,-1); (0,0) – (1,1\*12/5); at (0.6,2.5) [[$u$]{}]{}; at (2.9,1.1) [[$X$]{}]{}; at (-2.4,4.5) [[$\scriptstyle(x,u)=1$]{}]{};
The $n$-simplex $\Delta^n$ is of course a particular pyramid. It is obtained using the vector $u=(1,\ldots,1)$.
The Minkowski functional of the pyramid $X$ associated with $u$ is $${\ensuremath{\mu_{X}}}(x)=
\begin{cases}
(x,u) &\hspace*{-3mm},\quad x\geq 0,
\\
\infty &\hspace*{-3mm},\quad \text{otherwise}.
\end{cases}$$ Write $u=\sum_{j=1}^n\alpha_je_j$, where $e_j$ is the $j$-th canonical basis vector, and set $y_j=\frac 1{\alpha_j}e_j$. Clearly, $\{y_1,\ldots,y_n\}$ is linearly independent. Each vector $x=\sum_{j=1}^n\xi_je_j$ can be written as $x=\sum_{j=1}^n(\xi_j\alpha_j)y_j$, and this is a subconvex combination if and only if $\xi_j\geq 0$ and $\sum_{j=1}^n\xi_j\alpha_j\leq 1$, i.e., if and only if $x\in X$. Thus $X$ is generated by $\{y_1,\ldots,y_n\}$ as a .
The linear map given by the diagonal matrix made up of the $\alpha_j$’s induces a bijection of $X$ onto $\Delta^n$, and maps the $y_j$’s to the corner points of $\Delta^n$. Hence, $X$ is free with basis $\{y_1,\ldots,y_n\}$.
If $u$ is not strictly positive, the situation changes drastically. Then $X$ is not finitely generated as a , because it is unbounded whereas the subconvex hull of a finite set is certainly bounded.
(0,0) – (0,2) – (16,4) – (16,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); (-1,2-1/8) – (16,4); (0,0) – (-3/8,3); at (-0.8,3.2) [[$u$]{}]{}; at (9.5,1.7) [[$X$]{}]{}; at (14.8,5.0) [[$\scriptstyle(x,u)=1$]{}]{};
Now we return to the functor [[$\widehat F$]{}]{}.
\[Ghat-Minko\] Let $X\subseteq\mathbb R^n$ be a , and assume that $X$ is compact. Then $${{\ensuremath{\widehat F}}}X=\Big\{(o,\phi)\in\mathbb R\times(\mathbb R^n)^A\mid\
o\geq 0,\ o+\sum_{a\in A}{\ensuremath{\mu_{X}}}(\phi(a))\leq 1 \Big\}
.$$
In the following we use the elementary fact that every convex map has a linear extension.
\[LinExt\] Let $V_1,V_2$ be vector spaces, let $X\subseteq V_1$ be a , and let $c\colon X\to V_2$ be a convex map. Then $c$ has a linear extension $\tilde c\colon V_1\to V_2$. If $\operatorname{span}X=V_1$, this extension is unique.
Rescaling in this representation of ${{\ensuremath{\widehat F}}}X$ leads to a characterisation of [[$\widehat F$]{}]{}-coalgebra maps. We give a slightly more general statement; for the just said, use $X=Y$.
\[Ghat-Minko-Coalg\] Let $X,Y\subseteq\mathbb R^n$ be s, and assume that $X$ and $Y$ are compact. Further, let $c\colon X\to\mathbb R_+\times(\mathbb R^n)^A$ be a convex map, and let $\tilde c\colon \mathbb R^n\to\mathbb R\times(\mathbb R^n)^A$ be a linear extension of $c$.
Then $c(X)\subseteq{{\ensuremath{\widehat F}}}Y$, if and only if $$\label{2}
{{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}({{\tilde c}_{a}}(x))\leq{\ensuremath{\mu_{X}}}(x),\quad x\in\mathbb R^n
.$$
An extension theorem for ${{\ensuremath{\widehat F}}}$-coalgebras {#sec-ExtensionTheorem}
=================================================================
In this section we establish an extension theorem for ${{\ensuremath{\widehat F}}}$-coalgebras. It states that an ${{\ensuremath{\widehat F}}}$-coalgebra, whose carrier has a particular geometric form, can, under a mild additional condition, be embedded into an ${{\ensuremath{\widehat F}}}$-coalgebra whose carrier is free and finitely generated.
\[ExEx\] Let $(X,c)$ be an ${{\ensuremath{\widehat F}}}$-coalgebra whose carrier $X$ is a compact subset of a euclidean space $\mathbb R^n$ with $\Delta^n\subseteq X\subseteq\mathbb R_+^n$. Assume that the output map ${{c}_{\text{\rm o}}}$ does not vanish on invariant coordinate hyperplanes in the sense that ($e_j$ denotes again the $j$-th canonical basis vector in $\mathbb R^n$) $$\label{nv}
\begin{aligned}
& \nexists\,I\subseteq\{1,\ldots,n\}{{.\kern3pt}}\\
& I\neq\emptyset,\quad
{{c}_{\text{\rm o}}}(e_j)=0,j\in I,
\quad
{{c}_{a}}(e_j)\subseteq\operatorname{span}\{e_i\mid i\in I\},a\in A,j\in I.
\end{aligned}$$ Then there exists an ${{\ensuremath{\widehat F}}}$-coalgebra $(Y,d)$, such that $X\subseteq Y\subseteq\mathbb R_+^n$, the inclusion map $\iota\colon X\to Y$ is a ${{\sf Coalg}({{{\ensuremath{\widehat F}}}})}$-morphism, and $Y$ is the subconvex hull of $n$ linearly independent vectors (in particular, $Y$ is free with $n$ generators).
The idea of the proof can be explained by geometric intuition. Say, we have an ${{\ensuremath{\widehat F}}}$-coalgebra $(X,c)$ of the stated form, and let $\tilde c\colon \mathbb R^n\to\mathbb R\times(\mathbb R^n)^A$ be the linear extension of $c$ to all of $\mathbb R^n$, cf. Lemma \[LinExt\].
(0,0) – (0,2) to \[out=20,in=0\] (3,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); at (0,1.5) [[$\bullet$]{}]{}; at (-0.5,1.5) [[$e_2$]{}]{}; at (1.5,0) [[$\bullet$]{}]{}; at (1.5,-0.5) [[$e_1$]{}]{}; at (14,1.4) [${{\ensuremath{\widehat F}}}X$]{}; (0,2) to \[out=20,in=0\] (3,0); at (1.4,0.9) [[$X$]{}]{}; (3,1.4) to \[out=20,in=160\] (13,1.4); at (8,3) [$c=\tilde c|_X$]{};
Remembering that pyramids are free and finitely generated, we will be done if we find a pyramid $Y\supseteq X$ which is mapped into ${{\ensuremath{\widehat F}}}Y$ by $\tilde c$:
(0,0) – (0,2) to \[out=20,in=0\] (3,0) – (0,0); (0,0) – (0,4-5/12) – (11-12/5,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); at (0,1.5) [[$\bullet$]{}]{}; at (-0.5,1.5) [[$e_2$]{}]{}; at (1.5,0) [[$\bullet$]{}]{}; at (1.5,-0.5) [[$e_1$]{}]{}; at (14,1.4) [${{\ensuremath{\widehat F}}}X$]{}; (0,2) to \[out=20,in=0\] (3,0); at (1.4,0.9) [[$X$]{}]{}; (3,1.4) to \[out=20,in=160\] (13,1.4); at (8,3) [$c=\tilde c|_X$]{}; (-1,4) – (11,-1); at (4.2,1) [[$Y$]{}]{}; at (14,3.4) [${{\ensuremath{\widehat F}}}Y$]{}; at (14.1,2.15)
[90]{} $\subseteq$
; (2,3.4) to \[out=20,in=160\] (13,3.4); at (8,5) [$\tilde c|_Y$]{};
This task can be reformulated as follows: For each pyramid $Y_1$ containing $X$ let $P(Y_1)$ be the set of all pyramids $Y_2$ containing $X$, such that $\tilde c(Y_2)\subseteq{{\ensuremath{\widehat F}}}Y_1$. If we find $Y$ with $Y\in P(Y)$, we are done.
Existence of $Y$ can be established by applying a fixed point principle for set-valued maps. The result sufficient for our present level of generality is Kakutani’s generalisation [@kakutani:1941 Corollary] of Brouwers fixed point theorem.
Properness of [[$\widehat F$]{}]{} {#sec-PropernessSubcubic}
==================================
In this section we give the second main result of the paper.
\[GhatProp\] The functor ${{\ensuremath{\widehat F}}}$ is proper.
In fact, for each two given coalgebras with free finitely generated carrier and each two elements having the same trace, a zig-zag with free and finitely generated nodes relating those elements can be found, which has three intermediate nodes with the middle one forming a span.
We try to follow the proof scheme familiar from the cubic case. Assume we are given two ${{\ensuremath{\widehat F}}}$-coalgebras with free finitely generated carrier, say $(\Delta^{n_1},c_1)$ and $(\Delta^{n_2},c_2)$, and elements $x_1\in\Delta^{n_1}$ and $x_2\in\Delta^{n_2}$ having the same trace. Since ${{\ensuremath{\widehat F}}}\Delta^{n_j}\subseteq\mathbb R\times(\mathbb R^{n_j})^A$ we can apply Lemma \[LinExt\] and obtain ${{\ensuremath{F_{\,{\mathbb R}}}}}$-coalgebras $(\mathbb R^{n_j},\tilde c_j)$ with $\tilde c_j|_{\Delta^{n_j}}=c_j$. This leads to the basic diagram: $$\xymatrix@C=15pt@R=35pt@M=7pt{
\Delta^{n_1} \ar@{.>}[r] \ar@{.>}[d]_{c_1} & \mathbb R^{n_1} \ar[d]_{\tilde c_1}
&& Z \ar[ll]_{\pi_1} \ar[rr]^{\pi_2} \ar[d]^d
\save[]+<3pt,10pt>*\txt{\begin{rotate}{90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,27pt>*\txt{$\mathbb R^{n_1}\!\!\times\mathbb R^{n_2}$}\restore
&& \mathbb R^{n_2} \ar[d]_{\tilde c_2} & \Delta^{n_2} \ar@{.>}[l] \ar@{.>}[d]^{c_2}
\\
{{\ensuremath{\widehat F}}}\Delta^{n_1} \ar@{.>}[r] & {{\ensuremath{F_{\,{\mathbb R}}}}}\mathbb R^{n_1}
&& {{\ensuremath{F_{\,{\mathbb R}}}}}Z \ar[ll]_{{{\ensuremath{F_{\,{\mathbb R}}}}}\pi_1} \ar[rr]^{{{\ensuremath{F_{\,{\mathbb R}}}}}\pi_2}
\save[]+<-3pt,-10pt>*\txt{\begin{rotate}{-90}$\subseteq$\end{rotate}}\restore
\save[]+<2pt,-29pt>*\txt{$\mathbb R\!\!\times(\mathbb R^{n_1}\!\!\times\mathbb R^{n_2})^A$}\restore
&& {{\ensuremath{F_{\,{\mathbb R}}}}}\mathbb R^{n_2} & {{\ensuremath{\widehat F}}}\Delta^{n_2} \ar@{.>}[l]
}$$ At this point the line of argument known from the cubic case breaks: it is *not* granted that $Z\cap(\Delta^{n_1}\times \Delta^{n_2})$ becomes an ${{\ensuremath{\widehat F}}}$-coalgebra with the restriction of $d$.
The substitute for $Z\cap(\Delta^{n_1}\times \Delta^{n_2})$ suitable for proceeding one step further is given by the following lemma, where we tacitly identify $\mathbb R^{n_1}\times\mathbb R^{n_2}$ with $\mathbb R^{n_1+n_2}$.
\[GhatRestr\] We have $d(Z\cap 2\Delta^{n_1+n_2})\subseteq{{\ensuremath{\widehat F}}}(Z\cap 2\Delta^{n_1+n_2})$.
This shows that $Z\cap 2\Delta^{n_1+n_2}$ becomes an ${{\ensuremath{\widehat F}}}$-coalgebra with the restriction of $d$. Still, we cannot return to the usual line of argument: it is *not* granted that $\pi_j(Z\cap 2\Delta^{n_1+n_2})\subseteq\Delta^{n_j}$. This forces us to introduce additional nodes to produce a zig-zag in ${{\sf Coalg}({{{\ensuremath{\widehat F}}}})}$. These additional nodes are given by the following lemma. There $\operatorname{co}( -)$ denotes the convex hull.
\[GhatNodes\] Set $Y_j=\operatorname{co}(\Delta^{n_j}\cup\pi_j(Z\cap 2\Delta^{n_1+n_2}))$. Then $\tilde c_j(Y_j)\subseteq{{\ensuremath{\widehat F}}}Y_j$.
This shows that $Y_j$ becomes an ${{\ensuremath{\widehat F}}}$-coalgebra with the restriction of $\tilde c_j$. We are led to a zig-zag in ${{\sf Coalg}({{{\ensuremath{\widehat F}}}})}$: $$\xymatrix@C=25pt{
(\Delta^{n_1},c_1) \ar[r]^{\subseteq}
& (Y_1,\tilde c_1) & \big(Z\cap 2\Delta^{n_1+n_2},d\big) \ar[l]_{\pi_1\mkern40mu} \ar[r]^{\mkern40mu\pi_2}
& (Y_2,\tilde c_2) & (\Delta^{n_2},c_2) \ar[l]_{\supseteq}
}$$ This zig-zag relates $x_1$ and $x_2$ since $(x_1,x_2)\in Z\cap 2\Delta^{n_1+n_2}$.
Using Minkowski’s Theorem and the argument from Lemma \[RedLemPca\] (Appendix \[app:B\]) shows that the middle node has finitely generated carrier. The two nodes with incoming arrows are, as convex hulls of two finitely generated s, of course also finitely generated. But in general they will not be free (and this is essential, remember Remark \[fg\]). Now Theorem \[ExEx\] comes into play.
\[GhatFree\] Assume that each of $(\Delta^{n_1},c_1)$ and $(\Delta^{n_2},c_2)$ satisfies the following condition: $$\label{nv2}
\begin{aligned}
& \nexists\,I\subseteq\{1,\ldots,n\}{{.\kern3pt}}\\
& I\neq\emptyset,\
{{c_j}_{\text{\rm o}}}(e_k)=0,k\in I,
\
{{c_j}_{a}}(e_k)\subseteq\operatorname{co}(\{e_i\mid i\in I\}\cup\{0\}),a\in A,k\in I.
\end{aligned}$$ Then there exist free finitely generated s $U_j$ with $Y_j\subseteq U_j\subseteq\mathbb R_+^{n_j}$ which satisfy $\tilde c_j(U_j)\subseteq{{\ensuremath{\widehat F}}}U_j$.
This shows that $U_j$, under the additional assumption on $(\Delta^{n_j},c_j)$, becomes an ${{\ensuremath{\widehat F}}}$-coalgebra with the restriction of $\tilde c_j$. Thus we have a zig-zag in ${{\sf Coalg}({{{\ensuremath{\widehat F}}}})}$ relating $x_1$ and $x_2$ whose nodes with incoming arrows are free and finitely generated, and whose node with outgoing arrows is finitely generated: $$\xymatrix@C=25pt{
(\Delta^{n_1},c_1) \ar@{.>}[r]^{\subseteq} \ar[rd]
& (Y_1,\tilde c_1) \ar@{.>}[d]
\save[]+<6pt,-15pt>*\txt{\begin{rotate}{-90}$\scriptstyle\subseteq$\end{rotate}}\restore
& \big(Z\cap 2\Delta^{n_1+n_2},d\big) \ar@{.>}[l]_{\pi_1\mkern40mu} \ar@{.>}[r]^{\mkern40mu\pi_2} \ar[ld] \ar[rd]
& (Y_2,\tilde c_2) \ar@{.>}[d]
\save[]+<-9pt,-15pt>*\txt{\begin{rotate}{-90}$\scriptstyle\subseteq$\end{rotate}}\restore
& (\Delta^{n_2},c_2) \ar@{.>}[l]_{\supseteq} \ar[ld]
\\
& (U_1,\tilde c_1)
&& (U_2,\tilde c_2) &
}$$ Removing the additional assumption on $(\Delta^{n_j},c_j)$ is an easy exercise.
\[GhatRedLem\] Let $(\Delta^n,c)$ be an ${{\ensuremath{\widehat F}}}$-coalgebra. Assume that $I$ is a nonempty subset of $\{1,\ldots,n\}$ with $$\label{6}
{{c}_{\text{\rm o}}}(e_k)=0,\ k\in I\quad\text{and}\quad
{{c}_{a}}(e_k)\in\operatorname{co}\big(\{e_i\mid i\in I\}\cup\{0\}\big),\ a\in A,k\in I.$$ Let $X$ be the free with basis $\{e_k\mid k\in\{1,\ldots,n\}\setminus I\}$, and let $f\colon \Delta^n\to X$ be the -morphism with $$f(e_k)=
\begin{cases}
0 &\hspace*{-3mm},\quad k\in I,
\\
e_k &\hspace*{-3mm},\quad k\not\in I.
\end{cases}$$ Further, let $g\colon X\to[0,1]\times X^A$ be the -morphism with $$g(e_k)=\Big({{c}_{\text{\rm o}}}(e_k),\big(f({{c}_{a}}(e_k))\big)_{a\in A}\Big),\quad k\in\{1,\ldots,n\}\setminus I
.$$ Then $(X,g)$ is an ${{\ensuremath{\widehat F}}}$-coalgebra, and $f$ is an ${{\ensuremath{\widehat F}}}$-coalgebra morphism of $(\Delta^n,c)$ onto $(X,g)$.
\[GhatRedCor\] Let $(\Delta^n,c)$ be an ${{\ensuremath{\widehat F}}}$-coalgebra. Then there exists $k\leq n$, an ${{\ensuremath{\widehat F}}}$-coalgebra $(\Delta^k,g)$, such that $(\Delta^k,g)$ satisfies the assumption in Lemma \[GhatFree\] and such that there exists an ${{\ensuremath{\widehat F}}}$-coalgebra map $f$ of $(\Delta^n,c)$ onto $(\Delta^k,g)$.
The proof of Theorem \[GhatProp\] is now finished by putting together what we showed so far. Starting with ${{\ensuremath{\widehat F}}}$-coalgebras $(\Delta^{n_j},c_j)$ without any additional assumptions, and elements $x_j\in\Delta^{n_j}$ having the same trace, we first reduce by means of Corollary \[GhatRedCor\] and then apply Lemma \[GhatFree\]. This gives a zig-zag as required: $$\xymatrix@C=17pt@M=3pt{
(\Delta^{n_1},c_1) \ar@{.>}[d]_{\psi_1} \ar[rd]
&& \big(Z\cap 2\Delta^{k_1+k_2},d\big) \ar[ld] \ar[rd] &&
(\Delta^{n_2},c_2) \ar@{.>}[d]_{\psi_2} \ar[ld]
\\
(\Delta^{k_1},g_1) \ar@{.>}[r]
& (U_1,\tilde g_1) &
& (U_2,\tilde g_2) & (\Delta^{k_2},g_2) \ar@{.>}[l]
}$$ and completes the proof of properness of ${{\ensuremath{\widehat F}}}$.
\#1[0=]{}
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Category theory basics {#sec:app-basics}
======================
We start by recalling the basic notions of category, functor and natural transformation, so that all of the results in the paper are accessible also to non-experts.
A category ${\text{\tt{C}}}$ is a collection of objects and a collection of arrows (or morphisms) from one object to another. For every object $X \in {\text{\tt{C}}}$, there is an identity arrow $\operatorname{id}_X\colon X \to X$. For any three objects $X, Y, Z \in {\text{\tt{C}}}$, given two arrows $f\colon X \to Y$ and $g\colon Y\to Z$, there exists an arrow $g {\mathrel{\circ}}f\colon X \to Z$. Arrow composition is associative and $\operatorname{id}_X$ is neutral w.r.t. composition. The standard example is ${{\text{\tt{Set}}}}$, the category of sets and functions.
A functor $F$ from a category ${\text{\tt{C}}}$ to a category ${\text{\tt{D}}}$, notation $F\colon {\text{\tt{C}}} \to {\text{\tt{D}}}$, assigns to every object $X\in {\text{\tt{C}}}$, an object $FX \in {\text{\tt{D}}}$, and to every arrow $f\colon X \to Y$ in ${\text{\tt{C}}}$ an arrow $Ff\colon FX \to FY$ in ${\text{\tt{D}}}$ such that identity arrows and composition are preserved.
A concrete category is a category ${\text{\tt{C}}}$ equipped with a faithful functor $\mathcal U\colon {\text{\tt{C}}} \to {{\text{\tt{Set}}}}$. Intuitively, a concrete category has objects that are sets with some additional structure, e.g. algebras, and morphisms that are particular kind of functions, and $\mathcal U$ is a canonical forgetful functor. All categories that we consider are algebraic and hence concrete.
[r]{}[.1]{} @R-.5pc[ FX\^-[\_X]{}\^-[Ff]{} & FY\^-[\_Y]{}\
GX\^-[Gf]{} & GY ]{}
Let $F\colon {\text{\tt{C}}} \to {\text{\tt{D}}}$ and $G\colon {\text{\tt{C}}} \to {\text{\tt{D}}}$ be two functors. A natural transformation $\sigma\colon F \Rightarrow G$ is a family of arrows $\sigma_X\colon FX \to GX$ in ${\text{\tt{D}}}$ such that the diagram on the right commutes for all arrows $f\colon X \to Y$.
Monads and Algebras {#sec-app:monads}
-------------------
A monad is a functor $T\colon{\text{\tt{C}}} \rightarrow {\text{\tt{C}}}$ together with two natural transformations: a unit $\eta\colon \operatorname{id}_{{\text{\tt{C}}}}
\Rightarrow T$ and multiplication $\mu \colon T^{2} \Rightarrow
T$. These are required to make the following diagrams commute, for $X\in{\text{\tt{C}}}$. $$\[email protected]@R-.5pc{
T X\ar[rr]^-{\eta_{T X}}\ar@{=}[drr] & & T^{2}X\ar[d]^{\mu_X} & &
T X\ar[ll]_{T\eta_{X}}\ar@{=}[dll]
& & T^{3}X\ar[rr]^-{\mu_{T X}}\ar[d]_{T\mu_{X}}
& & T^{2}X\ar[d]^{\mu_X} \\
& & T X & &
& &
T^{2}X\ar[rr]_{\mu_X} & & T X
}$$ Given two monads $S,T$ with units and multiplications $\eta^S,\eta^T$ and $\mu^S,\mu^T$, respectively, and a natural transformation $\iota\colon S\Rightarrow T$, we say that $\iota$ is a monad morphism, and $S\leq T$ along $\iota$, if $\eta^T=\iota\circ\eta^S$ and $\iota\circ\mu^S=\mu^T\circ\iota\iota$ where $\iota\iota \stackrel{\text{def}}{=}T\iota{\mathrel{\circ}}\iota \stackrel{\text{nat.}}{=} \iota {\mathrel{\circ}}S\iota$.
We briefly describe some examples of monads on ${{\text{\tt{Set}}}}$.
- The finitely supported subprobability distribution monad ${\mathcal{D}}$ is defined, for a set $X$ and a function $f\colon X \to Y$, as $${\mathcal{D}}X
\,\,\, = \,\,\,
\{\varphi\colon X \to [0,1] \mid \sum_{x \in X} \varphi(x) \le 1,\, \operatorname{supp}(\varphi) \text{~is~finite}\}$$ and $${\mathcal{D}}f(\varphi)(y)
\,\,\, = \,\,\,
\sum\limits_{x\in f^{-1}(\{y\})} \varphi(x).$$ Here and below $\operatorname{supp}(\varphi) = \{x \in X \mid \varphi(x) \neq 0\}$. The unit of ${\mathcal{D}}$ is given by a Dirac distribution $\eta_X(x) = \delta_x = ( x \mapsto 1)$ for $x \in X$ and the multiplication by $\mu_X(\Phi)(x) = \sum\limits_{\varphi \in \operatorname{supp}(\Phi)}
\Phi(\varphi)\cdot \varphi(x)$ for $\Phi \in {\mathcal{D}}{\mathcal{D}}X$.
- For a semiring $\mathbb S$ the $\mathbb S$-valuations monad ${T_{\mathbb{S}}}$ is defined as ${T_{\mathbb{S}}}X = \{\varphi\colon X \to \mathbb S \mid \operatorname{supp}(\varphi) \text{ is finite}\}$ and on functions $f\colon X \to Y$ we have ${T_{\mathbb{S}}}f(\varphi)(y) = \sum_{x \in f^{-1}(\{y\})} \varphi(x)$. Its unit is given by $\eta_X(x) = ( x \mapsto 1)$ and multiplication by $\mu_X(\Phi)(x) = \sum_{\varphi \in \operatorname{supp}\Phi} \Phi(\varphi) \cdot \varphi(x)$ for $\Phi \in {T_{\mathbb{S}}}{T_{\mathbb{S}}}X$.
- To illustrate the connection between ${\mathcal{D}}$ and ${T_{\mathbb{S}}}$, consider yet another monad: For a semiring $\mathbb S$, and a (suitable) subset $S \subseteq \mathbb S$, the ($\mathbb S, S$)-valuations monad $T_{\mathbb S, S}$ is defined as follows. On objects it acts like $$T_{\mathbb S,S}X = \{\varphi\colon X \to \mathbb S \mid \operatorname{supp}(\varphi) \text{ is finite and }
\sum_{x \in X} \varphi(x) \in S\}$$ on functions it acts like ${T_{\mathbb{S}}}$. The unit and multiplication are defined as in ${T_{\mathbb{S}}}$. Note that ${\mathcal{D}}= T_{\mathbb R_+,[0,1]}$.
With a monad $T$ on a category ${\text{\tt{C}}}$ one associates the Eilenberg-Moore category ${\text{\tt{C}}}^{T}$ of Eilenberg-Moore algebras. Objects of ${\text{\tt{C}}}^{T}$ are pairs ${\mathbb{A}} = (A, \alpha)$ of an object $A \in {\text{\tt{C}}}$ and an arrow $\alpha\colon T A \rightarrow A$, making the first two diagrams below commute. $$\[email protected]{
A\ar@{=}[dr]\ar[r]^-{\eta_A} & T A\ar[d]^{\alpha}
&
T^{2}A\ar[d]_{\mu_A}\ar[r]^-{T\alpha} & T A\ar[d]^{\alpha}
& &
T A\ar[d]_{\alpha}\ar[r]^-{T h} & T B\ar[d]^{\beta} \\
& A
&
T A\ar[r]_-{\alpha} & A
& &
A\ar[r]_-{h} & B
}$$
A homomorphism from an algebra ${\mathbb{A}} = (A, \alpha)$ to an algebra ${\mathbb{B}} = (B, b)$ is a map $h\colon
A\rightarrow B$ in ${\text{\tt{C}}}$ between the underlying objects making the diagram above on the right commute.
From now on fix ${\text{\tt{C}}}={{\text{\tt{Set}}}}$. A free Eilenberg-Moore algebra for a monad $T$ generated by $X$ is $(TX, \mu_X)$ and we will often denote it simply by $TX$. A free finitely generated Eilenberg-Moore algebra for $T$ is an algebra $TX$ with $X$ a finite set. The diagram in the middle thus says that the map $\alpha$ is a homomorphism from $T A$ to ${\mathbb{A}}$.
Indeed, $(TX,\mu_X)$ is free on $X$ as for every $T$-algebra ${\mathbb{A}} = (A, \alpha)$ and any ${{\text{\tt{Set}}}}$-morphism $f\colon X \to A$ there is a unique ${{\text{\tt{Set}}}^{T}}$-morphism $f^\#\colon (TX,\mu_X) \to \mathbb A$ such that $f^\# {\mathrel{\circ}}\eta_A = f$ — it is easy to see that this unique extension $f^\#$ is the Kleisli extension of $f$, i.e., $f^\# = \alpha {\mathrel{\circ}}Tf$. Moreover, note that $(f^\# {\mathrel{\circ}}\eta_A)^\# = f^\#$, by the uniqueness of the extension.
Proof details for properness of cubic functors {#app:B}
==============================================
Since $\operatorname{tr}_{c_1} x_1=\operatorname{tr}_{c_2} x_2$, we have $${{c_1}_{\text{\rm o}}}({{c_1}_{w}}(x_1))=[\operatorname{tr}_{c_1} x_1](w)=[\operatorname{tr}_{c_2} x_2](w)={{c_2}_{\text{\rm o}}}({{c_2}_{w}}(x_2)),\quad w\in A^*,$$ and therefore $d_1|_Z=d_2|_Z$. Moreover, $${{c_j}_{a}}({{c_j}_{w}}(x_j))={{c_j}_{wa}}(x_j),\quad w\in A^*,$$ and therefore $d_j(Z)\subseteq\mathbb S\times Z^A$.
Remembering Remark \[ProperSemiring\], we have to show that the functor ${{\ensuremath{F_{\,{\mathbb S}}}}}$ is proper. We have the zig-zag , and the $\mathbb S$-semimodule $Z$ is, as a subsemimodule of the finitely generated $\mathbb S$-semimodule $\mathbb S^{n_1}\times\mathbb S^{n_2}$, itself finitely generated.
Proof of the extension lemma
----------------------------
The proof of the extension lemma follows directly from the following two abstract properties.
\[lem:ext-lem-1\] Assume ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$ via $\iota\colon {T_{\mathbb{S}}}\Rightarrow {T_{\mathbb{E}}}$ and let $X$ be a finite set. Let ${\mathbb{Y}} \in {{\text{\tt{Set}}}^{{T_{\mathbb{S}}}}}$ and ${\mathbb{Z}} \in {{\text{\tt{Set}}}^{{T_{\mathbb{E}}}}}$ and assume we are given an arrow $a_Y\colon {T_{\mathbb{S}}}X \to {\mathbb{Y}}$ in ${{\text{\tt{Set}}}^{{T_{\mathbb{S}}}}}$ and a ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$-homomorphism $h\colon {\mathbb{Y}} \to {\mathbb{Z}}$. Then there exists an arrow $a_Z\colon {T_{\mathbb{E}}}X \to {\mathbb{Z}}$ in ${{\text{\tt{Set}}}^{{T_{\mathbb{E}}}}}$ making the following diagram commute. $$\xymatrix@R=0.7em{
{T_{\mathbb{S}}}X \ar[rr]^{\iota} \ar[d]_{a_Y}&& {T_{\mathbb{E}}}X \ar[d]^{a_Z}\\
{\mathbb{Y}} \ar[rr]^{h}&& {\mathbb{Z}}
}$$
Consider the map $h {\mathrel{\circ}}a_Y {\mathrel{\circ}}\eta_{\mathbb S,X}\colon X \to Z$. Let $a_Z =(h {\mathrel{\circ}}a_Y {\mathrel{\circ}}\eta_{\mathbb S,X})^{\#_{\mathbb{E}}}$.
All morphisms in the square are ${{\text{\tt{Set}}}^{{T_{\mathbb{S}}}}}$-morphisms: $\alpha_Y$ by definition; $\iota_x$ as one of the monad morphism laws shows this; $h$ as it is a ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$-homomorphism; and $\alpha_Z = M_\iota(\alpha_Z)$. Clearly, then $h {\mathrel{\circ}}\alpha_Y$ and $\alpha_Z {\mathrel{\circ}}\iota_X$ are ${{\text{\tt{Set}}}^{{T_{\mathbb{S}}}}}$-morphisms from the free algebra $({T_{\mathbb{S}}}X, \mu_X)$ to ${\mathbb{Z}}$.
Therefore, for the commutativity of the square it suffices to show that $$h {\mathrel{\circ}}\alpha_Y {\mathrel{\circ}}\eta_{\mathbb S} = \alpha_Z {\mathrel{\circ}}\iota_X {\mathrel{\circ}}\eta_{\mathbb S}$$ as then, by the uniqueness of the extension, $$h {\mathrel{\circ}}\alpha_Y = (h{\mathrel{\circ}}\alpha_Y {\mathrel{\circ}}\eta_{\mathbb S})^{\#_{\mathbb{S}}} =
(\alpha_Z {\mathrel{\circ}}\iota_X {\mathrel{\circ}}\eta_{\mathbb S})^{\#_{\mathbb{S}}} = \alpha_Z {\mathrel{\circ}}\iota_X.$$
The last needed equality follows because ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$ along $\iota$ and so $$(h {\mathrel{\circ}}a_Y {\mathrel{\circ}}\eta_{\mathbb S,X})^{\#_{\mathbb{E}}} {\mathrel{\circ}}\iota_X {\mathrel{\circ}}\eta_{\mathbb{S},X} = (h {\mathrel{\circ}}a_Y {\mathrel{\circ}}\eta_{\mathbb S,X})^{\#_{\mathbb{E}}} {\mathrel{\circ}}\eta_{\mathbb E,X} = h {\mathrel{\circ}}a_Y {\mathrel{\circ}}\eta_{\mathbb S,X}. \vspace*{-7mm}$$
\[lem:ext-lem-2\] The map $\iota \times \iota^A$ is a ${T_{\mathbb{S}}}\le{T_{\mathbb{E}}}$-homomorphism from ${{\ensuremath{F_{\,{\mathbb{S}}}}}}({T_{\mathbb{S}}}X)$ to ${{\ensuremath{F_{\,{\mathbb{E}}}}}}({T_{\mathbb{E}}}X)$.
Since the functor $M_\iota$ induced by the monad morphism $\iota$ satisfies $\mathcal U_\mathbb{S} {\mathrel{\circ}}M_\iota = \mathcal U_\mathbb{E}$, it preserves all limits (as $\mathcal U_\mathbb{E}$ preserves them and $\mathcal U_\mathbb{S}$ reflects them). Hence, in particular, it preserves products. Since $\iota_X\colon ({T_{\mathbb{S}}}X, \mu_{\mathbb S,X}) \to M_\iota({T_{\mathbb{E}}}X, \mu_{\mathbb E,X})$ is a ${T_{\mathbb{S}}}$-algebra homomorphism by one of the monad morphism laws, we have $$\iota_1 \times \iota_X^A \colon {T_{\mathbb{S}}}1 \times {T_{\mathbb{S}}}X^A \to M_\iota({T_{\mathbb{E}}}1) \times M_\iota({T_{\mathbb{E}}}X)^A = M_\iota({T_{\mathbb{E}}}1 \times {T_{\mathbb{E}}}X^A)$$ is one as well.
Since $\mathbb S\subseteq\mathbb E$, we have $${{\ensuremath{F_{\,{\mathbb E}}}}}X\cap{{\ensuremath{F_{\,{\mathbb S}}}}}Y=(\mathbb E\times X^A)\cap(\mathbb S\times Y^A)
=\mathbb S\times(X\cap Y)^A={{\ensuremath{F_{\,{\mathbb S}}}}}(X\cap Y)
.$$ Assume now that $Y_j\subseteq X_j$. We have $$({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_1)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_1)=
\{(o,(({x_1}_a,{x_2}_a))_{a\in A})\in\mathbb E\times(X_1\times X_2)^A\mid o\in\mathbb S,{x_1}_a\in Y_1\}
,$$ and the analogous formula for $({{\ensuremath{F_{\,{\mathbb E}}}}}\pi_2)^{-1}({{\ensuremath{F_{\,{\mathbb S}}}}}Y_2)$. This shows that the intersection of these two inverse images is equal to $\mathbb S\times(Y_1\times Y_2)^A$.
Proof of the reduction lemma
----------------------------
Reducing from to {#reducing-from-to .unnumbered}
-------------------
The reduction lemma for passing from abelian groups to commutative monoids arises from a classical result of algebra. Namely, it is a corollary of the following theorem due to D.Hilbert, cf. [@hilbert:1890 Theorem II] see also [@bachem:1978 Theorem 1.1].
\[PolyhedralZ\] Let $W$ be a $n\times m$-matrix with integer entries, and let $X$ be the commutative monoid $$X=\big\{x\in\mathbb Z^n\mid x\cdot W\geq 0\big\}
,$$ where the monoid operation is the usual addition on $\mathbb Z^n$. Then $X$ is finitely generated as a commutative monoid.
The reduction lemma for passing from to is a corollary Since every finitely generated abelian group is also finitely generated as a commutative monoid, we obtain a somewhat stronger variant.
\[RedLemN\] Let $Z$ be a finitely generated abelian group, let $m\in\mathbb N$, and let $\varphi\colon Z\to\mathbb Z^m$ be a group homomorphism. Then $\varphi^{-1}(\mathbb N^m)$ is finitely generated as a commutative monoid.
Write $Z$, up to an isomorphism, as a direct sum of cyclic abelian groups $$\label{3}
Z=\mathbb Z^k\oplus\Big[\bigoplus_{j=1}^n\mathbb Z/a_j\mathbb Z\Big]$$ with $a_j\geq 2$. Since $\varphi$ maps into the torsionfree group $\mathbb Z^m$, we must have $$\varphi\Big(\bigoplus_{j=1}^n\mathbb Z/a_j\mathbb Z\Big)=\{0\}
.$$ Hence, an element $x\in Z$ satisfies $\varphi(x)\geq 0$, if and only if $\varphi(x_0)\geq 0$ where $x=x_0+x_1$ is the decomposition of $x$ according to the direct sum . The action of the map $\psi=\varphi|_{\mathbb Z^k}\colon \mathbb Z^k\to\mathbb Z^m$ is described as multiplication of $x_0=(\xi_1,\ldots,\xi_k)$ with some $k\times m$-matrix $W$ having integer coefficients. Thus $$\psi^{-1}(\mathbb N^m)=\big\{x_0\in\mathbb Z^k\mid x_0\cdot W\geq 0\big\}
,$$ and by Hilbert’s Theorem $\psi^{-1}(\mathbb N^m)$ is finitely generated as a commutative monoid.
The set $\bigoplus_{j=1}^n\mathbb Z/a_j\mathbb Z$ also has a finite set of generators as a monoid, for example the residue classes $1/a_j\mathbb Z$, $j=1,\ldots,n$. Together we see that $\varphi^{-1}(\mathbb N^m)$ has a finite set of generators as a commutative monoid.
Reducing from Q to {#reducing-from-qto .unnumbered}
--------------------
The reduction lemma for passing from vector spaces over $\mathbb Q$ to $\mathbb Q_+$-semimodules is a corollary of the one passing from to . Thus we have the corresponding stronger variant also in this case.
\[RedLemQ+\] Let $Z$ be a finite dimensional $\mathbb Q$-vector space, let $m\in\mathbb N$, and let $\varphi\colon Z\to\mathbb Q^m$ be $\mathbb Q$-linear. Then $\varphi^{-1}(\mathbb Q_+^m)$ is finitely generated as a $\mathbb Q_+$-semimodule.
Let $\{u_1,\ldots,u_k\}$ be a set of generators of $Z$ as a $\mathbb Q$-vector space. Write $$\varphi(u_j)=\Big(\frac{a_{j,1}}{b_{j,1}},\ldots,\frac{a_{j,m}}{b_{j,m}}\Big),\quad
j=1,\ldots,k
,$$ with $a_{j,i}\in\mathbb Z$ and $b_{j,i}\in\mathbb N\setminus\{0\}$. Set $b=\prod_{j=1}^k\prod_{i=1}^mb_{j,i}$, then $\varphi(bu_j)\in\mathbb Z^m$, $j=1,\ldots,k$.
Let $Z'\subseteq Z$ be the $\mathbb Z$-submodule generated by $\{bu_1,\ldots,bu_k\}$, and set $\psi=\varphi|_{Z'}$. Then $\psi$ is a $\mathbb Z$-linear map of $Z'$ into $\mathbb Z^m$. By Lemma \[RedLemN\], $\psi^{-1}(\mathbb N^m)$ is finitely generated as $\mathbb N$-semimodule, say by $\{v_1,\ldots,v_l\}\subseteq Z'$.
Given $x\in\varphi^{-1}(\mathbb Q_+^m)$, choose $\nu_1,\ldots,\nu_k\in\mathbb Q$ with $x=\sum_{j=1}^k\nu_ju_j$. Write $\nu_j=\frac{\alpha_j}{\beta_j}$ with $\alpha_j\in\mathbb Z$ and $\beta_j\in\mathbb N\setminus\{0\}$, and set $\beta=\prod_{j=1}^k\beta_j$. Then $$\beta b\cdot x=\sum_{j=1}^k(\beta\nu_j)\cdot bu_j\in Z'
,$$ and $$\psi(\beta b\cdot x)=\varphi(\beta b\cdot x)=\beta b\cdot\varphi(x)\in\mathbb Q_+^m\cap\mathbb Z^m=\mathbb N^m
.$$ Thus $\beta b\cdot x$ is an $\mathbb N$-linear combination of the elements $v_1,\ldots,v_l$, and hence $x$ is a $\mathbb Q_+$-linear combination of these elements. This shows that $\varphi^{-1}(\mathbb Q_+^m)$ is generated by $\{v_1,\ldots,v_l\}$ as a $\mathbb Q_+$-semimodule.
Reducing from R to {#reducing-from-rto .unnumbered}
--------------------
The reduction lemma for passing from vector spaces over $\mathbb R$ to convex cones arises from a different source than the previously studied. Namely, it is a corollary of the below classical theorem of H.Minkowski, cf. [@minkowski:1896] see also [@rockafellar:1970 Theorem 19.1].
Recall that a convex subset $X$ of $\mathbb R^n$ is called *polyhedral*, if it is a finite intersection of half-spaces, i.e., if there exist $l\in\mathbb N$, $u_1,\ldots,u_l\in\mathbb R^n$, and $\nu_1,\ldots,\nu_l\in\mathbb R$, such that $$X=\big\{x\in\mathbb R^n\mid (x,u_j)\leq\nu_j,j=1,\ldots,l\big\}
,$$ where $(\cdot,\cdot)$ denotes the euclidean scalar product on $\mathbb R^n$. On the other hand, $X$ is said to be *generated by points $a_1,\ldots,a_{l_1}$ and directions $b_1,\ldots,b_{l_2}$*, if $$X=\Big\{\sum_{j=1}^{l_1}\alpha_ja_j+\sum_{j=1}^{l_2}\beta_jb_j\mid
\alpha_j\in[0,1],\sum_{j=1}^{l_1}\alpha_j=1,\ \beta_j\geq 0,j=1,\ldots,l_2\Big\}
.$$ Note that a convex set generated by some points and directions is bounded, if and only if no (nonzero) directions are present. Further, a convex set is a cone, if and only if it allows a representation where only directions occur.
\[PolyhedralR\] Let $X$ be a convex subset of $\mathbb R^n$. Then $X$ is polyhedral, if and only if $X$ is generated by a finite set of points and directions.
The relevance of Minkowski’s Theorem in the present context is that it shows that the intersection of two finitely generated sets is finitely generated (since the intersection of two polyhedral sets is obviously polyhedral).
The reduction lemma for passing from R to is an immediate corollary. Since every finite dimensional $\mathbb R$-vector space is also finitely generated as a convex cone, we have the corresponding stronger version.
\[RedLemR+\] Let $Z$ be a finite dimensional $\mathbb R$-vector space, let $m\in\mathbb N$, and let $\varphi\colon Z\to\mathbb R^m$ be $\mathbb R$-linear. Then $\varphi^{-1}(\mathbb R_+^m)$ is finitely generated as a convex cone.
*Step 1:* The image $\varphi(Z)$ is a linear subspace of $\mathbb R^m$, in particular, polyhedral. The positive cone $\mathbb R_+^m$ is obviously also polyhedral. We conclude that the convex cone $\varphi(Z)\cap\mathbb R_+^m$ is generated by some finite set of directions.
*Step 2:* The kernel $\varphi^{-1}(\{0\})$ is, as a linear subspace of the finite dimensional vector space $Z$, itself finite dimensional (generated, say, by $\{u_1,\ldots,u_k\}$). Thus it is also finitely generated as a convex cone (in fact, $\{\pm u_1,\ldots,\pm u_k\}$ is a set of generators).
Choose a finite set of directions $\{a_1,\ldots,a_l\}$ generating $\varphi(Z)\cap\mathbb R_+^m$ as a convex cone, and choose $v_j\in Z$ with $\varphi(v_j)=a_j$, $j=1,\ldots,l$. We claim that $\{\pm u_1,\ldots,\pm u_k\}\cup\{v_1,\ldots,v_l\}$ generates $\varphi^{-1}(\mathbb R_+^m)$ as a convex cone. To see this, let $x\in \varphi^{-1}(\mathbb R_+^m)$. Choose $\alpha_1,\ldots,\alpha_l\geq 0$ with $\varphi(x)=\alpha_1a_1+\ldots+\alpha_la_l$. Then $$\varphi\big(x-(\alpha_1v_1+\ldots+\alpha_lv_l)\big)=
\varphi(x)-\big(\alpha_1\varphi(v_1)+\ldots+\alpha_l\varphi(v_l)\big)=0
,$$ and hence we find $\beta_1^\pm,\ldots,\beta_k^\pm\geq 0$ with $$x-(\alpha_1v_1+\ldots+\alpha_lv_l)=(\beta_1^+u_1+\ldots+\beta_k^+u_k)+(\beta_1^-(-u_1)+\ldots+\beta_k^-(-u_k))
.$$
Reducing from to {#reducing-from-to-1 .unnumbered}
-------------------
The reduction lemma for passing from vector spaces over $\mathbb R$ to positively convex algebras is again a corollary of Theorem \[PolyhedralR\]. However, in a sense the situation is more complicated. One, the corresponding strong version fails; in fact, no (nonzero) $\mathbb R$-vector space is finitely generated as a . Two, unlike in categories of semimodules, the direct product $T_{[0,1]}B_1\times T_{[0,1]}B_2$ does not coincide with $T_{[0,1]}(B_1\dot\cup B_2)$.
\[RedLemPca\] Let $n_1,n_2\in\mathbb N$, and let $Z$ be a linear subspace of $\mathbb R^{n_1}\times\mathbb R^{n_2}$. Then $Z\cap(\Delta^{n_1}\times\Delta^{n_2})$ is finitely generated as a positively convex algebra.
Obviously, $Z$ and $\Delta^{n_1}\times\Delta^{n_2}$ are both polyhedral. We conclude that $Z\cap(\Delta^{n_1}\times\Delta^{n_2})$ is generated by a finite set of points and directions. Since it is bounded, no direction can occur, and it is thus finitely generated as a .
Self-contained proof of Lemma \[RedLemN\]
=========================================
We provide a short and self-contained proof of the named reduction lemma. It proceeds via an argument very specific for $\mathbb N$; the essential ingredient is that the order of $\mathbb N$ is total and satisfies the descending chain condition. Note that the following argument also proves Hilbert’s Theorem.
First, a common fact about the product order on $\mathbb N^m$ (we provide an explicit proof since we cannot appoint a reference).
\[Incomparable\] Let $m\in\mathbb N$, and let $M\subseteq\mathbb N^m$ be a set of pairwise incomparable elements. Then $M$ is finite.
Assume that $M$ is infinite, and choose a sequence $(a_n)_{n\in\mathbb N}$ of different elements of $M$. Write $a_n=(\alpha_{n,1},\ldots,\alpha_{n,m})$. We construct, in $m$ steps, a subsequence $(b_n)_{n\in\mathbb N}$ of $(a_n)_{n\in\mathbb N}$ with the property that (we write $b_n=(\beta_{n,1},\ldots,\beta_{n,m})$) $$\label{5}
\forall k\in\{1,\ldots,m\}{{.\kern3pt}}L_k=\sup_{n\in\mathbb N}\beta_{n,k}<\infty
\ \vee\
\beta_{0,k}<\beta_{1,k}<\beta_{2,k}<\cdots$$ In the first step, extract a subsequence of $(a_n)_{n\in\mathbb N}$ according to the behaviour of the sequence of first components $(\alpha_{n,1})_{n\in\mathbb N}$. If $\sup_{n\in\mathbb N}\alpha_{n,1}<\infty$, take the whole sequence $(a_n)_{n\in\mathbb N}$ as the subsequence. If $\sup_{n\in\mathbb N}\alpha_{n,1}=\infty$, take a subsequence $(a_{n_j})_{j\in\mathbb N}$ with $$\alpha_{n_0,1}<\alpha_{n_1,1}<\alpha_{n_2,1}<\cdots
.$$ Repeating this step, always starting from the currently chosen subsequence, we succesively extract subsequences which after $l$ steps satisfy the property for the components up to $l$.
Denote $$I_1=\big\{k\in\{1,\ldots,m\}\mid \sup_{n\in\mathbb N}\beta_{n,k}<\infty\big\}
,\quad
I_2=\big\{k\in\{1,\ldots,m\}\mid \sup_{n\in\mathbb N}\beta_{n,k}=\infty\big\}$$ The map $n\mapsto(\beta_{n,k})_{k\in I_1}$ maps $\mathbb N$ into the finite set $\prod_{k\in I_1}\{0,\ldots,L_k\}$, and hence is not injective. Choose $n_1<n_2$ with $\beta_{n_1,k}=\beta_{n_2,k}$, $k\in I_1$. Since $\beta_{n_1,k}<\beta_{n_2,k}$, $k\in I_2$, we obtain $b_{n_1}\leq b_{n_2}$. However, by our choice of the elements $a_n$, $b_{n_1}\neq b_{n_2}$. Thus $M$ contains a pair of different but comparable elements.
If $\varphi^{-1}(\mathbb N^m)=\{0\}$, there is nothing to prove. Hence, assume that $\varphi^{-1}(\mathbb N^m)\neq\{0\}$.
*Step 1:* We settle the case that $Z\subseteq\mathbb Z^m$ and $\varphi$ is the inclusion map. Let $M$ be the set of minimal elements of $(Z\cap\mathbb N^m)\setminus\{0\}$. From the descending chain condition we obtain $$\forall x\in(Z\cap\mathbb N^m)\setminus\{0\}{{.\kern3pt}}\exists\,y\in M{{.\kern3pt}}y\leq x$$ By Lemma \[Incomparable\], $M$ is finite, say $M=\{a_1,\ldots,a_l\}$. Now we show that $M$ generates $Z$ as commutative monoid. Let $x\in Z$, and assume that $x-\sum_{j=1}^l\alpha_ja_j\neq 0$ for all $\alpha_j\in\mathbb N$. By the descending chain condition, the set of all elements of this form contains a minimial element, say, $x-\sum_{j=1}^l\tilde\alpha_ja_j$. Choose $y\in M$ with $y\leq x-\sum_{j=1}^l\tilde\alpha_ja_j$. Since $y\neq 0$, we have $x-\sum_{j=1}^l\tilde\alpha_ja_j-y<x-\sum_{j=1}^l\tilde\alpha_ja_j$ and we reached a contradiction.
*Step 2:* The kernel $\varphi^{-1}(\{0\})$ is, as a subgroup of the finitely generated abelian group $Z$, itself finitely generated (remember here that $\mathbb Z$ is a Noetherian ring). Let $\{u_1,\ldots,u_k\}$ be a set of generators of $\varphi^{-1}(\{0\})$ as abelian group. Then $\{\pm u_1,\ldots,\pm u_k\}$ is a set of generators of $\varphi^{-1}(\{0\})$ as a commutative monoid.
By Step 1 we find $\{a_1,\ldots,a_l\}\subseteq\mathbb Z^m$ generating $\varphi(Z)\cap\mathbb N^m$ as a commutative monoid. Choose $v_j\in Z$ with $\varphi(v_j)=a_j$, $j=1,\ldots,l$. Then we find, for each $x\in Z$, a linear combination of the $v_j$’s with nonnegative integer coefficients such that $$\varphi\Big(x-\sum_{j=1}^l\nu_jv_j\Big)=0
.$$ Hence, $\{\pm u_1,\ldots,\pm u_k\}\cup\{v_1,\ldots,v_l\}$ generates $\varphi^{-1}(Z)$ as commutative monoid.
Properties of ${{\ensuremath{\widehat F}}}$ {#app:D}
===========================================
Here the inclusion “$\supseteq$” is obvious. For the reverse inclusion, let $(o,\phi)\in{{\ensuremath{\widehat F}}}X$ and choose $p_{a,j}$ and $x_{a,j}$ according to Definition \[Ghat-def\]. Set $p_a=\sum_{j=1}^{n_a}p_{a,j}$. If $p_a=0$, set $x_a=0$. If $p_a>0$, set $x_a=\sum_{j=1}^n\frac{p_{a,j}}{p_a}x_{a,j}$. Then $x_a\in X$ and $f(a)=\sum_{j=1}^{n_a}p_{a,j}x_{a,j}=p_ax_a$.
\[GhSurj\] The functor ${{\ensuremath{\widehat F}}}$ preserves surjective algebra homomorphisms.
Let $X,Y$ be s, and $f:X\to Y$ a surjective algebra homomorphism. Let $(o,g)\in{{\ensuremath{\widehat F}}}Y$ be given. By Lemma \[Ghat-simple\] we can choose $p_a\in[0,1]$ and $y_a\in Y$ such that $o+\sum_{a\in A}p_a\leq 1$ and $g(a)=p_ay_a$, $a\in A$. Since $f$ is surjective, we find $x_a\in X$ with $f(x_a)=y_a$. Let $h:A\to X$ be the function $h(a)=p_ax_a$. Again using Lemma \[Ghat-simple\], we see that $(o,h)\in{{\ensuremath{\widehat F}}}X$. By our choice of $x_a$, it holds that $({{\ensuremath{\widehat F}}}f)(o,h)=(o,g)$.
Let $(o,\phi)\in{{\ensuremath{\widehat F}}}X$, and choose $p_a\in[0,1]$ and $x_a\in X$ as in Lemma \[Ghat-simple\]. Then ${\ensuremath{\mu_{X}}}(\phi(a))=p_a{\ensuremath{\mu_{X}}}(x_a)\leq p_a$, and hence $o+\sum_{a\in A}{\ensuremath{\mu_{X}}}(\phi(a))\leq 1$. Further, $o\in[0,1]$, in particular $o\geq 0$.
Conversely, assume that $o\geq 0$ and $o+\sum_{a\in A}{\ensuremath{\mu_{X}}}(\phi(a))\leq 1$. Let $a\in A$. Set $p_a={\ensuremath{\mu_{X}}}(\phi(a))$, then $p_a\in[0,1]$ since $\sum_{a\in A}p_a\leq 1$. To define $x_a$ consider first the case that ${\ensuremath{\mu_{X}}}(\phi(a))=0$. In this case $\phi(a)=0$ since $X$ is bounded, and we set $x_a=0$. If ${\ensuremath{\mu_{X}}}(\phi(a))>0$, set $x_a=\frac 1{{\ensuremath{\mu_{X}}}(\phi(a))}\phi(a)$. Since $X$ is closed, we have $x_a\in X$. In both cases, we obtained a representation $\phi(a)=p_ax_a$ with $p_a\in[0,1]$ and $x_a\in X$. Clearly, $o+\sum_{a\in A}p_a\leq 1$, and we conclude that $(o,\phi)\in{{\ensuremath{\widehat F}}}X$.
We build the extension in three stages.
We extend $c$ to the cone generated by $X$: Set $C=\bigcup_{t>0}tX$, and define $c_1\colon C\to V_2$ by the following procedure. Given $x\in C$, choose $t>0$ with $x\in tX$, and set $c_1(x)=t\cdot c\big(\frac 1tx\big)$. By this procedure the map $c_1$ is indeed well-defined. To see this, assume $x\in tX\cap sX$ where w.l.o.g. $s\leq t$. Then $\frac 1tx=\frac st\cdot\frac 1sx$. Since $\frac st\leq 1$, it follows that $c(\frac 1tx)=\frac st c(\frac 1sx)$, and hence $t\cdot c(\frac 1tx)=s\cdot c(\frac 1sx)$. Let us check that $c_1$ is cone-morphism, i.e., that $$c_1(x+y)=c_1(x)+c_2(y),\ x,y\in C,\qquad c_1(px)=pc_1(x),\ x\in C,p\geq 0
.$$ Given $x,y\in C$, choose $t>0$ such that $x,y,x+y\in tX$. Observe here that $C$ is a union of an increasing family of sets. Then $$\begin{aligned}
c_1(x+y)= &\, 2t\cdot c\Big(\frac 1{2t}(x+y)\Big)
=2t\cdot c\Big(\frac 12\cdot\frac 1tx+\frac 12\cdot\frac 1ty\Big)
\\[2mm]
= &\, 2t\cdot\Big[\frac 12c\big(\frac 1tx\big)+\frac 12c\big(\frac 1ty\big)\Big]
\\[2mm]
= &\, t\cdot c\Big(\frac 1tx\Big)+t\cdot c\Big(\frac 1ty\Big)
=c_1(x)+c_2(y)
.
\end{aligned}$$ Given $x\in C$ and $p>0$, choose $t>0$ with $x\in tX$. Then $px\in(pt)X$, and we obtain $$c_1(px)=pt\cdot c\Big(\frac 1{pt}(px)\Big)=pt\cdot c\Big(\frac 1tx\Big)=pt\cdot\frac 1t c_1(x)
=pc_1(x)
.$$ For $p=0$, the required equality is trivial. Finally, observe that $c_1$ extends $c$, since for $x\in X$ we can choose $t=1$ in the definition of $c_1$.
We extend $c_1$ to the linear subspace generated by $C$: Since $C$ is a cone, we have $\operatorname{span}C=C-C$. We define $c_2\colon \operatorname{span}C\to V_2$ by the following procedure. Given $x\in\operatorname{span}C$, choose $a_+,a_-\in C$ with $x=a_+-a_-$, and $c_2(x)=c_1(a_+)-c_2(a_-)$. By this procedure the map $c_2$ is indeed well-defined. Assume $x=a_+-a_-=b_+-b_-$. Then $a_++b_-=b_++a_-$, and we obtain $$c_1(a_+)+c_1(b_-)=c_1(a_++b_-)=c_1(b_++a_-)=c_1(b_+)+c_1(a_-)
,$$ which yields $c_1(a_+)-c_1(a_-)=c_1(b_+)-c_1(b_-)$. Let us check that $c_2$ is linear. Given $x,y\in\operatorname{span}C$, choose representations $x=a_+-a_-$, $y=b_+-b_-$. Then $x+y=(a_++b_+)-(a_-+b_-)$, and we obtain $$\begin{aligned}
c_2(x+y)= &\, c_1(a_++b_+)-c_1(a_-+b_-)
=\big[c_1(a_+)+c_1(b_+)\big]-\big[c_1(a_-)+c_1(b_-)\big]
\\
= &\, \big[c_1(a_+)-c_1(a_-)\big]+\big[c_1(b_+)-c_1(b_-)\big]
=c_2(x)+c_2(y)
.
\end{aligned}$$ Given $x\in\operatorname{span}C$ and $p\in\mathbb R$, choose a representation $x=a_+-a_-$ and distinguish cases according to the sign of $p$. If $p>0$, we have the representation $px=pa_+-pa_-$ and hence $$\begin{aligned}
c_2(px)= &\, c_1(pa_+)-c_1(pa_-)=pc_1(a_+)-pc_1(a_-)
\\
= &\, p\big[c_1(a_+)-c_1(a_-)\big]=pc_2(x)
.
\end{aligned}$$ If $p<0$, we have the representation $px=(-p)a_--(-p)a_+$ and hence $$\begin{aligned}
c_2(px)= &\, c_1((-p)a_-)-c_1((-p)a_+)=(-p)c_1(a_-)-(-p)c_1(a_+)
\\
= &\, p\big[c_1(a_+)-c_1(a_-)\big]=pc_2(x)
.
\end{aligned}$$ For $p=0$, the required equality is trivial. Finally, observe that $c_2$ extends $c_1$, since for $x\in C$ we can choose the representation $x=x-0$ in the definition of $c_2$.
We extend $c_2$ to $V_1$: By linear algebra a linear map given on a subspace can be extended to a linear map on the whole space.
The uniqueness statement is clear.
First assume that holds. Let $x\in X$. Then ${\ensuremath{\mu_{X}}}(x)\leq 1$, and we obtain $${{c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}({{c}_{a}}(x))=
{{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}({{\tilde c}_{a}}(x))\leq{\ensuremath{\mu_{X}}}(x)\leq 1
.$$ Further, ${{c}_{\text{\rm o}}}(x)\geq 0$ by assumption. Now Lemma \[Ghat-Minko\] gives $c(x)\in{{\ensuremath{\widehat F}}}Y$.
Conversely, assume $c(X)\subseteq{{\ensuremath{\widehat F}}}Y$, and let $x\in\mathbb R^n$ be given. If ${\ensuremath{\mu_{X}}}(x)=\infty$, the relation trivially holds. If ${\ensuremath{\mu_{X}}}(x)=0$, then $x=0$ since $X$ is bounded. Hence, the left side of equals $0$, and again holds. Assume that ${\ensuremath{\mu_{X}}}(x)\in(0,\infty)$. Since $X$ is closed, we have ${\ensuremath{\mu_{X}}}(x)^{-1}x\in X$, and hence $c({\ensuremath{\mu_{X}}}(x)^{-1}x)\in{{\ensuremath{\widehat F}}}Y$. From Lemma \[Ghat-Minko\], we get the estimate $$\begin{aligned}
{{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}({{\tilde c}_{a}}(x))= &\, {\ensuremath{\mu_{X}}}(x)
\Big(
{{\tilde c}_{\text{\rm o}}}\big(\frac 1{{\ensuremath{\mu_{X}}}(x)}x\big)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}\big({{\tilde c}_{a}}(\frac 1{{\ensuremath{\mu_{X}}}(x)}x)\big)
\Big)
\\
= &\, {\ensuremath{\mu_{X}}}(x)
\Big(
{{c}_{\text{\rm o}}}\big(\frac 1{{\ensuremath{\mu_{X}}}(x)}x\big)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}\big({{c}_{a}}(\frac 1{{\ensuremath{\mu_{X}}}(x)}x)\big)
\Big)
\leq{\ensuremath{\mu_{X}}}(x)
.
\end{aligned}$$
Proof details of the Extension Theorem
======================================
Recall Kakutani’s theorem [@kakutani:1941 Corollary].
\[Kak\] Let $M\subseteq\mathbb R^n$ and $P\colon M\to\mathcal P(M)$. Assume
1. $M$ is nonempty, compact, and convex,
2. for each $x\in M$, the set $P(x)$ is nonempty, closed, and convex,
3. the map $P$ has closed graph in the sense that, whenver $x_n\in M$, $x_n\to x$, and $y_n\in P(x_n)$, $y_n\to y$, it follows that $y\in P(x)$.
Then there exists $x\in M$ with $x\in P(x)$.
Note that $P$ having closed graph implies that $P(x)$ is closed for all $x$. To see this, let $y_n\in P(x)$, $y_n\to y$, and use the constant sequence $x_n=x$ in the closed graph property.
In the proof of Theorem \[ExEx\] we shall, as in Example \[Pyramid\], identify a pyramid $Y$ with the appropriately scaled normal vector $u$ of its inclined side. Then, for two pyramids $Y_1$ and $Y_2$ with corresponding normal vectors $u_1$ and $u_2$, the requirement that $X\subseteq Y_j$ becomes $(x,u_j)\leq{\ensuremath{\mu_{X}}}(x)$, $x\geq 0$, and the requirement $\tilde c(Y_2)\subseteq{{\ensuremath{\widehat F}}}Y_1$ becomes ${{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u_1)\leq(x,u_2)$, $x\geq 0$, cf. Corollary \[Ghat-Minko-Coalg\].
Let $M$ be the set $$M=\big\{u\in\mathbb R^n\mid u\geq 0\text{ and }(x,u)\leq{\ensuremath{\mu_{X}}}(x),x\geq 0\big\}
.$$ We have to include vectors $u$ with possibly vanishing components into $M$ to ensure closedness. It will be a step in the proof to show that a fixed point must be strictly positive.
Let $P\colon M\to\mathcal P(M)$ be the map $$P(u)=\big\{v\in M\mid {{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)\leq(x,v), x\geq 0\big\}
.$$ Here we again denote by $\tilde c\colon \mathbb R^n\to\mathbb R\times(\mathbb R^n)^A$ the linear extension of $c$. Observe that $\tilde c(x)\geq 0$ for all $x\geq 0$, since $\Delta^n\subseteq X$ and $c(x)\geq 0$ for $x\in X$.
It is easy to check that $M$ and $P$ satisfy the hypothesis of Kakutani’s Theorem, the crucial point being that $P(u)\neq\emptyset$.
$M$ is nonempty: We have $0\in M$.
$M$ is compact: To show that $M$ is closed let $u_n\in M$ with $u_n\to u$. Since $u_n\geq 0$ also $u\geq 0$, and for each fixed $x\geq 0$ continuity of the scalar product yields $(x,u)=\lim_{n\to\infty}(x,u_n)\leq{\ensuremath{\mu_{X}}}(x)$. Further, $M$ is bounded since $(e_j,u)\leq{\ensuremath{\mu_{X}}}(e_j)\leq 1$, $j=1,\ldots,n$, by our assumption that $\Delta^n\subseteq X$, and hence $u\in[0,1]^n$.
$M$ is convex: Let $u_1,u_2\in M$ and $p\in[0,1]$. First, clearly, $pu_1+(1-p)u_2\geq 0$. Second, for each $x\geq 0$, $$\begin{aligned}
(x,pu_1+(1-p)u_2)= &\, p(x,u_1)+(1-p)(x,u_2)
\\
\leq &\, p{\ensuremath{\mu_{X}}}(x)+(1-p){\ensuremath{\mu_{X}}}(x)={\ensuremath{\mu_{X}}}(x)
.
\end{aligned}$$
$P(u)$ is nonempty: Let $u\in M$ be given. The map $x\mapsto{{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)$ is a linear functional on $\mathbb R^n$. Thus we find $v\in\mathbb R^n$ representing it as $x\mapsto(x,v)$. Since $e_j\in X$, we have $$(e_j,v)={{\tilde c}_{\text{\rm o}}}(e_j)+\sum_{a\in A}({{\tilde c}_{a}}(e_j),u)\geq 0
.$$ Further, using that $u\in M$ and $\tilde c(X)\subseteq{{\ensuremath{\widehat F}}}X$, we obtain that for each $x\geq 0$ $$(x,v)={{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)
\leq{{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{X}}}({{\tilde c}_{a}}(x))
\leq{\ensuremath{\mu_{X}}}(x)
.$$ Together, we see that $v\in M$. By its definition, therefore, $v\in P(u)$.
$P(u)$ is convex: Let $v_1,v_2\in P(u)$ and $p\in[0,1]$. First, since $M$ is convex, $pv_1+(1-p)v_2$ belongs to $M$. Second, for each $x\geq 0$, $$\begin{aligned}
(x,pv_1+ &\, (1-p)v_2)=p(x,v_1)+(1-p)(x,v_2)
\\
\geq &\, p\Big({{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)\Big)+
(1-p)\Big({{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)\Big)
\\
= &\, {{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)
.
\end{aligned}$$
$P$ has closed graph: Let $u_n\in M$, $u_n\to u$, and $v_n\in P(u_n)$, $v_n\to v$. Then $u,v\in M$ since $M$ is closed. Now fix $x\geq 0$. Continuity of the scalar product allows to pass to the limit in the relation $${{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),v_n)\leq(x,u_n)
,$$ which holds for all $n\in\mathbb N$. This yields ${{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),v)\leq(x,u)$.
Having verified all necessary hypothesis, Theorem \[Kak\] can be applied and furnishes us with $u\in M$ satisfying $u\in P(u)$, explicitly, $u\in\mathbb R^n$ with $$\label{1}
u\geq 0,\quad (x,u)\leq{\ensuremath{\mu_{X}}}(x),x\geq 0,\quad {{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}({{\tilde c}_{a}}(x),u)\leq(x,u),x\geq 0
.$$ Set $Y=\{x\geq 0\mid (x,u)\leq 1\}$. Then $Y$ is a , and by definition contained in $\mathbb R_+^n$. It contains $X$ since $u\in M$, and since $u\in P(u)$ we have $\tilde c(Y)\subseteq{{\ensuremath{\widehat F}}}Y$. Thus $d=\tilde c|_Y$ turns $Y$ into an ${{\ensuremath{\widehat F}}}$-coalgebra, and since $c=\tilde c|_X=(\tilde c|_Y)|_X=d|_X$, the inclusion map $\iota\colon X\to Y$ is an ${{\ensuremath{\widehat F}}}$-coalgebra morphism.
It remains to show that $Y$ is generated by $n$ linearly independent vectors. Remembering again Example \[Pyramid\], this is equivalent to $u$ being strictly positiv. Let $I=\{j\in\{1,\ldots,n\}\mid (e_j,u)=0\}$. For each $j\in I$ the last relation in implies that ${{c}_{\text{\rm o}}}(e_j)=0$ and $({{c}_{a}}(e_j),u)=0$, $a\in A$. Since $u\geq 0$ and ${{c}_{a}}(e_j)\geq 0$, we conclude that the vector ${{c}_{a}}(e_j)$ can have nonzero components only in those coordinates where $u$ has zero component. In other words, ${{c}_{a}}(e_j)\in\operatorname{span}\{e_i\mid i\in I\}$. Now gives $I=\emptyset$.
Proof details for properness of ${{\ensuremath{\widehat F}}}$
=============================================================
We denote $v_j=(1,\ldots,1)\in\mathbb R^{n_j}$. By Example \[Pyramid\] $$\mu_{\Delta^{n_j}}(x_j)=(x_j,v_j),\ x_j\in\mathbb R_+^{n_j}
.$$ Since $(\Delta^{n_j},c_j)$ is an ${{\ensuremath{\widehat F}}}$-coalgebra, Corollary \[Ghat-Minko-Coalg\] yields $${{\tilde c{_j}}_{\text{\rm o}}}(x_j)+\sum_{a\in A}({{\tilde c{_j}}_{a}}(x_j),v_j)\leq(x_j,v_j),\quad x_j\in\mathbb R_+^{n_j}
,j=1,2
.$$ Summing up these two inequalities yields that for $x_1\in\mathbb R_+^{n_1}$ and $x_2\in\mathbb R_+^{n_2}$ $$\label{7}
\Big[{{\tilde c{_1}}_{\text{\rm o}}}(x_1)+{{\tilde c{_2}}_{\text{\rm o}}}(x_2)\Big]+
\sum_{a\in A}\Big[({{\tilde c{_1}}_{a}}(x_1),v_1)+({{\tilde c{_2}}_{a}}(x_2),v_2)\Big]
\leq(x_1,v_1)+(x_2,v_2)
.$$ Recall that $Z$ denotes the linear subspace of $\mathbb R^{n_1}\times\mathbb R^{n_2}$ constructed in the basic diagram (referring back to Lemma \[Prod\]). The definition of the map $d$ in the basic diagram ensures that for $(x_1,x_2)\in Z$ $${{d}_{\text{\rm o}}}((x_1,x_2))={{\tilde c{_1}}_{\text{\rm o}}}(x_1)={{\tilde c{_2}}_{\text{\rm o}}}(x_2),\quad
{{d}_{a}}((x_1,x_2))=\big({{\tilde c{_1}}_{a}}(x_2),{{\tilde c{_2}}_{a}}(x_2)\big)
.$$ Set $v=\frac 12(1,\ldots,1)\in\mathbb R^{n_1+n_2}$. Plugging the above into and dividing by $2$ yields $${{d}_{\text{\rm o}}}((x_1,x_2))+\sum_{a\in A}\big({{d}_{a}}((x_1,x_2)),v)\leq \big((x_1,x_2),v\big)
,\quad (x_1,x_2)\in Z\cap\mathbb R_+^{n_1+n_2}
.$$ We have $$\mu_{Z\cap 2\Delta^{n_1+n_2}}(x)=\max\big\{\mu_Z(x),\mu_{2\Delta^{n_1+n_2}}(x)\big\}
=
\begin{cases}
(x,v) &\hspace*{-3mm},\quad x\in Z\cap\mathbb R_+^{n_1+n_2},
\\
\infty &\hspace*{-3mm},\quad \text{otherwise}.
\end{cases}$$ Here the first equality holds by property 3. listed after Definition \[Minko\]. The second equality is based on Example \[MinkoExa\] and Example \[Pyramid\]: First, $2\Delta^{n_1+n_2}$ is the pyramid constructed with $v$, and hence $\mu_{2\Delta^{n_1+n_2}}(x) = (x,v)$ if $x\geq 0$, and $\infty$ otherwise. Second, $Z$ is a linear subspace, hence in particular a convex cone, and thus $\mu_Z(x) = 0$ if $x\in Z$, and $\infty$ otherwise.
The inclusion $d(Z\cap 2\Delta^{n_1+n_2})\subseteq{{\ensuremath{\widehat F}}}(Z\cap 2\Delta^{n_1+n_2})$ can now be deduced with help of Lemma \[Ghat-Minko\]. Start with $(x_1,x_2)\in Z\cap 2\Delta^{n_1+n_2}$. Then $((x_1,x_2),v)\leq 1$. Moreover, $(x_1,x_2)\geq 0$ and $(x_1,x_2)\in Z$ which allows to apply $d$. We obtain $d_o(x_1,x_2) + \sum_a (d_a(x_1,x_2),v) \leq 1$. Since $d(Z)\subseteq F_{\mathbb R}(Z)$, we have $d_a(x_1,x_2)\in Z$. Now remember the computation of $d_a(x_1,x_2)$. The map $\tilde c_j$ is the linear extension of $c_j$, hence $$\tilde c_j(\Delta^{n_j}) = c_j(\Delta^{n_j}) \subseteq \hat F(\Delta^{n_j}) \subseteq [0,1]\times [0,1]^A
.$$ In particular, $\tilde c_j$ takes nonnegative values on $\Delta^{n_j}$, and by linearity thus on all of $(\mathbb R_+)^{n_j}$. This shows $d_a(x_1,x_2)\geq 0$ and $d_o(x_1,x_2)\geq 0$. By the above computation of the Minkowski functional $\mu_{Z\cap 2\Delta^{n_1+n_2}}$, by now we know that we are in the first case, $(d_a(x_1,x_2),v)=\mu_{Z\cap 2\Delta^{n_1+n_2}}(d_a(x_1,x_2))$, and thus $$d_o(x_1,x_2) + \sum_a \mu_{Z\cap 2\Delta^{n_1+n_2}}(d_a(x_1,x_2)) \leq 1
.$$ Lemma 16 applies, and yields $d(x_1,x_2)\in \hat F(Z\cap 2\Delta^{n_1+n_2})$.
Using the basic diagram, we obtain $$\begin{aligned}
& \tilde c_j(\Delta^{n_j})\subseteq{{\ensuremath{\widehat F}}}\Delta^{n_j}\subseteq{{\ensuremath{\widehat F}}}Y_j,
\\
& \tilde c_j(\pi_j(Z\cap 2\Delta^{n_1+n_2}))\subseteq{{\ensuremath{\widehat F}}}(\pi_j(Z\cap 2\Delta^{n_1+n_2}))\subseteq{{\ensuremath{\widehat F}}}Y_j.
\end{aligned}$$ Since $\tilde c_j$ is linear, in particular convex, and ${{\ensuremath{\widehat F}}}Y_j$ is convex, it follows that $$\tilde c_j\big(\operatorname{co}(\Delta^{n_j}\cup\pi_j(Z\cap 2\Delta^{n_1+n_2}))\big)\subseteq{{\ensuremath{\widehat F}}}Y_j
.$$
We check that the $Y_j$ satisfies the hypothesis of Theorem \[ExEx\]. By its definition $\Delta^{n_j}\subseteq Y_j\subseteq\mathbb R_+^{n_j}$. Since $Y_j$ is finitely generated, recall that $Y_j$ is the convex hull of two finitely generated s, it is a compact subset of $\mathbb R^{n_j}$. Finally, since the coalgebra structure on $Y_j$ is an extension of the one on $\Delta^{n_j}$, the present assumption implies that the condition of Theorem \[ExEx\] is satisfied. Note here that $\Delta^{n_j}\cap\operatorname{span}\{e_i\mid i\in I\}=\operatorname{co}(\{e_i\mid i\in I\}\cup\{0\})$.
Applying Theorem \[ExEx\] we obtain extensions $U_j$ as required.
We show that the diagram $$\xymatrix@C=60pt{
\Delta^n \ar[r]^f \ar[d]_c & X \ar[d]^g
\\
{{\ensuremath{\widehat F}}}\Delta^n \ar[r]_{\operatorname{id}\times(f\circ -)} & [0,1]\times X^A
}$$ commutes. First, for $k\not\in I$, we have $((\operatorname{id}\times(f\circ -))\circ c)(e_k)=(g\circ f)(e_k)$ by the definition of $g$. Second, consider $k\in I$. Then $(g\circ f)(e_k)=0$ since $f(e_k)=0$. By , also $((\operatorname{id}\times(f\circ -))\circ c)(e_k)=0$.
Since ${{\ensuremath{\widehat F}}}f$ maps ${{\ensuremath{\widehat F}}}\Delta^n$ into ${{\ensuremath{\widehat F}}}X$, we have $g(X)\subseteq{{\ensuremath{\widehat F}}}X$. This says that $X$ indeed becomes an ${{\ensuremath{\widehat F}}}$-coalgebra with structure $g$. Revisiting the above diagram shows that $f$ is an ${{\ensuremath{\widehat F}}}$-coalgebra morphism.
Applying Lemma \[GhatRedLem\] repeatedly, we obtain after finitely many steps an ${{\ensuremath{\widehat F}}}$-coalgebra $(\Delta^k,g)$ such that no nonempty subset $I\subseteq\{1,\ldots,k\}$ with exists for $(\Delta^k,g)$, and that we have an ${{\ensuremath{\widehat F}}}$-coalgebra morphism $f\colon (\Delta^n,c)\to(\Delta^k,g)$. Note here that in each application of the lemma the number of generators decreases.
[^1]: The notions of monads and algebraic categories are central to this paper. We recall them in Appendix \[sec:app-basics\] to make the paper accessible to all readers.
[^2]: This functor was denoted $\hat G$ in [@silva.sokolova:2011] where it was first studied in the context of axiomatization of trace semantics.
[^3]: In [@beal.lombardy.sakarovitch:2006] only a sketch of the proof is given, cf. [@beal.lombardy.sakarovitch:2006 §3.3]. In this sketch one important point is not mentioned. Using the terminology of [@beal.lombardy.sakarovitch:2006 §3.3]: it could a priori be possible that the size of the vectors in $G$ and the size of $G$ both oscillate.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a detailed analysis of strongly driven spontaneous four-wave mixing in a lossy integrated microring resonator side-coupled to a channel waveguide. A nonperturbative, analytic solution within the undepleted pump approximation is developed for a cw pump input of arbitrary intensity. In the strongly driven regime self- and cross-phase modulation, as well as multi-pair generation, lead to a rich variety of power-dependent effects; the results are markedly different than in the low power limit. The photon pair generation rate, single photon spectrum, and joint spectral intensity (JSI) distribution are calculated. Splitting of the generated single photon spectrum into a doublet structure associated with both pump detuning and cross-phase modulation is predicted, as well as substantial narrowing of the generated signal and idler bandwidths associated with the onset of optical parametric oscillation at intermediate powers. Both the correlated and uncorrelated contributions to the JSI are calculated, and for sufficient powers the uncorrelated part of the JSI is found to form a quadruplet structure. The pump detuning is found to play a crucial role in all of these phenomena, and a critical detuning is identified which divides the system behaviour into distinct regimes, as well as an optimal detuning strategy which preserves many of the low-power characteristics of the generated photons for arbitrary input power.'
author:
- 'Z. Vernon'
- 'J.E. Sipe'
bibliography:
- 'CW\_draft.bib'
title: Strongly driven nonlinear quantum optics in microring resonators
---
Introduction {#sec:intro}
============
Integrated optical microresonators continue to develop as a promising platform for generating, controlling, and measuring quantum states of light [@Xu2008; @Xia2007; @Ferrera2008; @Ferrera2009; @Azzini2012; @Azzini2012a; @Preble2015; @Gentry2015; @Kumar2015; @Jung2013]. Advances in fabricating such chip-based structures is enabling the construction of micron-scale optical ring resonators with reported quality factors Q of over one million [@Ferrera2009; @Razzari2010]. By exploiting the nonlinear optical response of the ring medium, combined with the massive enhancement of intraring pump intensity made possible by the large Q values of the resonator, a wide variety of nonlinear optical phenomena can be realized using relatively modest input powers. Entangled photon pair generation in silicon microrings has been demonstrated using mere $\mathrm{\mu W}$ of pump power [@Azzini2012; @Gentry2015], and optical parametric oscillation in a silicon nitride microring has been observed using a pump power of only 50 mW [@Levy2010]. Arrays of coupled silicon microrings have also been investigated as a potential source of heralded single photons [@Davanco2012].
Such high-Q microrings are ideal for investigations of strongly driven nonlinear optical effects. Depending on the application, these effects can be undesirable or highly sought after: multi-pair production from an entangled photon pair source contaminates the sought after energy correlation, whereas optical parametric oscillation (OPO) *only* arises in the strongly driven regime. Theoretical studies of quantum nonlinear optics in integrated microresonators have typically treated the nonlinearity perturbatively [@Vernon2015; @Chen2011; @Camacho2012; @Helt2010; @Yang2007; @Yang2007a; @Helt2012; @Liscidini2012], which limits calculations to quantities relating to a single generated photon pair.
Recently we presented a general theoretical treatment of photon pair generation arising from spontaneous four-wave mixing (SFWM) in microring resonators, fully accounting for the quantum effects of scattering losses within the resonator [@Vernon2015]. As our focus was on the effects of such losses, we limited ourselves to the low power regime in which a perturbative solution of the relevant equations of motion provides an adequate description of the pair generation process. In this work we extend our analysis to a more strongly driven regime, where perturbative strategies are inadequate and competing nonlinear effects, including self-phase modulation (SPM) and cross-phase modulation (XPM), become important. We restrict ourselves to one of the most common pump states used in experiment, that of a coherent, narrowband continuous wave pump beam, for which a nonperturbative, analytic solution to the semiclassical equations of motion is achievable within the undepleted pump approximation. This approximation limits us to pump intensities below the onset of OPO, the threshold for which is clearly indicated by our equations; OPO in such structures will be the subject of a later communication. Even below the OPO threshold, the subtle interplay between the various nonlinear terms that couple the ring modes, as well as the effects of multiple photon pair generation, give rise to a rich variety of nonlinear optical phenomena that are accessible by varying only two input parameters, namely the pump intensity and detuning.
In Sec. \[sec:hamiltonian\_and\_fields\] we begin by assembling the relevant Hamiltonian and field operators for the ring-channel system. In Sec. \[sec:eqns\_of\_motion\] a brief review and summary of our earlier [@Vernon2015] theoretical framework is presented, wherein the system’s dynamics are reduced to a set of coupled ordinary differential equations for the ring operators alone. Steady state solutions for the pump mode, incorporating the effects of SPM and scattering losses, are developed in Sec. \[sec:pump\_dynamics\], and the stability of those solutions is studied. The equations of motion for the signal and idler modes are then solved in Sec. \[sec:signal\_idler\_dynamics\], enabling the calculation of physical quantities including the photon generation rate, single photon power spectrum, and joint spectral intensity distribution. For each of these measurable quantities the corresponding predictions at low and high pump powers are compared, and we identify a set of experimental features, or “smoking guns," that distinguish the qualitative behaviour at high pump powers from that at low pump powers.
Hamiltonian and fields {#sec:hamiltonian_and_fields}
======================
We consider an integrated microring resonator side-coupled to a channel waveguide, as illustrated in Fig. \[fig:ring\_diagram\]. We assume the ring size and quality factor Q have been chosen such that the ring accommodates individual resonant modes which are well separated in frequency; that is, we are in the high finesse limit. While a simple generalization of our framework can be used to treat arbitrarily many ring resonances, we restrict our model for the time being to contain only three ring modes of interest. The full system Hamiltonian can then be written [@Vernon2015] as $$\begin{aligned}
\label{main_hamiltonian}
H = H_{\mathrm{channel}} + H_{\mathrm{ring}} + H_{\mathrm{coupling}} + H_{\mathrm{bath}},\end{aligned}$$ wherein $H_{\mathrm{channel}}$ refers to the channel fields, $H_{\mathrm{ring}}$ to the ring modes, $H_{\mathrm{coupling}}$ to the coupling between the channel and ring, and $H_{\mathrm{bath}}$ to any modes into which ring photons may be lost, as well the coupling of those modes to the ring modes. Introducing channel fields $\psi_J(z)$, the channel Hamiltonian is $$\begin{aligned}
\label{H_channel_defn}
H_{\mathrm{channel}} &=& \sum_J \Bigg[ \hbar\omega_J \int dz\; \psi_J^\dagger(z)\psi_J(z) \nonumber \\
&+& \frac{i\hbar v_J}{2}\int dz\; \left( \frac{d\psi_J^\dagger(z)}{dz}\psi_J(z) - \mathrm{H.c.}\right)\Bigg],\end{aligned}$$ where the fields satisfy the usual commutation relations $$\begin{aligned}
\label{commutators}
\left[\psi_J(z),\psi_{J'}(z')\right] &=& 0,\nonumber \\
\left[\psi_J(z),\psi_{J'}^\dagger(z')\right] &=& \delta(z-z')\delta_{JJ'}.\end{aligned}$$ The index $J\in\{P,S,I\}$ runs over three fields of interest, respectively labelled $P$, $S$ and $I$ for pump, signal and idler, with corresponding reference frequencies $\omega_J$ and propagation speeds $v_J$. Each field $\psi_J$ contains frequency components centred at $\omega_J$, taken to be the resonant frequency of the corresponding ring mode, and ranges over a bandwidth that does not overlap with those of other fields $\psi_{J'}$, but involving excitation over sufficiently long distances that the Dirac $\delta$ function in (\[commutators\]) is a good approximation [@Vernon2015]. By allowing the fields to have different propagation speeds we include the possibility of group velocity dispersion between the different channel fields. The Hamiltonian (\[H\_channel\_defn\]) does assume group velocity dispersion within each channel field is negligible, but it is straightforward to include arbitrary dispersion. The spatial co-ordinate $z$ ranges from $z=-\infty$ to $z=+\infty$ with the coupling to the ring assumed to take place at a single point $z=0$. Within this point coupling approximation the coupling Hamiltonian becomes $$\begin{aligned}
H_{\mathrm{coupling}} = \sum_J\left(\hbar\gamma_J b_J^\dagger \psi_J(0) + \mathrm{H.c.}\right),\end{aligned}$$ in which we have introduced ring-channel coupling coefficients $\gamma_J$, as well as discrete ring mode annihilation operators $b_J$. In addition to the physical channel, to simulate scattering losses in the ring we include an extra “phantom channel" into which ring photons can be lost. The phantom channel similarly accommodates three fields $\phi_J(z)$ with respective propagation speeds $u_J$ and coupling coefficients $\mu_J$, and is represented as $H_\mathrm{bath}$ by a channel and coupling Hamiltonian identical to those for the physical channel [@Vernon2015].
The Hamiltonian for the ring modes can be written as $$\begin{aligned}
H_{\mathrm{ring}} = \sum_J \hbar\omega_J b_J^\dagger b_J + H_{\mathrm{NL}},\end{aligned}$$ where $H_{\mathrm{NL}}$ includes all the nonlinearity in the system. Since the fields will be most intense within the ring resonator, we neglect channel nonlinearities and take $H_{\mathrm{NL}}$ to contain only ring mode operators. In this work we consider effects arising from the third-order nonlinear susceptibility in the ring, taking $$\begin{aligned}
H_{\mathrm{NL}} &=& \left(\hbar\Lambda b_P b_P b_S^\dagger b_I^\dagger + \mathrm{H.c.}\right) + \hbar\eta b_P^\dagger b_P^\dagger b_P b_P \nonumber \\
&+& \hbar\zeta\left(b_S^\dagger b_P^\dagger b_S b_P + b_I^\dagger b_P^\dagger b_I b_P\right).\end{aligned}$$ The first term is responsible for SFWM, in which two pump photons are converted to a signal and idler photon pair. The second leads to SPM of the pump, while the latter two are responsible for XPM between the pump and signal and idler modes. It is safe to neglect SPM and XPM terms that involve only the signal and idler modes, since the power in those modes will be small compared to that in the pump mode. While we focus in this work on SFWM involving a single pump mode, it is straightforward to incorporate multiple pump modes into our model. The nonlinear coupling coefficients $\Lambda$, $\eta$ and $\zeta$ are not independent, as they arise from the same nonlinear susceptibility, but we formally leave them arbitrary for the time being so that the effects of each term in $H_{\mathrm{NL}}$ can more easily be identified. Obtaining expressions for these constants depends on the approximations used to derive the nonlinear sector of the ring Hamiltonian. We present our derivation of this Hamiltonian and the associated constants $\Lambda$, $\eta$ and $\zeta$ in Appendix \[appendix:lambda\], arriving at an estimate of $$\begin{aligned}
\Lambda \approx \frac{\hbar\omega_P^2cn_2}{n^2 V_{\mathrm{ring}}},\end{aligned}$$ with $\eta=\Lambda/2$ and $\zeta=2\Lambda$. In this expression $n_2$ refers to the nonlinear refractive index of the ring material, $n$ to the linear refractive index, and $V_{\mathrm{ring}}$ to the volume of the ring mode. For the silicon nitride rings used in typical experiments [@Levy2010], with $n_2\approx 2.4\times 10^{-19}\mathrm{m^2/W}$ [@Ikeda2008] this yields $\Lambda\sim 10\;\mathrm{Hz}$. For typical silicon rings [@Azzini2012], with $n_2 \approx 2.7\times 10^{-18}\mathrm{m^2/W}$ [@Boyd2008] this calculation predicts $\Lambda \sim 10^3\;\mathrm{Hz}$.
![Integrated ring-channel system geometry with labelled ring modes and incoming and outgoing outgoing channel fields. Photons generated in the ring may exit to the physical channel or be lost to the upper effective “phantom channel".[]{data-label="fig:ring_diagram"}](ringchannel.png){width="1.0\columnwidth"}
Equations of motion {#sec:eqns_of_motion}
===================
The Heisenberg equations of motion for the field operators $\psi_J(z,t)$ and $\phi_J(z,t)$ and the ring operators $b_J(t)$ follow from the Hamiltonian (\[main\_hamiltonian\]), and can be simplified by the introduction of auxiliary quantities[@Vernon2015]; here we summarize the results.
The equations of motion for the channel fields are $$\begin{aligned}
\label{basic_field_eqn}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_J(z,t) = -i\gamma_J b_J(t) \delta(z),\end{aligned}$$ with similar expressions obeyed by the phantom channel fields $\phi_J(z,t)$. Note that the solutions to these equations contain a discontinuity at $z=0$, which is a consequence of our point-coupling assumption. To avoid explicitly dealing with this discontinuity, it is helpful to introduce formal channel fields which we identify as those fields which are incoming and outgoing with respect to the coupling point. We define the incoming field $\psi_{J<}(z,t)$ by $$\begin{aligned}
\psi_{J<}(z,t) = \psi_J(z,t)\;\;\mathrm{for}\; z<0,\end{aligned}$$ and extend it to $z\geq 0$ by requiring everywhere that it satisfy the homogeneous version of (\[basic\_field\_eqn\]), $$\begin{aligned}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_{J<}(z,t) = 0.\end{aligned}$$ This confers a false future on $\psi_{J<}(z,t)$, corresponding to the free evolution of the incoming field without any coupling to the ring. We similarly define the outgoing field $\psi_{J>}(z,t)$ by taking $$\begin{aligned}
\psi_{J>}(0,t) = \psi_J(z,t)\;\;\mathrm{for}\; z>0, \end{aligned}$$ and demanding that for all $z$ $$\begin{aligned}
\left(\frac{\partial}{\partial t} + v_J\frac{\partial}{\partial z} + i\omega_J\right)\psi_{J>}(z,t) = 0,\end{aligned}$$ giving $\psi_{J>}(z,t)$ a false past to the left of the coupling point. By an identical procedure we may define the incoming and outgoing phantom channel fields $\phi_{J<}(z,t)$ and $\phi_{J>}(z,t)$. Since we will primarily be concerned with the properties of the photons generated in the ring, which exit to one of the channels and propagate to positive $z$, all calculations involving the ring’s output will be carried out on the outgoing fields $\psi_{J>}(z,t)$. Our goal is therefore to construct an explicit solution for these fields in terms of the incoming fields $\psi_{J<}(z,t)$. Indeed, since these fields freely propagate, the field at large positive $z$ (where any measurements on the generated photons would occur) is entirely determined by the outgoing field at $z=0$, $$\begin{aligned}
\psi_J(z,t) = e^{-i\omega_J z/v_J}\psi_{J>}(0,t-\nicefrac{z}{v_J})\;\;\mathrm{for}\;z>0.\end{aligned}$$ It therefore suffices to construct a solution for $\psi_{J>}(0,t)$, which can be very simply related to the incoming field $\psi_{J<}(0,t)$ and the corresponding ring operator $b_J(t)$ [@Vernon2015] via $$\begin{aligned}
\label{channel_transformation}
\psi_{J>}(0,t) = \psi_{J<}(0,t) - \frac{i\gamma_J}{v_J}b_J(t).\end{aligned}$$
For each operator $\mathcal{O}_J(t)$ it will be convenient to define the corresponding slowly-varying barred operator $\overline{\mathcal{O}}_J(t)$, $$\begin{aligned}
\overline{\mathcal{O}}_J(t) = e^{i\omega_J t}\mathcal{O}_J(t).\end{aligned}$$ In terms of these quantities and the incoming and outgoing fields, the equations for the ring mode annihilation operators ${\overline{b}}_J(t)$ are found to satisfy
\[ring\_eqns\_master\] $$\begin{aligned}
\left(\frac{d}{dt} + {\overline{\Gamma}}_P + 2i\eta{\overline{b}}_P^\dagger(t){\overline{b}}_P(t)\right){\overline{b}}_P(t) &=& -i\gamma_P^*{\overline{\psi}}_{P<}(0,t) - i\mu_{P}^*{\overline{\phi}}_{P<}(0,t) - 2i\Lambda^*{\overline{b}}_P^\dagger(t){\overline{b}}_S(t){\overline{b}}_I(t)e^{-i\Delta_\mathrm{ring} t}, \label{ring_pump_master} \\
\left(\frac{d}{dt} + {\overline{\Gamma}}_S + i\zeta{\overline{b}}_P^\dagger(t){\overline{b}}_P(t)\right){\overline{b}}_S(t) &=& -i\gamma_S^*{\overline{\psi}}_{S<}(0,t) - i\mu_S^*{\overline{\phi}}_{S<}(0,t) -i\Lambda{\overline{b}}_P(t){\overline{b}}_P(t){\overline{b}}_I^\dagger(t)e^{i\Delta_\mathrm{ring} t}, \label{ring_signal_master}\\
\left(\frac{d}{dt} + {\overline{\Gamma}}_I + i\zeta{\overline{b}}_P^\dagger(t){\overline{b}}_P(t)\right){\overline{b}}_I(t) &=& -i\gamma_I^*{\overline{\psi}}_{I<}(0,t) - i\mu_I^*{\overline{\phi}}_{I<}(0,t) -i\Lambda{\overline{b}}_P(t){\overline{b}}_P(t){\overline{b}}_S^\dagger(t)e^{i\Delta_\mathrm{ring} t}, \label{ring_idler_master}\end{aligned}$$
where we have introduced the ring mode detuning $$\begin{aligned}
\Delta_\mathrm{ring} = \omega_S + \omega_I - 2\omega_P,\end{aligned}$$ as well as the total effective linewidths ${\overline{\Gamma}}_J$, $$\begin{aligned}
{\overline{\Gamma}}_J=\Gamma_J + M_J,\end{aligned}$$ where $\Gamma_J$ and $M_J$ denote the damping rates associated with the physical channel and phantom channel couplings, respectively: $$\begin{aligned}
\Gamma_J &=& \frac{|\gamma_J|^2}{2v_J}, \nonumber \\
M_J &=& \frac{|\mu_J|^2}{2u_J}.\end{aligned}$$ These total damping rates can be simply related to the quality factors $Q_J$ of the resonator modes; for example, for the pump resonance $$\begin{aligned}
Q_P=\frac{\omega_P}{{\overline{\Gamma}}_P},\end{aligned}$$ which yields $Q_P\sim 10^6$ for a ring with ${\overline{\Gamma}}_P=1$ GHz given a pump with wavelength $\lambda=1550$ nm. The coupled set of driven, damped ordinary differential equations (\[ring\_eqns\_master\]) fully describes the nonlinear dynamics of the ring-channel system. Combined with the channel transformation (\[channel\_transformation\]), a solution to this system of equations permits the calculation of any measurable quantities on the outgoing photons in the channel.
It is important to note at this stage that our treatment neglects the effect of ring heating due to the large circulating pump power present in the ring. Such thermal effects are routinely observed in experimental investigations of microring systems, and typically manifest as an effective power-dependent drift in the resonant frequencies of the ring as it undergoes thermal expansion [@Levy2010]. For slowly varying and cw pumps a simple way to account for this is through the addition of a pump photon number-dependent correction to each resonance. Our model already incorporates a similar effect: SPM and XPM of each mode are represented by precisely such terms. The inclusion of thermal resonance drift can therefore be modelled by altering the coefficients $\eta$ and $\zeta$ in Eqs. (\[ring\_eqns\_master\]), which would be replaced by effective constants $\eta_{\mathrm{eff}}$ and $\zeta_\mathrm{eff}$, $$\begin{aligned}
\label{thermal_substitution}
\eta_\mathrm{eff} = \eta + \eta_\mathrm{thermal}, \nonumber \\
\zeta_\mathrm{eff} = \zeta + \zeta_\mathrm{thermal}.\end{aligned}$$ While $\eta$ and $\zeta$ are both positive, $\eta_\mathrm{thermal}$ and $\zeta_\mathrm{thermal}$ would be negative, since as the ring expands the resonant frequencies are typically lowered [@Almeida2004]. Depending on the relative magnitude of the thermal drift coefficients compared to the SPM and XPM strengths, in some circumstances $\eta_\mathrm{eff}$ and $\zeta_\mathrm{eff}$ may become negative. While for the remainder of this work we neglect thermal drift of the ring resonances, so that $\eta_\mathrm{thermal}=\zeta_\mathrm{thermal}=0$, we emphasize that our conclusions do not depend sensitively on this assumption unless otherwise stated.
Steady state pump solution {#sec:pump_dynamics}
==========================
The set of coupled equations (\[ring\_eqns\_master\]) treats both the pump and signal and idler modes quantum mechanically, retaining the operator nature of $b_J(t)$ for each $J$. While this is necessary if one wishes to fully account for the nonclassical properties of the pump mode, in typical experiments [@Azzini2012a; @Azzini2012] the system is pumped by a coherent laser beam or pulse. In such situations the initial pump state is described by setting each incoming pump mode to a coherent state. The pump field can then be well approximated by its expectation value, which is a classical function of time. To implement this semiclassical approximation we take $$\begin{aligned}
{\overline{b}}_P(t)\rightarrow {\overline{\beta}}_P(t) = \langle {\overline{b}}_P(t) \rangle.\end{aligned}$$ In addition to treating the pump classically, we also implement the undepleted pump approximation. In the equation for the ring pump mode (\[ring\_pump\_master\]) the term involving ${\overline{b}}_P^\dagger {\overline{b}}_S {\overline{b}}_I$ accounts for the effect on the pump mode when a signal-idler photon pair is produced. Neglecting such effects, we drop this term and instead take the semiclassical pump amplitude ${\overline{\beta}}_P(t)$ to satisfy $$\begin{aligned}
\label{intermediate_pump_eqn}
\bigg( \frac{d}{dt} + {\overline{\Gamma}}_P &+& 2i\eta|{\overline{\beta}}_P(t)|^2 \bigg){\overline{\beta}}_P(t)\nonumber \\ &=& -i\gamma_P^*\langle{\overline{\psi}}_{P<}(0,t)\rangle,\end{aligned}$$ in which we have assumed $\langle {\overline{\phi}}_{P<}(0,t) \rangle = 0$, so that there is no incoming pump energy in the phantom channel. Note that while this approximation amounts to neglecting pump depletion due to photon pair generation, linear pump losses are still accounted for in our model, as evidenced by the presence of the damping term ${\overline{\Gamma}}_P$ in Eq. (\[intermediate\_pump\_eqn\]).
In this work we consider the case of a continuous wave (cw) pump beam injected in to the channel, so that $$\begin{aligned}
\langle {\overline{\psi}}_{P<}(0,t) \rangle = \frac{p}{\gamma_P^*}e^{-i\Delta_P t},\end{aligned}$$ where $\Delta_P$ is the detuning of the injected pump from the ring pump resonance, and $p$ is a constant related to the input pump power $P_{\mathrm{in}}$ in the channel at the coupling point via $$\begin{aligned}
p = \sqrt{\frac{2\Gamma_P P_\mathrm{in}}{\hbar\omega_P}}.\end{aligned}$$ In steady state, after the ring pump mode has come to equilibrium with the channels, we expect there to be a constant average number of pump photons $N_P$ in the ring, where $$\begin{aligned}
N_P=\lim_{t\to\infty}|{\overline{\beta}}_P(t)|^2.\end{aligned}$$ Defining $\widetilde{\beta}_P(t) = e^{i\Delta_P t}{\overline{\beta}}_P(t)$, from Eq. (\[intermediate\_pump\_eqn\]) we have $$\begin{aligned}
\label{pump_tilde_eqn}
\left( \frac{d}{dt} + {\overline{\Gamma}}_P + i(2\eta|{\overline{\beta}}_P(t)|^2 - \Delta_P) \right)\widetilde{\beta}_P(t) = -ip.\end{aligned}$$ It is not difficult to show that $N_P$ will be constant only when $\widetilde{\beta}_P(t)$ has both constant amplitude and constant phase, so that $d\widetilde{\beta}_P(t)/dt=0$. Setting this time derivative to zero in the above equation and taking the modulus squared of the result, we find that in steady state $N_P$ must be a root of the cubic equation $$\begin{aligned}
\label{pump_cubic_root_eqn}
C_P(N_P)=0,\end{aligned}$$ where $$\begin{aligned}
\label{pump_cubic}
& &C_P(N_P) \equiv \nonumber \\ &4&\eta^2N_P^3 - 4\eta\Delta_P N_P^2 + ({\overline{\Gamma}}_P^2 + \Delta_P^2)N_P - |p|^2.\end{aligned}$$ In the absence of SPM (when $\eta \to 0$, or when the input power is very small), $N_P$ is related to the incoming power by a simple linear function, $$\begin{aligned}
N_P = \frac{|p|^2}{{\overline{\Gamma}}_P^2 + \Delta_P^2}.\end{aligned}$$ The presence of SPM, however, complicates the task of determining $N_P$ as a function of $|p|^2$ for a given detuning $\Delta_P$ and nonlinearity $\eta$. The cubic equation (\[pump\_cubic\_root\_eqn\]) has in general as many as three real, positive roots. Furthermore, only some of these may correspond to *stable* solutions of (\[intermediate\_pump\_eqn\]). Before solving for the roots of $C_P(N_P)$, we first derive a set of criteria to assess the stability of any such solution.
To determine whether or not a given root of (\[pump\_cubic\]) is stable, we conduct an analysis similar to that of Hoff, Nielsen and Andersen [@Andersen2015]. For a given constant solution $\widetilde{\beta}_P^{(0)}$ to (\[pump\_tilde\_eqn\]), we define the fluctuation amplitude $\delta\beta_P(t)$ via $$\begin{aligned}
\widetilde{\beta}_P(t) = \widetilde{\beta}_P^{(0)} + \delta\beta_P(t).\end{aligned}$$ Keeping terms up to first order in $\delta\beta_P$, the equations of motion satisfied by $\delta\beta_P(t)$ and $\delta\beta_P^*(t)$ can be written as $$\begin{aligned}
\frac{d}{dt}\begin{pmatrix}
\delta\beta_P(t) \\ \delta\beta_P^*(t)
\end{pmatrix} = F\begin{pmatrix}
\delta\beta_P(t) \\ \delta\beta_P^*(t)
\end{pmatrix},\end{aligned}$$ where $F$ is the $2\times 2$ matrix given by $$\begin{aligned}
\lefteqn{F =}\nonumber \\ &&\begin{pmatrix}
-{\overline{\Gamma}}_P - i(4\eta N_P - \Delta_P) & -2i\eta\left[\widetilde{\beta}_P^{(0)}\right]^2 \\
2i\eta\left[\widetilde{\beta}_P^{(0)*}\right]^2 & -{\overline{\Gamma}}_P + i(4\eta N_P - \Delta_P)
\end{pmatrix}.\end{aligned}$$ For a given solution to be stable, we require the real part of both eigenvalues of $F$ to be negative, so that the fluctuation term $\delta\beta_P(t)$ will decay with time. These eigenvalues are $$\begin{aligned}
f_\pm = -{\overline{\Gamma}}_P \pm \sqrt{4\eta^2N_P^2 - (4\eta N_P - \Delta_P)^2}.\end{aligned}$$ Now, $\mathrm{Re}(f_-)<0$ automatically; demanding that $\mathrm{Re}(f_+)<0$ yields the condition $$\begin{aligned}
4\eta^2 N_P^2 - (4\eta N_P - \Delta_P)^2 < {\overline{\Gamma}}_P^2.\end{aligned}$$ Solving this inequality, we find that any solution $N_P$ for Eq. (\[pump\_cubic\_root\_eqn\]) corresponds to a stable solution of Eq. (\[pump\_tilde\_eqn\]) if $|\Delta_P|$ is below a “critical detuning", $|\Delta_P| < \Delta_{\mathrm{critical}}$, where $$\begin{aligned}
\Delta_{\mathrm{critical}} = \sqrt{3}\;{\overline{\Gamma}}_P.\end{aligned}$$ When $|\Delta_P| > \Delta_{\mathrm{critical}}$, a solution $N_P$ of (\[pump\_cubic\_root\_eqn\]) corresponds to a stable solution of (\[pump\_tilde\_eqn\]) if and only if $N_P$ lies outside a certain interval, $N_P \notin (N_-,N_+)$, where $$\begin{aligned}
\label{stability_eqn}
N_{\pm} = \frac{1}{3\eta}\left(\Delta_P \pm \frac{1}{2}\sqrt{\Delta_P^2 - \Delta_\mathrm{critical}^2}\right).\end{aligned}$$
![(Colour online) Steady state average photon number in the ring pump mode as a function of channel input power with ${\overline{\Gamma}}_P=1\;\mathrm{GHz}$, $\eta=1\;\mathrm{Hz}$ for (a) zero detuning, (b) subcritical detuning, (c) supercritical detuning. Red and blue curves indicate stable solutions, green unstable. The dashed line respresents choice of optimal detuning to maximize $N_P$ at each input power ($\Delta_P=\Delta_P^\mathrm{opt}(N_P)=2\eta N_P$).[]{data-label="fig:photon_number"}](DZero.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Steady state average photon number in the ring pump mode as a function of channel input power with ${\overline{\Gamma}}_P=1\;\mathrm{GHz}$, $\eta=1\;\mathrm{Hz}$ for (a) zero detuning, (b) subcritical detuning, (c) supercritical detuning. Red and blue curves indicate stable solutions, green unstable. The dashed line respresents choice of optimal detuning to maximize $N_P$ at each input power ($\Delta_P=\Delta_P^\mathrm{opt}(N_P)=2\eta N_P$).[]{data-label="fig:photon_number"}](DSub.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Steady state average photon number in the ring pump mode as a function of channel input power with ${\overline{\Gamma}}_P=1\;\mathrm{GHz}$, $\eta=1\;\mathrm{Hz}$ for (a) zero detuning, (b) subcritical detuning, (c) supercritical detuning. Red and blue curves indicate stable solutions, green unstable. The dashed line respresents choice of optimal detuning to maximize $N_P$ at each input power ($\Delta_P=\Delta_P^\mathrm{opt}(N_P)=2\eta N_P$).[]{data-label="fig:photon_number"}](DSup.png "fig:"){width="1.0\columnwidth"}
Having established the stability criteria for a given $N_P$, we return to the task of finding real, positive roots of (\[pump\_cubic\_root\_eqn\]). While analytic expressions for the roots exist, it is more instructive to use indirect arguments to study their nature. Taking the derivative of $C_P$ with respect to $N_P$, we find that $dC_P/dN_P=0$ at $N_P=N_{\pm}$. Thus, for subcritical detunings ($|\Delta_P| < \Delta_\mathrm{critical})$, where the $N_{\pm}$ are not purely real, there are no local extrema – it is easy to show that a graph of $C_P(N_P)$ is monotonically increasing and intersects the $N_P$ axis only once, leading to a single real, positive root $N_P$ which corresponds to a stable solution. On the other hand, for supercritical detunings ($|\Delta_P| > \Delta_\mathrm{critical})$, the function $C_P(N_P)$ goes through a local maximum at $N_-$ and minimum at $N_+$. The number of times $C_P(N_P)$ intersects the $N_P$ axis is then determined by the power parameter $|p|^2$; varying it translates the graph of $C_P(N_P)$ vertically. If $C_P(N_-)>0$ and $C_P(N_+)<0$, the graph of the function intersects the $N_P$ axis three times, indicating the existence of three real, positive values of $N_P$. The outer two correspond to stable solutions, while the inner root is unstable. These multiple cases are illustrated in Fig. \[fig:photon\_number\], in which $N_P$ is plotted as a function of input power for various detunings. For values of $|\Delta_P|$ above the critical detuning of $\sqrt{3}\;{\overline{\Gamma}}_P$ there exists a region of optical bistability, in which two stable equilibrium average pump photon numbers for a given input power are permitted, a phenomenon that has been observed experimentally in microring systems [@Almeida2004]. In this region the two stable solutions are separated by an unstable (and therefore physically inaccessible) range of $N_P$. Also plotted in this figure is the case of “optimal detuning", $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$, in which $\Delta_P$ is not taken to be fixed, but chosen to exactly cancel the effect of SPM as $P_\mathrm{in}$ is increased, $$\begin{aligned}
\label{optimal_detuning}
\Delta_P=\Delta_P^{\mathrm{opt}}(N_P) = 2\eta N_P,\end{aligned}$$ which restores the simple linear relationship between $N_P$ and $|p|^2$, $$\begin{aligned}
N_P = \frac{|p|^2}{{\overline{\Gamma}}_P^2}.\end{aligned}$$ This behaviour is indicated by the dashed line in Fig. \[fig:photon\_number\], which corresponds to a stable pump solution for all input powers, always lies on or above the fixed-detuning curves, and at each input power corresponds to the choice of detuning that maximizes $N_P$.
Signal and idler dynamics {#sec:signal_idler_dynamics}
=========================
Having developed the steady state pump solution, we return to the signal and idler equations of motion. We first develop an exact solution to these equations, valid for a cw pump of arbitrary intensity, and then use this solution to calculate the photon pair generation rate, as well as the one- and two-photon spectra of the generated photons.
Exact solution
--------------
We begin by writing the equations (\[ring\_signal\_master\]–\[ring\_idler\_master\]) for the signal and idler ring operators in the presence of a classically described cw pump that leads to a ring pump amplitude of the form ${\overline{\beta}}_P(t) = {\overline{\beta}}_P e^{-i\Delta_P t}$, where ${\overline{\beta}}_P$ is a constant. Letting $\widetilde{b}_x(t) = e^{i\Delta_P t}{\overline{b}}_x(t)$ for $x=S,I$ we obtain $$\begin{aligned}
\frac{d}{dt}\begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} = M \begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} + D(t),\end{aligned}$$ where $M$ is the $2\times 2$ coupling matrix defined by $$\begin{aligned}
\label{M_defn}
\lefteqn{M =} \nonumber \\
& &\begin{pmatrix}
-{\overline{\Gamma}}_S - i(\zeta|{\overline{\beta}}_P|^2 - \Delta_P) & -i\Lambda {\overline{\beta}}_P^2 \\
i\Lambda{\overline{\beta}}_P^{*2} & -{\overline{\Gamma}}_I + i(\zeta|{\overline{\beta}}_P|^2 - \Delta_P)
\end{pmatrix},\;\;\;\end{aligned}$$ and $D(t)$ the driving term responsible for quantum fluctuations from the physical and phantom channels, $$\begin{aligned}
D(t) = \begin{pmatrix}
-ie^{i\Delta_P t}(\gamma_S^*{\overline{\psi}}_{S<}(0,t) + \mu_S^*{\overline{\phi}}_{S<}(0,t)) \\
ie^{-i\Delta_P t}(\gamma_I{\overline{\psi}}_{I<}^\dagger(0,t) + \mu_I{\overline{\phi}}_{I<}^\dagger(0,t)) \\
\end{pmatrix}.\end{aligned}$$ In obtaining $M$ we have assumed the ring resonances are equally spaced, so that $\Delta_\mathrm{ring}=\omega_S + \omega_I - 2\omega_P=0$; the pump detuning, however, is left arbitrary. Previously [@Vernon2015] we employed a perturbative approach in the frequency domain to solve these equations, while ignoring the effects of SPM and XPM. While this provides an adequate description of the pair generation process for low pump powers, a nonperturbative strategy is needed to treat the strongly driven case. In the cw regime, where $M$ is time-independent, this coupled set of linear ordinary differential equations can be solved exactly in the time domain for arbitrary pump intensities by taking $$\begin{aligned}
\begin{pmatrix}
\widetilde{b}_S(t) \\ \widetilde{b}_I^\dagger(t)
\end{pmatrix} = \int\limits_{-\infty}^{\;\;t} dt' G(t,t') D(t'),\end{aligned}$$ where the $2\times 2$ matrix Green function $G(t,t')$ is given by $$\begin{aligned}
G(t,t') &=& e^{\int_{t'}^t M dt''} = e^{M\cdot(t-t')} \nonumber \\
&=&\begin{pmatrix}
g_D(t,t') & g_A(t,t') \\ g_A^*(t,t') & g_D^*(t,t')
\end{pmatrix}.\end{aligned}$$ For simplicity we henceforth assume the ring-channel coupling constants and propagation speeds for each mode are the same, $\gamma_J=\gamma$, $v_J=u_J=v$ and $\mu_J=\mu$ so ${\overline{\Gamma}}_J={\overline{\Gamma}}$ for each $J$. The matrix elements $g_D$ and $g_A$ are then given by $$\begin{aligned}
&g_D&(t,t') = e^{-{\overline{\Gamma}}(t-t')} \\
\;\;\;\;&\times& \bigg(\cosh[{\overline{\rho}}(t-t')] - i\frac{\zeta N_P - \Delta_P}{{\overline{\rho}}}\sinh[{\overline{\rho}}(t-t')]\bigg) \nonumber\end{aligned}$$ and $$\begin{aligned}
g_A(t,t') = \frac{-i\Lambda{\overline{\beta}}_P^2}{{\overline{\rho}}}\sinh[{\overline{\rho}}(t-t')],\end{aligned}$$ in which we have introduced the dynamical parameter ${\overline{\rho}}$, $$\begin{aligned}
\label{rhobar_defn}
{\overline{\rho}}= \sqrt{\Lambda^2 N_P^2 - (\zeta N_P - \Delta_P)^2}.\end{aligned}$$ Depending on the pump photon number $N_P$ and detuning $\Delta_P$, ${\overline{\rho}}$ may be either purely real, purely imaginary, or exactly zero. Indeed, as will become clear in the following sections, ${\overline{\rho}}$ serves as an important parameter in characterizing the system’s behaviour.
With explicit solutions written down for the ring operators $\widetilde{b}_J(t)$, we can make use of the incoming-outgoing channel field relation (\[channel\_transformation\]) to determine ${\overline{\psi}}_{J>}(0,t)$. We find for the signal $$\begin{aligned}
\lefteqn{{\overline{\psi}}_{S>}(0,t) =} \\
\int dt' &\bigg[&q_{SS}(t,t'){\overline{\psi}}_{S<}(0,t') + p_{SS}(t,t'){\overline{\phi}}_{S<}(0,t') \nonumber \\
&+& q_{SI}(t,t'){\overline{\psi}}_{I<}^\dagger(0,t') + p_{SI}(t,t'){\overline{\phi}}_{I<}^\dagger(0,t')\bigg], \nonumber\end{aligned}$$ where we have introduced the temporal response functions $q_{xx'}(t,t')$ for the physical channel and $p_{xx'}(t,t')$ for the phantom channel: $$\begin{aligned}
& &q_{SS}(t,t') = \delta(t-t') \nonumber \\
&-& \frac{|\gamma|^2}{v}\theta(t-t')e^{-({\overline{\Gamma}}+ i\Delta_P)(t-t')} \\
&\times& [\cosh[{\overline{\rho}}(t-t')] - i\frac{\zeta|{\overline{\beta}}_P|^2 - \Delta_P}{{\overline{\rho}}}\sinh[{\overline{\rho}}(t-t')], \nonumber\end{aligned}$$ and $$\begin{aligned}
\lefteqn{q_{SI}(t,t') =} \\
& &\frac{-\gamma^2\Lambda{\overline{\beta}}_P^2}{v{\overline{\rho}}}\theta(t-t')e^{-i\Delta_P(t+t')}e^{-{\overline{\Gamma}}(t-t')}\sinh[{\overline{\rho}}(t-t')]. \nonumber\end{aligned}$$ The phantom channel response functions are related to these via $$\begin{aligned}
p_{SS}(t,t') = \frac{\mu^*}{\gamma^*}(q_{SS}(t,t') - \delta(t-t'))\end{aligned}$$ and $$\begin{aligned}
p_{SI}(t,t') = \frac{\mu}{\gamma}q_{SI}(t,t').\end{aligned}$$ Similar response functions $p_{Ix}(t,t')$ and $q_{Ix}(t,t')$ can be introduced for the idler fields, which, due to our assumption of equal coupling coefficients and propagation speeds for the signal and idler fields, are identical to those for the signal: $p_{IS} = p_{SI}$, $p_{II}=p_{SS}$, $q_{IS}=q_{SI}$ and $q_{II}=q_{SS}$.
Photon generation rate {#sec::pair_gen_rates}
----------------------
Armed with explicit expressions for the outgoing fields ${\overline{\psi}}_{S>}$ and ${\overline{\psi}}_{I>}$, we can calculate any measurable quantity related to the generated signal and idler photon pairs. Of particular interest is the photon pair generation rate, one of the primary figures of merit used in assessing the practical utility of the ring-channel system. The steady state outgoing flux of signal photons $J_S$ into the physical channel can be calculated via $$\begin{aligned}
J_S &=& \lim_{t\to\infty} v \langle {\overline{\psi}}_{S>}^\dagger(0,t){\overline{\psi}}_{S>}(0,t)\rangle \nonumber \\
&=& \lim_{t\to\infty} \frac{2{\overline{\Gamma}}}{\Gamma}\int dt' |q_{SI}(t,t')|^2.\end{aligned}$$ Computing the integral, we find $$\begin{aligned}
J_S = \frac{2\Gamma\Lambda^2 N_P^2}{{\overline{\Gamma}}^2 - {\overline{\rho}}^2}.\end{aligned}$$ The nature of the scaling of $J_S$ with pump photon number $N_P$ depends intimately on the character of ${\overline{\rho}}$, the behaviour of which as a function of $N_P$ for various detunings is illustrated in Fig. \[fig::rhobar\_plot\]. Recalling (\[rhobar\_defn\]), we find that ${\overline{\rho}}$ is real when $N_P \in [\Delta_P/3\Lambda,\Delta_P/\Lambda]$, with ${\overline{\rho}}=0$ at the endpoints of this interval, and imaginary otherwise.
![(Colour online) Real and imaginary parts of ${\overline{\rho}}$ as a function of $N_P$ for zero, subcritical and supercritical detunings, indicating that ${\overline{\rho}}$ is always either purely real or purely imaginary. The red line indicates ${\overline{\rho}}={\overline{\Gamma}}$. The transition between ${\overline{\rho}}$ being purely imaginary and purely real occurs at the points $N_P=\Delta_P/3\Lambda$ and $N_P=\Delta_P/\Lambda$. For supercritical detunings, there exist points where ${\overline{\rho}}={\overline{\Gamma}}$ (represented on the plot by red diamonds), indicating the onset of OPO behaviour. The nonlinear parameter $\Lambda$ taken as $\Lambda=10$ Hz.[]{data-label="fig::rhobar_plot"}](Rhobar_plot.png){width="1.0\columnwidth"}
![(Colour online) Photon pair generation rate as a function of channel input power for various detunings. The dashed curve indicates the optimal detuning case. System parameters for this plot are $\eta=1\;\mathrm{Hz}$, ${\overline{\Gamma}}=1\;\mathrm{GHz}$.[]{data-label="fig:pair_rate_plot"}](JSplot.png){width="1.0\columnwidth"}
For low enough $N_P$, when ${\overline{\rho}}\approx i|\Delta_P|$, $J_S$ scales quadratically with the number of pump photons $N_P$. Since in this regime $N_P$ is directly proportional to the channel input power $P_{\mathrm{in}}$, the overall scaling of $J_S$ with $P_{\mathrm{in}}$ remains quadratic, in agreement with experiment [@Azzini2012]. As the pump power increases, however, the scaling of $J_S$ is affected by several separate power-dependent processes.
First, for a fixed detuning $\Delta_P$, the SPM-induced drift of the pump resonance slows the scaling of $N_P$ with channel input power $P_\mathrm{in}$, as demonstrated in Fig. \[fig:photon\_number\]. Second, XPM between the pump, signal and idler modes effectively shifts the resonance lines of the signal and idler modes, compromising the resonance enhancement of the pair generation process. Finally, for supercritical detunings $|\Delta_P|>\Delta_\mathrm{critical}$, ${\overline{\rho}}\to{\overline{\Gamma}}$ when $N_P\to N_{\pm}$, where $N_{\pm}$ are the same two photon numbers that define the stability of the pump solution (\[stability\_eqn\]). In that limit the photon flux $J_S$ formally diverges. This unphysical prediction corresponds to the onset of optical parametric oscillation[@Levy2010; @Razzari2010]. As this threshold is approached, stimulated emission leads to photon pairs being generated faster than the rate at which they are removed from the ring, preventing the system from reaching a steady state within our model. We are prevented from treating this case by our assumption of an undepleted pump. Our results are expected to be valid when the intraring conversion efficiency $E_\mathrm{ring}$ is much less than unity; this efficiency, defined as the ratio between the steady state signal (or idler) and pump photon numbers, can be expressed as $$\begin{aligned}
E_\mathrm{ring} = \frac{J_S}{2\Gamma N_P} = \frac{\Lambda^2N_P}{{\overline{\Gamma}}^2-{\overline{\rho}}^2}.\end{aligned}$$ While in future work we intend to investigate the OPO regime and the associated effects of pump depletion, for the time being we restrict ourselves to regimes where $E_\mathrm{ring} \ll 1 $; in all examples presented below this inequality is satisfied.
Perhaps most remarkable is the special regime of optimal detuning, wherein $\Delta_P$ is chosen to maximize $N_P$ at each channel input power, $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$ as defined in Eq. (\[optimal\_detuning\]). For this choice ${\overline{\rho}}=0$ identically for all $N_P$, $$\begin{aligned}
\label{rhobar_special}
{\overline{\rho}}&=& \sqrt{\Lambda^2 N_P^2 - (\zeta N_P - \Delta_P)^2} \nonumber \\
&=& \sqrt{\Lambda^2 N_P^2 - \left(2\Lambda N_P - 2\frac{\Lambda}{2}N_P\right)^2} \nonumber \\
&=& 0.\end{aligned}$$ The photon pair flux then maintains its quadratic scaling with both $N_P$ and channel input power $P_{\mathrm{in}}$ over its entire domain: $$\begin{aligned}
J_S = \frac{8\Gamma^3\Lambda^2}{(\hbar\omega_P)^2{\overline{\Gamma}}^6}P_{\mathrm{in}}^2.\end{aligned}$$ This cancellation between the effects arising from photon pair generation, XPM, and the SPM-dependent detuning strategy $\Delta_P^\mathrm{opt}(N_P)$ arises from the simple fixed relationship between the associated nonlinear coupling strengths $\Lambda$, $\eta$ and $\zeta$. Crucial for this phenomenon is that the strength of the photon pair generation process scale quadratically with the pump photon number $N_P$. This cancellation effect would therefore not be possible using, for example, spontaneous parametric downconversion, the strength of which would scale linearly with $N_P$. The presence of thermal resonance drift would not compromise the existence of an $N_P$-dependent detuning strategy that yields ${\overline{\rho}}=0$ over all $N_P$, though such a strategy would no longer correspond to that which also linearizes and maximizes the relationship between $N_P$ and channel input power.
The photon pair generation rate as a function of channel input power is plotted in Fig. \[fig:pair\_rate\_plot\] for various values of $\Delta_P$ alongside this optimal detuning case. For lower powers, when SPM and XPM are negligible, a pump beam with $\Delta_P=0$ gives the best scaling of $J_S$. For intermediate powers the detuning may be tweaked to combat SPM and XPM in order to maximize $J_S$, while for high powers the optimal detuning strategy of $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$ beats any fixed subcritical detuning. The behaviour of these curves suggests a simple experiment to identify the presence of nonperturbative, strongly driven effects: one could simply measure the outgoing signal or idler power as a function of pump input power for a set of fixed, subcritical pump detunings. For fixed nonzero detunings $\Delta_P<\Delta_\mathrm{critical}$, strongly driven effects are indicated by the presence of a global maximum of generated signal power at intermediate pump input power, followed by decreasing signal power approaching an asymptotic value of $$\begin{aligned}
\hbar\omega_S\lim_{N_P\to\infty} J_S(N_P) = \hbar\omega_S\frac{2\Gamma}{3}.\end{aligned}$$ For critically coupled ring systems $\Gamma\approx{\overline{\Gamma}}/2$ [@Vernon2015], so the asymptotic signal power can be related to the total effective ring linewidth ${\overline{\Gamma}}$ as simply $\hbar\omega_S{\overline{\Gamma}}/3$. If thermal detuning of the ring resonances is included this asymptotic power will be different; however, the qualitative behaviour of the signal power as a function of pump power will be unchanged.
Single photon spectrum
----------------------
Another physical quantity of interest is the spectral lineshape of signal and idler photons that are emitted from the ring. For low power cw pumps these single photon spectra typically exhibit a Lorentzian lineshape [@Azzini2012; @Azzini2012a; @Vernon2015] with a characteristic width determined by the total effective linewidths of the microring cavity resonances. As we now demonstrate, these spectral characteristics are significantly different in the strongly pumped regime. We develop results for the signal field spectrum; the idler field will have identical properties.
![(Colour online) Origin of splitting in the signal and idler lineshapes. Green, blue, and red curves indicate pump, signal and idler resonances, respectively, and could respresent the enhancement factor [@Heebner2008] that would characterize the ratio of the intensity in the ring to the incident channel intensity in a linear experiment. The dashed green line indicates a pump detuned by $\Delta_P$, which leads to a generated pair having either its signal or idler photon detuned by $\sim 2\Delta_P$ from the corresponding resonance, as indicated by the signal and idler pairs connected by dashed black lines. The presence of pairs from both cases leads to a doublet structure for both the signal and idler lineshapes.[]{data-label="fig::splitting"}](Splitting.png){width="1.0\columnwidth"}
We define the power spectrum [@Mandel1995] for the signal channel field as $$\begin{aligned}
\label{nu_S_defn1}
\lefteqn{\nu_S(\omega_s; t) =} \\
& & \lim_{T\to\infty}\frac{1}{T}\int\displaylimits_{t-T/2}^{t+T/2} dt \int \frac{d\tau}{\sqrt{2\pi}} g^{(1)}(t,t+\tau) e^{i\omega_s \tau} \nonumber\end{aligned}$$ where the first-order temporal coherence function $g^{(1)}(t_1,t_2)$ is defined by $$\begin{aligned}
\label{g1_defn}
g^{(1)}(t_1,t_2) &=& v \langle {\overline{\psi}}_{S>}^\dagger(0,t_1){\overline{\psi}}_{S>}(0,t_2)\rangle \nonumber \\
&=& \frac{{\overline{\Gamma}}}{\Gamma}\int dt' q_{SI}^*(t_1,t')q_{SI}(t_2,t').\end{aligned}$$ In writing (\[nu\_S\_defn1\]) we have introduced the relative frequency co-ordinate $\omega_s$, which corresponds to a frequency offset from the ring reference $\omega_S$. The physical frequency $\overline{\omega}_s$ associated with $\omega_s$ is therefore $$\begin{aligned}
\overline{\omega}_s = \omega_S + \omega_s.\end{aligned}$$ In the remaining sections we adopt this notation of lowercase subscripts for frequency offsets: $\omega_s$ for the signal, $\omega_p$ for the pump and $\omega_i$ for the idler.
{width="1.0\columnwidth"} {width="1.0\columnwidth"} {width="1.0\columnwidth"} {width="1.0\columnwidth"}
Evaluating (\[g1\_defn\]) and setting $t_1=t$, $t_2=t+\tau$ we obtain $$\begin{aligned}
\lefteqn{g^{(1)}(t,t+\tau) =}\\
& & \frac{\Gamma\Lambda^2 N_P^2}{{\overline{\rho}}}e^{i\Delta\tau}e^{-{\overline{\Gamma}}|\tau|}\frac{{\overline{\rho}}\cosh[{\overline{\rho}}|\tau|] + {\overline{\Gamma}}\sinh[{\overline{\rho}}|\tau|]}{{\overline{\Gamma}}^2 - {\overline{\rho}}^2}\nonumber,\end{aligned}$$ which is independent of $t$, depending only on the relative time difference $\tau$, as would be expected for a cw pump. Taking the Fourier transform, we arrive at an expression for the lineshape, $$\begin{aligned}
\label{nu_S_defn}
\lefteqn{\nu_S(\omega_s)=} \\
& & \frac{4\Gamma{\overline{\Gamma}}\Lambda^2 N_P^2}{\sqrt{2\pi}|{\overline{\Gamma}}- {\overline{\rho}}+i(\omega_s - \Delta_P)|^2|{\overline{\Gamma}}+{\overline{\rho}}+i(\omega_s-\Delta_P)|^2} \nonumber,\end{aligned}$$ with an identical equation for the idler lineshape $\nu_I(\omega_i)$. This expression takes the form of a product of two Lorentzians. We consider first subcritical detunings. When ${\overline{\rho}}$ is imaginary, these Lorentzians have identical characteristic widths $\delta\omega={\overline{\Gamma}}$ and are centred on $\omega_s=\Delta_P\pm|{\overline{\rho}}|$. For low powers, when ${\overline{\rho}}\approx i|\Delta_P|$ the spectrum is therefore peaked at $\omega_s=0$ and $\omega_s=2\Delta_P$, in agreement with the perturbative calculation [@Vernon2015]. This splitting is easily understood as a consequence of the tradeoff between energy conservation and resonance enhancement of the pair generation process. As illustrated in Fig. \[fig::splitting\], when a photon pair is produced with a detuned pump, either the signal photon *or* idler photon in a pair, but not both, can be generated within a ring resonance; energy conservation then requires the other to be generated with a frequency that lies away from its corresponding resonance. This is seen in the $N_P\to 0$ limit of Fig. \[fig::lineshapes\](b). At sufficient pump photon number $N_P$ a similar splitting can arise from the effective XPM-induced detuning of the signal and idler ring resonances even for a pump with $\Delta_P=0$, as seen in Fig. \[fig::lineshapes\](a) for large $N_P$. When $\Delta_P=0$ the lineshape begins as a singly-peaked Lorentzian, eventually splitting to a doublet structure when ${\overline{\rho}}$ becomes imaginary as a consequence of XPM.
For nonzero $\Delta_P$, as $N_P$ increases, XPM effectively counters the pump detuning and the extent of this splitting is reduced as $|{\overline{\rho}}|$ decreases, eventually vanishing when $N_P=\Delta_P/3\Lambda$. If $N_P$ is increased further, ${\overline{\rho}}$ becomes real and ceases to contribute to spectral splitting, resulting instead in an effective correction to the linewidth. The lineshape then takes the form of a product of two Lorentzians, both centred on $\omega_s=\Delta_P$, with respective widths $\delta\omega_\pm = {\overline{\Gamma}}\pm {\overline{\rho}}$. As ${\overline{\rho}}$ becomes comparable to ${\overline{\Gamma}}$ the smaller of these two widths becomes dominant, leading to a lineshape with overall effective width $\delta\omega\approx{\overline{\Gamma}}-{\overline{\rho}}$. For subcritical detunings, as demonstrated at large $N_P$ in Fig. \[fig::lineshapes\](b), the spectral splitting is then resumed as ${\overline{\rho}}$ once again becomes imaginary. For supercritical detunings, as the threshold for optical parametric oscillation is approached ${\overline{\rho}}\to{\overline{\Gamma}}$ and the bandwidth of the emitted signal and idler photons becomes arbitrarily narrow, as seen in Fig. \[fig::lineshapes\](c). This follows from our idealization of the pump as an indefinitely coherent cw beam; in actual experiments the bandwidth of the generated photons will become comparable to that of the pump, a phenomenon that has been observed in strongly pumped experiments on silicon nitride microrings [@Levy2010].
Finally, as shown in Fig. \[fig::lineshapes\](d), in the special case of optimal detuning when $\Delta_P=\Delta_P^\mathrm{opt}(N_P)$, so that ${\overline{\rho}}=0$, the lineshape remains peaked at a single $N_P$-dependent frequency for each $N_P$, with unchanging characteristic width $\delta\omega={\overline{\Gamma}}$, precisely mimicking the low-power result at zero detuning.
Experimentally, measuring the signal or idler lineshape as a function of input power for a nonzero, subcritical detuning as in Fig. \[fig::lineshapes\](b) would reveal the richness of the strongly driven regime, and illustrate the behaviour of the ${\overline{\rho}}$ paramater, which incorporates the effects of both XPM and pair generation.
Joint spectral intensity
------------------------
To assess the degree of spectral correlation between the signal and idler modes, it is instructive to study the joint spectral intensity distribution of the generated photon pairs. While it is straightforward to define this quantity for a system driven by a train of weak pump pulses, in which multi-pair generation can be neglected, it is a more subtle task to craft a sensible measure of spectral correlation in the strongly driven cw regime. In particular, there is no single function that characterizes a joint probability amplitude of signal and idler photons, since in general there will be far more than two photons in the quantum state of the signal and idler modes. Furthermore, even for weak cw pumps, if one introduces outgoing channel annihilation operators $c_J(\omega_j)$ via $$\begin{aligned}
\label{amplitude_defn}
{\overline{\psi}}_{J>}(0,t) = \int \frac{d\omega_j}{\sqrt{2\pi}}c_J(\omega_j)e^{-i\omega_j t}\end{aligned}$$ and naively calculates expectation values of the form $\langle c_S^\dagger(\omega_s)c_I^\dagger(\omega_i)c_S(\omega_s)c_I(\omega_i)\rangle$, the idealization of a zero-bandwidth cw pump leads to ill-defined expressions involving the square of Dirac $\delta$ distributions.
To resolve these difficulties, in Appendix \[appendix:JSI\] we develop a model of a typical experiment used to characterize the JSI for weakly driven systems, in which the coincidence count rate of signal and idler photons at respective frequencies $\omega_s$ and $\omega_i$ is measured. We the extend the definition of the JSI to strongly driven systems by defining the JSI to equal the calculated outcome of such an experiment for arbitrary input power. This definition reduces to the usual result for single-pair output states, and serves as a sensible measure of spectral correlation between the signal and idler fields. This coincidence rate can be written as $$\begin{aligned}
\label{JSI_definition}
I_\mathrm{corr}(\omega_s,\omega_i) &=& \frac{v^2\delta t}{(2\pi)^2}\int dt_1...\int dt_4 \\
&\bigg[& e^{i\omega_s(t_3-t_1)}e^{i\omega_i(t_4-t_2)}T(t_1)T(t_2)T(t_3)T(t_4)\nonumber \\
&\times&\langle {\overline{\psi}}_{S>}^\dagger(t_1){\overline{\psi}}_{I>}^\dagger(t_2){\overline{\psi}}_{S>}(t_3){\overline{\psi}}_{I>}(t_4)\rangle\bigg],\nonumber\end{aligned}$$ where $T(t)$ is the Fourier transform of a transmission function $\hat{T}(\omega)$ that resolves the frequencies of the signal and idler photons prior to detection, $$\begin{aligned}
\label{T_defn}
T(t) = \int \frac{d\omega}{\sqrt{2\pi}}\hat{T}(\omega)e^{-i\omega t},\end{aligned}$$ and $\delta t$ is the temporal resolution of the coincidence counter. In this expression the spatial dependence of the field operators ${\overline{\psi}}_{J>}(z,t)$ has been suppressed; the signal and idler arms of the JSI measurement are assumed to occur at balanced distances from the ring-channel coupling point.
The four-time expectation value $\langle {\overline{\psi}}_{S>}^\dagger(t_1){\overline{\psi}}_{I>}^\dagger(t_2){\overline{\psi}}_{S>}(t_3){\overline{\psi}}_{I>}(t_4)\rangle$ is found to naturally split into two parts, $$\begin{aligned}
&v&^2\langle {\overline{\psi}}_{S>}^\dagger(t_1){\overline{\psi}}_{I>}^\dagger(t_2){\overline{\psi}}_{S>}(t_3){\overline{\psi}}_{I>}(t_4)\rangle = \\
& &A^*(t_1,t_2)A(t_3,t_4) + g^{(1)}(t_1,t_3)g^{(1)}(t_2,t_4),\nonumber\end{aligned}$$ where $$\begin{aligned}
A(t_1,t_2) = \int dt'& \bigg[&q_{SI}(t_1,t')q_{II}(t_2,t') \\
&+& p_{SI}(t_1,t')p_{II}(t_2,t')\bigg].\nonumber\end{aligned}$$ The function $g^{(1)}$ is precisely the first-order coherence function defined in Eq. (\[g1\_defn\]) used to calculate the single photon spectrum, $$\begin{aligned}
g^{(1)}(t_1,t_3) &=& \frac{\Gamma\Lambda^2 N_P^2}{{\overline{\rho}}}e^{i\Delta_P(t_3-t_1)}e^{-{\overline{\Gamma}}|t_3-t_1|} \\
&\times&\frac{{\overline{\rho}}\cosh[{\overline{\rho}}|t_3-t_1|] + {\overline{\Gamma}}\sinh[{\overline{\rho}}|t_3-t_1|]}{{\overline{\Gamma}}^2-{\overline{\rho}}^2}.\nonumber\end{aligned}$$ The $A(t_1,t_2)$ term, after computing the integrals, is given by $$\begin{aligned}
A(t_1,t_2)&=&\frac{\gamma^2\Lambda{\overline{\beta}}_P^2}{2v}e^{-{\overline{\Gamma}}|t_2-t_1|}e^{-i\Delta_P(t_1+t_2)} \\
&\times&\frac{[a_1\sinh[{\overline{\rho}}|t_2-t_1|] + a_2\cosh[{\overline{\rho}}|t_2-t_1|}{{\overline{\Gamma}}^2-{\overline{\rho}}^2},\nonumber\end{aligned}$$ where the constants $a_1$ and $a_2$ are defined by $$\begin{aligned}
a_1 &=& {\overline{\rho}}- i\frac{\zeta N_P -\Delta_P}{{\overline{\rho}}}{\overline{\Gamma}}, \nonumber \\
a_2 &=& {\overline{\Gamma}}-i(\zeta N_P - \Delta_P).\end{aligned}$$ The JSI can therefore be expressed as the sum of correlated and uncorrelated terms, $$\begin{aligned}
I(\omega_s,\omega_i) = I_{\mathrm{corr}}(\omega_s,\omega_i) + I_\mathrm{uncorr}(\omega_s,\omega_i),\end{aligned}$$ where $$\begin{aligned}
\label{phi_corr_defn}
\lefteqn{I_{\mathrm{corr}} (\omega_s,\omega_i) =} \\
& & \frac{\delta t}{(2\pi)^2}\bigg\vert\int d\nu_1 \int d\nu_2\hat{A}(\nu_1,\nu_2) \hat{T}(\omega_s-\nu_1)\hat{T}(\omega_i-\nu_2)\bigg \vert^2 \nonumber \end{aligned}$$ and $$\begin{aligned}
\label{phi_uncorr_defn}
\lefteqn{I_\mathrm{uncorr}(\omega_s,\omega_i) =\frac{\delta t}{(2\pi)^2}} \\
&\times&\int d\nu_1 \int d\nu_2\left[\hat{g}^{(1)}(\nu_1,-\nu_2)\hat{T}(\omega_s-\nu_1)\hat{T}(\omega_s - \nu_2)\right] \nonumber \\
&\times& \int d\nu_1' \int d\nu_2'\left[\hat{g}^{(1)}(\nu_1',-\nu_2')\hat{T}(\omega_i-\nu_1')\hat{T}(\omega_i - \nu_2')\right]. \nonumber\end{aligned}$$ As indicated by their labels, $I_\mathrm{uncorr}$ can be expressed as a separable product of functions of $\omega_s$ and $\omega_i$, while $I_\mathrm{corr}$ cannot. Each is expressed as a convolution of the Fourier transforms $\hat{A}(\nu_1,\nu_2)$ and $\hat{g}^{(1)}(\nu_1,\nu_2)$ of the $A(t_1,t_2)$ and $g^{(1)}(t_1,t_2)$ functions, $$\begin{aligned}
\hat{A}(t_1,t_2)=\int \frac{dt_1}{\sqrt{2\pi}} \int \frac{dt_2}{\sqrt{2\pi}}A(t_1,t_2)e^{i\nu_1t_1}e^{i\nu_2t_2},\end{aligned}$$ and similarly for $g^{(1)}(\nu_1,\nu_2)$, with the transmission filter function $\hat{T}(\nu_1)\hat{T}(\nu_2)$. The $A$ and $g^{(1)}$ functions are determined by the dynamics of the signal and idler modes in the ring, while their convolution with the $T$ functions reflects the frequency averaging that arises from the finite resolution of a realistic JSI measurement.
Computing the Fourier transform, we find for $\hat{A}$
$$\begin{aligned}
\hat{A}(\nu_1,\nu_2) = \Gamma^2\Lambda^2 N_P^2 \delta(\nu_1+\nu_2-2\Delta_P)\left(\frac{1-i\frac{\zeta N_P-\Delta_P}{{\overline{\rho}}}}{(i\Delta\nu - {\overline{\Gamma}}+ {\overline{\rho}})(-i\Delta\nu-{\overline{\Gamma}}+{\overline{\rho}})} + \frac{1+i\frac{\zeta N_P-\Delta_P}{{\overline{\rho}}}}{(i\Delta\nu - {\overline{\Gamma}}- {\overline{\rho}})(-i\Delta\nu-{\overline{\Gamma}}-{\overline{\rho}})}\right),\end{aligned}$$
with $\Delta\nu = (\nu_1 - \nu_2)/2$. The term in parentheses multiplying the $\delta$ function varies on the scale of ${\overline{\Gamma}}$. Assuming that the measurement frequency resolution $\delta\omega_\mathrm{trans}$ is much narrower than this, the slowly varying term can be pulled out of the integrals in (\[phi\_corr\_defn\]), leaving $$\begin{aligned}
\label{phi_corr_expression}
I_\mathrm{corr}(\omega_s,\omega_i) &\approx& \delta t\Gamma^2\Lambda^2N_P^2 [D(\omega_s-\Delta_P,\omega_i-\Delta_P)]^2 \\
&\times& \bigg\vert\frac{1-i\frac{\zeta N_P-\Delta_P}{{\overline{\rho}}}}{(i(\omega_i-\Delta_P) - {\overline{\Gamma}}+ {\overline{\rho}})(-i(\omega_i-\Delta_P)-{\overline{\Gamma}}+{\overline{\rho}})} + \frac{1+i\frac{\zeta N_P-\Delta_P}{{\overline{\rho}}}}{(i(\omega_i-\Delta_P) - {\overline{\Gamma}}- {\overline{\rho}})(-i(\omega_i-\Delta_P)-{\overline{\Gamma}}-{\overline{\rho}})}\bigg\vert^2 \nonumber\end{aligned}$$
where $$\begin{aligned}
\lefteqn{D(\omega_s,\omega_i) =} \\
& &\frac{1}{2\pi}\int d\nu_1 \int d\nu_2 \delta(\nu_1+\nu_2)\hat{T}(\omega_s-\nu_1)\hat{T}(\omega_i-\nu_2).\nonumber\end{aligned}$$ The function $D(\omega_s,\omega_i)$ can be interpreted as the “smoothed" version of the Dirac $\delta(\omega_s + \omega_i)$ distribution, and arises from the finite bandwidth of the JSI measurement scheme; $D(\omega_s-\Delta_P,\omega_i-\Delta_P)$ is sharply peaked and uniform along the energy-conserving antidiagonal line $\omega_s+\omega_i-2\Delta_P=0$ with characteristic width $\delta\omega_\mathrm{trans}$ (the measurement resolution) in the direction orthogonal to that line.
Finally, taking the Fourier transform of $g^{(1)}(t_1,t_3)$, we find $$\begin{aligned}
\lefteqn{\hat{g}^{(1)}(\nu_1,-\nu_2) = \delta(\nu_1-\nu_2) }\\
&\times& \frac{4{\overline{\Gamma}}\Gamma\Lambda^2N_P^2}{|{\overline{\Gamma}}-{\overline{\rho}}+i(\nu_1-\Delta_P)|^2|{\overline{\Gamma}}+{\overline{\rho}}+i(\nu_1-\Delta_P)|^2}.\nonumber\end{aligned}$$ As with $\hat{A}$, apart from the $\delta$ function this is slowly varying compared to the measurement resolution; the term multiplying the $\delta$ function can be pulled out of the integral in Eq. (\[phi\_uncorr\_defn\]). The uncorrelated contribution to the JSI $I_\mathrm{uncorr}$ is therefore well approximated by $$\begin{aligned}
\label{phi_uncorr_expression}
I_\mathrm{uncorr}(\omega_s,\omega_i)\approx \frac{\delta t}{2\pi}\bigg\vert \int d\omega|\hat{T}(\omega)|^2\bigg\vert^2\nu_S(\omega_s)\nu_I(\omega_i),\;\;\;\end{aligned}$$ where $\nu_S(\omega_s)$ and $\nu_I(\omega_i)$ are precisely the single-photon lineshape functions given by Eq. (\[nu\_S\_defn\]) as derived in the previous section. The uncorrelated part of the JSI is therefore proportional to the simple product the signal and idler lineshapes.
For low power cw pumps, wherein multi-pair generation is insignificant, the uncorrelated part of the JSI $I_\mathrm{uncorr}$ is negligible and $I_\mathrm{corr}$ dominates. The JSI then takes the form of a narrow antidiagonal line corresponding to the energy-conserving condition $\omega_s+\omega_i-2\Delta_P=0$. For $\Delta_P=0$, the line is singly peaked, as illustrated in Fig. \[fig::JSI\_splitting\_plots\](a). For nonzero $\Delta_P$ the line is distributed among two peaks separated by $2\Delta_P$, as evident in Fig. \[fig::JSI\_splitting\_plots\](b), consistent with the single photon spectrum derived in the previous section. At higher powers, such as in Fig. \[fig::JSI\_splitting\_plots\](c), this splitting can also arise from XPM-induced signal and idler detuning even for a pump with $\Delta_P=0$. When the splitting is due to XPM-induced signal and idler detuning, the JSI remains centred on the unperturbed ring resonances. On the other hand, when pump detuning is responsible for the splitting, the JSI is translated by $\Delta_P$ along both frequency axes.
In Fig. \[fig::JSI\_uncorrelated\_piece\] the uncorrelated contribution $I_\mathrm{uncorr}$ to the JSI is plotted for the same pump parameters as in Fig. \[fig::JSI\_splitting\_plots\]. The weight of the uncorrelated contribution is extremely small compared to the correlated contribution at low powers, as indicated by the scales in Figs. \[fig::JSI\_splitting\_plots\] and \[fig::JSI\_uncorrelated\_piece\], but grows to an appreciable level at high powers. For $\Delta_P=0$, as in Fig. \[fig::JSI\_uncorrelated\_piece\](a), at low $N_P$ the uncorrelated part of the JSI displays a single peak centred at the origin. In the regimes that give rise to split lineshapes, as illustrated in Fig. \[fig::JSI\_uncorrelated\_piece\](b) and \[fig::JSI\_uncorrelated\_piece\](c), the uncorrelated contribution takes the form of four distinct peaks, symmetrically placed about the centre of the overall distribution. Two of these peaks lie on the antidiagonal, overlapping with the correlated contribution. The remaining two lie on the diagonal, and would therefore appear to violate energy conservation if assumed to correspond to signal and idler photons that originated from the same pair. It is therefore natural to interpret these peaks as corresponding to signal and idler photons that are detected from *separate* pairs. As these uncorrelated, “non-energy conserving" peaks are well separated from the correlated part of the JSI, they are uncontaminated by the correlated contribution to the JSI. The properties of photon pairs detected in these peaks would therefore be expected to differ from those detected in the antidiagonal peaks. We intend to investigate such properties in future work.
The form of the JSI depends qualitatively on whether ${\overline{\rho}}$ is imaginary or real, a behaviour we saw earlier in the single photon spectrum. When ${\overline{\rho}}$ is imaginary, and thus contributes to the frequency terms in the denominators of Eqs. () and (), a splitting in the JSI appears. When ${\overline{\rho}}$ is real, and acts as an effective correction to the linewidth ${\overline{\Gamma}}$, the JSI is localized to a single line arising from the overlap of $I_\mathrm{corr}$ with a single peak in $I_\mathrm{uncorr}$. For sufficiently detuned pumps, in the regime of real ${\overline{\rho}}$ the uncorrelated contribution can be large enough to be visible on the JSI plot without exaggeration or scaling. Indeed, for supercritical detunings $|\Delta_P|>\Delta_\mathrm{critical}$, as the OPO threshold is approached and ${\overline{\rho}}\to{\overline{\Gamma}}$ the uncorrelated contribution vastly dominates over the correlated contribution, as seen in Fig. \[fig::JSI\_real\_rhobar\]. This is an expected consequence of the rapid growth in the photon pair generation rate in this regime – multiple photon pairs are generated in sufficiently large quantities that joint detection of a signal and idler photon originating from the same energy-conserving pair is unlikely relative to the probability of detecting a signal and idler photon which originated from separate pairs and thus obey no relationship in energy. Another effect seen as ${\overline{\rho}}\to{\overline{\Gamma}}$ is the narrowing of the entire JSI distribution to a small point-like peak centred on $(\omega_s,\omega_i)=(\Delta_P,\Delta_P)$. Within our idealization of a zero-bandwidth cw pump the area of this point would be limited only by the frequency resolution of the JSI measurement scheme, though in actual experiments the finite pump bandwidth would serve as a fundamental lower bound for the overall extent of the JSI.
Perhaps the most definitive experimental indication of strongly driven effects lies in the top-right and bottom-left uncorrelated peaks of the JSI distribution for a detuned pump as in Fig. \[fig::JSI\_uncorrelated\_piece\](b). For sufficient detunings these peaks are well separated from the antidiagonal and thus easily distinguished from the correlated part of the JSI. For low powers, wherein only one photon pair is generated in the ring at any given time, they would be entirely absent from the measured JSI. As the power increases, *any* non-spurious coincidence detection of photons in these regions indicates multi-pair generation, as photons generated in those peaks do not conserve energy and therefore must be associated with separate, independently produced pairs.
![(Colour online) Correlated part $I_\mathrm{corr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for a pump with (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz. The splitting evident in (b) arises from the pump detuning, whereas in (c) the XPM-induced detuning of the signal and idler ring modes is responsible.[]{data-label="fig::JSI_splitting_plots"}](JSI_corr_0_10.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Correlated part $I_\mathrm{corr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for a pump with (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz. The splitting evident in (b) arises from the pump detuning, whereas in (c) the XPM-induced detuning of the signal and idler ring modes is responsible.[]{data-label="fig::JSI_splitting_plots"}](JSI_corr_3Gamma_10.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Correlated part $I_\mathrm{corr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for a pump with (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz. The splitting evident in (b) arises from the pump detuning, whereas in (c) the XPM-induced detuning of the signal and idler ring modes is responsible.[]{data-label="fig::JSI_splitting_plots"}](JSI_corr_0_2e8.png "fig:"){width="1.0\columnwidth"}
![(Colour online) Uncorrelated part $I_\mathrm{uncorr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs. Pump parameters are (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz.[]{data-label="fig::JSI_uncorrelated_piece"}](JSI_uncorr_0_10.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Uncorrelated part $I_\mathrm{uncorr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs. Pump parameters are (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz.[]{data-label="fig::JSI_uncorrelated_piece"}](JSI_uncorr_3Gamma_10.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Uncorrelated part $I_\mathrm{uncorr}$ of joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs. Pump parameters are (a) $\Delta_P=0, N_P=10$, (b) $\Delta_P=3{\overline{\Gamma}}, N_P=10$, and (c) $\Delta_P=0, N_P=2\times 10^8$. Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz.[]{data-label="fig::JSI_uncorrelated_piece"}](JSI_uncorr_0_2e8.png "fig:"){width="1.0\columnwidth"}
![(Colour online) Joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for (a) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=9.8\times 10^7$ (90% of OPO threshold) and (b) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=1\times 10^8$ (95% of OPO threshold). Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz.[]{data-label="fig::JSI_real_rhobar"}](JSI_supercrit_90.png "fig:"){width="1.0\columnwidth"} ![(Colour online) Joint spectral intensity distribution, scaled to unit maximum, of signal and idler photon pairs for (a) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=9.8\times 10^7$ (90% of OPO threshold) and (b) $\Delta_P=1.5\Delta_\mathrm{critical}$ with $N_P=1\times 10^8$ (95% of OPO threshold). Ring parameters are taken as ${\overline{\Gamma}}=1$ GHz and $\Lambda=10$ Hz.[]{data-label="fig::JSI_real_rhobar"}](JSI_supercrit_95.png "fig:"){width="1.0\columnwidth"}
Conclusion
==========
We have investigated the strongly driven regime of spontaneous four-wave mixing in microring resonators for a cw pump input. A nonperturbative, exact analytic solution to the semiclassical equations of motion within the undepleted pump approximation was developed, which permits the calculation of any physical quantity related to the outgoing signal and idler fields while fully taking into account intraring scattering losses. The effects of self- and cross- phase modulation, as well as multi-pair generation, were found to drastically alter the nature of the photon pair generation process at high powers. A critical pump detuning of $\Delta_\mathrm{critical}=\sqrt{3}\;{\overline{\Gamma}}$, where ${\overline{\Gamma}}$ is the total effective linewidth of the ring resonances, was found to divide the behaviour of the system into two regimes. For supercritically detuned pumps, a region of optical bistability of the pump mode is predicted, and a threshold emerges for optical parametric oscillation of the signal and idler modes. Pump power-dependent splitting of the generated signal and idler photon spectra was uncovered, arising from both pump detuning and cross-phase modulation. In certain intermediate-power regimes, dramatic narrowing of the spectral linewidth of generated signal and idler photons associated with the approach to optical parametric oscillation was found. The joint spectral intensity distribution (JSI) was analysed, and found to consist of separate uncorrelated and correlated contributions. The correlated contribution is negligible at low powers, but becomes significant as multi-pair generation becomes appreciable at higher powers. In the regime of spectral splitting, the uncorrelated part of the JSI displays an intriguing quadruplet of peaks, two of which are well separated from the correlated part. An optimal detuning strategy was derived in which the pump detuning is chosen to exactly cancel the effect of self-phase modulation at each input power, maximizing the intraring pump intensity. By detuning the pump in this manner the effects of both spectral splitting and bandwidth reduction are eliminated, and the photon pair generation rate continues to scale quadratically with the pump input even for arbitrarily high powers.
Three simple experimental tests of our predictions in the strongly driven regime were proposed:
1. For fixed subcritical nonzero detunings the photon pair generation rate as a function of input pump power is predicted to have a local maximum at intermediate powers, followed by a decreasing approach to an asymptotic level at high powers.
2. The single photon spectra of the outgoing signal and idler fields are predicted to show spectral splitting proportional to the pump detuning at low powers, followed by a regime of a singly peaked spectrum with pump power-dependent narrowing of bandwidth at intermediate powers, and finally resuming a doublet structure at high powers.
3. The presence of two non-energy conserving peaks lying on the diagonal of the joint spectral intensity distribution, which are a consequence of multi-pair generation, is predicted to occur for sufficiently large pump powers.
Our analysis was restricted to cw pump inputs; studying how these strongly driven phenomena are altered for short pulses requires a numerical approach. Additionally, a slightly more sophisticated solution is required to fully study the regime of optical parametric oscillation, in which the undepleted pump approximation breaks down. We intend to extend our techniques to treat these regimes future publications.
Calculation of nonlinear coupling constants {#appendix:lambda}
===========================================
To estimate the nonlinear coupling constants $\Lambda$, $\eta$ and $\zeta$, we present in this section a derivation of the nonlinear sector $H_\mathrm{NL}$ of the ring Hamiltonian. For the moment we imagine the ring has been decoupled from both the physical and phantom channels, so that it is idealized as a perfect, isolated cavity. We expand the electric field $\mathbf{E}(\mathbf{r})$ and electric displacement field $\mathbf{D}(\mathbf{r})$ in the ring in terms of discrete ring modes $\mathbf{E}_\alpha(\mathbf{r})$ and $\mathbf{D}_\alpha(\mathbf{r})$ as $$\begin{aligned}
\label{modes_defn}
\mathbf{E}(\mathbf{r})&=&\sum_\alpha \sqrt{\frac{\hbar\omega_\alpha}{2}}b_\alpha \mathbf{E}_\alpha(\mathbf{r}) + \mathrm{H.c.}, \nonumber \\
\mathbf{D}(\mathbf{r})&=&\sum_\alpha \sqrt{\frac{\hbar\omega_\alpha}{2}}b_\alpha \mathbf{D}_\alpha(\mathbf{r}) + \mathrm{H.c.},\end{aligned}$$ where $\omega_\alpha$ are the mode frequencies and $b_\alpha$ the associated annihilation operators. The contribution to the ring Hamiltonian arising from the third-order nonlinear susceptibility can be written [@Sipe2004] as $$\begin{aligned}
\lefteqn{H_\mathrm{NL} =}\\
& & -\frac{1}{4\epsilon_0}\int d\mathbf{r} \Gamma^{ijkl}_{(3)}(\mathbf{r})D^i(\mathbf{r})D^j(\mathbf{r})D^k(\mathbf{r})D^l(\mathbf{r}) \nonumber\end{aligned}$$ with implied summation over repeated lowercase Roman indices, where $\epsilon_0$ is the permittivity of vacuum and $\Gamma^{ijkl}_{(3)}(\mathbf{r})$ represents the nonlinear response coefficients. Within the rotating wave approximation, only keeping relevant terms for the pump, signal and idler modes, we obtain $$\begin{aligned}
\lefteqn{H_\mathrm{NL}=} \\
&-&\frac{1}{4\epsilon_0}\left(\frac{4!}{2!1!1!}\right)\frac{\hbar\omega_P}{2}\sqrt{\frac{\hbar\omega_S}{2}\frac{\hbar\omega_I}{2}} Q_{SIPP} b_S^\dagger b_I^\dagger b_P b_P \nonumber \\
&-&\frac{1}{4\epsilon_0}\left(\frac{4!}{2!1!1!}\right)\frac{\hbar\omega_P}{2}\sqrt{\frac{\hbar\omega_S}{2}\frac{\hbar\omega_I}{2}} Q_{PPIS} b_P^\dagger b_P^\dagger b_I b_S \nonumber \\
&-&\frac{1}{4\epsilon_0}\left(\frac{4!}{2!2!}\right)\left(\frac{\hbar\omega_P}{2}\right)^2 Q_{PPPP} b_P^\dagger b_P^\dagger b_P b_P \nonumber \\
&-&\frac{1}{4\epsilon_0}\left(\frac{4!}{1!1!1!1!}\right)\left(\frac{\hbar\omega_P}{2}\frac{\hbar\omega_S}{2}\right) Q_{SPSP} b_S^\dagger b_P^\dagger b_S b_P \nonumber \\
&-&\frac{1}{4\epsilon_0}\left(\frac{4!}{1!1!1!1!}\right)\left(\frac{\hbar\omega_P}{2}\frac{\hbar\omega_I}{2}\right) Q_{IPIP} b_I^\dagger b_P^\dagger b_I b_P \nonumber,\end{aligned}$$ where the constants $Q_{IJKL}$ are given by $$\begin{aligned}
\lefteqn{Q_{IJKL} = } \\
& &\int d\mathbf{r}\left(\Gamma^{ijkl}_{(3)}(\mathbf{r}) (D_I^i(\mathbf{r}))^*(D_J^j(\mathbf{r}))^*D_K^k(\mathbf{r})D_L^l(\mathbf{r})\right)\nonumber.\end{aligned}$$ As is typically done for dispersive media we take [@Bhat2006] $$\begin{aligned}
\lefteqn{\Gamma^{ijkl}_{(3)}(\mathbf{r})} \\
&=&\frac{\chi_{(3)}^{ijkl}(\mathbf{r})}{\epsilon_0^2 n^2(\mathbf{r};\omega_1)n^2(\mathbf{r};\omega_2)n^2(\mathbf{r};\omega_3)n^2(\mathbf{r};\omega_4)},\nonumber\end{aligned}$$ where $n(\mathbf{r};\omega)$ is the linear refractive index of the ring medium at frequency $\omega$, and $\chi_{(3)}^{ijkl}(\mathbf{r})$ is the frequency-dependent nonlinear susceptibility. To evaluate the coefficients $Q_{IJKL}$ we introduce co-ordinates for the ring $\mathbf{r}_\perp$ and $l_\phi$, such that the volume element $$\begin{aligned}
d\mathbf{r} = \rho d\rho d\phi dz\end{aligned}$$ can be written as $$\begin{aligned}
d\mathbf{r} = \frac{\rho d\rho dz}{R}dl_\phi = d\mathbf{r}_\perp dl_\phi,\end{aligned}$$ where $R$ is the nominal ring radius and $l_\phi=R\phi$, which varies from $0$ to $2\pi R \equiv L$, the nominal ring circumference. The co-ordinate $\mathbf{r}_\perp$ is understood as shorthand for the pair $(\rho,z)$. Writing the mode fields $\mathbf{E}_\alpha(\mathbf{r})$ as $$\begin{aligned}
\mathbf{E}_\alpha(\mathbf{r}) = \frac{\mathbf{e}_\alpha(\mathbf{r}_\perp)e^{i k_\alpha l_\phi}}{\sqrt{L}},\end{aligned}$$ where $k_\alpha=2\pi n_\alpha/L$ for integer $n_\alpha$, we can simplify $H_\mathrm{NL}$ to $$\begin{aligned}
\label{H_expansion}
\lefteqn{H_\mathrm{NL} =} \\
&-& \frac{3}{L^2}\frac{\hbar\omega_P}{2}\sqrt{\frac{\hbar\omega_S}{2}\frac{\hbar\omega_I}{2}} Q_{SIPP}'b_S^\dagger b_I^\dagger b_P b_P \nonumber \\
&-& \frac{3}{L^2}\frac{\hbar\omega_P}{2}\sqrt{\frac{\hbar\omega_S}{2}\frac{\hbar\omega_I}{2}} Q_{PPIS}'b_P^\dagger b_P^\dagger b_S b_S
\nonumber \\
&-& \frac{3}{2L^2}\left(\frac{\hbar\omega_P}{2}\right)^2 Q_{PPPP}'b_P^\dagger b_P^\dagger b_P b_P \nonumber \\
&-& \frac{6}{L^2}\left(\frac{\hbar\omega_P}{2}\frac{\hbar\omega_S}{2}\right)^2 Q_{SPSP}'b_S^\dagger b_P^\dagger b_S b_P \nonumber \\
&-& \frac{6}{L^2}\left(\frac{\hbar\omega_P}{2}\frac{\hbar\omega_I}{2}\right)^2 Q_{IPIP}'b_I^\dagger b_P^\dagger b_I b_P \nonumber,\end{aligned}$$ in which the reduced constants $Q_{IJKL}'$ are given by $$\begin{aligned}
\label{Q_reduced}
\lefteqn{Q_{IJKL}' = \frac{1}{\sqrt{Z_I Z_J Z_K Z_L}}} \\
&\times& \bigg[\int d\mathbf{r}_\perp d l_\phi \epsilon_0\chi_{(3)}^{ijkl}(\mathbf{r}_\perp,l_\phi)(e^i_I(\mathbf{r}_\perp))^*(e^j_J(\mathbf{r}_\perp))^* \nonumber \\
& &\;\;\;\;\times
e^k_K(\mathbf{r}_\perp)e^l_L(\mathbf{r}_\perp)e^{i(k_K + k_L - k_I - k_J) l_\phi}\bigg].\end{aligned}$$ The modes (\[modes\_defn\]) are normalized [@Bhat2006] such that $$\begin{aligned}
\lefteqn{Z_\alpha =} \\
& & \frac{1}{L}\int d\mathbf{r}_\perp dl_\phi \epsilon_0 n^2(\mathbf{r}_\perp;\omega_\alpha)\mathbf{e}^*_\alpha(\mathbf{r}_\perp)\cdot \mathbf{e}_\alpha(\mathbf{r}_\perp)\gamma_\mathrm{gp}(\mathbf{r}_\perp;\omega_\alpha),\nonumber \\
&=& 1,\end{aligned}$$ where $\gamma_\mathrm{gp}(\mathbf{r}_\perp ;\omega_\alpha)$ is the ratio of the group and phase velocities of the ring medium at each frequency and spatial point. However, we display the $Z_\alpha$ explicitly in (\[Q\_reduced\]) so that the expression can be used regardless of whether or not the $\mathbf{e}_\alpha(\mathbf{r}_\perp)$ are normalized such that $Z_\alpha=1$.
To estimate the constants $Q'_{IJKL}$ we approximate the ratio $\gamma_{gp}\approx 1$ everywhere, and assume the $\mathbf{e}_\alpha(\mathbf{r}_\perp)$ to be of uniform magnitude within the ring and vanish elsewhere. We take this uniform magnitude to be unity and assume that for the modes of interest $(2k_P - k_I - k_S)L\ll 1$. For modes with polarization mainly perpendicular to the ring plane, the relevant susceptibility will be $\chi_{(3)}^{zzzz}\equiv \chi_{(3)}$, independent of position in the ring; this can be immediately generalized to treat other mode polarizations. We then have $$\begin{aligned}
Z_\alpha = \epsilon_0 n^2 A,\end{aligned}$$ where A is the cross-sectional area of the ring. Taking $\omega_S\approx\omega_I\approx\omega_P$ in the prefactors of (\[H\_expansion\]), we finally obtain $$\begin{aligned}
H_{\mathrm{NL}} &\approx& \left(\hbar\Lambda b_P b_P b_S^\dagger b_I^\dagger + \mathrm{H.c.}\right) + \hbar\eta b_P^\dagger b_P^\dagger b_P b_P \nonumber \\
&+& \hbar\zeta\left(b_S^\dagger b_P^\dagger b_S b_P + b_I^\dagger b_P^\dagger b_I b_P\right),\end{aligned}$$ where $$\begin{aligned}
\Lambda = \frac{3\hbar\omega_P^2\chi_{(3)}}{4\epsilon_0 n^4 LA}\end{aligned}$$ and $\eta=\Lambda/2$, $\zeta=2\Lambda$. In terms of the more experimentally accessible nonlinear refractive index $n_2=3\chi_{(3)}/4\epsilon_0cn^2$ this becomes $$\begin{aligned}
\Lambda = \frac{\hbar\omega_P^2 c n_2}{n^2 L A},\end{aligned}$$ which is in line with the results of similar derivations [@Andersen2015].
An operational definition of the joint spectral intensity {#appendix:JSI}
=========================================================
When characterizing a source of entangled photon pairs, the *joint spectral intensity* (JSI) distribution is often introduced in the low power limit, when the state of the signal and idler modes is well approximated by $$\begin{aligned}
\label{two_photon_JSI}
|\psi_{SI}\rangle &=& p_\mathrm{vac}|\mathrm{vac}\rangle \\
&+& \int d\omega_s\int d\omega_i f(\omega_s,\omega_i)a_S^\dagger(\omega_s) a_I^\dagger(\omega_i)|\mathrm{vac}\rangle,\nonumber\end{aligned}$$ where $|p_\mathrm{vac}|^2<1$ is a constant and the $a_J^\dagger(\omega)$ refer to the creation operators at frequency $\omega$ for the signal and idler modes[@Grice2001]. The unsymmetrized and unnormalized JSI for the such a state is defined as $|f(\omega_s,\omega_i)|^2$, and is proportional to the probability density per unit time of jointly detecting a signal and idler photon pair with respective frequencies $\omega_s$ and $\omega_i$. For strongly pumped sources, when multiple photon pairs are generated in significant quantities so that higher order terms involving more than two creation operators appear in the state, it is less straightforward to define a single function that characterizes the energy relationship between simultaneously detected signal and idler photons. Instead, one can operationally extend the definition of the JSI to strongly pumped sources by calculating for arbitrary input power the outcome of experiments designed to measure $|f(\omega_s,\omega_i)|^2$ in the low power limit. In this section we develop such a calculation for a typical measurement scheme employed to measure the JSI of photon pairs produced in a microring resonator.
![(Colour online) Schematic of experimental setup for measuring the joint spectral intensity distribution. Pump, signal and idler $\{P,S,I\}$ outputs from the ring resonator are incident on a filter $F$ that removes the pump component from the beam. Signal and idler fields with modes $c_S$ and $c_I$ are then separated by dichroic beamsplitter DBS, and independently filtered by monochromators, which are implemented by frequency-dependent beamsplitters BS1 and BS2. Each monochromator-beamsplitter transmits in a small window $\delta\omega_\mathrm{trans}$ about $\omega_i$ and $\omega_s$, respectively. Broadband photodetectors D1 and D2 measure the detector modes $c_{S,\mathrm{det}}$ and $c_{I,\mathrm{det}}$, and are connected to coincidence counter CC to register joint detection events within a temporal resolution of $\delta t$. Vacuum is input to the empty ports of BS1, BS2 and DBS.[]{data-label="fig:JSI_schematic"}](JSI_schematic.png){width="1.0\columnwidth"}
We consider a standard experimental setup [@Kim2005] to measure coincidence rates between signal and idler photons of particular frequencies as illustrated in Fig. \[fig:JSI\_schematic\]. The signal and idler fields are separated, and each field is sent through a separate monochromator set to transmit photons in some small range $\delta\omega_\mathrm{trans}$ about a centre frequency $\omega_s$ for the signal and $\omega_i$ for the idler. Placed after each monochromator are broadband photodetectors connected to a coincidence counter to identify simultaneously detected signal and idler photons. The transmission frequencies $\omega_s$ and $\omega_i$ are independently controllable, and correspond to a single point (or, more accurately, single bin) on the JSI plot, which is produced by scanning through $\omega_s$ and $\omega_i$ and measuring the corresponding coincidence rate. The transmission width is chosen to be much smaller than the linewidth of the measured photons, $\delta\omega_\mathrm{trans}\ll{\overline{\Gamma}}$, so that the full 2D spectrum can be resolved.
The monochromators can be simply modelled as frequency-dependent beamsplitters. Provided both the signal and idler arms of the experiment are balanced, the spatial dependence of the fields after the ring can be suppressed; all fields in this section are understood to be evaluated immediately after the ring-channel coupling point. We introduce annihilation operators $c_S(\omega_s)$ and $c_I(\omega_i)$ for the ring output fields ${\overline{\psi}}_{S>}(t)$ and ${\overline{\psi}}_{I>}(t)$ as in Eq. (\[amplitude\_defn\]). We can then apply the appropriate transformation to obtain the annihilation operators $c_{J,\mathrm{det}}(\omega_j)$ for the fields seen by the detectors placed after the monochromators. For the signal, we have $$\begin{aligned}
c_{S,\mathrm{det}}(\omega) = \hat{T}(\omega-\omega_s)c_S(\omega) + \hat{R}(\omega-\omega_s)c_{S,\mathrm{vac}}(\omega),\;\;\;\;\;\end{aligned}$$ in which $c_{S,\mathrm{vac}}$ refer to the modes on the other input port of the monochromator-beamsplitter, into which only vacuum is present. The transmission and reflection functions $\hat{T}(\omega)$ and $\hat{R}(\omega)$ determine which frequencies are transmitted by the the monochromator. For example, a simple filter may be modelled by a transmission function with a rectangular frequency profile, $$\begin{aligned}
\hat{T}(\omega) = \begin{cases}
1, & -\frac{\delta\omega_\mathrm{trans}}{2} < \omega < \frac{\delta\omega_\mathrm{trans}}{2} \\
0, & \mathrm{otherwise}. \end{cases}\end{aligned}$$ The exact choice of $\hat{T}(\omega)$ is not important for our purposes; for simplicity we only assume $\hat{T}(\omega)$ is a real, sufficiently narrow, symmetric function of $\omega$. The reflection function will be irrelevant, though it will satisfy the usual restrictions to correctly model a beamsplitter.
In exactly the same manner, modes $c_{I,\mathrm{det}}(\omega)$ seen by the detectors of the idler arm can be introduced. We can then write down the fields measured by each detector in the usual way, $$\begin{aligned}
{\overline{\psi}}_{J,\mathrm{det}}(t) = \int\frac{d\omega_j}{\sqrt{2\pi}}c_{J,\mathrm{det}}(\omega_j)e^{-i\omega_j t}.\end{aligned}$$ In a typical coincidence measurement the signal detector is continuously activated, and detection of a signal photon at time $t$ is used to trigger the activation of the idler detector (which is placed at a small delay relative to the signal detector) for a very short time $\delta t$, so that the idler detector samples the idler field during the time interval $[t-\delta t/2,t+\delta t/2]$. The average rate $I(\omega_s,\omega_i; t)$ at time $t$ of coincident detection events at $\omega_s$ and $\omega_i$ of the signal and idler detectors is given by the standard Glauber formula involving the fields at each detector [@GerryKnight2004], $$\begin{aligned}
\lefteqn{I(\omega_s,\omega_i;t) =}\\
& &\lim_{T\to\infty}\Bigg[\frac{1}{T} \int\displaylimits_{t-T/2}^{t+T/2}dt'\int\displaylimits_{t'-\delta t/2}^{t'+\delta t/2} dt'' \nonumber \\
& &v^2 \langle {\overline{\psi}}_{S,\mathrm{det}}^\dagger(t'){\overline{\psi}}_{I,\mathrm{det}}^\dagger(t''){\overline{\psi}}_{S,\mathrm{det}}(t'){\overline{\psi}}_{I,\mathrm{det}}(t'') \rangle\Bigg].\end{aligned}$$ In steady state the expectation value depends only on time difference $|t''-t'|$, which in the integrand is at most $\delta t$. Provided the coincidence resolution time $\delta t$ is much smaller than the timescale on which the expectation value varies (in our case ${\overline{\Gamma}}^{-1}$), this expression for $I(\omega_s,\omega_i;t)$ is then well approximated by $$\begin{aligned}
\lefteqn{I(\omega_s,\omega_i;t) \approx} \\
& & v^2 \delta t\langle {\overline{\psi}}_{S,\mathrm{det}}^\dagger(t){\overline{\psi}}_{I,\mathrm{det}}^\dagger(t){\overline{\psi}}_{S,\mathrm{det}}(t){\overline{\psi}}_{I,\mathrm{det}}(t) \rangle.\nonumber\end{aligned}$$
Proceeding to expand the detector fields in terms of their constituent modes, we find $$\begin{aligned}
\lefteqn{I(\omega_s,\omega_i;t) =} \\
& &\frac{v^2\delta t}{(2\pi)^2}\int d\nu_1 ...\int d\nu_4\bigg[e^{i(\nu_1+\nu_2-\nu_3-\nu_4)} \nonumber \\ &\times&\hat{T}^*(\nu_1-\omega_s)\hat{T}^*(\nu_2-\omega_i) \hat{T}(\nu_3-\omega_s)\hat{T}(\nu_4-\omega_i)\nonumber \\
&\times&\langle c_S^\dagger(\nu_1)c_I^\dagger(\nu_2)c_S(\nu_3)c_I(\nu_4)\rangle\bigg].\nonumber\end{aligned}$$ By writing the operators $c_J(\nu_i)$ in terms of their respective parent fields and then carrying out the integration over each $\nu_i$, we arrive at Eq. (\[JSI\_definition\]), where $T(t)$ is the Fourier transform of the transmission function $\hat{T}(\omega)$; see Eq. (\[T\_defn\]).
In obtaining (\[JSI\_definition\]) we have again used the fact that in steady state the expectation value is invariant with respect to time translations by $t$ in each argument. The expression (\[JSI\_definition\]) is manifestly real, independent of time, and in the limit of small $\delta\omega_\mathrm{trans}$ indeed reproduces the single-pair JSI $|f(\omega_s,\omega_i)|^2$ when calculated with an initial state of the form (\[two\_photon\_JSI\]).
This work was financially supported by the Natural Sciences and Engineering Research Council of Canada.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gaseous stars is analyzed in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler-Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed.\
Key Words and Phrases. Gaseous star. Adiabatic Oscillation. Self-adjoint operator. Friedrichs extension. Spectrum of Sturm-Liouville type. Brunt-Va̋isa̋la̋ frequency. g-Mode.
2010 Mathematical Subject Classification Numbers. 35P05, 35L51, 35Q31, 35Q85, 46N20, 76N15.
author:
- 'Tetu Makino [^1]'
title: 'On Linear Adiabatic Perturbations of Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations '
---
Introduction
============
We consider the adiabatic hydrodynamic evolution of a self-gravitating gaseous star governed by the Euler-Poisson equations
$$\begin{aligned}
&\frac{\partial\rho}{\partial t}+\sum_{k=1}^3\frac{\partial}{\partial x^k}(\rho v^k)=0, \label{EPa} \\
&\rho\Big(\frac{\partial v^j}{\partial t}+\sum_{k=1}^3 v^k\frac{\partial v^j}{\partial x^k}\Big)
+\frac{\partial P}{\partial x^j}+\rho\frac{\partial\Phi}{\partial x^j}=0, \quad j=1,2,3,
\label{EPb} \\
&\rho\Big(\frac{\partial S}{\partial t}+\sum_{k=1}^3 v^k\frac{\partial S}{\partial x^k}\Big)=0, \label{EPc} \\
&\triangle \Phi =4\pi\mathsf{G}\rho. \label{EPd}\end{aligned}$$
Here $t\geq 0, \mathbf{x}=(x^1,x^2,x^3) \in \mathbb{R}^3$. The unknowns $\rho \geq 0, P, S, \Phi \in \mathbb{R}$ are the density, the pressure, the specific entropy, the gravitational potential, and $\mathbf{v}=(v^1,v^2,v^3) \in \mathbb{R}^3$ is the velocity fields. $\mathsf{G}$ is a positive constant, the gravitation constant.
In this article the pressure $P$ is supposed to be a prescribed function of $\rho, S$. But for the sake of simplicity, we assume the equation of state of the ideal fluid, that is, we assume
$P$ is the function of $(\rho, S) \in [0,+\infty[ \times \mathbb{R}$ given by $$P=\rho^{\gamma}\exp\Big(\frac{S}{\mathsf{C}_V}\Big), \label{DefP}$$ where $\gamma$ and $\mathsf{C}_V$ are positive constants such that $$1<\gamma <2.$$
The constant $\gamma$ is the adiabatic exponent and $\mathsf{C}_V$ is the specific heat per unit mass at constant volume.
Since we are concerned with compactly supported density distribution $\rho$ in this article, the Poisson equation will be replaced by the Newtonian potential $$\Phi(t,\mathbf{x})=-4\pi\mathsf{G}\mathcal{K}\rho(t,\cdot)(\mathbf{x}), \label{NPot}$$ where $$\mathcal{K}f(\mathbf{x}):=\frac{1}{4\pi}\int\frac{f(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}d\mathbf{x}'. \label{defK}$$
We suppose that there is fixed a spherically symmetric equilibrium $\bar{\rho}, \bar{P},
\bar{S}, \bar{\Phi}$, which satisfy , , , , such that $\bar{\rho}(\mathbf{x})>0 \Leftrightarrow r=|\mathbf{x}| <R$ with a finite positive number $R$, the radius of the equilibrium.
We consider the perturbation $\mbox{\boldmath$\xi$}=\delta\mathbf{x}, \delta\rho, \delta P, \delta S, \delta\Phi$ at this fixed equilibrium. We use the Lagrangian co-ordinate which will be dented by the diversion of the letter $\mathbf{x}$ of the Eulerian co-ordinate. So, $\mathbf{x}$ runs on the fixed domain $B_R:=\{ \mathbf{x} \in \mathbb{R}^3 | r=|\mathbf{x}| < R\}$, while $\{\rho >0\} $ described by the Eulerian co-ordinate may move along $t$.
Then the linearized equation which governs the perturbations turns out to be $$\frac{\partial^2\mbox{\boldmath$\xi$}}{\partial t^2}+\mathbf{L}\mbox{\boldmath$\xi$}=0,
\label{Eqxi}$$ where $$\begin{aligned}
\mathbf{L}\mbox{\boldmath$\xi$}&=\frac{1}{\bar{\rho}}\mathrm{grad}\delta P+\frac{\delta\rho}{\bar{\rho}}\mathrm{grad}\bar{\Phi}+
\mathrm{grad}\delta\Phi \nonumber \\
&=\frac{1}{\bar{\rho}}\mathrm{grad}\delta P
-\frac{\delta\rho}{\bar{\rho}^2}\mathrm{grad}\bar{P}+
\mathrm{grad}\delta\Phi. \label{L}\end{aligned}$$
We have $$\begin{aligned}
&\delta\rho=-\mathrm{div}(\bar{\rho}\mbox{\boldmath$\xi$}), \label{drho} \\
&\delta\Phi=-4\pi\mathsf{G}\mathcal{K}(\delta\rho). \label{dPhi}\end{aligned}$$
Here $\delta$ denotes the Eulerian perturbation, while $\Delta$ will denote the Lagrangian perturbation. Recall their definition $$\begin{aligned}
&\Delta Q(t,\mathbf{x})=Q(t, \mbox{\boldmath$\varphi$}(t,\mathbf{x}))-\bar{Q}(\mathbf{x}), \\
&\delta Q(t,\mathbf{x})=Q(t,\mbox{\boldmath$\varphi$}(t,\mathbf{x}))-\bar{Q}
(\mbox{\boldmath$\varphi$}(t,\mathbf{x})),\end{aligned}$$ where $\mbox{\boldmath$\varphi$}(t,\mathbf{x})=\mathbf{x}+\mbox{\boldmath$\xi$}(t,\mathbf{x})$ is the steam line given by $$\frac{\partial}{\partial t}\mbox{\boldmath$\varphi$}(t,\mathbf{x})=\mathbf{v}(t,\mbox{\boldmath$\varphi$}(t,\mathbf{x})),
\quad \mbox{\boldmath$\varphi$}(0,\mathbf{x})=\mathbf{x}.$$ Thus it holds that $$\Delta Q=\delta Q+(\mbox{\boldmath$\xi$}|\mathrm{grad}\bar{Q})$$ in the linearized approximation, for any quantity $Q$.
Supposing that the initial perturbation of the density vanishes, that is, $\Delta\rho|_{t=0}=0$, the equation implies $\Delta\rho=0$ always, which is . Supposing $\Delta S|_{t=0} =0$, the equation implies $\Delta S=0$ always, therefore $$\Delta P=\overline{\frac{\gamma P}{\rho}}\Delta\rho.$$ This implies $$\delta P=\overline{\frac{\gamma P}{\rho}}\delta\rho+
\gamma\mathscr{A}\bar{P}(\mbox{\boldmath$\xi$}|\mathbf{e}_r) \label{dP}.$$\
Here we denote by $\mathbf{e}_r$ the unit vector $\partial/\partial r$ and we define the ‘Schwarzschild’s discriminant of convective stability’ $\mathscr{A}$ with the ‘Brunt-Va̋isa̋la̋ frequency’ $\mathscr{N}$ by
We put $$\mathscr{A}:=\overline{\frac{1}{\rho}\frac{d\rho}{dr}}-\overline{\frac{1}{\gamma P}\frac{dP}{dr}}\quad \Big(=-\frac{1}{\gamma \mathsf{C}_V}\frac{d\bar{S}}{dr}\Big)$$ and $$\mathscr{N}^2:=\mathscr{A}\overline{\frac{1}{\rho}\frac{dP}{dr}}
=-\mathscr{A}\overline{\frac{d\Phi}{dr}}.$$
For the physical meaning of these quantities, see [@LedouxW] or [@Cox Chapter III, Section 17]. When $\overline{dP/dr}<0$, $\mathscr{N}$ is real if and only if $\mathscr{A} \leq 0$. The condition $\mathscr{A}<0$ is that of the convective stability.\
Thus by , , we can see the right-hand side of is an integro-differential operator acting on the unknown $\mbox{\boldmath$\xi$}$, provided that the spherically symmetric equilibrium $(\bar{\rho}, \bar{S}, \bar{P})$ is fixed.
For the derivation of $\mathbf{L}$, see e.g., [@LedouxW], [@Cox] or [@Unno].\
The purpose of this article is to clarify the functional analysis properties of this integro-differential operator $\mathbf{L}$.\
Nonlinear evolution of spherically symmetric perturbations has been investigated sufficiently well in [@TM.OJM] and [@JJ.APDE]. In these studies spectral properties of the linearized operator for spherically symmetric perturbations, which was established by [@Beyer1995] and independently by [@Lin], are fully presupposed. Its spectrum was proved to be actually of the Sturm-Liouville type, and it was not obvious because of the singularity of the coefficients, caused by the physical vacuum boundary of the equilibrium. Therefore if we want to study nonlinear evolution of not necessarily spherically symmetric perturbations around a spherically symmetric equilibrium, we should prepare a sufficiently strong functional analysis study of spectral properties of the linearized operator for general, not necessarily spherically symmetric, perturbations. As for barotropic case, we have attacked this task, and have gotten sufficiently strong results in [@JJTM]. Thus here we consider the case of not necessarily barotropic motions. Unfortunately the results which we have established is little bit weaker than the barotropic case. There remains some open problems. But mathematically rigorous treatment of the problem is quiet new.\
This article is organized as follows.
In Section 2, we discuss on the existence of spherically symmetric equilibrium for prescribed entropy distribution. The concept of the ‘admissible’ equilibrium will play a crucial rôle throughout the mathematically rigorous investigations of this article. In Section 3, we prove the self-adjoint realization of the operator $\mathbf{L}$ as the Friedrichs extension. Astrophysical texts lacked mathematically rigorous proof. But such a strong assertion on the concrete form of the spectrum as that of the barotropic case given in [@JJTM] is not yet obtained. In order to investigate the specified concrete form of the spectrum we investigate eigenfunctions represented by spherical harmonics in Section 4. The situation is clarified to be quite different from the barotropic case. That is, it may be impossible to reduce the problem to that of Sturm-Liouville type. But the justification of the self-adjoint realization of the associated operator $\vec{L}_l$ for each degree $l$ of the harmonics $Y_{lm}$ can be done with success. A strong guess that the form of the spectrum of $\vec{L}_l$ is quite different from that of the barotropic case is suggested by the so-called ‘g-modes’. Section 5 is devoted to the study of the ‘g-modes’ and ‘p-modes’ proposed by astrophysicists. Justification of the Sturm-Liouville properties of the operators associated to ‘g-modes’, ‘p-modes’ can be done with success. But it may be difficult to prove that the sequence of eigenvalues which accumulates both to 0 and to $+\infty$ give a good approximation of the real eigenvalues of the original operators $\vec{L}_l$. This is a task to be done in a future work. The last Section 6 is devoted to examination of the arguments in the work [@Eisenfeld] by J. Eisenfeld. Besides the lack of the explicit announcement of the assumption that $\displaystyle \frac{1}{\gamma-1}$ be a rational number, because of the serious lack of proof of the identification of the spectra and the eigenvalues, the argument of [@Eisenfeld] does not succeed to prove the completeness of eigenfunctions as claimed. Therefore there is an open problem here concerning the proof of the absence of continuous spectrum.\
We shall use the following notations:
We denote
$$\begin{aligned}
&B_R:=\{ \mathbf{x}\in\mathbb{R}^3 \ |\ r=|\mathbf{x}| <R \}, \\
& \overline{B_R}:=\{ \mathbf{x}\in\mathbb{R}^3 \ |\ r=|\mathbf{x}| \leq R \}.\end{aligned}$$
\[Not.1\] 1) A function $F$ on a subset of $\mathbb{R}^3$ is said to be spherically symmetric if there exists a function $f$ on a subset of $[0,+\infty[$ such that $F(\mathbf{x})=f(|\mathbf{x}|)$ for $\forall \mathbf{x}$ in the domain of $F$. Then we shall denote $f=F^{\sharp}$.
2\) For a function $f$ on a subset of $[0,+\infty[$, we shall denote by $f^{\flat}$ the function on a subset of $\mathbb{R}^3$ such that $f^{\flat}(\mathbf{x})=
f(|\mathbf{x}|)$ for $\forall \mathbf{x}$ such that $r=|\mathbf{x}|$ is in the domain of $f$.
3\) When it is expected that no confusion may occur, we shall divert the symbols $f$ or $F$ instead of $f^{\flat}$ or $F^{\sharp}$.
Here let us note the following lemma, which can be verified easily:
\[Lem.0\] If a function $f$ defined on $[0,R[$ satisfies $f^{\flat}\in C^{k+2}(B_R), k=0,1,2,\cdots$, then $f \in C^{k+2}([0,R[), \quad \displaystyle \frac{df}{dr}\Big|_{r=+0}=0$, and $\displaystyle \frac{1}{r}\frac{df}{dr} \in C^k([0,R[)$.
Proof. We can show inductively that $$D^kh(r)=\frac{1}{r^{k+1}}\int_0^rD^{k+2}f(s)s^kds$$ for $\displaystyle h=\frac{1}{r}\frac{df}{dr}$. Therefore $$D^kh(r) \rightarrow \frac{D^{k+2}f(0)}{k+1}\quad\mbox{as}\quad r \rightarrow +0.$$ $\square$\
We denote the unit vectors $$\mathbf{e}_r=\frac{\partial}{\partial r},\quad \mathbf{e}_{\vartheta}=\frac{1}{r}\frac{\partial}{\partial\vartheta},
\quad \mathbf{e}_{\phi}=\frac{1}{r\sin\phi}\frac{\partial}{\partial\phi}$$ for the spherical polar co-ordinates $$\begin{aligned}
&x^1=r\sin\vartheta\cos\phi, \nonumber \\
&x^2=r\sin\vartheta\sin\phi, \nonumber \\
&x^3=r\cos\vartheta.\end{aligned}$$
Existence of spherically symmetric equilibrium for prescribed entropy distribution
==================================================================================
In this section we establish the existence of spherically symmetric equilibria which enjoy good properties used in the following consideration on $\mathbf{L}$.\
Let us put the following
\[Def.2\] A pair of $t$-independent spherically symmetric functions $(\bar{\rho}, \bar{S})
\in C_0^1(\mathbb{R}^3; [0,+\infty[)\times C^1(\mathbb{R}^3; \mathbb{R})$ which satisfies with $\mathbf{v}=0$ and $\Phi$, $P$ determined by , is called an admissible spherically symmetric equilibrium, if there is a finite positive number $R$ such that
1\) $\{\bar{\rho} >0\}=B_R $;
2\) $\bar{\rho}^{\gamma-1}, \bar{S} \in C^{\infty}(B_R) \cap
C^{2,\alpha}(\overline{B_R})$, $\alpha$ being a positive number such that $0<\alpha <\min(\frac{\gamma}{\gamma-1}-2, 1)$ ;
3\) $\displaystyle \frac{d\bar{\rho}^{\sharp}}{dr}, \frac{dP^{\sharp}}{dr} <0$ for $0<r<R$ and $$\frac{1}{r}\frac{d\bar{\rho}^{\sharp}}{dr}\Big|_{r=+0}<0,
\quad \frac{1}{r}\frac{dP^{\sharp}}{dr}\Big|_{r=+0} <0; \label{PON}$$
4\) The boundary $\partial B_R$, on which $\bar{\rho}=0$, is a physical vacuum boundary, that is, $$-\infty <\frac{d}{dr}(\bar{\rho}^{\gamma-1})^{\sharp}\Big|_{r=R-0}<0, \label{PhysVacBd}$$ which means $$-\infty < \frac{d}{dr}\overline{\frac{\gamma P}{\rho}}^{\sharp}\Big|_{r=R-0} <0,$$ where $\overline{\gamma P/\rho}=\overline{(\partial P/\partial\rho)_{S=\mbox{Const}}}$ is the square of the sound speed.
Note that $\gamma <2$ implies $2<\frac{\gamma}{\gamma-1}$, therefore such an $\alpha$ exists. Moreover we see that $\bar{\rho}^{\gamma-1}\in C^{2,\alpha}(\overline{B_R})$ implies $\bar{\rho}^{\gamma}, \bar{P}, \bar{S} \in
C^{2,\alpha}(\overline{B_R})$. However $\bar{\rho}\in C^{1,\alpha}(\overline{B_R})$ and $\bar{\rho} \not\in C^2(\overline{B_R})$ unless $\frac{1}{\gamma-1}\geq 2 \Leftrightarrow \gamma \leq \frac{3}{2}$.\
Note that 4) of Definition \[Def.2\] is equivalent to $$-\infty <\frac{1}{\bar{\rho}^{\sharp}}\frac{d\bar{P}^{\sharp}}{dr}\Big|_{r=R-0} <0.$$ In fact, we have $$\frac{1}{\rho}\frac{dP}{dr}=\Big[\frac{\gamma}{\gamma-1}\frac{d}{dr}\rho^{\gamma-1}+
\frac{\rho^{\gamma-1}}{\mathsf{C}_V}\frac{dS}{dr}\Big]\exp\Big(\frac{S}{\mathsf{C}_V}\Big),$$ and $\rho=\bar{\rho}^{\sharp}\rightarrow 0$ as $r\rightarrow R-0$ and $S=\bar{S}\in C^1(\bar{B}_R)$.\
We claim
\[Th.1\] Let a smooth function $\Sigma$ on $\mathbb{R}$ and a positive number $\rho_{\mathsf{O}}$ be given. Assume that it holds, for $\eta>0$, that $$\gamma +\frac{\gamma-1}{\mathsf{C}_V}\eta\frac{d}{d\eta}\Sigma(\eta) >0. \label{pDP}$$ Either if $\displaystyle\frac{4}{3}<\gamma <2$ or if $\displaystyle \frac{6}{5}<\gamma \leq \frac{4}{3}$ and $\rho_{\mathsf{O}}$ is sufficiently small, then there exists an admissible spherically symmetric equilibrium $(\bar{\rho}, \bar{S})$ such that $\bar{S}=\Sigma(\bar{\rho}^{\gamma-1})$ and $\rho(O)=\rho_{\mathsf{O}}$.
Proof . Consider the functions $f^P, f^u$ defined by $$\begin{aligned}
&f^P(\rho):=\rho^{\gamma}\exp\Big[\frac{\Sigma(\rho^{\gamma-1})}{\mathsf{C}_V}\Big], \\
&f^u(\rho):=\int_0^{\rho}
\frac{Df^P(\rho')}{\rho'}d\rho'\end{aligned}$$ for $\rho >0$. Thanks to the assumption we have $$Df^P(\rho) >0$$ for $\rho >0$, and there exists a smooth function $\Lambda$ on $\mathbb{R}$ such that $\Lambda(0)=0$ and $$f^P(\rho)=\mathsf{A}\rho^{\gamma}(1+\Lambda(\rho^{\gamma-1}))$$ for $\rho >0$. Here $\mathsf{A}:=\exp(\Sigma(0)/\mathsf{C}_V)$ is a positive constant. Then $$u=f^u(\rho)=\frac{\gamma\mathsf{A}}{\gamma-1}\rho^{\gamma-1}(1+\Lambda_u(\rho^{\gamma-1}))$$ and the inverse function $f^{\rho}$ of $f^u$ $$\rho=f^{\rho}(u)=\Big(\frac{\gamma-1}{\gamma\mathsf{A}}\Big)^{\frac{1}{\gamma-1}}
u^{\frac{1}{\gamma-1}}(1+\Lambda_{\rho}(u))$$ are given. Here $\Lambda_u, \Lambda_{\rho}$ are smooth functions on $\mathbb{R}$ such that $\Lambda_u(0)=0, \Lambda_{\rho}(0)=0$.
Therefore the problem is reduced to that for barotropic case to solve $$-\frac{1}{r^2}\frac{d}{dr}r^2\frac{du}{dr}=4\pi\mathsf{G}f^{\rho}(u), \quad
u=u_{\mathsf{O}}+O(r^2) \qquad ( r \rightarrow +0 )$$ by the shooting method. Here $u_{\mathsf{O}}=f^u(\rho_{\mathsf{O}})$ is given. The proof of the asserted result can be found in [@TM1984] and [@RendallS]. $\square$
In the barotropic case, the quantity $u$ means the specific enthalpy. But in the general baroclinic case, $u$ is not the specific enthalpy $\chi$ which should be defined as $$\chi:=\mathsf{C}_VT+\frac{P}{\rho}=\frac{\gamma}{\gamma-1}\frac{P}{\rho},$$ $T=P/((\gamma-1)\mathsf{C}_V\rho)$ being the absolute temperature. In fact we have $$\frac{d}{dr}(u-\chi)=-\frac{1}{\mathsf{C}_V}\eta\frac{d}{d\eta}\Sigma(\eta)
\Big|_{\eta=\bar{\rho}^{\gamma-1}}\overline{\frac{P}{\rho^2}\frac{d\rho}{dr} }$$ does not vanish if $\Sigma$ is not constant, that is, $\bar{S}$ is not constant, for $d\bar{\rho}/dr <0$.
Hereafter in this article we fix an admissible spherically symmetric equilibrium $(\bar{\rho}, \bar{S})$, and denote $$\rho_{\mathsf{O}}=\bar{\rho}(O)(>0),
\quad
\rho_{\mathsf{O}1}=-\lim_{r\rightarrow +0}\frac{1}{r}\frac{d\bar{\rho}}{dr}(>0),
\quad
P_{\mathsf{O}1}=-\lim_{r\rightarrow +0}\frac{1}{r}\frac{dP}{dr}(>0).$$\
As for the Schwarzschild’s discriminant, since $\displaystyle \mathscr{A}=-\frac{1}{\gamma \mathsf{C}_V}\frac{d\bar{S}}{dr} $ and we are assuming $ \bar{S} \in
C^{2,\alpha}(\overline{B_R})$, we have $$\mathscr{A}=O(r)$$ as $r\rightarrow +0$. (Recall Lemma \[Lem.0\].)
If $\bar{S}=\Sigma(\bar{\rho}^{\gamma-1})$, then we have $$\mathscr{A}=-\frac{1}{\gamma \mathsf{C}_V}\frac{d\bar{S}}{dr}
=-\frac{1}{\gamma \mathsf{C}_V}\frac{d}{d\eta}\Sigma(\eta)\Big|_{\eta=\bar{\rho}^{\gamma-1}}
\frac{d}{dr}\bar{\rho}^{\gamma-1}.$$ Therefore we have the following
\[Prop.1\] Suppose $\bar{S}=\Sigma(\bar{\rho}^{\gamma-1})$. If $$\frac{d}{d\eta}\Sigma(\eta) <0 \quad [\!( \frac{d}{d\eta}\Sigma(\eta) \leq 0 )\!]$$ for $\eta \geq 0$, then it holds $$\mathscr{A} <0 \quad [\!( \mathscr{A} \leq 0 )\!]$$ for $0<r\leq R$.
Self-adjoint realization of $\mathbf{L}$
========================================
We are considering the integro-differential operator $$\mathbf{L}\mbox{\boldmath$\xi$}=\frac{1}{\rho}\mathrm{grad}\delta P-
\frac{\delta\rho}{\rho^2}\mathrm{grad} P +\mathrm{grad}\delta\Phi,$$ where
$$\begin{aligned}
\delta\rho&=-\mathrm{div}(\rho\mbox{\boldmath$\xi$}), \\
\delta P&=\frac{\gamma P}{\rho}\delta \rho+\gamma \mathscr{A}P(\mbox{\boldmath$\xi$}|\mathbf{e}_r) ,\\
\delta \Phi&=-4\pi\mathsf{G}\mathcal{K}(\delta\rho). \end{aligned}$$
Here and hereafter the bars to denote the quantities evaluated at the fixed equilibrium are omitted, that is, $\rho, P$ etc stand for $\bar{\rho}, \bar{P}$ etc.\
Let us consider the operator $\mathbf{L}$ in the Hilbert space $\mathfrak{H}=L^2((B_R, \rho dx), \mathbb{C}^3)$ endowed with the norm $\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}$ defined by $$\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2=\int_{B_R}|\mbox{\boldmath$\xi$}(\mathbf{x})|^2\rho(\mathbf{x})d\mathbf{x}$$
We shall use
For complex number $z=x+\sqrt{-1}y, x, y \in \mathbb{R}$, the complex conjugate is denoted by $z^*=x-\sqrt{-1}y$. Thus, for $\displaystyle \mbox{\boldmath$\xi$}=\sum \xi^k\frac{\partial}{\partial x^k}, \xi^k \in
\mathbb{C}$, we denote $\displaystyle \mbox{\boldmath$\xi$}^*=\sum (\xi^k)^*\frac{\partial}{\partial x^k}$.
First we observe $\mathbf{L}$ restricted on $C_0^{\infty}(B_R, \mathbb{C}^3)$. Let us write $$\begin{aligned}
\mathbf{L}\mbox{\boldmath$\xi$}&=\mathbf{L}_0\mbox{\boldmath$\xi$} +4\pi\mathsf{G}\mathbf{L}_1\mbox{\boldmath$\xi$}, \\
\mathbf{L}_0\mbox{\boldmath$\xi$}&=
\mathrm{grad}\Big(-\frac{\gamma P}{\rho^2}\mathrm{div}(\rho\mbox{\boldmath$\xi$})+
\frac{\gamma\mathscr{A}P}{\rho}(\mbox{\boldmath$\xi$}|\mathbf{e}_r)\Big) + \nonumber \\
&+\frac{\gamma\mathscr{A}P}{\rho^2}\Big(-\mathrm{div}(\rho\mbox{\boldmath$\xi$})+
\frac{d\rho}{dr}(\mbox{\boldmath$\xi$}|\mathbf{e}_r)\Big)\mathbf{e}_r, \\
\mathbf{L}_1\mbox{\boldmath$\xi$}&=\mathrm{grad}\mathcal{K}(\mathrm{div}(\rho\mbox{\boldmath$\xi$})).\end{aligned}$$
Using this expression for $\mbox{\boldmath$\xi$}_{(\mu)} \in C_0^{\infty}(B_R), \mu=1,2$, we have the following formula by integration by parts: $$\begin{aligned}
(\mathbf{L}_0\mbox{\boldmath$\xi$}_{(1)}|\mbox{\boldmath$\xi$}_{(2)})_{\mathfrak{H}}&=
\int\frac{\gamma P}{\rho^2}\mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(1)})\mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(2)}^*) + \\
&+\int\frac{\gamma\mathscr{A}P}{\rho}
\Big[(\mbox{\boldmath$\xi$}_{(1)}|\mathbf{e}_r)\cdot\mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(2)}^*)
-\mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(1)})\cdot (\mbox{\boldmath$\xi$}_{(2)}|\mathbf{e}_r)^*\Big] + \\
&+\int\frac{\gamma\mathscr{A}P}{\rho}\frac{d\rho}{dr}(\mbox{\boldmath$\xi$}_{(1)}|\mathbf{e}_r)
(\mbox{\boldmath$\xi$}_{(2)}|\mathbf{e}_r)^*, \\
(\mathbf{L}_1\mbox{\boldmath$\xi$}_{(1)}|\mbox{\boldmath$\xi$}_{(2)})_{\mathfrak{H}}&=
-\int\mathcal{K}(\mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(1)})\cdot \mathrm{div}(\rho\mbox{\boldmath$\xi$}_{(2)}^*).\end{aligned}$$
Thanks to the symmetry of $\mathcal{K}$, we have $$(\mathbf{L}\mbox{\boldmath$\xi$}_{(1)}|\mbox{\boldmath$\xi$}_{(2)})_{\mathfrak{H}}=
(\mbox{\boldmath$\xi$}_{(1)}|\mathbf{L}\mbox{\boldmath$\xi$}_{(2)})_{\mathfrak{H}},$$ that is, $\mathbf{L}$ restricted on $C_0^{\infty}(B_R)$ is a symmetric operator. Of course $C_0^{\infty}(B_R)$ is dense in $\mathfrak{H}$.
Moreover we have $$\begin{aligned}
(\mathbf{L}_0\mbox{\boldmath$\xi$}|\mbox{\boldmath$\xi$})&=
\int\frac{\gamma P}{\rho^2}|\mathrm{div}(\rho\mbox{\boldmath$\xi$})|^2 + \\
&+2\sqrt{-1}\mathfrak{Im}\Big[
\int\frac{\gamma\mathscr{A}P}{\rho}(\mbox{\boldmath$\xi$}|\mathbf{e}_r)\cdot\mathrm{div}(\rho\mbox{\boldmath$\xi$}^*)\Big] +
\int\frac{\gamma\mathscr{A}P}{\rho}\frac{d\rho}{dr}|(\mbox{\boldmath$\xi$}|\mathbf{e}_r)|^2.\end{aligned}$$
Since $\mathscr{A} \in C^{1,\alpha}(\overline{B_R})$, we have $$|\mathscr{A}|\sqrt{\frac{\gamma P}{\rho}}\leq C_1$$ on $0<r\leq R$, for $P/\rho =O(R-r)$. Therefore $$\begin{aligned}
\Big|\int
\frac{\gamma\mathscr{A}P}{\rho}(\mbox{\boldmath$\xi$}|\mathbf{e}_r)
\mathrm{div}(\rho\mbox{\boldmath$\xi$})^*\Big|
&\leq C_1\int
\sqrt{ \frac{\gamma P}{\rho} }|(\mbox{\boldmath$\xi$}|\mathbf{e}_r)|
|\mathrm{div}(\rho\mbox{\boldmath$\xi$})| \\
&\leq \frac{C_1}{2}\Big[
\frac{1}{\epsilon}\int \rho |(\mbox{\boldmath$\xi$}|\mathbf{e}_r)|^2+
\epsilon\int
\frac{\gamma P}{\rho^2}|\mathrm{div}(\rho\mbox{\boldmath$\xi$})|^2\Big] \\
&\leq \frac{C_1}{2}\Big[
\frac{1}{\epsilon}\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2+
\epsilon\int\frac{\gamma P}{\rho^2}|\mathrm{div}\rho\mbox{\boldmath$\xi$})|^2\Big].\end{aligned}$$
Since $\displaystyle \frac{P}{\rho}\frac{d\rho }{dr}=O(\rho)$, we have $$\Big|\frac{\gamma\mathscr{A}P}{\rho}\frac{d\rho}{dr}\Big|\leq C \rho$$ Therefore we have $$\Big|\int\frac{\gamma\mathscr{A}P}{\rho}\frac{d\rho}{dr}|(\mbox{\boldmath$\xi$}|\mathbf{e}_r)|^2\Big|\leq C_2\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2.$$
Thus $$(\mathbf{L}_0\mbox{\boldmath$\xi$}|\mbox{\boldmath$\xi$})_{\mathfrak{H}}\geq
\Big(1-\frac{\epsilon C_1}{2}\Big)\int
\frac{\gamma P}{\rho^2}|\mathrm{div}(\rho\mbox{\boldmath$\xi$})|^2
-\Big(\frac{C_1}{2\epsilon}+C_2\Big)\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2.$$ Taking $\epsilon$ so small that $1-\frac{\epsilon C_1}{2} \geq 0$, we get $$(\mathbf{L}_0\mbox{\boldmath$\xi$}|\mbox{\boldmath$\xi$})_{\mathfrak{H}}\geq
-\Big(\frac{C_1}{2\epsilon}+C_2\Big)\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2.$$
On the other hand, it is known that $$(\mathbf{L}_1\mbox{\boldmath$\xi$}|\mbox{\boldmath$\xi$})_{\mathfrak{H}} \geq -\rho_{\mathsf{O}}\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}
^2.$$ For a proof , see [@JJTM Proof of Proposition 2].
Summing up, $\mathbf{L}$ is bounded from below in $\mathfrak{H}$. Therefore, thanks to [@Kato Chapter VI, Section 2.3], we have
The integro-differential operator $\mathbf{L}$ on $C_0^{\infty}(B_R, \mathbb{C}^3)$ admits the Friedrichs extension, which is a self-adjoint operator, in $\mathfrak{H}$.
We want to clarify the spectral property of the self-adjoint operator $\mathbf{L}$. But this task has not yet been completely done.
At least we can claim that the spectrum of $\mathbf{L}$ cannot be of the Sturm-Liouville type in the sense defined in [@JJTM], (that is, the spectrum $\sigma(T)$ of a self-adjoint operator $T$ in a Hilbert space $\mathsf{X}$ is said to be of the Sturm-Liouville type if $\sigma(T)$ consists of isolated eigenvalues with finite multiplicities,) since $\mathrm{dim}\mathsf{N}(\mathbf{L})=\infty$, where $\mathsf{N}(\mathbf{L})$ denotes the kernel of $\mathbf{L}$. In fact, if we consider a scalar field $a$ on $B_R$ given by a function $a^{\sharp}: (r,\vartheta,\phi) \mapsto a(\mathbf{x})$ which belongs to $C_0^{\infty}(]0,R[\times ]0, \pi[\times \mathbb{S}^1)$, then the field $$\mbox{\boldmath$\xi$}=\frac{1}{\sin\vartheta}\frac{\partial a}{\partial \phi}\mathbf{e}_r
-\frac{\partial a}{\partial\vartheta}\mathbf{e}_{\phi}$$ belongs to $C_0^{\infty}(B_R)$ and satisfies $\mathrm{div}(\rho\mbox{\boldmath$\xi$})=0$ and $(\mbox{\boldmath$\xi$}|\mathbf{e}_r)=0$, therefore it belongs to $\mathsf{N}(\mathbf{L})$. Since the dimension of spaces of such $a^{\sharp}$ is infinite, we see $\mathrm{dim}\mathsf{N}(\mathbf{L})=\infty$.
In the work [@JJTM], we proved that, when $S$ is constant so that $\mathscr{A}=0$, then the spectrum of the operator $\mathbf{L}$ is $\{0\}\cup \{\lambda_n , n=1,2,\cdots \}$, where 0 is an essential spectrum and $\lambda_n$ are eigenvalues of finite multiplicities, $\lambda_n \rightarrow +\infty$ as $n \rightarrow \infty$, provided that $\mathbf{L}$ is considered in the Hilbert space $$\mathfrak{F}=\mathfrak{H}\cap\{ \mbox{\boldmath$\xi$} |
\mathrm{div}(\rho\mbox{\boldmath$\xi$})\in \mathfrak{G}\},$$ while $$\mathfrak{G}=
L^2(B_R, \frac{1}{\rho}\frac{dP}{d\rho}d\mathbf{x})\cap\{g | \int gd\mathbf{x}=0\}.$$
So, we have the following
\[Q.1\] Even if $\mathscr{A}$ does not vanish somewhere, the spectrum of the operator $\mathbf{L}$ considered in a suitable Hilbert space, e.g. $\mathfrak{F}$, or something like it, which is dense in $\mathfrak{H}$ and in which $\mathbf{L}$ is self-adjoint, is of the form $\{0\}\cup \{\lambda_j\}, \lambda_1\leq \lambda_2 \leq \cdots \leq \lambda_n \leq \rightarrow +\infty$, where the multiplicities of $\lambda_j$ are finite?
We guess that the answer is ‘No’ when $S$ is not constant and $\mathscr{A}<0$ for $0<r<R$. In such a situation, it is expected that the so called g-modes may appear, that is, there may exist a sequence of eigenvalues which accumulates to 0. See Section 6. But a rigorous mathematical justification of this observation has not been found.
Solutions represented by spherical harmonics
============================================
In this section we consider the perturbation $\mbox{\boldmath$\xi$}$ of the particular form $$\mbox{\boldmath$\xi$}=V^r(r)Y_{lm}(\vartheta,\phi)\mathbf{e}_r+
V^h(r)\nabla_sY_{lm}+V^v(r)\nabla_s^{\perp}Y_{lm}.$$ Here $l, m \in \mathbb{Z}, 0 \leq l, |m|\leq l$, and $Y_{lm}$ is the spherical harmonics: $$\begin{aligned}
&Y_{lm}(\vartheta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}
P_l^m(\cos\vartheta)e^{\sqrt{-1}m\phi}, \\
&Y_{l,-m}=(-1)^mY_{lm}^*\end{aligned}$$ for $m \geq 0$, while $P_l^m$ is the associated Legendre function given by $$P_l^m(\zeta)=\frac{(-1)^m}{2^ll!}(1-\zeta^2)^{m/2}\Big(\frac{d}{d\zeta}\Big)^{m+l}(\zeta^2-1)^l.$$ See [@Jackson]. We use
We denote
$$\begin{aligned}
&\nabla_sf:=\frac{\partial f}{\partial\vartheta}\mathbf{e}_{\vartheta}+
\frac{1}{\sin\vartheta}
\frac{\partial f}{\partial\phi}\mathbf{e}_{\phi}, \\
&\nabla_s^{\perp}f:=\frac{1}{\sin\vartheta}
\frac{\partial f}{\partial\phi}\mathbf{e}_{\vartheta}-\frac{\partial f}{\partial\vartheta}\mathbf{e}_{\phi}.\end{aligned}$$
Note that
$$\begin{aligned}
&\mathrm{div}(\psi(r)Y_{lm}\mathbf{e}_r)=\Big(
\frac{1}{r^2}\frac{d}{dr}r^2\frac{d\psi}{dr}\Big)Y_{lm}\mathbf{e}_r, \\
&\mathrm{div}(\psi(r)\nabla_sY_{lm})=-\frac{l(l+1)}{r}\psi(r)Y_{lm}\mathbf{e}_r, \\
&\mathrm{div}(\psi(r)\nabla_s^{\perp}Y_{lm})=0.\end{aligned}$$
\
Then , , read
$$\begin{aligned}
\delta\rho&=\delta\check{\rho}(r)Y_{lm}(\vartheta,\phi), \\
\delta P&=\delta\check{P}(r)Y_{lm}(\vartheta,\phi), \\
\delta\Phi&=\delta\check{\Phi}(r)Y_{lm}(\vartheta, \phi),\end{aligned}$$
where
$$\begin{aligned}
\delta\check{\rho}&=-\frac{1}{r^2}\frac{d}{dr}(r^2\rho V^r)+\frac{l(l+1)}{r}\rho V^h, \label{4.6a}\\
\delta\check{P}&=\frac{\gamma P}{\rho}\delta\check{\rho}+\gamma
\mathscr{A}P V^r \nonumber \\
&=-\frac{\gamma P}{r^2\rho}\frac{d}{dr}(r^2\rho V^r)+\gamma \mathscr{A}PV^r+
l(l+1)\frac{\gamma P}{r}V^h, \label{4.6b}\\
\delta\check{\Phi}&=-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho}) \nonumber \\
&=4\pi\mathsf{G}\mathcal{H}_l\Big(
\frac{1}{r^2}\frac{d}{dr}(r^2\rho V^r)-\frac{l(l+1)}{r}\rho V^h\Big).\label{4.6c}\end{aligned}$$
Here the integral operator $\mathcal{H}_l$ is defined by
We put $$\mathcal{H}_lf(r)=\frac{1}{2l+1}\Big[
\int_r^{\infty}f(r')\Big(\frac{r}{r'}\Big)^lr'dr'+
\int_0^rf(r')\Big(\frac{r}{r'}\Big)^{-(l+1)}r'dr'\Big],$$ provided that $f \in L^2([0,+\infty[, r^2dr)$ and $f(r)=0$ for $r\geq R$.
The equation reads $$\frac{\partial^2V^r}{\partial t^2}+L_l^r=0, \quad \frac{\partial^2V^h}{\partial t^2}+L_l^h=0, \quad \frac{\partial^2V^v}{\partial t^2} =0, \label{EqVl}$$ where
$$\begin{aligned}
&L_l^r=\frac{1}{\rho}\frac{d}{dr}\delta\check{P}-\frac{1}{\rho^2}\frac{dP}{dr}\delta\check{\rho}+\frac{d}{dr}\delta\check{\Phi}, \\
&L_l^h=\frac{1}{r}\Big(\frac{\delta\check{P}}{\rho}+\delta\check{\Phi}\Big).\end{aligned}$$
We mean $$\mathbf{L}(V^rY_{lm}\mathbf{e}_r+V^h\nabla_sY_{lm}
+V^v\nabla_s^{\perp}Y_{lm})=L_l^rY_{lm}\mathbf{e}_r+
L_l^h\nabla_sY_{lm}.$$
We are going to analyze the operator $$\vec{L}_l=\begin{bmatrix}
L_l^r \\
\\
L_l^h
\end{bmatrix}$$ which acts on $$\vec{V}=\begin{bmatrix}
V^r \\
V^h
\end{bmatrix}.$$
Actually we can neglect the component $V^v(r)$, which should be an arbitrary affine function of $t$ in order to satisfy , and we are going to consider the eigenvalue problem $$\vec{L}_l\vec{V}=\lambda\vec{V}.$$\
As for the integral operator $\mathcal{H}_l$, we shall keep in mind the following lemma, which is easy to prove:
\[Lem.2\] 1) Let $l =0$ and $f \in L^2([0,+\infty[, r^2dr), f(r)=0$ for $r \geq R$. Then $H=\mathcal{H}_0f \in C^1(]0,+\infty[)$ and satisfies $$\begin{aligned}
&H(r)=O(1), \quad \frac{d}{dr}H(r) =O(r^{-\frac{1}{2}})\quad \mbox{as}\quad r \rightarrow +0, \\
&H(r)=\frac{C}{r}\quad\mbox{as}\quad r \geq R,\end{aligned}$$ $$-\frac{1}{r^2}\frac{d}{dr}\Big(r^2\frac{dH}{dr}\Big)=f. \label{EqH0}$$ Here $C$ is the constant given by $$C=\int_0^Rf(r)r^{2}dr.$$
Conversely, if $H$ is absolutely continuous and satisfies on $]0,+\infty[$, there exist constants $C_1,C_2$ such that $$H=\mathcal{H}_0f +C_1 +\frac{C_2}{r},$$ therefore, $H\in L^{\infty}([0,+\infty[)$and $ H \rightarrow 0 $ as $ r \rightarrow +\infty$ if and only if $H=\mathcal{H}_0f$.
2\) Let $l \geq 1$ and $f \in L^2([0,+\infty[, r^2dr), f(r)=0$ for $r \geq R$. Then $H=\mathcal{H}_lf \in C^1(]0,+\infty[)$ and satisfies $$\begin{aligned}
&H(r)=O(r^{\frac{1}{2}}), \quad \frac{d}{dr}H(r)=O(r^{-\frac{1}{2}})\quad \mbox{as}\quad r \rightarrow +0, \\
&H(r)=Cr^{-(l+1)}\quad\mbox{as}\quad r \geq R,\end{aligned}$$ $$-\frac{1}{r^2}\frac{d}{dr}\Big(r^2\frac{dH}{dr}\Big)+
\frac{l(l+1)}{r^2}H=f. \label{EqHl}$$ Here $C$ is the constant given by $$C=\frac{1}{2l+1}\int_0^Rf(r)r^{l+2}dr.$$
Conversely, if $H$ is absolutely continuous and satisfies on $]0,+\infty[$, there exists constants $C_1,C_2$ such that $$H=\mathcal{H}_lf +C_1 r^l +C_2 r^{-(l+1)},$$ and $H\in L^2([0,+\infty[)$ if and only if $H=\mathcal{H}_lf$.
Case $l=0$
----------
First let us consider the case $l=0$, when only $m=0$ is possible, and $\displaystyle Y_{00}=\frac{1}{\sqrt{4\pi}}$. We are considering $$\mbox{\boldmath$\xi$}=\frac{V(r)}{\sqrt{4\pi}}\mathbf{e}_r,$$ where we write $V$ instead of $V^r$, while we need not consider $V^h$.
We are concerned with the operator $$L_0=\frac{1}{\rho}\frac{d}{dr}\delta\check{P}-\frac{1}{\rho^2}\frac{dP}{dr}\delta\check{\rho}+\frac{d}{dr}\delta\check{\Phi}$$ with
$$\begin{aligned}
\delta\check{\rho}&=-\frac{1}{r^2}\frac{d}{dr}(r^2\rho V), \\
\delta\check{P}&=\frac{\gamma P}{\rho}\delta\check{\rho}+\gamma
\mathscr{A}P V \nonumber \\
&=-\frac{\gamma P}{r^2\rho}\frac{d}{dr}(r^2\rho V)+\gamma \mathscr{A}PV, \\
\delta\check{\Phi}&=-4\pi\mathsf{G}\mathcal{H}_0(\delta\check{\rho}) \nonumber \\
&=4\pi\mathsf{G}\mathcal{H}_0\Big(
\frac{1}{r^2}\frac{d}{dr}(r^2\rho V)\Big)\end{aligned}$$
so that $$\frac{d}{dr}\delta\check{\Phi}=-4\pi\mathsf{G}\rho V.$$ We mean $$\mathbf{L}\Big(\frac{V}{\sqrt{4\pi}}\mathbf{e}_r\Big)=\frac{L_0V}{\sqrt{4\pi}}\mathbf{e}_r.$$\
Introducing the variable $\psi$ by $$V=r\psi,$$ and putting $$L^{\mathsf{ss}}\psi=\frac{1}{r}L_0(r\psi),$$ we analyze the differential operator operator $L^{\mathsf{ss}}$ in the Hilbert space $L^2([0,R], \rho r^4dr), \mathbb{C})$, since $$\|\mbox{\boldmath$\xi$}\|_{\mathfrak{H}}^2=\int_0^R|\psi(r)|^2\rho(r)r^4dr$$ for $\displaystyle \mbox{\boldmath$\xi$}=\frac{r\psi}{\sqrt{4\pi}}\mathbf{e}_r$.\
We claim
The operator $L^{\mathsf{ss}}$ on $C_0^{\infty}(]0,R[)$ admits the Friedrichs extension, a self-adjoint operator bounded from below in $L^2([0,R], \rho r^4dr)$, and its spectrum consists of simple eigenvalues $\lambda_1^{\mathsf{ss}}<\lambda_2^{\mathsf{ss}}<\cdots <
\lambda_n^{\mathsf{ss}}<\cdots \rightarrow +\infty$.
Proof. First we write $L^{\mathsf{ss}}$ as $$L^{\mathsf{ss}}\psi=-\frac{1}{r^4\rho}
\frac{d}{dr}\Big(\gamma r^4P\frac{d\psi}{dr}\Big)+q_{00}(r)\psi,$$ where $$\begin{aligned}
q_{00}(r)&=-\frac{\gamma P}{\rho^2}\frac{d^2\rho}{dr^2}
-\frac{\gamma-1}{\rho^2}\frac{dP}{dr}\frac{d\rho}{dr}
+\frac{\gamma P}{\rho^3}\Big(\frac{d\rho}{dr}\Big)^2 \nonumber \\
&-\frac{\gamma P}{r\rho^2}\frac{d\rho}{dr}
-\frac{3(\gamma-1)}{r\rho}\frac{dP}{dr}
+\frac{1}{r\rho}\frac{d}{dr}(\gamma r\mathscr{A}P)-4\pi\mathsf{G}\rho.
\label{q00}\end{aligned}$$
We see that $$|q_{00}(r)|\leq C\quad\mbox{for}\quad 0<r<R.$$ In fact, although each term in the first line of the right-hand side of is of order $(R-r)^{-1}$, these singularities are canceled after the summation, which turns out to be $$-\frac{\gamma}{\gamma-1}\Big[
\frac{d^2\eta}{dr^2}+\frac{\gamma^1}{\gamma \mathsf{C}_V}\eta^{\frac{2-\gamma}{\gamma-1}}
\frac{dS}{dr}\frac{d\eta}{dr}\Big]\exp\Big(\frac{S}{\mathsf{C}_V}\Big)=O(1),$$ where $\eta:=\rho^{\gamma-1} \in C^{2,\alpha}(\overline{B_R})$.
So, we perform the Liouville transformation of $L^{\mathsf{ss}}\psi=\lambda\psi$ to $$-\frac{d^2y}{dx^2}+q_0(x)y=\lambda y,$$ where $$\begin{aligned}
x&=\int\sqrt{\frac{c}{a}}dr,\qquad y=(ac)^{\frac{1}{4}}\psi, \\
q_0&=q_{00}+\frac{1}{4}\frac{a}{c}Q, \\
Q&=\frac{d^2}{dr^2}\log(ac)-\frac{1}{4}\Big(\frac{d}{dr}
\log(ac)\Big)^2+\Big(\frac{d}{dr}\log a\Big)\Big(
\frac{d}{dr}\log(ac)\Big)\end{aligned}$$ with $$a=\gamma r^4P,\qquad c=r^4\rho.$$ (See [@BirkhoffR p. 275, Theorem 6] or [@Yosida p.110].)
Since $$\sqrt{\frac{c}{a}}=\sqrt{\frac{\rho}{\gamma P}},$$ we can put $$x=\int_0^r\sqrt{\frac{\rho}{\gamma P}}(r')dr'$$ so that $x$ runs on the interval $[0,x_+]$, where $x_+=\int_0^R\sqrt{\rho/\gamma P}dr$. We have $$\begin{aligned}
&x \sim \sqrt{\frac{\rho}{\gamma P}}\Big|_Or\quad\mbox{as}\quad r \rightarrow +0 \\
&x_+-x \sim C_R(R-r)^{\frac{1}{2}}\quad\mbox{as}\quad r \rightarrow R-0\end{aligned}$$ with a positive constant $C_R$.
It can be verified that $$\begin{aligned}
&q_0 \sim \frac{2}{x^2} \quad\mbox{as}\quad x\rightarrow +0 \\
&q_0 \sim
\frac{(\gamma+1)(3-\gamma)}{4(\gamma-1)^2}\frac{1}{(x_+-x)^2}
\quad\mbox{as}\quad x \rightarrow x_+-0.\end{aligned}$$ Hence the assertion follows from [@ReedS p.159, Theorem X.10]. $\square$
Case $l \geq 1$
---------------
Suppose $\l \geq1$.\
Let us consider the Hilbert space $\mathfrak{X}_l$ of functions $\vec{f}=(f^r, f^h)^{\top}$ defined on $[0,R[$ endowed with the norm $\|\vec{f}\|_{\mathfrak{X}_l}$ given by
$$\|\vec{f}\|_{\mathfrak{X}_l}^2=\int_0^R
(|f^r(r)|^2+l(l+1)|f^h(r)|^2)\rho r^2dr.$$
Of course $\vec{f} =(f^r, f^h)^{\top} \in \mathfrak{X}_l$ if and only if $$\mbox{\boldmath$\xi$}=\vec{f}^{\flat/lm}:=
f^r(r)Y_{lm}\mathbf{e}_r+f^h(r)\nabla_sY_{lm} \in \mathfrak{H}$$ for $|m|\leq l$, and $$\|\vec{f}^{\flat/lm}\|_{\mathfrak{H}}=\sqrt{4\pi}\|\vec{f}\|_{\mathfrak{X}_l}.$$ we consider the operator $\vec{L}_l$ in $\mathfrak{X}_l$. We claim
\[Th.3\] The integro-differential operator $\vec{L}_l$ on $C_0^{\infty}([0,R[, \mathbb{C}^2)$ admits the Friedrichs extension, which is a self-adjoint operator bounded from below, in $\mathfrak{X}_l$.
Proof. First we look at $L_l^r, L_l^h$ by writing them as
$$\begin{aligned}
L_l^r&=-\frac{\gamma P^{\frac{1}{\gamma}}}{\rho}\frac{d}{dr}\Big(\frac{P^{1-\frac{2}{\gamma}}}{r^2}
\frac{d}{dr}(r^2P^{\frac{1}{\gamma}}V^r)\Big)
+\frac{\mathscr{A}}{\rho}\frac{dP}{dr}V^r+ \nonumber \\
&+l(l+1)\frac{\gamma P^{\frac{1}{\gamma}}}{\rho}\frac{d}{dr}
\Big(\frac{P^{1-\frac{1}{\gamma}}}{r}V^h\Big)+\frac{d\Psi}{dr}, \\
L_l^h&=-\frac{\gamma P^{1-\frac{1}{\gamma}}}{r^3\rho}
\frac{d}{dr}(r^2P^{\frac{1}{\gamma}}V^r)+
\frac{l(l+1)}{r^2}\frac{\gamma P}{\rho}V^h+\frac{\Psi}{r},\end{aligned}$$
where $$\begin{aligned}
\Psi=\delta\check{\Phi}&=-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho}) \nonumber \\
&=4\pi\mathsf{G}\mathcal{H}_l
\Big(\frac{1}{r^2}\frac{d}{dr}(r^2\rho V^r)-\frac{l(l+1)}{r}\rho V^h\Big).\end{aligned}$$
Using this expression, we see that the operator $\vec{L}_l$ restricted on $C_0^{\infty}([0,R[, \mathbb{C}^2)$ is symmetric and bounded from below in $\mathfrak{X}_l$.
In fact, if $\vec{V}_{(\mu)} \in C_0^{\infty}, \mu=1,2$, then the integration by parts leads us to $$\begin{aligned}
(\vec{L}_l\vec{V}_{(1)}|\vec{V}_{(2)})_{\mathfrak{X}_l} &=\gamma \int W_{(1)}W_{(2)}^*dr +\int\mathscr{A}\frac{dP}{dr}V_{(1)}^rV_{(2)}^{r*}r^2dr + \nonumber \\
&-4\pi\mathsf{G}\int\mathcal{H}_l(\delta\check{\rho}_{(1)})(\delta\check{\rho}_{(2)})^*r^2dr,\end{aligned}$$ where $$W_{(\mu)}:=\frac{P^{\frac{1}{2}-\frac{1}{\gamma}}}{r}
\frac{d}{dr}(r^2P^{\frac{1}{\gamma}}V_{(\mu)}^r)
-l(l+1)P^{\frac{1}{2}}V_{(\mu)}^h.$$ Since the integral operator $\mathcal{H}_l$ is symmetric, we see that the restriction of $\vec{L}_l$ onto $C_0^{\infty}$ is symmetric.
Let us estimate $$(\vec{L}_l\vec{V}|\vec{V})_{\mathfrak{X}_l}
=\gamma\int |W|^2dr +\int\mathscr{A}\frac{dP}{dr}|V^r|^2r^2dr
-4\pi\mathsf{G}\int\mathcal{H}_l(\delta\check{\rho})(\delta\check{\rho})^*r^2dr$$ from below for $\vec{V} \in C_0^{\infty}$.
Since $$\Big|\mathscr{A}\frac{dP}{dr}\Big|\leq C_1 \rho ,$$ we have $$\Big|\int
\mathscr{A}\frac{dP}{dr}|V^r|^2r^2dr\Big|
\leq C_1\|\vec{V}\|_{\mathfrak{X}_l}^2.$$ On the other hand, we know $$\Big|\int\mathcal{H}_l(\delta\check{\rho})(\delta\check{\rho})^*r^2dr\Big|\leq
\rho_{\mathsf{O}}\|\vec{V}\|_{\mathfrak{X}_l}^2.$$ For a proof, see [@JJTM Section 5.2]. Therefore we have $$(\vec{L}_l\vec{V}|\vec{V})_{\mathfrak{X}_l}\geq
\gamma \int |W|^2dr -(C_1+4\pi\mathsf{G}\rho_{\mathsf{O}})\|\vec{V}\|_{\mathfrak{X}_l}^2,$$ that is, $\vec{L}_l$ is bounded from below.
Therefore, thanks to [@Kato Chapter VI, Section 2.3], the restriction of $\vec{L}_l$ onto $C_0^{\infty}$ admits the Friedrichs extension. This completes the proof of Theorem \[Th.3\]. $\square$\
We see $$W=-\frac{1}{\gamma}rP^{-\frac{1}{2}}\delta\check{P}$$ by a direct calculation.\
Hereafter we denote by $\vec{L}_l$ the self-adjoint operator in $\mathfrak{X}_l$.
Note that the domain $\mathsf{D}(\vec{L}_l)$ of the Friedrichs extension is given by $$\mathsf{D}(\vec{L}_l)=\overset{\circ}{\mathfrak{W}}_l\cap\{\vec{V} \ | \ \vec{L}_l\vec{V} \in \mathfrak{X}_l\quad\mbox{in distribution sense}\quad \}.$$ Here $\overset{\circ}{\mathfrak{W}}_l$ is the closure of $C_0^{\infty}([0,R[,\mathbb{C}^2)$ in the Hilbert space $\mathfrak{W}_l$ endowed with the norm $\|\cdot\|_{\mathfrak{W}_l}$ given by $$\|\vec{V}\|_{\mathfrak{W}_l}^2=
\|\vec{V}\|_{\mathfrak{X}_l}^2+\|
\delta\check{\rho}\|_{L^2(\frac{\gamma P}{\rho^2}r^2dr)}^2, \label{NormWl}$$ where $$\|\delta\check{\rho}\|_{L^2(\frac{\gamma P}{\rho^2}r^2dr)}^2=
\int_0^R
\Big|-\frac{1}{r^2}\frac{d}{dr}(r^2\rho V^r)+
\frac{l(l+1)}{r}\rho V^h\Big|^2\frac{\gamma P}{\rho^2}r^2dr.$$
Here let us note that we can claim
\[Prop.2\] If $\vec{V}\in \mathfrak{W}_l$ satisfies $|V^r| \leq C$, then $\vec{V}$ belongs to $\overset{\circ}{\mathfrak{W}}_l$.
Proof can be done by taking $\varphi_n\in C^{\infty}([0,R])$ such that $\varphi_n(r)=1$ for $0 \leq r \leq R-\frac{1}{n},
\varphi_n(r)=0$ for $R-\frac{1}{2n} \leq r \leq R$, and $0 \leq \varphi_n\leq 1, |d\varphi_n/dr|\leq Cn$, and considering $\varphi_n\cdot \vec{V}$ for $\vec{V} \in \mathfrak{W}_l$. Let us omit the details.\
Now, actually is equivalent to $$\|\vec{V}\|_{\mathfrak{X}_l}^2+\gamma \int_0^R|W|^2dr$$ with $$W=\frac{P^{\frac{1}{2}-\frac{1}{\gamma}}}{r}
\frac{d}{dr}(r^2P^{\frac{1}{\gamma}}V^r)
-l(l+1)P^{\frac{1}{2}}V^h
=-\frac{1}{\gamma}rP^{-\frac{1}{2}}\delta\check{P},$$ that is, is equivalent to $((\vec{L}_l+\kappa)\vec{V}|\vec{V})_{\mathfrak{X}_l}$ with a sufficiently large constant $\kappa$.\
On a parallel with the Question \[Q.1\] we have
\[Q.2\] Let $l \geq 1$. Even if $S$ is not constant, the spectral properties of $\vec{L}_l$ can be reduced to the study of an operator with spectrum of the Sturm-Liouville type as discussion of [@JJTM Section 5.2]?
We guess that the answer is ‘No’.\
As for the dimension of the kernel of $\vec{L}_l$, we have the following
1\) Let $l \geq1$. Suppose that $\mathscr{A}=0$ identically on $]0,R[$. Then $\mathrm{dim}\mathsf{N}(\vec{L}_l)=\infty$.
2\) Suppose that $\mathscr{A}<0$ everywhere on $]0,R[$. Then $\mathrm{dim}\mathsf{N}(\vec{L}_l)=0$ when $l \geq 2$ and $\mathrm{dim}\mathsf{N}(\vec{L}_1)=1$ when $l=1$.
Proof. 1) Let $l \geq 1$ and suppose that $\mathscr{A}=0$ identically. Then, if $$-\frac{1}{r^2}\frac{d}{dr}(r^2\rho V^r)+\frac{l(l+1)}{r}\rho V^h=0, \label{4.29}$$ then $\delta\check{\rho}=\delta\check{P}=\delta\check{\Phi}=0$ by , , , therefore $L_l^r=L_l^h=0$, that is, $\vec{V}\in \mathsf{N}(\vec{L}_l)$. But, if $a \in C_0^{\infty}(]0,R[)$, then $\vec{V}=(V^r,V^h)^T$ given by $$V^r(r)=\frac{1}{\rho}\cdot\frac{l(l+1)}{r^2}\int_0^ra(r')r'dr',\quad
V^h(r)=\frac{1}{\rho}a(r)$$ belongs to $C_0^{\infty}(]0,R[)$ and satisfies so that $\vec{L}_l\vec{V}=0$. Since $a \in C_0^{\infty}$ is arbitrary, we see $\mathrm{dim}\mathsf{N}(\vec{L}_l)=\infty$.
2\) Suppose that $\mathscr{A} <0$ everywhere. Let us consider $\vec{V}\in \mathsf{N}(\vec{L}_l)$, $l\geq 1$. Of course $\vec{V} \in \mathfrak{X}_l$, and moreover, since $\vec{V} \in
\mathsf{D}(\vec{L}_l)$, we have $$W=-\frac{1}{\gamma}rP^{-\frac{1}{2}}\delta\check{P}\in L^2([0,R],dr),$$ therefore $\delta\check{P}\in L^2([0,R], P^{-1}r^2dr)$. It follows that $$\frac{\gamma P}{\rho}\delta\check{\rho}=\delta\check{P}-\gamma\mathscr{A}PV^r
\in L^2([0,R], P^{-1}r^2dr),$$ therefore $$\delta\check{\rho} \in L^2([0,R], P\rho^{-2}r^2dr) \subset L^2([0,R], r^2dr),$$ since $P\rho^{-2}\geq 1/C$. Thus $\delta\check{\Phi}=-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho})$ enjoys the properties listed in Lemma \[Lem.2\]. Now $\vec{V}\in \mathsf{N}(\vec{L}_l)$ means
$$\begin{aligned}
&\frac{1}{\rho}\frac{d}{dr}\delta\check{P}-\frac{1}{\rho^2}\frac{dP}{dr}\check{\rho}+
\frac{d}{dr}\delta\check{\Phi}=0, \label{N4.33a}\\
&\frac{\delta\check{P}}{\rho}+\delta\check{\Phi}=0. \label{N4.33b}\end{aligned}$$
It follows from that $$\Phi'\delta\check{\rho}=\frac{d\rho}{dr}\delta\check{\Phi}, \label{4.31}$$ where we put $$\Phi':=\frac{d\Phi}{dr}=-\frac{1}{\rho}\frac{dP}{dr}.$$ On the other hand, $\delta\check{\Phi}=-4\pi\mathsf{G}
\mathcal{H}_l(\delta\check{\rho})$ implies $$\Big[\frac{1}{r^2}\frac{d}{dr}r^2\frac{d}{dr}-\frac{l(l+1)}{r^2}\Big]\delta\check{\Phi}=
4\pi\mathsf{G}\delta\check{\rho}. \label{4.33}$$ Let us consider $$X:=\frac{\delta\check{\Phi}}{\Phi'},$$ keeping in mind that $$\Phi'
=\frac{d\Phi}{dr}=\frac{4\pi\mathsf{G}}{r^2}\int_0^r\rho(r')(r')^2dr'>0
\quad\mbox{for}\quad r>0.$$ Then $\delta\check{\Phi}=\Phi'X$ and reads $\displaystyle \delta\check{\rho}=\frac{d\rho}{dr}X$. Thus, eliminating $\delta\check{\rho}$ from and , we can derive the equation $$-\frac{d}{dr}\Big(r^2(\Phi')^2\frac{dX}{dr}\Big)+(l+2)(l-1)(\Phi')^2X=0, \label{4.35}$$ using $\displaystyle \frac{1}{r^2}\frac{d}{dr}\Big(r^2\Phi'\Big)=4\pi\mathsf{G}\rho$. We owe this trick to N. R. Lebovitz, [@Lebovitz], but we are considering that the equation holds for $0<r<+\infty$ in view of $\delta\check{\rho}=\frac{d\rho}{dr}=0$ for $r \geq R$ so that we do not care the ‘boundary condition’ of $\delta\check{\Phi}$ at $r=R$.
Let us multiply by $X$ and integrate it on $[0,+\infty[$. We are going to perform the integration by parts using the following observations: For $r \geq R$ we have $$\begin{aligned}
&\Phi'=\frac{C_0}{r^2},\quad C_0=\mathrm{Const.}>0, \\
&\delta\check{\Phi}=\frac{C_1}{r^{l+1}},\quad C_1=\mathrm{Const.}, \\
&X=\frac{C_1}{C_0}\frac{1}{r^{l-1}},\end{aligned}$$ and, on the other hand, as $r\rightarrow +0$, we have $$\begin{aligned}
&\Phi'=O(r), \quad\frac{d\Phi'}{dr}=\frac{d^2\Phi}{dr^2}=O(1), \\
&\delta\check{\Phi}=O(r^{\frac{1}{2}}), \quad \frac{d}{dr}\delta\check{\Phi}=O(r^{-\frac{1}{2}}), \\
\mbox{therefore}& \\
&\frac{dX}{dr}=O(r^{-\frac{3}{2}}),
\quad
r^2(\Phi')^2\frac{dX}{dr}X=O(r^2).\end{aligned}$$ Thus the contributions from the boundaries vanish, that is, $$r^2(\Phi')^2\frac{dX}{dr}X \rightarrow 0$$ both as $r \rightarrow +0$ and as $r \rightarrow +\infty$. So the integration by parts gives $$\int_0^{+\infty}r^2(\Phi')^2\Big(\frac{dX}{dr}\Big)^2dr+
(l+2)(l-1)\int_0^{+\infty}(\Phi')^2X^2dr=0. \label{4.36}$$
Let $l \geq 2$, that is, $(l+2)(l-1)>0$. Then implies $X=0$, therefore, $\delta\check{\Phi}=\delta\check{\rho}=\delta\check{P}=0$. Then we have $$\gamma\mathscr{A}PV^r=\delta\check{P}-\frac{\gamma P}{\rho}\delta\check{\rho}=0$$ by . Since $\mathscr{A}\not=0$ everywhere, we have $V^r=0$. Since $\delta\check{\rho}=0$, this implies $V^h=0$ in view of . Thus $\vec{V}=\vec{0}$, and $\mathrm{dim}\mathsf{N}(\vec{L}_l)=0$.
Let $l=1$. Then $\displaystyle \frac{dX}{dr}=0$, that is, $X$ is a constant $\kappa$. Then we have $$\delta\check{\Phi}=-\kappa\frac{1}{\rho}\frac{dP}{dr},\quad
\delta\check{P}=\kappa\frac{dP}{dr}, \quad
\delta\check{\rho}=\kappa\frac{d\rho}{dr},$$ which imply $V^r=V^h=-\kappa$. Conversely, if $V^r=V^h=1$, then we have $$\delta\check{\rho}=-\frac{d\rho}{dr}, \quad
\delta\check{P}=-\frac{dP}{dr}, \quad
\delta\check{\Phi}=4\pi\mathsf{G}\mathcal{H}_1\Big(-\frac{d\rho}{dr}\Big)=-\frac{d\Phi}{dr},$$ and $L_1^r=L_1^h=0$. But $(1,1)^{\top} \in \mathsf{D}(\vec{L}_1)$, since $(1,1)^{\top} \in \mathfrak{W}_l$. Recall Proposition \[Prop.2\].
Summing up, we can claim $\mathrm{dim}\mathsf{N}(\vec{L}_1)=1$. This completes the proof. $\square$\
As noted in [@Lebovitz], the identity $$Y_{10}\mathbf{e}_r+\nabla_sY_{10}=\sqrt{\frac{3}{4\pi}}\frac{\partial}{\partial x^3}$$ leads us to the interpretation that the eigenfunction $\vec{V}=(1,1)^{\top}$ for $l=1$ means a uniform translation, and it can be eliminated by requiring that the center of mass remains fixed in space, or, by requiring $\displaystyle \int_{B_R}(\delta\rho)x^3d\mathbf{x}=0$.
Cowling approximation, g-modes, p-modes
=======================================
We are considering the eigenvalue problem $$\vec{L}_l\vec{V}=\lambda\vec{V}.$$ In this section we suppose $ l \geq 1 $ and consider the case $\lambda \not=0$.
Cowling approximation
---------------------
The Cowling approximation is done by neglecting the gravitational perturbations. Thus we consider the eigenvalue problem $$\vec{L}_{[\mathrm{C}]l}\vec{V}=\lambda \vec{V}, \label{5.2}$$ where $$\vec{L}_{[\mathrm{C}]l}=
\begin{bmatrix}
L_{[\mathrm{C}]l}^r \\
\\
\\
L_{[\mathrm{C}]l}^h
\end{bmatrix}
=
\begin{bmatrix}
\displaystyle \frac{1}{\rho}\frac{d}{dr}\delta\check{P} -\frac{1}{\rho^2}\frac{dP}{dr}\delta\check{\rho} \\
\\
\displaystyle \frac{1}{r\rho}\delta\check{P}
\end{bmatrix}.$$
We are going to rewrite the problem by introducing the variables $$v=r^2P^{\frac{1}{\gamma}}V^r, \qquad w=P^{-\frac{1}{\gamma}}\delta\check{P}.$$
Then we have the following expressions:
$$\begin{aligned}
V^r&=\frac{1}{r^2}P^{-\frac{1}{\gamma}}v, \label{5.5a}\\
V^h&=\frac{1}{l(l+1)}\Big[
\frac{1}{r}P^{-\frac{1}{\gamma}}\frac{dv}{dr}+\frac{1}{\gamma}rP^{-1+\frac{1}{\gamma}}
w\Big], \\
\delta\check{P}&=P^{\frac{1}{\gamma}}w, \\
\delta\check{\rho}&=\frac{1}{\gamma}\rho P^{-1+\frac{1}{\gamma}}w
-\frac{1}{r^2}\mathscr{A}\rho P^{-\frac{1}{\gamma}}v, \\
\frac{d}{dr}\delta\check{P}-\frac{1}{\rho}\frac{dP}{dr}\delta\check{\rho}&=
P^{\frac{1}{\gamma}}\frac{dw}{dr}+\frac{1}{r^2}\mathscr{A}P^{-\frac{1}{\gamma}}
\frac{dP}{dr}v. \label{5.5e}\end{aligned}$$
\
Since we are considering the case in which $\lambda \not=0$, using the expressions - , we can write the problem as
$$\begin{aligned}
&\frac{dv}{dr}=\Big(\frac{\mathscr{S}_l^2}{\lambda}-1\Big)\frac{r^2\mathscr{B}}{\mathsf{c}^2}w, \label{5.7a} \\
&\frac{dw}{dr}=(\lambda-\mathscr{N}^2)\frac{1}{r^2\mathscr{B}}v. \label{5.7b}\end{aligned}$$
Here we denote $$\mathscr{B}:=\frac{P^{\frac{2}{\gamma}}}{\rho},$$ and we use
We put $$\mathscr{S}_l^2:=\frac{l(l+1)}{r^2}\frac{\gamma P}{\rho}=\frac{l(l+1)\mathsf{c}^2}{r^2},
\qquad \mathsf{c}^2:=\frac{\gamma P}{\rho}.$$
Recall $$\mathscr{N}^2=\frac{\mathscr{A}}{\rho}\frac{dP}{dr}.$$
$\mathscr{S}_l$ is called the Lamb frequency, and $\displaystyle \mathsf{c}=\sqrt{\frac{\gamma P}{\rho}}=\sqrt{\frac{dP}{d\rho}}$ is the speed of sound when $P \propto \rho^{\gamma}$.\
Eigenvalue problem for g-modes
------------------------------
Now, so called ‘g-modes’ are given by putting $\lambda=0$ in . Then we get the approximation $$\frac{dw}{dr}+\frac{\mathscr{N}^2}{r^2\mathscr{B}}v=0 \label{5.12}$$
Now let us assume
\[A.3\] There exists a positive numbers $\delta$ such that $$\frac{1}{r}\frac{d\bar{S}}{dr}
\geq \delta$$ for $0<r <R$.
Note that, when $\bar{S}=\Sigma(\bar{\rho}^{\gamma-1})$, this Assumption \[A.3\] is fulfilled if
$$\frac{d\Sigma}{d\eta}(\rho_{\mathsf{O}}^{\gamma-1})<0$$ for $ 0\leq \eta \leq \rho_{\mathsf{O}}^{\gamma-1}$, thanks to and .\
The Assumption \[A.3\] implies $\displaystyle \mathscr{A} =-\frac{1}{\mathsf{C}_V}\frac{d\bar{S}}{dr} < 0$ on $\overline{B_{R}} \setminus \{ O\}$. So, we can take $\mathscr{N} >0$ on $\overline{B_{R}}\setminus\{ O\}$.\
Under this assumption, which guarantees $\mathscr{N}^2>0$ for $0<r<R$, inserting this into , we get the following eigenvalue problem for eigenvalue $1/\lambda$: $$-\frac{d}{dr}\Big(\frac{r^2\mathscr{B}}{\mathscr{N}^2}\frac{dw}{dr}\Big)+
\frac{r^2\rho}{\gamma P}\mathscr{B}w=
\frac{1}{\lambda}l(l+1)\mathscr{B}w. \label{5.13}$$\
Let us consider the eigenvalue problem : $$L^{\mathsf{g}}w=\frac{1}{\lambda}w,$$ where $$L^{\mathsf{g}}w=
-\frac{1}{l(l+1)\mathscr{B}}\frac{d}{dr}\Big(\frac{r^2\mathscr{B}}{\mathscr{N}^2}\frac{dw}{dr}\Big)+
\frac{1}{\mathscr{S}_l^2}w.$$\
First we note
That there is a constant $C$ such that $$\frac{\rho}{C}\leq \mathscr{B} \leq C\rho, \label{ineqB}$$ and $$\frac{r^2}{C}\leq \mathscr{N}^2 \leq C r^2. \label{ineqN}$$ Moreover $\mathscr{B}/\rho$ and $r^2/\mathscr{N}^2$ as functions of $r^2$ belong to $C^2([0, R])$.
In fact, the estimate is clear, since $$\mathscr{B}=\rho\exp\Big[\frac{2S}{\gamma \mathsf{C}_V}\Big],$$ and the estimate can be shown by the expression $$\mathscr{N}^2=-\frac{1}{\gamma \mathsf{C}_V}\frac{dS}{dr}\frac{1}{\rho}\frac{dP}{dr}
=\frac{1}{\gamma \mathsf{C}_V}\frac{dS}{dr}\frac{d\Phi}{dr}$$ under the Assumption $2^{\sharp}$ and the physical vacuum boundary condition . Moreover recall Lemma \[Lem.0\].\
We perform the Liouville transformation of to $$-\frac{d^2y}{dx^2}+q^{\mathsf{g}}y=\frac{1}{\lambda}y,$$ where $$\begin{aligned}
&x=\int \sqrt{\frac{c}{a}}dr,\qquad y=(ac)^{\frac{1}{4}}w, \\
&q^{\mathsf{g}}=\frac{b}{c}+\frac{1}{4}\frac{a}{c}Q, \\
&Q=\frac{d^2}{dr^2}\log(ac) -\frac{1}{4}\Big(\frac{d}{dr}\log(ac)\Big)^2
+\Big(\frac{d}{dr}\log a\Big)\Big(\frac{d}{dr}\log(ac)\Big),\end{aligned}$$ with $$a=\frac{r^2\mathscr{B}}{\mathscr{N}^2},\quad
b=\frac{r^2\rho \mathscr{B}}{\gamma P},
\quad c=l(l+1)\mathscr{B}.$$ (See [@BirkhoffR p. 275, Theorem 6] or [@Yosida p.110].)
Noting that $$\frac{1}{C}\leq \sqrt{\frac{c}{a}}=\sqrt{l(l+1)}\frac{\mathscr{N}}{r}\leq C,$$ we can put $$x=\int_0^r\sqrt{\frac{c}{a}}$$ so that $x$ runs on the finite interval $[0, x_+^{\mathsf{g}}]$, where $x_+^{\mathsf{g}}:=\int_0^R\sqrt{c/a}$.
Look at $$\begin{aligned}
q^{\mathsf{g}}&=\frac{r^2}{l(l+1)}\Big(\frac{\rho}{\gamma P} +\frac{Q}{4\mathscr{N}^2}\Big),\\
Q&=\frac{d^2}{dr^2}\log\frac{r^2\mathscr{B}^2}{\mathscr{N}^2}
-\frac{1}{4}\Big(\frac{d}{dr}\log\frac{r^2\mathscr{B}^2}{\mathscr{N}^2}\Big)^2
+\Big(\frac{d}{dr}\log\frac{r^2\mathscr{B}}{\mathscr{N}^2}\Big)
\Big(\frac{d}{dr}\log\frac{r^2\mathscr{B}^2}{\mathscr{N}^2}\Big).\end{aligned}$$
It can be shown that $$q^{\mathsf{g}}=O(1)\quad\mbox{as}\quad x\rightarrow +0,$$ keeping in mind that $r^2/\mathscr{N}^2 \in C^2([0,R[)$.
On the other hand, we see $$q^{\mathsf{g}} \sim \frac{2\gamma-1}{4(\gamma-1)^2}\frac{1}{(x_+^{\mathsf{g}}-x)^2}
\quad\mbox{as}\quad x \rightarrow x_+^{\mathsf{g}}-0.$$ Here, under the assumption $1<\gamma <2$, we have $\displaystyle \frac{2\gamma-1}{4(\gamma-1)^2}
>0$, and $\displaystyle \frac{2\gamma-1}{4(\gamma-1)^2}\gtrless\frac{3}{4}$ if and only if $\displaystyle \gamma \gtrless \frac{3}{2}$, when the boundary point $x_+^{\mathsf{g}}$ is of limit point/circle type. See [@ReedS p.159, Theorem X.10].
Anyway there exists a constant $C$ such that $$q^{\mathsf{g}} \geq -C \quad \mbox{for}\quad 0<x<x_+^{\mathsf{g}}.$$
Hence we can claim that the operator $\displaystyle -\frac{d^2}{dx^2}+q^{\mathsf{g}}$ defined on $C_0^{\infty}(]0,x_+^{\mathsf{g}}[)$ admits the Friedrichs extension, a self-adjoint operator in $L^2([0,x_+^{\mathsf{g}}])$, whose spectrum consists of simple or double eigenvalues. In other words we can claim the following
The operator $L^{\mathsf{g}}$ defined on $C_0^{\infty}(]0,R[)$ admits the Friedrichs extension, a self-adjoint operator in $L^2([0,R], l(l+1)\mathscr{B}dr)$, whose spectrum consists of simple or double eigenvalues $\displaystyle \frac{1}{\lambda_n^{\mathsf{g}}},
n=1,2,\cdots$ : $\displaystyle \frac{1}{\lambda_1^{\mathsf{g}}}\leq \frac{1}{\lambda_2^{\mathsf{g}}} \leq \cdots
\leq\frac{1}{\lambda_n^{\mathsf{g}}} \leq \rightarrow +\infty$. The eigenvalues are simple if $\gamma >3/2$.
Note that $1/\lambda_1^{\mathsf{g}} >0$. In fact, if $y_1$ is an associated eigenfunction such that $\|y_1\|_{L^2([0,x_+^{\mathsf{g}}])}=1$, then $w_1=(ac)^{-\frac{1}{4}}y_1$ satisfies $$\int_0^R\Big(a(r)\Big|\frac{dw_1}{dr}\Big|^2+b(r)|w_1|^2\Big)dr=\frac{1}{\lambda_1^{\mathsf{g}}}
\int_0^R|w_1|^2c(r)dr
=\frac{1}{\lambda_1^{\mathsf{g}}}.$$ Since $ a=r^2\mathscr{B}/\mathscr{N}^2>0, b=r^2\rho \mathscr{B}/\gamma P>0$, we see $1/\lambda_1^{\mathsf{g}} >0$. Thus $$\lambda_1^{\mathsf{g}} \geq \lambda_2^{\mathsf{g}} \geq \cdots \geq \lambda_n^{\mathsf{g}} \geq \cdots
\rightarrow +0.$$\
Eigenvalue problem for p-modes
------------------------------
So called ‘p-modes’ are given by putting $1/\lambda =0$ in . Then we get the approximation $$\frac{dv}{dr}+\frac{r^2\rho}{\gamma P}\mathscr{B}w=0. \label{5.14}$$ Inserting this into , we get the following eigenvalue problem for eigenvalue $\lambda$: $$-\frac{d}{dr}\Big(\frac{\gamma P}{r^2\rho \mathscr{B}}\frac{dv}{dr}\Big)
+\frac{\mathscr{N}^2}{r^2\mathscr{B}}v=
\lambda\frac{1}{r^2\mathscr{B}}v. \label{5.15}$$\
Let us consider the eigenvalue problem : $$L^{\mathsf{p}}v=\lambda v,$$ where $$L^{\mathsf{p}}v=
-r^2\mathscr{B}\frac{d}{dr}\Big(\frac{\gamma P}{r^2\rho \mathscr{B}}\frac{dv}{dr}\Big)
+\mathscr{N}^2v.$$
We perform the Liouville transformation of to $$-\frac{d^2y}{dx^2}+q^{\mathsf{p}}y=\lambda y,$$ where $$\begin{aligned}
&x=\int\sqrt{\frac{c}{a}}dr, \qquad y=(ac)^{\frac{1}{4}}v, \\
&q^{\mathsf{p}}=\frac{b}{c}+\frac{1}{4}\frac{a}{c}Q, \\
&Q=\frac{d^2}{dr^2}\log(ac) -\frac{1}{4}\Big(\frac{d}{dr}\log(ac)\Big)^2
+\Big(\frac{d}{dr}\log a\Big)\Big(\frac{d}{dr}\log(ac)\Big),\end{aligned}$$ with $$\begin{aligned}
&a=\frac{\gamma P}{r^2\rho \mathscr{B}}=\frac{\gamma P^{1-\frac{2}{\gamma}}}{r^2\rho},\qquad
b=\frac{\mathscr{N}^2}{r^2\mathscr{B}}=\frac{\mathscr{N}^2}\rho P^{-\frac{2}{\gamma}}, \\
&c=l(l+1)\frac{1}{r^2\mathscr{B}}=l(l+1)\frac{\rho P^{-\frac{2}{\gamma}}}{r^2}.\end{aligned}$$ Noting that $$\sqrt{\frac{c}{a}}=\sqrt{\frac{\rho}{\gamma P}}\qquad
\begin{cases}
\quad\rightarrow C_0=\sqrt{\frac{\rho}{\gamma P}}(O) \quad\mbox{as}\quad r\rightarrow +0 \\
\quad \sim C_1(R-r)^{-\frac{1}{2}}\quad\mbox{as}\quad r\rightarrow R-0,
\end{cases}$$ where $C_0, C_1$ are positive constants, we can put $$x=\int_0^r\sqrt{\frac{c}{a}}$$ so that $x$ runs over the finite interval $[0,x_+^{\mathsf{p}}]$, where $x_+^{\mathsf{p}}=\int_0^R\sqrt{c/a}$, and $$\begin{aligned}
&x \sim C_0r \quad \mbox{as}\quad r \rightarrow +0 \\
&x_+^{\mathsf{p}}-x \sim 2C_1(R-r)^{\frac{1}{2}} \quad\mbox{as}\quad r \rightarrow R-0.\end{aligned}$$
Look at $$\begin{aligned}
&q^{\mathsf{p}}=\mathscr{N}^2+\frac{1}{4}\frac{\gamma P}{\rho}Q, \\
&Q=\frac{d^2}{dr^2}\log\frac{\rho P^{1-\frac{4}{\gamma}}}{r^4}
-\frac{1}{4}\Big(\frac{d}{dr}\log\frac{\rho P^{1-\frac{4}{\gamma}}}{r^4}\Big)^2 +\Big(\frac{d}{dr}\log\frac{P^{1-\frac{2}{\gamma}}}{r^2}\Big)
\Big(\frac{d}{dr}\log\frac{\rho P^{1-\frac{4}{\gamma}}}{r^4}\Big).\end{aligned}$$ It can be shown that $$q^{\mathsf{p}} \sim \frac{2}{x^2}\quad\mbox{as}\quad x \rightarrow +0$$ and $$q^{\mathsf{p}} \sim \frac{(3-\gamma)(\gamma+1)}{4(\gamma-1)^2}\frac{1}{(x_+^{\mathsf{p}}-x)^2}
\quad\mbox{as}\quad x \rightarrow x_+^{\mathsf{p}} -0.$$ Here note that $$\frac{(3-\gamma)(\gamma+1)}{4(\gamma-1)^2}>\frac{3}{4}$$ for $1<\gamma <2$. Anyway there is a constant $C$ such that $$q^{\mathsf{p}} \geq -C \quad\mbox{for}\quad 0<x<x_+^{\mathsf{p}}.$$ The boundary points $0, x_+^{\mathsf{p}}$ are of limit point type. Hence, thanks to [@ReedS p.159, Theorem X.10], we can claim that the operator $\displaystyle -\frac{d^2}{dx^2}+q^{\mathsf{p}}$ defined on $C_0^{\infty}(]0,x_+^{\mathsf{p}}[)$ admits the Friedrichs extension, a self-adjoint operator in $L^2([0,x_+^{\mathsf{p}}])$, whose spectrum consists of simple eigenvalues. In other words we can claim the following
The operator $L^{\mathsf{p}}$ defined on $C_0^{\infty}(]0,R[)$ admits the Friedrichs extension, a self-adjoint operator in $\displaystyle L^2([0, R], \frac{1}{r^2\mathscr{B}}dr)$, whose spectrum consists of simple eigenvalues $\lambda_n^{\mathsf{p}}, n=1,2,\cdots,$ : $\lambda_1^{\mathsf{p}}<\lambda_2^{\mathsf{p}} < \cdots <\lambda_n^{\mathsf{p}}<\cdots \rightarrow +\infty$.
Note that $\lambda_1^{\mathsf{p}} >0$, since $a>0, b>0$.
Behavior of $\mathscr{S}_l^2$ and $\mathscr{N}^2$ as functions of $r$ and related speculations
----------------------------------------------------------------------------------------------
Summing up the above discussions, we have two sequences of eigenvalues $\lambda_n^{\mathsf{g}}$ and $\lambda_n^{\mathsf{p}}$, of which the former accumulates to 0 and the latter accumulates to $+\infty$. Astrophysicists believe that they give, at least approximately, eigenvalues of the problem : $\vec{L}_{[\mathrm{C}]l}\vec{V}=\lambda\vec{V}$. See, e.g., [@LedouxW], [@Cox], and so on. The priority of this observation may go back to the work by T. G. Cowling, [@Cowling], on November 3, 1941. So, we have the following
\[Q.3\] Can we perform a mathematically rigorous justification of the observation that this sequence of eigenvalues which runs to two directions and accumulates both to 0 and to $+\infty$ gives a good landscape of the real eigenvalues of $\vec{L}_l$ ?
Mathematically rigorous affirmative answer to the Question \[Q.3\] has not yet been found. But we guess that the Cowling approximation by $\vec{L}_{[\mathrm{C}]l}$ of $\vec{L}_l$ by neglecting the gravitational perturbation may be relatively easily justified. However it may be not so easy to justify the approximation of the eigenvalue problem by the g-modes / p-modes argument.
Let us reconsider the way of approximations leading the g-modes and p-modes.\
### Behavior of $\mathscr{S}_l^2$ and $\mathscr{N}^2$
First let us consider the behavior of the quantity $\mathscr{N}^2$ as a function of $r$.\
Here let us weaken the Assumption \[A.3\] as the following\
[**Assumption $\mathbf{2}^{\flat}$** ]{} [*There exist positive numbers $\delta $ and $r_0 (<R)$ such that $$\frac{1}{r}\frac{d\bar{S}}{dr} \geq \delta$$ for $0<r\leq r_0$.* ]{}\
Namely we do not require that $d\bar{S}/dr > 0$ throughout $0<r<R$ but admit that $d\bar{S}/dr < 0$ so that $ \mathscr{N}^2 < 0$ for $0< R-r \ll 1$ in accordance with the more realistic view due to the helioseismology.\
Let us recall how we derived the equation of g-modes. We put $\lambda=0$ in . If $\mathscr{N}^2$ were a positive constant, $\mathscr{N}^2-\lambda$ could be replaced by $\mathscr{N}^2$ for very small $\lambda$. But now $\mathscr{N}^2$ is a function of $r$. The limit $\displaystyle \mathscr{N}_{\mathsf{O}1}^2=\lim_{r\rightarrow +0}\frac{1}{r}
\frac{d\mathscr{N}^2}{dr}$ exists. Since $$\mathscr{N}^2=-\frac{1}{\gamma \mathsf{C}_V}\frac{dS}{dr}\frac{1}{\rho}\frac{dP}{dr},$$ the Assumption $\mathbf{2}^{\flat}$ and $P_{\mathsf{O}1}>0$ imply that $\mathscr{N}_{\mathsf{O}1}^2$ is positive. Thus $\mathscr{N}^2(r)$ is strictly increasing on an interval $]0, \delta_0], \delta_0 \ll 1$ and $\mathscr{N}^2 \sim \frac{1}{2}\mathscr{N}_{\mathsf{O}1}^2r^2$ as $r\rightarrow +0$. Therefore we have positive numbers $\lambda_{\mathscr{N}}$ and $r_{\mathscr{N}}^*$ with the following property: For any $\lambda \in ]0, \lambda_{\mathscr{N}}]$ there is a unique $r_{\mathscr{N}}(\lambda) \in ]0, r_{\mathscr{N}}^*[$ such that, for $r \in ]0, r_{\mathscr{N}}^*]$, $$\lambda < \mathscr{N}^2(r) \quad \Leftrightarrow \quad
r_{\mathscr{N}}(\lambda)<r.$$
Then $\mathscr{N}^2-\lambda$ changes the sign near $0$. In other words, no matter how small $\lambda$ is, we still have $\mathscr{N}^2(r)
<\lambda$ for sufficiently small $r$. Thus the way of approximation is strongly doubtful.\
However, since $\mathscr{N}^2$ is bounded on $]0, R[$, we can generally claim at least that there exists sufficiently large $\lambda_{\mathscr{N}}^+$ such that, for any $\lambda \geq \lambda_{\mathscr{N}}^+$, it holds that $\mathscr{N}^2(r) <\lambda$ for $0<r<R$.\
On the other hand, let us consider the behavior of $\mathscr{S}_l^2$ as a function of $r$.
We see that, for $\lambda > 0$ fixed, $\displaystyle \frac{\mathscr{S}_l^2}{\lambda} -1 > 0$ for $0<r\ll 1$ and $\displaystyle \frac{\mathscr{S}_l^2}{\lambda} -1<0$ for $0<R-r \ll 1$, since $$\mathscr{S}_l^2 \sim \frac{l(l+1)\gamma P(O)}{\rho_{\mathsf{O}}}\frac{1}{r^2}
\quad\mbox{as}\quad r\rightarrow +0,$$ and $$\mathscr{S}_l^2 \sim l(l+1)C(R-r)\quad\mbox{as}\quad
r\rightarrow R-0,$$ where $C$ is a positive constant. Looking at $$\frac{1}{\mathscr{S}_l^2}\frac{d}{dr}\mathscr{S}_l^2 =\varphi(r)
+\frac{1}{\mathsf{C}_V}\frac{dS}{dr},$$ where $$\varphi(r):=-\frac{2}{r}+(\gamma-1)\frac{1}{\rho}\frac{d\rho}{dr},$$ we see that $ \varphi(r) <0$ for $0<r<R$, $\displaystyle \varphi(r) \sim -\frac{2}{r}$ as $r\rightarrow +0$, and $\displaystyle \varphi(r)\sim -\frac{\gamma(\gamma-1)}{R-r}$ as $r \rightarrow R-0$. Therefore we have $\displaystyle \frac{d}{dr}\mathscr{S}_l^2 <0$ for $0<R-r\ll 1$, since $dS/dr$ is bounded. Hence there exists $\lambda_{\mathscr{S}}>0$ such that, for any $0<\lambda \leq \lambda_{\mathscr{S}}$, there exists a unique $r_{\mathscr{S}}(\lambda) \in ]0, R[$ such that $$\mathscr{S}_l^2(r)<\lambda \quad \Leftrightarrow \quad
r_{\mathscr{S}}(\lambda)<r.$$ Then $\displaystyle \frac{\mathscr{S}_l^2}{\lambda}-1$ changes the sign near the surface $r=R$.
On the other hand, since $\displaystyle \frac{d}{dr}\mathscr{S}_l^2 <0$ for $0<r\ll 1$ and $\mathscr{S}_l^2 \rightarrow +\infty$ as $r\rightarrow +0$, there exists sufficiently large $\lambda_{\mathscr{S}}^+$ such that, for any $\lambda \geq \lambda_{\mathscr{S}}^+$, there exists unique $r_{\mathscr{S}}(\lambda) \in ]0,R[$ such that $$\mathscr{S}_l^2(r) >\lambda\quad \Leftrightarrow r< r_{\mathscr{S}}(\lambda).$$ Then $\displaystyle \frac{\mathscr{S}_l^2}{\lambda}-1$ changes the sign near the center $r=0$.
Moreover, we can claim that $\mathscr{S}_l^2$ is monotone decreasing from $+\infty$ to $0$ as $r$ varies from $0$ to $R$, provided that $dS/dr$ is sufficiently small uniformly. In fact, we see $\displaystyle \varphi(r) <-\frac{2}{R}$ for $0<r<R$, and, if $\displaystyle \sup_{0<r<R}\frac{dS}{dr} \leq \frac{2\mathsf{C}_V}{R}$, then $\displaystyle \frac{d}{dr}\mathscr{S}_l^2 <0$ for $0<r<R$. Actually this can be the case if, $4/3 < \gamma <2$ and $\rho_{\mathsf{O}}$ being fixed, we take $S=\Sigma(\rho^{\gamma-1})=-\varepsilon\rho^{\gamma-1}$ with $0<\varepsilon \ll 1$.\
Summing up, we can claim the following
\[Prop.NS\] There exist sufficiently small positive numbers $\lambda_0$ and $r_{\mathscr{N}}^*$ and a sufficiently large positive number $\lambda^+$, say, $0<\lambda_0<\lambda^+<+\infty$, such that
1\) if $0<\lambda \leq \lambda_0$, then there exist $r_{\mathscr{N}}(\lambda)
$ and $
r_{\mathscr{S}}(\lambda) $, while $0<r_{\mathscr{N}}(\lambda) <r_{\mathscr{N}}^* <
r_{\mathscr{S}}(\lambda) <R $, such that, for $0<r \leq r_{\mathscr{N}}^*$, $$\lambda -\mathscr{N}^2 (r)>0\quad\Leftrightarrow
\quad 0<r<r_{\mathscr{N}}(\lambda),$$ and $$\frac{\mathscr{S}_l^2(r)}{\lambda}-1<0\quad\Leftrightarrow
\quad r_{\mathscr{S}}(\lambda) < r < R;$$
2\) if $\lambda^+<\lambda$, then $$\lambda -\mathscr{N}^2(r) >0 \quad \mbox{for}\quad 0<r<R,$$ and there exists $r_{\mathscr{S}}^+(\lambda) \in
]0,R[$ such that $$\frac{\mathscr{S}_l^2(r)}{\lambda}-1>0\quad\Leftrightarrow
\quad
0<r<r_{\mathscr{S}}^+(\lambda).$$
Moreover $r_{\mathscr{N}}(\lambda) \rightarrow +0, r_{\mathscr{S}}(\lambda)\rightarrow R-0$ as $\lambda \rightarrow +0$, $r_{\mathscr{S}}^+(\lambda) \rightarrow +0$ as $ \lambda \rightarrow +\infty $, and we can suppose $$\frac{d}{dr}\mathscr{N}^2 \Big|_{r=r_{\mathscr{N}}(\lambda)}>0,\qquad
\frac{d}{dr}\mathscr{S}_l^2\Big|_{r=r_{\mathscr{S}}(\lambda)} <0 \quad\mbox{for}\quad 0<\lambda<\lambda_0$$ and $$\frac{d}{dr}\mathscr{S}_l^2\Big|_{r=r_{\mathscr{S}}^+(\lambda)} <0.
\quad\mbox{for} \quad \lambda >\lambda^+.$$
### Analysis according to H. Shibahashi
Section 16 of [@Unno2] discusses the properties of oscillations precisely for various modes by the asymptotic methods using Airy functions. Let us examine the argument found there.\
We are studying the system of equations ,, which we write as
$$\begin{aligned}
\frac{dv}{dr}&=G(r, \lambda)w, \\
\frac{dw}{dr}&=H(r, \lambda)v,\end{aligned}$$
where
$$\begin{aligned}
G&:=\Big(\frac{\mathscr{S}_l^2}{\lambda}-1\Big)\frac{r^2\mathscr{B}}{\mathsf{c}^2}, \\
H&:=(\lambda-\mathscr{N}^2)\frac{1}{r^2\mathscr{B}}.\end{aligned}$$
We put $$\mathsf{k}^2(r, \lambda):=-GH=\frac{1}{\mathsf{c}^2\lambda}(\mathscr{S}_l^2-\lambda)(\mathscr{N}^2-\lambda). \label{DefK}$$
By Proposition \[Prop.NS\], when $0<\lambda \ll 1$, $G(r)$ changes the sign at $r_{\mathscr{S}}(\lambda)$ and $H(r)$ changes the sign at $r_{\mathscr{N}}(\lambda)$, and $\mathsf{k}^2(r)<0$ for $ 0<r<r_{\mathscr{N}}(\lambda)$, $\mathsf{k}^2(r)>0$ for $r_{\mathscr{N}}(\lambda)<r
<r_{\mathscr{N}}^*$ and $\mathsf{k}^2(r) <0$ for $r_{\mathscr{S}}(\lambda)<r<r_{\mathscr{N}}^*$. On the other hand, when $1 \ll \lambda$, $\mathsf{k}^2(r)<0$ for $0<r<r_{\mathscr{S}}^+(\lambda)$ and $\mathsf{k}^2(r) >0$ for $r_{\mathscr{S}}^+(\lambda) <r <R$.
The change of variables $$v=|G|^{\frac{1}{2}}y, \qquad w=|H|^{\frac{1}{2}}z$$ leads us to the equations
$$\begin{aligned}
&-\frac{d^2y}{dr^2}+(\mathsf{F}[G]-\mathsf{k}^2)y=0, \\
&-\frac{d^2z}{dr^2}+(\mathsf{F}[H]-\mathsf{k}^2)z=0. \label{524b}\end{aligned}$$
Here the nonlinear differential operator $\mathsf{F}$ is defined by the following
For any function $Q$ of $r$ in an open interval, we put $$\begin{aligned}
\mathsf{F}[Q](r)&=|Q|^{\frac{1}{2}}\frac{d^2}{dr^2}|Q|^{-\frac{1}{2}} \nonumber \\
&=\frac{3}{4}\Big(\frac{1}{Q}\frac{dQ}{dr}\Big)^2-\frac{1}{2}\frac{1}{Q}\frac{d^2Q}{dr^2}\end{aligned}$$ for $r \in I$ such that $Q(r)\not=0$.
By Proposition \[Prop.NS\], we see $$\begin{aligned}
&\mathsf{F}[G](r) \quad\sim\quad \frac{3}{4}\frac{1}{(r-r_{\mathscr{S}}(\lambda))^2}
\quad\mbox{as}\quad r \rightarrow r_{\mathscr{S}}(\lambda), \\
&\mathsf{F}[H](r)\quad\sim\quad \frac{3}{4}\frac{1}{(r-r_{\mathscr{N}}(\lambda))^2}
\quad\mbox{as}\quad r \rightarrow r_{\mathscr{N}}(\lambda)\end{aligned}$$ when $ 0<\lambda \ll 1$, and $$\mathsf{F}[G](r)\quad\sim\quad
\frac{3}{4}\frac{1}{(r-r_{\mathscr{S}}^+(\lambda))^2}
\quad\mbox{as}\quad r\rightarrow r_{\mathscr{S}}^+(\lambda)$$ when $1 \ll \lambda$.
From Line 8 from bottom of Page 133, [@Unno2] considers the case of $\lambda \gg 1$, and, to obtain the function $z$ in the region containing the turning point $r_{\mathscr{S}}^+(\lambda)$, introduces the transformation of variables $(r, z) \mapsto (\zeta, Z)$ defined by
$$\begin{aligned}
\zeta&=\Big(\frac{dr}{d\zeta}\Big)^2(\mathsf{k}^2-\mathsf{F}[H]), \label{536a} \\
Z&=\Big|\frac{dr}{d\zeta}\Big|^{-\frac{1}{2}}z.\end{aligned}$$
In view of [@Unno2 (16.22)], it seems that is replaced by $$\zeta=\Big(\frac{dr}{d\zeta}\Big)^2\mathsf{k}^2 \label{537}$$ as a plausible approximation, in which $ \mathsf{F}[H]$ is neglected in comparison to $\mathsf{k}^2$. If we adopt this approximation, the equation for $Z$ turns out to be $$\frac{d^2Z}{d\zeta^2}+\Big(\zeta -\mathsf{F}\Big[\frac{dr}{d\zeta}\Big]\Big) Z=0,\label{528}$$ provided that the equation is replaced by $$-\frac{d^2z}{dr^2}-\mathsf{k}^2z=0 \label{529}$$ in view of the approximation of neglecting $\mathsf{F}[H]$ in comparison to $\mathsf{k}^2$. Moreover again $\displaystyle \mathsf{F}\Big[\frac{dr}{d\zeta}\Big] $ is neglected in comparison to $\zeta$ in . Then we are lead to the equation $$\frac{d^2Z}{d\zeta^2}+\zeta Z=0$$ and its solutions can be written by the Airy functions. Thus [@Unno2] continues the discussion of WBKJ asymptotic analysis.
However, it seems difficult to justify the approximation by $\mathsf{k}^2$ of $\mathsf{k}^2-\mathsf{F}[H]$ near the turning point $r_{\mathscr{S}}^+(\lambda)$ by the following reason.
Note that Proposition \[Prop.NS\] says $$\mathsf{k}^2 \quad \sim\quad C(r-r_{\mathscr{S}}^+(\lambda))$$ as $ r\rightarrow r_{\mathscr{S}}^+(\lambda)$, $C$ being a positive constant, since $\displaystyle \frac{d}{dr}\mathscr{S}_l^2 <0$ and $\mathscr{N}^2-\lambda >0$ near the point $r=r_{\mathscr{S}}^+(\lambda)$. On the other hand, since $H <0$ near the turning point, we have, at $r=r_{\mathscr{S}}^+(\lambda)$, $$\begin{aligned}
\mathsf{F}[H]&=\frac{1}{2}(\mathscr{N}^2-\lambda)^{-2}
\Big(\frac{d}{dr}\mathscr{N}^2\Big)^2
-\frac{1}{2}(\mathscr{N}^2-\lambda)^{-1}\frac{d^2}{dr^2}\mathscr{N}^2 + \\
&-\frac{1}{2}\frac{1}{r^2} +\frac{1}{2}\frac{d}{dr}\Big(\frac{1}{\mathscr{B}}\frac{d\mathscr{B}}{dr}\Big) + \\
&+\Big\{
-\frac{1}{2}(\mathscr{N}^2-\lambda)^{-1}\frac{d}{dr}\mathscr{N}^2
+\frac{1}{2}\frac{1}{r}+
\frac{1}{2}\frac{1}{\mathscr{B}}\frac{d\mathscr{B}}{dr}\Big\}^2 \\
&=\frac{1}{2}(\mathscr{N}^2-\mathscr{S}_l^2)^{-2}
\Big(\frac{d}{dr}\mathscr{N}^2\Big)^2
-\frac{1}{2}(\mathscr{N}^2-\mathscr{S}_l^2)^{-1}\frac{d^2}{dr^2}\mathscr{N}^2 + \\
&-\frac{1}{2}\frac{1}{r^2} +\frac{1}{2}\frac{d}{dr}\Big(\frac{1}{\mathscr{B}}\frac{d\mathscr{B}}{dr}\Big) + \\
&+\Big\{
-\frac{1}{2}(\mathscr{N}^2-\mathscr{S}_l^2)^{-1}\frac{d}{dr}\mathscr{N}^2
+\frac{1}{2}\frac{1}{r}+
\frac{1}{2}\frac{1}{\mathscr{B}}\frac{d\mathscr{B}}{dr}\Big\}^2 \\
&\sim\quad -\frac{1}{4r^2}\end{aligned}$$ as $r=r_{\mathscr{S}}^+(\lambda)\rightarrow +0$ for $\lambda \rightarrow +\infty$, while $\mathscr{N}^2 \rightarrow +0, \mathscr{S}_l^2\rightarrow +\infty$. Thus we see that $-\mathsf{F}[H](r_{\mathscr{S}}^+(\lambda)) \gg 1$, and it could not be neglected in comparison to $\mathsf{k}^2$ which vanishes at $r=r_{\mathscr{S}}^+(\lambda)$. In other words, the approximation [@Unno2 (16.22)], that is, $$\zeta=\frac{\mathsf{k}^2(r)}{|\mathsf{k}^2(r)|}\Big|\frac{3}{2}\int_{r_{\mathscr{S}}^+(\lambda)}^r
|\mathsf{k}^2(r')|^{\frac{1}{2}}dr'\Big|^{\frac{2}{3}}, \label{531}$$ is strongly doubtful.
Can repeating neglects of $\mathsf{F}[H]$, two times, and $\mathsf{F}[dr/d\zeta]$ produce a justifiable good approximation as the result by canceling errors in combination?
For trial let us observe the situation when we avoid the neglect of $\mathsf{F}[H]$ in comparison to $\mathsf{k}^2$. Even if we do not replace by and if we do not replace by , we get the same equation , that is, is the exact equation without approximation by neglecting $\mathsf{F}[H]$ in comparison to $\mathsf{k}^2$. But, if we use the exact , we should use it only for $r -r_{\mathscr{S}}^+>0$ which should be mapped to $\zeta >0$, since, as we know, $(\mathsf{k}^2-\mathsf{F}[H])(r) > 0$ for $|r-r_{\mathscr{S}}^+| \ll 1$. Then instead of , we have, for $0 < r- r_{\mathscr{S}}^+ \ll 1$, $$\zeta=
\Big(\frac{3}{2}\int_{r_{\mathscr{S}}^+}^r
\sqrt{\mathsf{k}^2-\mathsf{F}[H]}(r')dr'\Big)^{\frac{2}{3}}.$$ Therefore we can claim $$\begin{aligned}
& \zeta \sim \Big(\frac{3}{2}\Big)^{\frac{2}{3}}
|\mathsf{F}[H](r_{\mathscr{S}}^+)|^{\frac{1}{3}}
(r-r_{\mathscr{S}}^+)^{\frac{2}{3}}, \\
&\frac{dr}{d\zeta} \sim |\mathsf{F}[H](r_{\mathscr{S}}^+)|^{-\frac{1}{2}}
\zeta^{\frac{1}{2}}, \\
& \mathsf{F}\Big[\frac{dr}{d\zeta}\Big](\zeta) \sim \frac{5}{16}\frac{1}{\zeta^2}\end{aligned}$$ as $\zeta \rightarrow +0$. Thus the quantity $\mathsf{F}[dr/d\zeta]$ could not be neglected in comparison to $\zeta$. In other words, the approximation by for the equation is still strongly doubtful.\
Here let us remark that the corresponding argument of the first edition [@Unno] in its Section 15, which is the same as that of [@Shibahashi], considers the case of $0<\lambda \ll 1$ to investigate the g-modes near the turning point $r_{\mathscr{N}}(\lambda)$. However the same difficulty can be observed also in this discussion. The transformation $(r, y) \mapsto (\xi, Y)$ is defined by $$\begin{aligned}
\xi&=\Big(\frac{dr}{d\xi}\Big)^2(\mathsf{k}^2-\mathsf{F}[G]), \label{5.33}\\
Y&=\Big|\frac{dr}{d\xi}\Big|^{-\frac{1}{2}}y. \label{5.34}\end{aligned}$$ The term $\mathsf{F}[G]$ is neglected in comparison to $\mathsf{k}^2$ at $r=r_{\mathscr{N}}(\lambda)$ to deduce [@Unno (15.22)] or [@Shibahashi (15)], say, $$\xi=\frac{\mathsf{k}^2(r)}{|\mathsf{k}^2(r)|}
\Big|\frac{3}{2}\int_{r_{\mathscr{N}}(\lambda)}^r
|\mathsf{k}^2(r')|^{\frac{1}{2}}dr'\Big|^{\frac{2}{3}}.$$ But $\mathsf{k}^2$ vanishes at $r=r_{\mathscr{N}}(\lambda)$, while $\mathsf{F}[G](r_{\mathscr{N}}(\lambda)) \gg 1$, since we can show that, at $r=r_{\mathscr{N}}(\lambda)$, $$\mathsf{F}[G] \quad \sim\quad \frac{3}{4}\frac{1}{r^2}$$ as $r=r_{\mathscr{N}}(\lambda)\rightarrow +0$ for $\lambda \rightarrow +0$. Thus the approximation proposed here is doubtful, too. Let us observe the situation by exact use of the equation . Now, since $\mathsf{k}^2-\mathsf{F}[G] < 0$ near $r_{\mathscr{N}}$, used for $r > r_{\mathscr{N}}$ should give the mapping of $ r >r _{\mathscr{N}}$ to $\xi <0$. Therefore we have, for $0< r-r_{\mathscr{N}} \ll 1$, $$\xi =-\Big(\frac{3}{2}\int_{r_{\mathscr{N}}}^r
\sqrt{\mathsf{F}[G]-\mathsf{k}^2}(r')dr'\Big)^{\frac{2}{3}}.$$ We have $$\begin{aligned}
& \xi \sim -\Big(\frac{3}{2}\Big)^{\frac{2}{3}}(\mathsf{F}[G](r_{\mathscr{N}}))^{\frac{1}{3}}
(r-r_{\mathscr{N}})^{\frac{2}{3}}, \\
& \frac{dr}{d\xi} \sim -(\mathsf{F}[G](r_{\mathscr{N}}))^{-\frac{1}{2}}|\xi|^{\frac{1}{2}}, \\
&\mathsf{F}\Big[\frac{dr}{d\xi}\Big]=\mathsf{F}\Big[-\frac{dr}{d\xi}\Big]
\sim \frac{5}{16}\frac{1}{|\xi|^2}\end{aligned}$$ as $\xi \nearrow 0 \quad (\Leftrightarrow r \searrow r_{\mathscr{N}})$. Thus the approximation by $\displaystyle \frac{d^2Y}{d\xi^2}+\xi Y=0$ for the equation $$\frac{d^2Y}{d\xi^2}+\Big(\xi-\mathsf{F}\Big[\frac{dr}{d\xi}\Big]\Big)Y=0$$ is doubtful, too.\
### Analysis according to D. O. Gough
Another formulation for the asymptotic analysis of the eigenvalue problem is done by D. O. Gough [@Gough]. Let us introduce it.\
We start with the eigenvalue problem under the Cowling approximation. Let us consider $\lambda \not=0$. Then the system to be considered is
$$\begin{aligned}
&\frac{1}{r^2}\frac{d}{dr}(r^2V^r)-\frac{g}{\mathsf{c}^2}V^r+
\Big(1-\frac{l(l+1)\mathsf{c}^2}{\lambda r^2}\Big)\frac{\delta\check{P}}{\rho \mathsf{c}^2}=0 \label{543.1a} \\
&\frac{d}{dr}\delta\check{P}+\frac{g^2}{\mathsf{c}^2}\delta\check{P}+
(\mathscr{N}^2-\lambda)\rho V^r=0. \label{543.1b}\end{aligned}$$
Here we put $$g:=-\frac{1}{\rho}\frac{dP}{dr}=\frac{d\Phi}{dr},\qquad \mathsf{c}^2=\frac{\gamma P}{\rho}.$$ Let us recall $$\mathscr{N}^2=-g\mathscr{A}=\frac{1}{\mathsf{C}_{V}}g\frac{dS}{dr}=g\Big(
\frac{1}{\mathsf{H}[\rho]}-\frac{g}{\mathsf{c}^2}\Big),$$ where and hereafter we use the following
For any function $Q$ of $r$ in an open interval $I$, we put $$\frac{1}{\mathsf{H}[g]}:=-\frac{1}{Q}\frac{dQ}{dr}$$ for $r \in I$ such that $Q(r)\not= 0$.
$\mathsf{H}[Q]$ is called the ‘scale hight of the quantity $Q$’. We note the relation $$\mathsf{F}[Q]=\frac{1}{4\mathsf{H}[Q]^2}\Big(1-2\frac{d}{dr}\mathsf{H}[Q]\Big).$$
Anyway the system for $(\xi, \eta)$, where $$\xi= V^r,\qquad \eta=\Delta\check{P}=\delta\check{P}+\frac{dP}{dr}V^r,$$ reads
$$\begin{aligned}
&\frac{d\xi}{dr}+A_{11}\xi+A_{12}\eta =0, \label{543.6a} \\
&\frac{d\eta}{dr}+A_{21}\xi+A_{22}\eta=0 \label{543.6b}\end{aligned}$$
with $$\begin{aligned}
&A_{11}=\frac{2}{r}-\frac{l(l+1)g}{\lambda r^2},\qquad A_{12}=
\Big(1-\frac{l(l+1)\mathsf{c}^2}{\lambda r^2}\Big)\frac{1}{\rho \mathsf{c}^2}, \nonumber \\
&A_{21}=\frac{g\rho F}{\lambda r},\qquad A_{22}\frac{l(l+1)g}{\lambda r^2},\end{aligned}$$ where we put $$F=F(\lambda):=l(l+1)\frac{g}{r}
-\Big(2+\frac{r}{\mathsf{H}[g]}\Big)\lambda
-\frac{r}{g}\lambda^2.$$
Now the system turns out to be the single second order equation for $\eta$: $$\frac{d^2\eta}{dr^2}+\frac{1}{\mathsf{H}[\mathfrak{Q}]}\frac{d\eta}{dr}+
\Big[\frac{1}{\mathsf{c}^2}\Big(\lambda+g\Big(\frac{1}{\mathsf{H}[g]}+
\frac{2}{r}\Big)\Big)-\frac{l(l+1)}{r^2}\Big(1-\frac{\mathfrak{N}^2}{\lambda}\Big)\Big]\eta=0, \label{543.9}$$ where we put $$\mathfrak{Q}:=\frac{g\rho F}{r^3},$$ and $$\mathfrak{N}^2:=g\Big(\frac{1}{\mathsf{H}[\mathfrak{Q}]}-\frac{g}{\mathsf{c}^2}-\frac{2}{\mathsf{H}[g]}
-\frac{4}{r}\Big) \quad
\Big( =\mathscr{N}^2+\frac{g}{\mathsf{H}[rF/g]}\Big). \label{543.47}$$
The second order equation is transformed to the standard form $$-\frac{d^2\Psi}{dr^2}+\mathfrak{V}(r)\Psi=0 \label{543.12}$$ by the change of variable $$\eta=\mathfrak{Q}^{\frac{1}{2}}\Psi.$$ It can be calculated that $$\mathfrak{V}(r)=\frac{1}{\mathsf{c}^2}(\lambda_c-\lambda)+\frac{l(l+1)}{r^2}
\Big(1-\frac{\mathfrak{N}^2}{\lambda}\Big),$$ where we put $$\lambda_c:=
\frac{\mathsf{c}^2}{4\mathsf{H}[\mathfrak{Q}]^2}
\Big(1-2\frac{d}{dr}\mathsf{H}[\mathfrak{Q}]\Big)
-g\Big(\frac{1}{\mathsf{H}[g]}+\frac{2}{r}\Big). \label{543.51}$$
So, D. O. Gough [@Gough] performs asymptotic analysis of this equation , which is [@Gough (5.4.7)], while $K^2$ of [@Gough] is nothing but $-\mathfrak{V}(r)$ here.
The reduction of the problem to the above standard form is exact, and, in order to perform the WKB analysis, D. O. Gough rewrites $\mathfrak{V}(r)$ as $$\begin{aligned}
-\lambda\mathsf{c}^2\mathfrak{V}(r)&=\lambda(\lambda-\lambda_c)-\mathscr{S}_l^2(\lambda-\mathfrak{N}^2) \nonumber \\
&=(\lambda-\lambda_+)(\lambda-\lambda_-),\end{aligned}$$ where $$\lambda_{\pm}=\frac{1}{2}\Big(
(\mathscr{S}_l^2+\lambda_c)\pm\sqrt{(\mathscr{S}_l^2+\lambda_c)^2-4\mathfrak{N}^2\mathscr{S}_l^2}\Big).$$ See [@Gough (5.5.1)(5.5.2)]. But we should note that $\lambda_c$ and $\mathfrak{N}$ depends not only on $r$ but also on $\lambda$. In fact, using the quantity $F(\lambda)=F(r,\lambda)$, we can write $$\begin{aligned}
\frac{1}{\mathsf{H}[\mathfrak{Q}]}&=\frac{1}{\mathsf{H}[g]}+
\frac{1}{\mathsf{H}[\rho]}+\frac{1}{\mathsf{H}[F(\lambda)]}+\frac{3}{r} \nonumber \\
&=\frac{1}{\mathsf{H}[\rho]}+\frac{1}{\mathsf{H}[gF(\lambda)/r^3]}, \\
\mathfrak{N}^2&=g\Big(-\frac{1}{\mathsf{H}[g]}+
\frac{1}{\mathsf{H}[\rho]}+\frac{1}{\mathsf{H}[F(\lambda)]}-\frac{1}{r}-\frac{g}{\mathsf{c}^2}\Big) \nonumber \\
&=\mathscr{N}^2+\frac{g}{\mathsf{H}[rF(\lambda)/g]}.\end{aligned}$$ (Here let us note that $g/r$ and $r/g$ are bounded on $0<r<R$. ) So, it is not clear whether the term $1/\mathsf{H}[F(\lambda)]$ can be neglected or, at least, can be replaced by a quantity independent of $\lambda$, or not, in order to perform the asymptotic analysis to justify the existence of g-modes.
In fact, in the preceding review [@DeubnerG] by D. O. Gough himself and in the later book [@Aerts] by C. Aerts, J. Christensen-Dalsgaard and D. W. Kurtz, the approximation $${\mathsf{H}[\mathfrak{Q}]} \approx {\mathsf{H}[\rho]}, \quad
\mathfrak{N}^2 \approx \mathscr{N}^2, \quad
\lambda_c \approx \frac{\mathsf{c}^2}{4\mathsf{H}[\rho]}\Big(1-2\frac{d}{dr}\mathsf{H}[\rho]\Big)$$ is adopted. If we use this approximation, we can forget the dependence on $\lambda$ of $\mathfrak{N}^2$ and $\lambda_c$. As D. O. Gough clarifies in [@Gough p. 440], this approximation is done by ‘not taking the spherical geometry fully into account’. In other words, , ‘reduce to the above approximation if $\displaystyle \frac{1}{\mathsf{H}[g]}+\frac{2}{r} \rightarrow 0$ and $\displaystyle \frac{1}{\mathsf{H}[\mathfrak{Q}]}\rightarrow\frac{1}{\mathsf{H}[\rho]}$’. It is said that ‘ $\mathfrak{N}^2$ and $\lambda_c$ are approximated well except very close to the center of the star’. So D. O.Gough calls this limits of $\lambda_c$ and $\mathfrak{N}^2$ the ‘planer values’.
Formulation as a first order system of four ordinary differential equations
===========================================================================
Let us consider the eigenvalue problem $$\vec{L}_l\vec{V}=\lambda \vec{V} \label{6.1}$$ for $ l\geq 1$. We consider $\lambda \not=0$.\
According to [@Dziembowski], we introduce the variables
$$\begin{aligned}
y_1&=\frac{V^r}{r}, \\
y_2&=-\lambda\Big(\frac{1}{\rho}\frac{dP}{dr}\Big)^{-1}V^h, \\
y_3&=-\frac{1}{r}\Big(\frac{1}{\rho}\frac{dP}{dr}\Big)^{-1}\delta\check{\Phi}, \\
y_4&=-\Big(\frac{1}{\rho}\frac{dP}{dr}\Big)^{-1}\frac{d}{dr}
\delta\check{\Phi}.\end{aligned}$$
Then the eigenvalue problem reads $$r\frac{d\vec{y}}{dr}=A(r, \lambda)\vec{y}, \label{6.3}$$ where $$\vec{y}=\begin{bmatrix}
y_1 \\
y_2 \\
y_3 \\
y_4
\end{bmatrix},
\quad
A(r,\lambda)=
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{bmatrix}$$ with $$\begin{aligned}
&a_{11}=-3-\frac{r}{\gamma P}\frac{dP}{dr}, \quad a_{12}=
\Big(1-\frac{\mathscr{S}_l^2}{\lambda}\Big)\frac{r}{\gamma P}\frac{dP}{dr},
\quad a_{13}=-\frac{r}{\gamma P}\frac{dP}{dr}, \quad a_{14}=0 \nonumber \\
&a_{21}=(\mathscr{N}^2-\lambda)\Big(\frac{1}{r\rho}\frac{dP}{dr}\Big)^{-1},
\quad a_{22}=-\Big(\frac{dP}{dr}\Big)^{-1}\frac{d}{dr}\Big(r\frac{dP}{dr}\Big)+
\frac{r}{\gamma P}\frac{dP}{dr}, \nonumber \\
&a_{23}=r\mathscr{A}, \quad a_{24}=0 \nonumber \\
&a_{31}=0, \quad a_{32}=0, \quad a_{33}=\frac{r^2}{\rho}\frac{dP}{dr}\frac{d}{dr}\Big(\frac{r}{\rho}\frac{dP}{dr}\Big)^{-1} , \quad a_{34}=1, \nonumber \\
&a_{41}=4\pi\mathsf{G}r^2\rho^2\Big(\frac{dP}{dr}\Big)^{-1}\mathscr{A},
\quad a_{42}=
4\pi\mathsf{G}\frac{r^2\rho^2}{\gamma P}, \nonumber \\
&a_{43}=-4\pi\mathsf{G}\frac{r^2\rho^2}{\gamma P}+l(l+1),
\quad a_{44}=-\Big(\frac{r}{\rho}\frac{dP}{dr}\Big)^{-1}\frac{d}{dr}
\Big(\frac{r^2}{\rho}\frac{dP}{dr}\Big).\end{aligned}$$
Here recall $$\mathscr{A}=\frac{1}{\rho}\frac{d\rho}{dr}-\frac{1}{\gamma P}\frac{dP}{dr}, \quad
\mathscr{N}^2=\mathscr{A}\frac{1}{\rho}\frac{dP}{dr}, \quad
\mathscr{S}_l^2=\frac{l(l+1)}{r^2}\frac{\gamma P}{\rho}.$$
The boundary conditions at $r=+0$, which correspond to $\vec{V}\in {\mathfrak{W}}_l$ and $\delta\check{\Phi}=-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho})$, read
$$\begin{aligned}
&\int_0^{\epsilon}|y_1|^2\rho r^4dr +
\frac{l(l+1)}{|\lambda|^2}\int_0^{\epsilon}
|y_2|^2\frac{1}{\rho}\Big(\frac{dP}{dr}\Big)^2r^2dr + \nonumber \\
&+\frac{1}{\gamma}\int_0^{\epsilon}
|y_2-y_3|^2\frac{1}{P}\Big(\frac{dP}{dr}\Big)^2r^4dr < \infty, \\
&y_3=O(r^{-\frac{3}{2}}),\quad y_4=O(r^{-\frac{3}{2}})\quad\mbox{as}\quad r \rightarrow +0.\end{aligned}$$
Here $0< \epsilon \ll 1$.
On the other hand the boundary conditions at $r=R-0$ read
$$\begin{aligned}
&\int_{R-\epsilon}^{R}|y_1|^2\rho r^4dr +
\frac{l(1+1)}{|\lambda|^2}\int_{R-\epsilon}^{R}
|y_2|^2\frac{1}{\rho}\Big(\frac{dP}{dr}\Big)^2r^2dr + \nonumber \\
&+\frac{1}{\gamma}\int_{R-\epsilon}^{R}
|y_2-y_3|^2\frac{1}{P}\Big(\frac{dP}{dr}\Big)^2r^4dr < \infty, \\
& y_3(R)=\lim_{r\rightarrow R-0}y_3(r),
y_4(R)=\lim_{r\rightarrow R-0}y_4(r)\quad\mbox{exist and} \nonumber \\
&y_4(R)+(l+1)y_3(R)=0. \label{6.9b}\end{aligned}$$
Note that the condition , which means $$r\frac{d}{dr}\delta\check{\Phi}=-(l+1)\delta\check{\Phi}\quad\mbox{at}\quad r=R,$$ comes from that $\delta\check{\Phi}=-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho})$ should satisfy $$\delta\check{\Phi}(r)=\frac{C}{r^{l+1}} \quad\mbox{for}\quad r \geq R.$$ Otherwise, the corresponding $\delta\check{\Phi}$ for which $\displaystyle -r\frac{1}{\rho}\frac{dP}{d\rho}y_3$ is the candidate might be equal to $-4\pi\mathsf{G}\mathcal{H}_l(\delta\check{\rho})+C'r^{l}$ with $C'\not=0$.\
Now let us suppose
\[Ass.3\] $\rho$ and $S$ are analytic functions of $r^2$ near $r=0$.
This assumption holds if $S=\Sigma(\rho^{\gamma-1})$ with a function $\Sigma(\eta)$ which is analytic near $\eta=\rho_{\mathsf{O}}^{\gamma-1}$.\
Let us consider the system at $r=+0$. Under the Assumption \[Ass.3\], we see reads $$r\frac{d\vec{y}}{dr}=\Big(K_0+\sum_{m\geq 1}r^{2m}K_m\Big)\vec{y}, \label{6.10}$$ where $$K_0=
\begin{bmatrix}
-3 & \frac{l(l+1)\beta}{\lambda} & 0 & 0 \\
\frac{\lambda}{\beta} & -2 & 0 & 0 \\
0 & 0 & -2 & 1 \\
0 & 0 & l(l+1) & -3
\end{bmatrix}$$ and $\sum r^{2m}K_m$ is a convergent matrix-valued power series in $r^2$ with positive radius of convergence. Here $\beta=P_{\mathsf{O}1}/\rho_{\mathsf{O}}$ with a positive number $$P_{\mathsf{O}1}:=-\lim_{r\rightarrow +0}\frac{1}{r}\frac{dP}{dr}.$$ The eigenvalues of $K_0$ are $l-2, -(l+3)$, which are double.
Thus, putting $$z=r^2,\quad \vec{y}=
\begin{bmatrix}
l\beta & 0 & (l+1)\beta & 0 \\
\lambda & 0 & -\lambda & 0 \\
0 & 1 & 0 & 1 \\
0 & l & 0 & -(l+1)
\end{bmatrix}
\vec{w},$$ we have a system $$z\frac{d\vec{w}}{dz}=\Big(\mathbf{R}+
\sum_{m\geq 0}z^{m+1}A_m\Big)\vec{w} \label{6.14}$$ with $$\mathbf{R}=\mathrm{diag}(\rho_1, \rho_1, \rho_2, \rho_2),$$ where $$\rho_1=\frac{l-2}{2}, \qquad \rho_2=-\frac{l+3}{2}.$$ Since $\rho_1-\rho_2=l+\frac{1}{2}$ is not an integer, [@CoddingtonL Chapter 4, Theorem 4.1] gives a fundamental matrix $\Phi_w$ of the system of the form $$\Phi_w(z)=\Big(I_4+\sum_{m\geq 1}z^mP_m\Big)z^{\mathbf{R}},$$ where $\sum z^mP_m $ is a convergent matrix-valued power series. Note that $$z^{\mathbf{R}}=\mathrm{diag}(z^{\rho_1}, z^{\rho_1}, z^{\rho_2}, z^{\rho_2}).$$
As result, we have a fundamental system of solutions $\vec{y}=\mbox{\boldmath$\varphi$}_{0j}(r), j=1,2,3,4,$ of the system of the form
$$\begin{aligned}
& \mbox{\boldmath$\varphi$}_{01}=
\begin{bmatrix}
l{\beta} \\
\lambda \\
0 \\
0
\end{bmatrix}
r^{l-2}(1+O(r^2)), \\
&\mbox{\boldmath$\varphi$}_{02}=
\begin{bmatrix}
0 \\
0 \\
1 \\
l
\end{bmatrix}
r^{l-2}(1+O(r^2)), \\
&\mbox{\boldmath$\varphi$}_{03}=
\begin{bmatrix}
(l+1){\beta} \\
-\lambda \\
0 \\
0
\end{bmatrix}
r^{-(l+3)}(1+O(r^2)), \\
&\mbox{\boldmath$\varphi$}_{04}=
\begin{bmatrix}
0 \\
0 \\
1 \\
-(l+1)
\end{bmatrix}
r^{-(l+3)}(1+O(r^2)).\end{aligned}$$
It is easy to see that only $\mbox{\boldmath$\varphi$}_{01}, \mbox{\boldmath$\varphi$}_{02}$ satisfy the boundary conditions. Therefore we have $$\vec{y}=C_{01}\mbox{\boldmath$\varphi$}_{01}+C_{02}\mbox{\boldmath$\varphi$}_{02},$$ with constants $C_{01}, C_{02}$ in order that $\vec{y}$ gives $\vec{V} \in \mathfrak{W}_l$.\
Let us consider the system at the boundary point $r=R-0$. Here we suppose
\[Ass.4\] 1) As $r \rightarrow R-0$, it holds that
$$\begin{aligned}
&\rho=C(R-r)^{\nu}\Big(1+
\sum_{k_1+k_2\geq 1}a_{k_1k_2}
(R-r)^{k_1}(R-r)^{(\nu+1)k_2}\Big), \\
&S=\sum_{k_1+k_2\geq 0}b_{k_1k_2}(R-r)^{k_1}(R-r)^{(\nu+1)k_2},\end{aligned}$$
where $C$ is a positive constant, $$\nu=\frac{1}{\gamma-1},$$ and $\sum a_{k_1k_2}X_1^{k_1}X_2^{k_2}, \sum b_{k_1k_2}X_1^{k_1}X_2^{k_2}$ are double power series with positive radii of convergence.
2\) The index $\displaystyle \nu=\frac{1}{\gamma-1}$ is a rational number.
The first part 1) of the Assumption \[Ass.4\] holds if $S=\Sigma(\rho^{\gamma-1})$ with a function $\Sigma(\eta)$ which is analytic near near $\eta=0$. For a proof, see Appendix.\
Let $\nu=\nu_N/\nu_D, \nu_N, \nu_D $ being mutually prime natural numbers.
Putting $$s=(R-r)^{1/\nu_D},$$ the Assumptions \[Ass.4\] gives that the system reads $$s\frac{d\vec{y}}{ds}=\Big(K_0^R+\sum_{m\geq 1}s^m
K_m^R\Big)\vec{y}, \label{6.22}$$ where $$K_0^R=
\begin{bmatrix}
-\nu_N & \nu_N & \nu_N & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}.$$ So, putting $$z=s,\qquad \vec{y}=
\begin{bmatrix}
1 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\vec{w},$$ we have a system $$z\frac{d\vec{w}}{dz}=\Big(\mathbf{R}^R+\sum_{m\geq 0}z^{m+1}A_m^R\Big)\vec{w},$$ where $$\mathbf{R}^R=\mathrm{diag}(0, 0, 0, -\nu_N).$$ Here recall that $\nu_N$ is supposed to be an integer $\geq 2$. Hence we should apply the recipe prescribed in the proof of [@CoddingtonL Chapter 4, Theorem 4.2]. The result is a fundamental matrix $\Phi_y^R(s)$ of of the form $$\Phi_y^R(s)=
\begin{bmatrix}
1 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
I_3 & \sum_{j=1}^{\nu_N}s^{j-\nu_N}\mathbf{c}_j \\
\mathbf{0}^{\top} & s^{-\nu_N}
\end{bmatrix}
(I_4+\sum_{m\geq 1}s^mP_m)
\begin{bmatrix}
I_3 & (\log s)\mathbf{b} \\
\mathbf{0}^{\top} & 1
\end{bmatrix},$$ where $\sum s^mP_m$ is a convergent matrix-valued power series, $\mathbf{c}_j$ and $\mathbf{b}$ are 3-vectors.
Thus we have solutions $\vec{y}=\mbox{\boldmath$\varphi$}_{Rj}, j=1,2,3,4$ of the form
$$\begin{aligned}
&\mbox{\boldmath$\varphi$}_{R1}=
\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}
+O(s), \\
&\mbox{\boldmath$\varphi$}_{R2}=
\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}
+O(s), \\
&\mbox{\boldmath$\varphi$}_{R3}=
\begin{bmatrix}
0 \\
0 \\
0 \\
1
\end{bmatrix}
+O(s), \\
&\mbox{\boldmath$\varphi$}_{R4}=
\begin{bmatrix}
1 \\
0 \\
0 \\
0
\end{bmatrix}
s^{-\nu_N}(1+O(s\log s)).\end{aligned}$$
By the boundary conditions, we have $$\vec{y}=C_{R1}\mbox{\boldmath$\varphi$}_{R1}+C_{R2}(\mbox{\boldmath$\varphi$}_{R2}-(l+1)\mbox{\boldmath$\varphi$}_{R3}).$$\
Summing up, $\vec{y}$ should be $$\vec{y}=C_{01}\mbox{\boldmath$\varphi$}_{01}+C_{02}\mbox{\boldmath$\varphi$}_{02}=
C_{R1}\mbox{\boldmath$\varphi$}_{R1}+C_{R2}
(\mbox{\boldmath$\varphi$}_{R2}-(l+1)\mbox{\boldmath$\varphi$}_{R3}) \label{6.30}$$ in order that $\vec{y}$ gives $\vec{V}\in \mathfrak{W}_l$. Conversely, if so, then the corresponding $\vec{V} $ belongs to $\mathfrak{W}_l$ and we see that $V^r= ry_1$ is bounded, therefore, thanks to Proposition \[Prop.2\], $\vec{V}$ belongs to $\overset{\circ}{\mathfrak{W}}_l$ so to $\mathsf{D}(\vec{L}_l)$ as an eigenfunction.\
The condition reads $$D(r,\lambda):=\mathrm{det}(\mbox{\boldmath$\varphi$}_{01}(r,\lambda),
\mbox{\boldmath$\varphi$}_{02}(r,\lambda),
\mbox{\boldmath$\varphi$}_{R1}(r,\lambda),
\mbox{\boldmath$\varphi$}_{R2}
(r,\lambda)-(l+1)\mbox{\boldmath$\varphi$}_{R3}(r,\lambda)) =0.$$ But the condition $D(r,\lambda)=0$ is independent of $r$. So, fixing $r_0 \in ]0,R[$, we can consider $f(\lambda)=D(r_0,\lambda)$, which is a holomorphic function of $\lambda \in \mathbb{C}\setminus \{0\}$. Now we have that $f(\lambda)=0$ if and only if $\lambda$ is an eigenvalue of . Since $f(\lambda)\not=0$ for $ \lambda \notin [ -C, +\infty[ $, which $\lambda$ belongs to the resolvent set of $\vec{L}_l$, $f$ is not an identical $0$. Therefore the zeros of $f$ cannot accumulate to a value in $\mathbb{C}\setminus \{0\}$. Thus we can claim
Suppose the Assumptions \[Ass.3\],\[Ass.4\]. If there exist eigenvalues to the eigenvalue problem , then they are at most countably many eigenvalues located on the real axis, and cannot accumulate to a value $\not=0$.
Note that the above argument does not give the answer to the following
\[Q.ex\] Do exist actually eigenvalues to the problem ? In other words, does the function $f=D(r_0,\cdot)$ actually attains zeros on $\mathbb{R}\setminus \{0\}$ ?
In order to answer to this Question \[Q.ex\] we should examine the connection formula between the regular singular points $r=0, r=R$ as follows.
Since $(\mbox{\boldmath$\varphi$}_{01}, \mbox{\boldmath$\varphi$}_{02}, \mbox{\boldmath$\varphi$}_{03}, \mbox{\boldmath$\varphi$}_{04} )$ is a fundamental matrix of solutions, there are coefficients $a_{1j}(\lambda), a_{2j}(\lambda), j=1,2,3,4,$ of the connection formula such that $$\begin{aligned}
&\mbox{\boldmath$\varphi$}_{R1}=\sum_{j=1}^4a_{1j}\mbox{\boldmath$\varphi$}_{0j}, \\
&\mbox{\boldmath$\varphi$}_{R2}-(l+1)\mbox{\boldmath$\varphi$}_{R3}=
\sum_{j=1}^4a_{2j}\mbox{\boldmath$\varphi$}_{0j}.\end{aligned}$$ Then we have $$D(r,\lambda)=
\mathrm{det}
\begin{bmatrix}
a_{13}(\lambda) & a_{23}(\lambda) \\
a_{14}(\lambda) & a_{24}(\lambda)
\end{bmatrix}
\mathrm{det}(\mbox{\boldmath$\varphi$}_{01},\mbox{\boldmath$\varphi$}_{02},\mbox{\boldmath$\varphi$}_{03}, \mbox{\boldmath$\varphi$}_{04}).$$ Since $\mathrm{det}(\mbox{\boldmath$\varphi$}_{01},\mbox{\boldmath$\varphi$}_{02},\mbox{\boldmath$\varphi$}_{03}, \mbox{\boldmath$\varphi$}_{04})\not=0$, we have $$D(r,\lambda)=0 \quad \Leftrightarrow \quad
\mathrm{det}
\begin{bmatrix}
a_{13}(\lambda) & a_{23}(\lambda) \\
a_{14}(\lambda) & a_{24}(\lambda)
\end{bmatrix}
=0,$$ which is the condition for that $\lambda$ is an eigenvalue. Thus we should examine the dependence of the coefficients $a_{1j}, a_{2j}, j=3,4,$ of the global connection formula on $\lambda$. But this task seems not to be so easy.\
Anyway, suppose the answer to the Question \[Q.ex\] is ‘Yes’, and let $\{\lambda_j\}$ be the eigenvalues, which are actually infinitely many. J. Eisenfeld [@Eisenfeld] claims that the spectrum $\sigma(\vec{L}_l)$ of the self-adjoint operator $\vec{L}_l$ can be exhausted by them, that is, $\sigma(\vec{L}_l)\setminus \{0\}=\{\lambda_j\}$. However the proof of [@Eisenfeld] seems to be incomplete. Let us examine it.
Let $\lambda\not=0 $ be not an eigenvalue. Then $(\vec{L}_l-\lambda)^{-1}$ exists and the range $\mathsf{R}(\vec{L}_l-\lambda)$ is dense in $\mathfrak{X}_l$, since the residual spectrum is empty for the self-adjoint $\vec{L}_l$. The problem is:\
\[Q.4\] Does it hold that $\mathsf{R}(\vec{L}_l-\lambda) =\mathfrak{X}_l$? Or, is $(\vec{L}_l-\lambda)^{-1}$ bounded with respect to the norm $\|\cdot\|_{\mathfrak{X}_l}$ ?
If the answer is always ‘Yes’, then $\lambda \in \varrho(\vec{L}_l)$ for $\lambda$ which is neither $0$ nor an eigenvalue,, and we can claim $\sigma(\vec{L}_l)\setminus \{0\}=\{\lambda_j\}$ as J. Eisenfeld claims. Then the answer to the following question is ‘Yes’:
\[Q.5\] Do the eigenfunctions associated with the eigenvalues $\{\lambda_j\}$ form a complete orthonormal system of $\mathfrak{X}_l$ ?
Actually J. Eisenfeld [@Eisenfeld] claims the answer ‘Yes’ to the Question \[Q.5\] by referring [@DunfordS p.905, Theorem X.3.4]. However, in order to apply (the unbounded version of ) [@DunfordS Theorem X.3.4], it is necessary to prove that $\sigma(\vec{L}_l)\setminus\{0\}=
\{\lambda_j\}$, that is, to prove the answer ‘Yes’ to the Question \[Q.4\]. So let us consider the Question \[Q.4\]
Of course, if the Question \[Q.5\] is affirmatively answered, it guarantees the very existence of the eigenvalues, say affirmative answer to the Question \[Q.ex\]. But, for us, this is a kind of circular argument.
However, we should note that, maybe we should consider the spectral property of the operator $\vec{L}_l$ in a suitable Hilbert space $\mathfrak{Y}$, e.g., $\mathfrak{W}_l$, which is not equal to but dense in $\mathfrak{X}_l$ such that the restriction of $\vec{L}_l$ to $\mathfrak{Y}$ is self-adjoint. Then we should consider the Questions \[Q.4\], \[Q.5\] in this situation. In fact, for the barotropic case, according to [@JJTM], the spectral property of $\mathbf{L}$ has been clarified not in $\mathfrak{H}=L^2(B_R, \rho d\mathbf{x}) $ but in $\mathfrak{F}=\mathfrak{H}\cap
\{ \mbox{\boldmath$\xi$} | \mathrm{div}(\rho\mbox{\boldmath$\xi$}) \in \mathfrak{G}\} $ with $\mathfrak{G}=
L^2(B_R,\frac{1}{\rho}\frac{dP}{d\rho}d\mathbf{x})\cap\{g|\int gd\mathbf{x}=0\}$ and the situation of $\vec{L}_l$ is the same. Thus maybe we should modify the Questions \[Q.4\], \[Q.5\] by replacing $\mathfrak{X}_l$ by this suitable $\mathfrak{Y}$.
In order to try to answer the Question \[Q.4\], maybe we should look at a concrete representation of the solution of the equation $$\vec{L}_l\vec{V}-\lambda\vec{V}=\vec{f}, \label{6.31}$$ where $\vec{f}=(f^r, f^h)^{\top} $ is given in $\mathsf{R}(\vec{L}_l-\lambda)$. Here we suppose that $\lambda\not=0$ and $\lambda$ is not an eigenvalue.
It can be shown that reads $$r\frac{d\vec{y}}{dr}=A(r, \lambda)\vec{y}+\vec{h}, \label{6.32}$$ where $\vec{h}=(h_1, h_2, h_3, h_4)^{\top}$ with
$$\begin{aligned}
h_1=&\frac{r\rho}{\gamma P}f^h, \\
h_2=&\Big(\frac{1}{\rho}\frac{dP}{dr}\Big)^{-1}
\Big[-f^r+
r\frac{d}{dr}f^h + (1+r\mathscr{A})f^h \Big], \\
h_3=&0, \\
h_4=&4\pi\mathsf{G}r^2\frac{\rho^2}{P}\Big(\frac{1}{\rho}\frac{dP}{d\rho}\Big)^{-1}f^h.\end{aligned}$$
Put $$\mbox{\boldmath$\Phi$}(r)=(\mbox{\boldmath$\varphi$}_{01}, \mbox{\boldmath$\varphi$}_{02},\mbox{\boldmath$\varphi$}_{R1}, \mbox{\boldmath$\varphi$}_{R2}
-(l+1)\mbox{\boldmath$\varphi$}_{R3}).$$
Since $\lambda\not=0$ is not an eigenvalue, $\mbox{\boldmath$\Phi$}(r)$ is non-singular, that is, $\mathrm{det}\mbox{\boldmath$\Phi$}(r)\not=0$. The solution of should be of the form $$\vec{y}(r)=\mbox{\boldmath$\Phi$}(r)\int_{R/2}^r\mbox{\boldmath$\Phi$}(r')^{-1}
\vec{h}(r')\frac{dr'}{r'}+\mbox{\boldmath$\Phi$}(r)\mathbf{c},$$ where $\mathbf{c}$ is a suitable constant vector, which should be determined from $\vec{h}$ of $\vec{f}$ so that the corresponding $\vec{V}$ to $\vec{y}$ belong to $\mathsf{D}(\vec{L}_l)$.
In [@Eisenfeld p.365] J. Eisenfeld claims that the constant vector $\mathbf{c}=(c_1,c_2, c_3, c_4)^{\top}$ should be chosen as $$\begin{aligned}
& c_1=-\int_{R/2}^Rk_1(r)\frac{dr}{r},\qquad
c_2=-\int_{R/2}^Rk_2(r)\frac{dr}{r}, \\
&c_3=\int_0^Rk_3(r)\frac{dr}{r},\qquad
c_4=\int_0^Rk_4(r)\frac{dr}{r},\end{aligned}$$ where $k_j=k_i(r), i=1,2,3,4,$ is the $i$-th component of the vector $\mbox{\boldmath$\Phi$}(r)^{-1}\vec{h}(r)$, that is, $$\mbox{\boldmath$\Phi$}(r)^{-1}\vec{h}(r)=
\begin{bmatrix}
k_1 \\
k_2 \\
k_3 \\
k_4
\end{bmatrix}.$$ In other words, it is claimed that the solution $\vec{y}$ should be given as $$\begin{aligned}
\vec{y}(r)&=\Big(-\int_r^Rk_1(r')\frac{dr'}{r'}\Big)\mbox{\boldmath$\varphi$}_{01}(r) +
\Big(-\int_r^Rk_2(r')\frac{dr'}{r'}\Big)\mbox{\boldmath$\varphi$}_{02}(r) + \\
&+\Big(\int_0^rk_3(r')\frac{dr'}{r'}\Big)\mbox{\boldmath$\varphi$}_{R1}(r) +
\Big(\int_0^rk_4(r')\frac{dr'}{r'}\Big)(\mbox{\boldmath$\varphi$}_{R2}(r)-(l+1)\mbox{\boldmath$\varphi$}_{R3}(r)).\end{aligned}$$ However there we can find no persuasive argument on why the constants $c_j$ should be chosen so ?, and why $c_j$ can be chosen so?, that is, why the definite integrals $\displaystyle \int_{R/2}^Rk_1(r)\frac{dr}{r},\int_{R/2}^Rk_2(r)\frac{dr}{r},\int_0^Rk_3(r)\frac{dr}{r},
\int_0^Rk_4(r)\frac{dr}{r} $ are well-determined as finite numbers?
In order to determine $\mathbf{c}$, maybe we need asymptotic behaviors of $\mbox{\boldmath$\Phi$}(r)$ and $\mbox{\boldmath$\Phi$}(r)^{-1}$ as $r \rightarrow +0$ and as $r\rightarrow R-0$. Let us consider $r \rightarrow R-0$. Since $\mbox{\boldmath$\Phi$}(r)$ and $$\mbox{\boldmath$\Phi$}_R=(\mbox{\boldmath$\varphi$}_{R1}, \mbox{\boldmath$\varphi$}_{R2}, \mbox{\boldmath$\varphi$}_{R3}, \mbox{\boldmath$\varphi$}_{R4})$$ are fundamental matrices of the homogeneous system, there exists a constant matrix $C=(c_{ij})_{ij}$ such that $$\mbox{\boldmath$\Phi$}(r)=\mbox{\boldmath$\Phi$}_R(r)C.$$ See [@CoddingtonL Chapter 1, Theorem 7.3]. Of course $C$ is of the form $$C=
\begin{bmatrix}
c_{11} & c_{12} & 1 & 0 \\
c_{21} & c_{22} & 0 & 1 \\
c_{31} & c_{32} & 0 &-(l+1) \\
c_{41} & c_{42} & 0 & 0
\end{bmatrix}.$$ We should determine $c_{ij}, i=1,2,3,4, j=1,2$. But J. Eisenfeld claims a relation which can be interpreted as $$\begin{aligned}
\mbox{\boldmath$\varphi$}_{01}&=c_{31}\mbox{\boldmath$\varphi$}_{R3}+c_{41}\mbox{\boldmath$\varphi$}_{R4}, \\
\mbox{\boldmath$\varphi$}_{02}&=c_{32}\mbox{\boldmath$\varphi$}_{R3}+c_{42}\mbox{\boldmath$\varphi$}_{R4}\end{aligned}$$ at [@Eisenfeld p. 366, line 7]. In other words it is claimed that $c_{ij}=0$ for $i,j=1,2$. However we cannot find its proof there. It seems to be doubtful. Thus the asymptotic behaviors of the matrix given by [@Eisenfeld (5.11), (5.12)], which can be interpreted as the asymptotic behaviors of $\mbox{\boldmath$\Phi$}(r)^{-1}$ as $r \rightarrow +0$ and as $r\rightarrow R-0$, seem to be not well-grounded.
As conclusion, the argument by J. Eisenfeld [@Eisenfeld] is too weak to determine the answer to the Question \[Q.4\]. The problem is still open.\
[**Supplementary Remark** ]{}\
The same speculation can be done for the system of equations , under the Cowling approximation. Namely let us consider the system
$$\begin{aligned}
&\frac{dv}{dr}=\frac{1}{\lambda}(\mathscr{S}_l^2-\lambda)
\frac{r^2\mathscr{B}}{\mathsf{c}^2}w \label{SR201a} \\
&\frac{dw}{dr}=(\lambda-\mathscr{N}^2)\frac{1}{r^2\mathscr{B}}v \label{SR201b}\end{aligned}$$
for the variables $$v=r^2P^{\frac{1}{\gamma}}V^r,\quad w=P^{-\frac{1}{\gamma}}\delta\check{P}.$$ Recall $$\begin{aligned}
&\mathscr{B}=\frac{P^{\frac{2}{\gamma}}}{\rho},\quad
\mathscr{S}_l^2=\frac{l(l+1)}{r^2}\frac{\gamma P}{\rho},
\quad \mathsf{c}^2=\frac{\gamma P}{\rho}, \\
&\mathscr{N}^2=\frac{\mathscr{A}}{\rho}\frac{dP}{d\rho}=
-\frac{1}{\gamma \mathsf{C}_V}\frac{dS}{dr}\frac{1}{\rho}\frac{dP}{d\rho}=
\frac{1}{\gamma\mathsf{C}_V}\frac{dS}{dr}\frac{d\Phi}{dr}.\end{aligned}$$
The boundary conditions for $(v,w)$ at the boundaries $r=0, r=R$ can be derived from the condition that the corresponding $\vec{V}=(V^r,V^h)^{\top}$ belong to $\overset{\circ}{\mathfrak{W}}_l$. It reads $$\|\vec{V}\|_{\mathfrak{X}_l}^2=
\int_0^R|v|^2\frac{dr}{\mathscr{B}}+
\frac{1}{l(l+1)}\int_0^R\Big|\frac{dv}{dr}+
\frac{r^2\mathscr{B}}{\mathsf{c}^2}w\Big|^2\frac{dr}{\mathscr{B}} <\infty,$$ and $$\gamma\int_0^R|W|^2dr=
\int_0^R|w|^2\frac{r^2\mathscr{B}}{\mathsf{c}^2}dr <\infty,$$ provided that $r^2P^{-\frac{1}{\gamma}}|v|$ is bounded on $]0,R[$.
Let us consider the boundary $r=0$. The system , can be written $$r\frac{d\vec{u}}{dr}=
\begin{bmatrix}
-1 & \frac{1}{\lambda}(\mathscr{S}_l^2-\lambda)\frac{r^2\mathscr{B}}{\mathsf{c}^2} \\
\frac{\lambda-\mathscr{N}^2}{\mathscr{B}} & 0
\end{bmatrix}
\vec{u},$$ where $\displaystyle \vec{u}=(u,w)^{\top}, u=\frac{v}{r}$. Under the Assumption \[Ass.3\], we see $$\begin{bmatrix}
-1 & \frac{1}{\lambda}(\mathscr{S}_l^2-\lambda)\frac{r^2\mathscr{B}}{\mathsf{c}^2} \\
\frac{\lambda-\mathscr{N}^2}{\mathscr{B}} & 0
\end{bmatrix}
=
\begin{bmatrix}
-1 & \frac{l(l+1)B_{\mathsf{O}}}{\lambda} \\
\frac{\lambda}{B_{\mathsf{O}}} & 0
\end{bmatrix}
+[r^2]_1,$$ where $B_{\mathsf{O}}=\mathscr{B}\Big|_{r=0}=P_{\mathsf{O}}^{\frac{2}{\gamma}}/\rho_{\mathsf{O}}$. Therefore there is a fundamental system of solutions $\vec{u}_{01}=(u_{01},w_{01})^{\top},
\vec{u}_{02}=(u_{02}, w_{02})^{\top}$ such that $$\vec{u}_{01}\sim
\begin{bmatrix}
lB_{\mathsf{O}} \\
\lambda
\end{bmatrix}
r^l,\qquad
\vec{u}_{02}\sim
\begin{bmatrix}
(l+1)B_{\mathsf{O}} \\
-\lambda
\end{bmatrix}
r^{-(l+1)}$$ as $r\rightarrow +0$. Put $\mbox{\boldmath$\varphi$}_{0j}(r)=(ru_{0j}(r), w_{0j}(r))^{\top}, j=1,2$. Clearly only $\mbox{\boldmath$\varphi$}_{01}$ is the solution of which satisfies the boundary condition.
As for the boundary $r=R$, we consider the system $$s\frac{d\vec{U}}{ds}=
\begin{bmatrix}
-\nu_N &
-\nu_D\Big(\frac{\mathscr{S}_l^2}{\lambda}-1\Big)s^{-\nu_N+\nu_D}\frac{r^2\mathscr{B}}{\mathsf{c}^2} \\
-\nu_Ds^{\nu_N+\nu_D}(\lambda-\mathscr{N}^2)\frac{1}{r^2\mathscr{B}}& 0
\end{bmatrix}
\vec{U}$$ for $s=(R-r)^{1/\nu_D}, \vec{U}=(U, w)^{\top}, U=s^{-\nu_N}v$. Under the Assumptions \[Ass.3\] and \[Ass.4\], we see $$\begin{bmatrix}
-\nu_N &
-\nu_D\Big(\frac{\mathscr{S}_l^2}{\lambda}-1\Big)s^{-\nu_N+\nu_D}\frac{r^2\mathscr{B}}{\mathsf{c}^2} \\
-\nu_Ds^{\nu_N+\nu_D}(\lambda-\mathscr{N}^2)\frac{1}{r^2\mathscr{B}}& 0
\end{bmatrix}
=\begin{bmatrix}
-\nu_N & b \\
0 & 0
\end{bmatrix}
+[s]_1,$$ where $$b:=\lim_{s\rightarrow +0}\nu_Ds^{-\nu_N+\nu_D}
\frac{r^2\mathscr{B}}{\mathsf{c}^2}\Big|_{r=R-s} >0$$ Thus there is a fundamental system of solutions $\vec{U}_{Rj}=(U_{Rj}, w_{Rj})^{\top}, j=1,2$ such that $$\vec{U}_{R1}(r) \sim
\begin{bmatrix}
b \\
\nu_N
\end{bmatrix}
,
\quad
\vec{U}_{R2}\sim
\begin{bmatrix}
1 \\
0
\end{bmatrix}
s^{-\nu_N}$$ as $s \rightarrow +0$. Put $\mbox{\boldmath$\varphi$}_{Rj}=((R-r)^{\nu_N}U_{Rj}, w_{Rj})^{\top}, j=1,2$. Clearly only $\mbox{\boldmath$\varphi$}_{R1}$ is the solution of which satisfies the boundary condition.
Therefore the eigenfunction, if exists, should be $$\mbox{\boldmath$\varphi$}=C_1\mbox{\boldmath$\varphi$}_{01}=C_2\mbox{\boldmath$\varphi$}_{R1}.$$ That is, if and only if $$D(r,\lambda)=\mathrm{det}(\mbox{\boldmath$\varphi$}_{01}(r,\lambda),\mbox{\boldmath$\varphi$}_{R1}(r,\lambda))=0
\quad\mbox{for}\quad \forall / \exists r \in ]0,R[,$$ $\lambda$ is an eigenvalue. So, the question is
Do there exist a sequence of infinitely many zeros of $D(r_0,\cdot)$ which accumulates to $0$ and $+\infty$? To begin with, does $D(r_0,\cdot)$ admit at least one zero?
Of course there are determined the coefficients of connection of the two regular singular points $c_j(\lambda), j=1,2$ such that $$\mbox{\boldmath$\varphi$}_{01}(r,\lambda)=c_1(\lambda)\mbox{\boldmath$\varphi$}_{R1}(r,\lambda)+
c_2(\lambda)\mbox{\boldmath$\varphi$}_{R2}(r,\lambda),$$ and then we have $$D(r_0,\lambda)=0\quad\Leftrightarrow\quad c_2(\lambda)=0.$$ The dependence of the connection coefficient $c_2(\lambda)$ on $\lambda$ is problematic. Moreover, if $\lambda\not=0$ is not an eigenvalue, then
Does it hold $\mathsf{R}(\vec{L}_{[\mathrm{C}]l}-\lambda)=\mathfrak{X}_l$?
[**Acknowledgment**]{}\
This work is financially supported by JSPS KAKENHI Grant Number JP18K03371.\
[**Appendix**]{}\
Let us consider a function $f$ of the form $$f(u)=Ku^{\nu}(1+\sum_{k\geq 1}f_ku^k)$$ as $u \rightarrow +0$, where $K$ is a positive constant, $1<\nu <+\infty$, $\sum f_ku^k$ is a convergent power series, while $f(u)>0$ for $u >0$. Suppose $u=u(r), 0<r <R,$ satisfies $u >0, du/dr <0,$ $$\frac{d^2u}{dr^2}+\frac{2}{r}\frac{du}{dr}+f(u)=0$$ on $0<r<R$, $u \rightarrow +0$ as $r\rightarrow R-0$, and the limit $\displaystyle \lim_{r\rightarrow R-0}\frac{du}{dr} $ exists to be finite and strictly negative.
Then we have the expansion $$u=C\frac{R-r}{R}
\Big[
1+
\sum_{k_1+k_2+k_3\geq 1}b_{k_1k_2k_3}
\Big(\frac{R-r}{R}\Big)^{k_1}
\Big(C'\Big(\frac{R-r}{R}\Big)^{\nu+1}\Big)^{k_2}
\Big(C\frac{R-r}{R}\Big)^{k_3}\Big],$$ where $C$ is a positive constant, $C'=R^2KC^{\nu-1}$, and $\sum b_{k_1k_2k_3}X_1^{k_1}X_2^{k_2}X_3^{k_3}$ is a convergent triple power series.\
Let us sketch the proof. First note that there is a convergent power series $$G(u)=\sum_{k\geq 1}G_ku^k$$ such that $$u\frac{d}{du}\log f(u)=\nu+G(u).$$
Putting $$x_1=-\Big(\frac{r}{u}\frac{du}{dr}\Big)^{-1},
\quad
x_2=\frac{r^2f(u)}{u}\Big(\frac{r}{u}\frac{du}{dr}\Big)^{-2},
\quad x_3=u,$$ we get the autonomous system $$\begin{aligned}
&u\frac{dx_1}{du}=(1-x_1+x_2)x_1, \\
&u\frac{dx_2}{du}=(\nu+1-4x_1+2x_2+G(x_3))x_2, \\
&u\frac{dx_3}{du}=x_3.\end{aligned}$$ Here $(x_1,x_2, x_3)$ is considered as functions of $u >0$, and we have $(x_1,x_2,x_3) \rightarrow (0,0,0)$ as $u \rightarrow +0$.
Then there is a transformation of variables $(x_1,x_2,x_3) \leftrightarrow (\xi_1,\xi_2,\xi_3)$ of the form $$x_j=\xi_j(1+[\xi_1,\xi_2,\xi_3]_1),\quad j=1,2,3,$$ which reduce the system to $$u\frac{d\xi_1}{du}=\xi_1,\quad
u\frac{d\xi_2}{du}=(\nu+1)\xi_2 \quad u\frac{d\xi_3}{du}=\xi_3.$$ Here and hereafter $[X_1, X_2, X_3]_1$ generally stands for various convergent triple power series of the form $\sum_{k_1+k_2+k_3\geq 1}a_{k_1k_2k_3}X_1^{k_1}X_2^{k_2}X_3^{k_3}$. Take a general solution $$\xi_1=C_1u,\quad \xi_2=C_2u^{\nu+1},\quad \xi_3=u.$$ putting $C=1/C_1$, we get the desired expansion, since the integration of $$-\frac{u}{r}\frac{dr}{du}=x^1$$ gives $$\begin{aligned}
\log \frac{R}{r}&=-\log \Big(1-\frac{R-r}{R}\Big)=\frac{R-r}{R}\Big(1+
\Big[\frac{R-r}{R}\Big]_1\Big) \\
&=C_1u(1+[C_1u,C_2u^{\nu+1},u]_1).\end{aligned}$$
[99]{} C. Aerts, J. Christensen-Dalsgaard and D. W. Kurtz, Astroseismology, Springer, Dordrecht-Heidelberg-London-New York, 2010. H. R. Beyer, The spectrum of radial adiabatic stellar oscillations, J. Math. Physics, 36(1995), 4815-4825. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 3rd Ed., John Wiley and Sons, New York, 1959. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawhill, New York-Toronto-London, 1955. T. G. Cowling, The non-radial oscillations of polytropic stars, Monthly Notices Roy. Astronom. Soc. London, 101(1941), 367-375 . J. Cox, Theory of Stellar Pulsation, Princeton University Press, Princeton, 1980. F. -L. Deubner and D. O. Gough, Helioseismology: Oscillations as a diagnostic of the star interior, Ann. Rev. Astron. Astrophys., 22(1984), 594-619. N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, New York, 1963. W. Dziembowski, Nonradial oscillations of evolved stars. I. Quasiadiabatic approximation, Acta Astronomica, 21(1971), 289-306. J. Eisenfeld, A complete theorem for an integro-differential operator, J. Math. Anal. and Appl., 26(1969), 357-375. D. O. Gough, Linear adiabatic stellar pulsation, in J. -P. Zahn et al eds, Les Houches Session XLVII 1987, North-Holland, Amsterdam-London-New York-Tokyo, 1993, 1-32. J. D. Jackson, Classical Electrodynamics, Wiley and Sons, New York, 1962. Juhi Jang, Time periodic approximations of the Euler-Poisson system near Lane-Emden stars, Analysis and PDE., 9(2016), 1043-1078. Juhi Jang and T. Makino, Linearized analysis of barotropic perturbations aroud spherically symmetric gaseous stars governed by the Euler-Poisson equations, arXiv:1810.08294. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg-New York, 1980. N. R. Lebovitz, On the onset of convective instability, Astrophys. J., 142(1965), 1257-1260. P. Ledoux and Th. Walraven, Variable stars, in S. Flügge ed., Handbuch der Physik, Band LI, Springer, Berlin-Göttingen-Heidelberg, 1958, 353-604. S.-S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28(1997), 539-569. T. Makino, On the existence of positive solutions at infinity for ordinary differential equations of Emden type, Funkcialaj Ekvacioj, 27(1984), 319-329. T. Makino, On spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations, Osaka J. Math., 52(2015), 545-580. M. Reed and B. Simon, Methods of Modern Mathematical Physics, II, Fourier Analysis, Self-Adjointness, AcademicPress, New York, 1975. A. D. Rendall and B. G. Schmidt, Existence and properties of spherically symmetric static fluid bodies with a given equation of state, Classical Quantum Gravity, 8(1991), 985-1000. H. Shibahashi, Modal analysis of stellar nonradial oscillations by an asymptotic method, Publ. Astron. Soc. Japan, 31(1979), 87-104. W. Unno, Y. Osaki, H. Ando and H. Shibahashi, Nonradial Oscillations of Stars, University of Tokyo Press, Tokyo, 1979. W. Unno, Y. Osaki, H. Ando and H. Shibahashi, Nonradial Oscillations of Stars, 2nd Ed., University of Tokyo Press, Tokyo, 1989. K. Yosida, Lectures on Differential and Integral Equations, Interscience, New York, 1960.
[^1]: Professor Emeritus at Yamaguchi University, Japan E-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
-2.5cm -1.5cm =15.5cm
[Radiative corrections to $b\to c\tau\bar\nu_\tau$]{}
1.0cm
[**Andrzej Czarnecki**]{}\
0.3cm [*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}\
1.0cm [**Marek Jeżabek**]{}\
0.3cm [*Institute of Nuclear Physics, Kawiory 26a, PL-30055 Cracow, Poland*]{}\
[*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}\
0.7cm [and]{}\
0.7cm [**Johann H. Kühn**]{}\
0.3cm [*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}
1.5cm
[Abstract]{}
Analytical calculation is presented of the QCD radiative corrections to the rate of the process $b\to c\tau\bar\nu_\tau$ and to the $\tau$ lepton longitudinal polarization in $\tau\bar\nu_\tau$ rest frame. The results are given in the form of one dimensional infrared finite integrals over the invariant mass of the leptons. We argue that this form may be optimal for phenomelogical applications due to a possible breakdown of semilocal hadron-parton duality in decays of heavy flavours.
The semileptonic decay rate of $B$ mesons is one of the key ingredients in the determination of weak mixing angles. Transitions of the $b$ quark to the charmed as well as to the up quark have been analysed in great detail, exploiting either the inclusive decay rate or exclusive channels. The analysis of the inclusive rate is, however, affected by the uncertainty in the $b$ mass and by bound state corrections. This problem is only partly circumvented by fixing $m_b-m_c$ through the difference of bottom and charmed meson masses and by relating the bound state effects to phenomenological constants that can be determined from other observables in the context of the heavy quark effective theory (HQET). The decay rate is, furthermore, affected by perturbative QCD corrections, which have been calculated analytically up to order $\alpha_s$ for arbitrary $b$ and $c$ masses and massless leptons [@CM; @JK; @Nir]. Numerical results for perturbative QCD corrections to the partial decay rate $b\to \tau\bar\nu X$ have been recently obtained in Ref.[@FLNN]. The bound state corrections to this decay chanel up to order ${1/ m_b^2}$ are also known [@FLNN; @Koyrakh; @BKPS] in the context of the HQET. The comparison between decays into light ($\mu$, $e$) and heavy ($\tau$) leptons may furthermore help to test the theoretical approach and allow to fix some of the free parameters. Including $b\to u$ transitions, four kinematically different leptonic decay modes are thus available for the comparison.
In this paper analytical results for the decay rate $b\to c\tau\bar\nu_\tau$ are presented in the form of an one dimensional integral over the invariant mass squared ${\rm w}^2$ of the leptonic system. This formulation allows, at least in principle, the separation of the region of relatively small ${\rm w}^2$, where the inclusive parton model description based on local parton-hadron duality should work, from the region of large ${\rm w}^2$ where only one or few resonances are produced and the duality between the parton and hadron description may be doubtful[^1]. In the region of large ${\rm w}^2$ i.e. close to the Shifman-Voloshin limit[@VS] ${\rm w}^2= {\rm w}^2_{max}$ the theoretical description of lepton spectra based on summation over exclusive channels is particularly simple and reliable. The HQET approach for exclusive decays[@IW; @Neubert] on the other hand becomes quite involved if not impractical in the region of small ${\rm w}^2$ which is dominated by multiparticle final states.
The calculation of the lowest order rate as well as corrections can be related in a straightforward way to the corresponding calculations for the decay into a virtual $W$ boson with the subsequent integration over the mass of the $l\bar\nu$ system. The differential decay rate is proportional to $${\cal H}^{\alpha\beta}\,{\cal L}_{\alpha\beta}\,
{\rm dPS}(b\to c\tau\bar\nu)$$ ${\cal H}_{\alpha\beta}$ depends on quark and gluon fields and $$\begin{aligned}
{\cal L}_{\alpha\beta}(\tau ;\nu) &\sim&
\sum_{s}
\left[\,\bar u_\tau\gamma_\alpha(1
-\gamma_5) v_\nu\,\right]\,
\left[\,\bar u_\tau\gamma_\beta (1-\gamma_5)v_\nu\,\right]^\dagger
\nonumber\\
&\sim& \nu_\alpha \tau_\beta +
\tau_\alpha \nu_\beta - \nu\cdot\tau g_{\alpha\beta}
- {\rm i}\varepsilon_{\alpha\beta\gamma\delta}\nu^\gamma\tau^\delta
\label{eq:Lab}\end{aligned}$$ where $\tau^\alpha$ and $\nu^\alpha$ are the four-momenta of $\tau$ and $\bar\nu_\tau$. The phase space for the decay of $b$ into $c\tau\bar\nu$ is, in the standard way, decomposed into a sequence of two-particle final states $${\rm dPS}(b\to c\tau\bar\nu)\, \sim\,
{\rm d}{\rm w}^2\, {\rm dPS}(b\to c{\rm w})\,
{\rm dPS}({\rm w}\to \tau\bar\nu)$$ where ${\rm w}^\alpha=\tau^\alpha +\nu^\alpha$. Then, it is straightforward to show that $$\begin{aligned}
\lefteqn{
\int {\rm dPS}({\rm w}\to \tau\bar\nu){\cal L}_{\alpha\beta}\, \sim
{\cal A}\left(m_\tau^2/{{\rm w}}^2\right)\,T^{(0)}_{\alpha\beta}\,+
{\cal B}\left(m_\tau^2/{{\rm w}}^2\right)\,T^{(1)}_{\alpha\beta} =
}
\nonumber\\
&& {\textstyle
\left(\, 1 - {m_\tau^2/ {{\rm w}}^2}\, \right)^2
\left[\,\left(\,1 + {2m_\tau^2/ {{\rm w}}^2}\,\right)
{\rm w}_\alpha {\rm w}_\beta
- \left(\,{{\rm w}}^2+ {1\over 2} m_\tau^2\, \right)\,g_{\alpha\beta}
\right]
}
\label{eq:intL}\end{aligned}$$ where $$\begin{aligned}
T^{(0)}_{\alpha\beta} &=&
{\rm w}_\alpha {\rm w}_\beta
\nonumber\\
T^{(1)}_{\alpha\beta} &=&
{\rm w}_\alpha {\rm w}_\beta - {\rm w}^2 g_{\alpha\beta}\end{aligned}$$ It follows that the decay rate can be split accordingly into two incoherent terms which are related to weak decays of a heavy quark $Q$ into another quark $q$ and a real spin one or a spin zero boson. The relative weight of spin one versus spin zero contributions is governed by their respective spectral functions and it can be derived from eq.(\[eq:intL\]). For massless leptons ${\cal A}(0)=0$ and ${\cal B}(0)=1$, and therefore only the spin one (transversal) component ($\sim T^{(1)}_{\alpha\beta}$) contributes. The result is given in [@JK]. For fixed ${\rm w}^2$ this contribution to the rate can be obtained from the formula for $t\to bW$, the top quark decay into $b$ quark and a real $W$ boson [@JK][^2]. Multiplying this formula by ${\cal B}\left(m_\tau^2/{\rm w}^2\right)$ one obtains the contribution of the spin one component for $m_\tau\ne 0$. The other (longitudinal) contribution ($\sim T^{(1)}_{\alpha\beta}$) can be in an analogous way related to the (yet unobserved) decay $t\to bH^+$ where $H^+$ denotes a charged Higgs boson. Let $Q$ and $q$ denote the four-momenta of the quarks in the initial and in the final state. For a two-body decay mode the momentum of the $W$ boson is $W= Q-q$. In Born approximation $$\begin{aligned}
W_\mu \bar u(q)\,\gamma^\mu(1-\gamma_5)\, u(Q) &=&
\bar u(q)\,[\,(\hat Q - \hat q)(1-\gamma_5)\,]\, u(Q)
\nonumber\\
&=& \bar u(q)\,[\,(M-m)+(M+m)\gamma_5\,]\, u(Q)
\label{eq:redu}\end{aligned}$$ where the equations of motion $$\begin{aligned}
(\hat Q -M)\,u(Q)=0
\qquad\quad {\rm and} \qquad\quad
\bar u(q)\,(\hat q-m)=0\end{aligned}$$ have been used. The last line in (\[eq:redu\]) can be interpreted as the amplitude of the decay $Q\to qH$ where $H$ is a spin zero boson whose coupling to the weak quark current is given by $$g = (M-m) + (M-m)\gamma_5$$ Although not applicable for individual Feynman diagrams, the same relation holds true for the longitudinal contribution ($\sim W_\mu$) to the decay amplitude when ${\cal O}(\alpha_s)$ QCD corrections are included [@CD1]. Therefore this contribution to the rate $b\to c\tau\bar\nu_\tau$ can be extracted from a formula describing $t\to bH^+$ which has been given in [@CD]; cf. Model I therein.
Let us define now the following dimensionless quantities $$\textstyle{
\rho= {m_c^2/ m_b^2}\qquad\quad
\eta = {m_\tau^2/ m_b^2}\qquad\quad
{\rm and} \qquad\quad
t = {{\rm w}^2/m_b^2 } }$$ and the kinematic functions $$\begin{aligned}
p_0(t) &=& (1-t+\rho)/2
\nonumber\\
p_3(t) &=& \sqrt{p_0^2 - \rho}
\nonumber\\
p_\pm(t) &=& p_0 \pm p_3 = 1 - w_\mp(t)
\nonumber\\
Y_p(t) &=& {\textstyle{1\over 2}}\ln\left(p_+/ p_-\right)
=\ln\left(p_+/\sqrt{\rho}\right)
\nonumber\\
Y_w(t) &=& {\textstyle{1\over 2}}\ln\left(w_+/w_-\right)
=\ln\left(w_+/\sqrt{t}\right)
\label{eq:rapid}\end{aligned}$$ where ${\rm w}^\alpha=\tau^\alpha +\nu^\alpha$ and ${\rm w}^2$ denotes the effective mass of $\tau\bar\nu_\tau$. The partial rate of the decay $b\to c\tau\bar\nu_\tau$ is given by $$\Gamma(b\to c\tau\bar\nu_\tau)\:=\:
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma\over {\rm d}t}\,{\rm d}t
\label{eq:Gamtot}$$ with the differential rate $$\begin{aligned}
\lefteqn{
{{\rm d}\Gamma\over {\rm d}t} = \Gamma_{bc}\,
\left( 1-{\eta \over t}\right)^2 \, \left\{
\left( 1+{\eta \over 2t}\right)
\left[ {\cal F}_0(t) - {2\alpha_s\over 3 \pi} {\cal F}_1(t)\right]
+ {3\eta \over 2t}
\left[ {\cal F}_0^s(t) - {2\alpha_s\over 3 \pi} {\cal F}_1^s(t)\right]
\right\} }
\nonumber\\
\label{main} \\
\lefteqn{
\Gamma_{bc} =
{G_F^2 m_b^5 |V_{cb}|^2\over 192 \pi^3}
}\end{aligned}$$ $$\begin{aligned}
{\cal F}_0(t) &=& 4 p_3\,
\left[ \, (1-\rho)^2 + t(1+\rho) - 2 t^2\, \right]
\\
{\cal F}_0^s(t)
&=& 4p_3\,\left[\, (1-\rho)^2 -t ( 1+\rho)\, \right]
\\
{\cal F}_1(t)&=& {\cal A}_1 \Psi + {\cal A}_2 Y_w
+ {\cal A}_3 Y_p + {\cal A}_4 p_3 \ln\rho + {\cal A}_5 p_3
\\
{\cal F}_1^s(t)&=& {\cal B}_1 \Psi + {\cal B}_2 Y_w
+ {\cal B}_3 Y_p + {\cal B}_4 p_3 \ln\rho + {\cal B}_5 p_3
\label{massive}\end{aligned}$$ $$\begin{aligned}
\Psi &=& 8 \ln (2 p_3) -2\ln t\, +\,
\left[ 2{\rm Li}_2 (w_-) -2{\rm Li}_2 (w_+)
+4 {\rm Li}_2 ({2p_3/ p_+}) \right.
\nonumber \\ && \left. \hskip65pt
-4Y_p \ln({2p_3/ p_+})
-\ln p_- \ln w_+ + \ln p_+ \ln w_- \right]\, 2p_0/p_3\end{aligned}$$ $$\begin{aligned}
{\cal A}_1 &=& {\cal F}_0(t)
\nonumber\\
{\cal A}_2 &=&
- 8 (1-\rho) \left[ 1 +t -4 t^2-
\rho (2-t) +\rho^2 \right]
\nonumber\\
{\cal A}_3 &=&
- 2 \left[ 3 + 6 t -21 t^2 + 12 t^3
-\rho (1+12t+5t^2) + \rho^2(11+2t)- \rho^3 \right]
\nonumber\\
{\cal A}_4 &=&
- 6 \left[ 1 + 3 t - 4 t^2
-\rho (4-t) + 3 \rho^2 \right]
\nonumber\\
{\cal A}_5 &=&
- 2 \left[ 5+9t-6t^2 -\rho( 22 - 9t) + 5 \rho^2 \right]
\\
{\cal B}_1 &=& {\cal F}^s_0(t)
\nonumber\\
{\cal B}_2 &=& -8 (1-\rho)
\left[(1-\rho)^2-t (1+\rho)\right]
\nonumber\\
{\cal B}_3 &=& - 4(1 - \rho)^4/t
-2 (-1 + 3 \rho + 15 \rho^2 - 5 \rho^3)
+ 8 (1 + \rho) t
- 6 (1 + \rho) t^2
\nonumber\\
{\cal B}_4 &=& -4 (1-\rho)^3 /t
-2 (1-\rho) (1-11 \rho)
+6 (1 + 3 \rho) t
\nonumber\\
{\cal B}_5 &=& -6 (1-3 \rho) (3-\rho)
+18 t (1+\rho)\end{aligned}$$ The integral in eq.(\[eq:Gamtot\]) can be easily performed for the Born contribution. It reads:\
$$\begin{aligned}
\lefteqn{
\Gamma_0(b\to c\tau\bar\nu_\tau) =
\Gamma_{bc}\,
\left\{\,
24\,\left[\,\eta^2(1-\rho^2)\,{\cal Y}_w
+ \rho^2(1-\eta^2)\,{\cal Y}_p\,\right]\, +\,
\right. }
\nonumber\\ &&
\left.
2{\cal P}_3\,\left(1-7\eta-7\eta^2+\eta^3-7\rho
+12\eta\rho -7\eta^2\rho
-7\rho^2 -7\eta\rho^2+\rho^3 \right)
\,\right\}
\label{eq:Gam0tot}\end{aligned}$$ where $${\cal P}_3 = p_3(\eta),\qquad\quad
{\cal Y}_p = Y_p(\eta)\qquad\quad {\rm and} \qquad\quad
{\cal Y}_w = Y_w(\eta).$$ In principle the first order QCD correction can be also expressed in terms of polylogarithms. In particular for $\eta=0$ the formula (12) of [@Nir] is obtained. However, the complete result is lenghty. From the practical point of view it is much simpler to evaluate the integral in eq.(\[eq:Gamtot\]) numerically. Moreover, as it has been explained, for $t$ close to the Shifman-Voloshin limit the exclusive description is preferable, thus, for applications, it may be better to perform this integral only over a part of the available phase space. In recent articles [@SV; @LSW] the size of ${\cal O}(\alpha_s^2)$ corrections has been estimated using the scheme of Brodsky, Lepage and Mackenzie [@BLM] for fixing the scale $\mu$ of $\alpha_s$. It has turned out that this scale is rather low. This suggests that next-to-leading QCD corrections are large. The most serious problems arise in the region close to the no-recoil $t=t_{max}$ point. It is well known, cf. [@ACCMM; @JK1], that at the boundaries of the available phase space logarithmic divergences may appear as remnants of infrared divergences. This phenomenon arises when real radiation becomes supressed relative to virtual corrections. This is exactly what happens for $t$ in the neighbour of $t_{max}$ and once again one is led to the conclusion that the inclusive parton-like description may break down there.
In the massless limit $\rho\to 0$ which corresponds to $b\to u\tau\bar\nu_\tau$ transition the functions in eq.(\[massive\]) simplify considerably $$\begin{aligned}
{\cal F}_0(t) &=& 2 (1-t)^2 (1+2t)
\\
{\cal F}_0^s(t)
&=& 2(1-t)^2
\\
{\cal F}_1(t) &=&
{\cal F}_0(t)\,
{\textstyle
\left[\,
{2\over3}\pi^2
+4{\rm Li}_2(t)+2\ln t\ln(1-t)\,\right]
}
- (1-t)(5+9t - 6t^2)
\nonumber\\ &&
+\, 4t(1-t-2t^2)\ln t
+ 2(1-t)^2(5+4t)\ln(1-t)
\label{eq:F1y0}
\\
{\cal F}_1^s(t)&=&
\textstyle{
{\cal F}_0^s(t)\,
\left[{2\over 3}\pi^2
+ 4{\rm Li}_2(t) -{9\over2}
+\ln(1-t)\left(2\ln t-{2\over t}+5\right) \right]
}
+ 4(1-t)t\ln t
\nonumber\\ &&
\label{eq:F1sy0}\end{aligned}$$ After integration over $t$ one derives the following expression for the total partial rate of $b\to u\tau\bar\nu_\tau$ $$\begin{aligned}
\lefteqn{{1\over \Gamma_{bu}}\Gamma(b\to u\tau\bar \nu_\tau)=
1-8\eta+8\eta^3-\eta^4-12\eta^2\ln\eta}
\nonumber\\&&
-{2\alpha_s\over 3\pi}\left\{
(-1 + \eta) (75 - 539 \eta - 476 \eta^2 + 18 \eta^3)/12
\right.
\nonumber\\&& \qquad \quad
+ (3 - 24 \eta - 36 \eta^2 + 16 \eta^3 - 2 \eta^4) \pi^2/3
\nonumber\\&& \qquad \quad
+ 72 \eta^2 \left[\zeta(3)-{\rm Li}_3(\eta)\right]
+ 2 (1 - 8 \eta + 36 \eta^2 + 16 \eta^3 - 2 \eta^4) {\rm Li}_2(\eta)
\nonumber\\&& \qquad \quad
+ (1 - \eta^2) (31 - 320 \eta + 31 \eta^2) \ln(1 - \eta)/6
\nonumber\\&& \qquad \quad
+ \left[2 \eta + 15 \eta^2 - 94 \eta^3/3 + 31 \eta^4/6
- 8 \eta^2 \pi^2 + 24 \eta^2 {\rm Li}_2(\eta)
\right. \qquad \quad
\nonumber\\&& \left.\left. \qquad \qquad
+ 2 (1 - \eta^2) (1 - 8 \eta +\eta^2) \ln(1 - \eta)\right] \ln(\eta)
\right\}\end{aligned}$$ $$\begin{aligned}
\lefteqn{
\Gamma_{bu} =
{G_F^2 m_b^5 |V_{ub}|^2\over 192 \pi^3} }\end{aligned}$$ which agrees with eqs.(2.21) and (3.6) in a recent preprint [@BBBG].
It has been argued [@kalinowski] that the $\tau$ polarization in $b\to c\tau\bar\nu_\tau$ is particularly sensitive to deviations from the Standard Model. The longitudinal component of $\tau$ polarization can either be measured with reference to its direction of flight in the $b$ rest frame or, alternatively, relative to its direction of flight in the $\tau\bar\nu$ rest frame. To evaluate analytically QCD corrections to the longitudinal $\tau$ polarization in the $b$ rest frame is a demanding task. Previous experience from a similar calculation for the decay of polarized top quarks indicates that these corrections are typically quite small through most of the kinematical range. To substantiate this claim, the longitudinal $\tau$ polarization in the $\tau\bar\nu$ rest frame is evaluated including QCD corrections.
For V-A weak current the leptonic tensor $$\begin{aligned}
{\cal L}_{\alpha\beta}(\tau,s;\nu) &\sim&
\left[\,\bar u_\tau\gamma_\alpha(1
-\gamma_5) v_\nu\,\right]\,
\left[\,\bar u_\tau\gamma_\beta (1-\gamma_5)v_\nu\,\right]^\dagger
\nonumber\\
&\sim& \nu_\alpha {\cal T}_\beta +
{\cal T}_\alpha \nu_\beta - \nu\cdot{\cal T} g_{\alpha\beta}
- {\rm i}\varepsilon_{\alpha\beta\gamma\delta}\nu^\gamma
{\cal T}^\delta
\label{eq:Labs}\end{aligned}$$ where $${\cal T}^\alpha = {\textstyle{1\over 2}}
\left(\,\tau^\alpha -m_\tau s^\alpha\,\right)$$ and $s^\alpha$ is the $\tau$ polarization fourvector. The helicity states in the $\tau\bar\nu$ rest frame are obtained for $$m_\tau s^\alpha = \pm \left( \tau^\alpha
- {\textstyle{2\eta\over t-\eta}} \nu^\alpha \right)$$ It is evident that the net polarization can be calculated in the same way as the total rate, decomposing again the spin dependent term into longitudinal and transversal parts. For the positive helicity of $\tau$ one derives the following expression for the differential rate $$\begin{aligned}
\lefteqn{
{1\over \Gamma_{bc}}\,{{\rm d}\Gamma^{(+)}\over {\rm d}t} =
{\eta\over 2t}
\left( 1-{\eta \over t}\right)^2 \, \left\{
{\cal F}_0(t) + 3 {\cal F}_0^s(t)
- {2\alpha_s\over 3 \pi}
\left[\,
{\cal F}_1(t) + 3 {\cal F}_1^s(t)\,\right]
\right\} }
\label{polar}\end{aligned}$$ For the negative helicity one has $$\begin{aligned}
\lefteqn{
{{\rm d}\Gamma^{(-)}\over {\rm d}t} =
{{\rm d}\Gamma\over {\rm d}t} -
{{\rm d}\Gamma^{(+)}\over {\rm d}t}
}
\label{polar1}\end{aligned}$$ and the net helicity in the $\tau\nu_\tau$ rest frame is equal to $${\rm P} = - 1 \, + \,
{ 2
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma^{(+)}\over {\rm d}t}\,{\rm d}t
\over
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma\over {\rm d}t}\,{\rm d}t
}$$
In the subsequent discussion the mass difference $m_b-m_c$ will be fixed through the HQET relation $$m_b-m_c = [ (3m_{B^*}+m_{B})-(3m_{D^*}+m_{D})]/4 + \ldots$$ The most important corrections to the above relation arise from the kinetic energy of heavy quarks in $B$ and $D$ mesons. On physical grounds one expects this contribution to increase the difference between $m_b$ and $m_c$. We take $m_b- m_c=$ 3.4 GeV in numerical calculations and $m_b$ is varied between 4.5 and 5.0 GeV. In order to estimate effects of QCD corrections on measurable quantities we neglects all the problems with the parton-hadron duality and the scale ambiguity. The following results have been obtained: $$\begin{aligned}
&&\Gamma(b\to c\tau\bar\nu_\tau) =
\Gamma_{0}(b\to c\tau\bar\nu_\tau) \,
\left[\, 1 + ( -0.450\pm 0.016)\,\alpha_s\, \right]
\nonumber\\
&&\Gamma(b\to c e\bar\nu_e) =
\Gamma_{0}(b\to c e\bar\nu_e) \,
\left[\, 1 + ( -0.545\pm 0.025)\,\alpha_s\, \right]
\nonumber\\
&&R=BR(b\to\tau X)/BR(b\to eX) = R_0
\left[\, 1 + ( 0.094\mp 0.009)\,\alpha_s\, \right]
\nonumber\\
&&{\rm P} = -(0.293\mp 0.002)
\left[\, 1 + ( 0.140\mp 0.015)\,\alpha_s\, \right]
\nonumber\\
&&<\, t \,>(b\to\tau X)= (0.34\pm 0.03)
\left[\, 1 + ( 0.016\pm 0.003)\,\alpha_s\, \right]
\nonumber\\
&&<\, t \,>(b\to e X)= (0.20\pm 0.02)
\left[\, 1 + ( 0.035\pm 0.007)\,\alpha_s\, \right]
\nonumber\end{aligned}$$ where $<\, t \,>(\ldots)$ denote the average values of $t$ for the corresponding decay chanels. It is evident that the ${\cal O}(\alpha_s)$ corrections practically cancel in the ratio $R$ of the branching ratios as well as in the result for the polarization $P$. The moments $<\, t \,>(b\to\tau X)$ and $<\, t \,>(b\to e X)$ are also insensitive to $\alpha_s$ corrections. On the other hand all these quantities are sensitive to the quark masses and therefore they may be used for fixing $m_b$ and $m_c$.
Acknowledgements {#acknowledgements .unnumbered}
================
MJ would like to thank Kostya Chetyrkin for helpful discussions. This work was supported in part by KBN grant 2P30225206, by DFG contract 436POL173193S and by Graduiertenkolleg Elementarteilchenphysik at the University of Karlsruhe.
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[^1]: In a recent preprint [@BDS] the breakdown of the local parton-hadron duality has been invoked as the origin of problems with the semileptonic decay rate of $D$ mesons in the framework of HQET. Let us remark that for large ${\rm w}^2$ the kinetic energy of the hadronic system in $B$ decays can be similar to that in $D$ decays. Thus the semileptonic branching ratios for $b$ decays may be also affected for the effective mass of the hadronic system close to the resonance region.
[^2]: Note that the rates of up and down type quark decays into their respective isospin partners are of course identical, in contrast to the shapes of the lepton spectra.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show the influence of surface plasmons on the Casimir effect between two plane parallel metallic mirrors at arbitrary distances. Using the plasma model to describe the optical response of the metal, we express the Casimir energy as a sum of contributions associated with evanescent surface plasmon modes and propagative cavity modes. In contrast to naive expectations, the plasmonic modes contribution is essential at all distances in order to ensure the correct result for the Casimir energy. One of the two plasmonic modes gives rise to a repulsive contribution, balancing out the attractive contributions from propagating cavity modes, while both contributions taken separately are much larger than the actual value of the Casimir energy. This also suggests possibilities to tailor the sign of the Casimir force via surface plasmons.'
author:
- 'F. Intravaia'
- 'A. Lambrecht'
title: Surface plasmon modes and the Casimir energy
---
When H. Casimir first predicted the existence of a force between neutral mirrors in vacuum [@Casimir48], he considered two plane parallel perfect reflectors and found an interaction energy $E_{\text{Cas}}$ depending only on geometrical parameters, the mirrors distance $L$ and surface $A\gg L^{2}$, and two fundamental constants, the speed of light $c$ and Planck constant $\hbar $ $$E_{\text{Cas}}=-\frac{\hbar c\pi^2 A}{720L^{3}}. \label{CasimirForce}$$The signs have been chosen to fit the thermodynamical convention with the minus sign of the energy $E_{\text{Cas}}$ corresponding to a binding energy. The Casimir energy for perfect mirrors is usually obtained by summing the zero-point energies $\frac{\hbar \omega }{2}$ of the cavity eigenmodes, substracting the result for finite and infinite separation, and extracting the regular expression (\[CasimirForce\]) by inserting a formal high-energy cutoff and using the Euler-McLaurin formula [@qft].
In his seminal paper [@Casimir48], Casimir noticed that the energy should be a finite expression, without the need of any regularization, provided one takes into account the high frequency transparency of real mirrors. The idea was implemented by Lifshitz who calculated the Casimir energy for mirrors characterized by dielectric functions [@Lifshitz56]. For metallic mirrors he recovered expression (\[CasimirForce\]) for separations $L$ much larger than the plasma wavelength $\lambda _{\mathrm{p}}$ associated with the metal, as metals are very good reflectors at frequencies much smaller than the plasma frequency $\omega _{\mathrm{p}}$. At shorter separations in contrast, the Casimir effect probes the optical response of metals at frequencies where they are poor reflectors and the Casimir energy is reduced with respect to (\[CasimirForce\]). This reduction has been studied in great detail recently ([@Jaekel91; @GenetPRA03] and references therein) since it plays a central role in the comparison of theoretical predictions ([@compar] and references therein) with experimental results [@expts].
In the limit of small separations $L \ll\lambda _{\mathrm{p}}$, the Casimir effect has another interpretation establishing a bridge between quantum field theory of vacuum fluctuations and condensed matter theory of forces between two metallic bulks. It can indeed be understood as resulting from the Coulomb interaction between surface plasmons, that is the collective electron excitations propagating on the interface between each bulk and the intracavity vacuum [@plasmon; @Barton79; @Schram73]. The corresponding field modes are evanescent waves and have an imaginary longitudinal wavevector. We will call them plasmonic modes at arbitrary distances as they coincide with the surface plasmon modes at small distances. Plasmonic modes have to be seen in contrast to ordinary propagating cavity modes, which have a real longitudinal wavevector. For simplicity we will call those in the following photonic modes. Photonic modes are usually considered in quantum field theory of the Casimir effect [@qft] and are thought to determine the Casimir effect at large distances where the mirrors can be treated as perfect reflectors. At short distances, plasmonic modes are known to dominate the interaction [@GenetAFLB03; @Henkel03].
The purpose of the present letter is to show the singular behavior of one of the two plasmonic modes, which gives rise to a repulsive contribution to the Casimir energy at all distances, ensuring in this way that the correct value for the Casimir energy is recovered, in particular the ideal Casimir energy at large distances. Plasmonic modes have therefore a much greater importance than usually appreciated. To show this, we will use the decomposition of the Casimir energy as a sum of zero-point energies $\frac{\hbar \omega }{2}$ over the whole set of modes of the cavity with its two mirrors described by a plasma model. This set contains plasmonic as well as photonic modes. As expected from [@GenetAFLB03; @Henkel03], the contributions of plasmonic modes will be found to dominate the Casimir effect for small separations corresponding to Coulomb interaction between surface plasmons. But, contrary to naive expectations, they do not vanish for large separations. For distances larger than about $\lambda_{\mathrm{p}}/4\pi$ ($\sim$10nm for typical metals) they even give rise to a contribution having simultaneously a negative sign and a too large magnitude with respect to the Casimir formula (\[CasimirForce\]). The repulsive character can be attributed to one of the two plasmonic modes. The photonic modes as well as the second plasmonic mode give rise to an attractive contribution much larger than (\[CasimirForce\]). It is therefore the repulsive contribution of a single plasmonic mode which renders the total plasmonic mode contribution to the Casimir energy repulsive outside the short distance limit while assuring at the same time that the sum over all modes reproduces (\[CasimirForce\]) at large distances. This repulsive character may open interesting possibilities to tailor surface plasmons via nanostructuration of metallic surfaces in order to change the sign of the total Casimir force.
In this letter, we restrict our attention to the situation of two infinitely large plane mirrors at zero temperature so that the only modification of Casimir formula (\[CasimirForce\]) is due to the metals finite conductivity. This modification is calculated by evaluating the radiation pressure of vacuum fields upon the two mirrors [@GenetPRA03] $$\begin{aligned}
E &=&-\sum_{\epsilon}\sum_{\mathbf{k}}\sum_{\omega }\frac{i\hbar}{2} \ln({1-r_{\mathbf{k}}^{\epsilon}[\omega
]^{2}e^{2ik_{z}L}}) + c.c. \label{realForce} \\
\sum_{\mathbf{k}} &\equiv &A\int \frac{\mathrm{d}^{2}\mathbf{k}}{4\pi ^{2}}\quad ,\quad \sum_{\omega }\equiv \int_{0}^{\infty }\frac{\mathrm{d}\omega }{2\pi }. \notag\end{aligned}$$The energy $E$ is obtained by summing over polarization $\epsilon$=(TE,TM), transverse wavevector $\mathbf{k}\equiv \left(
k_{x},k_{y}\right) $ (with $z$ the longitudinal axis of the cavity) and frequency $\omega $; $k_{z}$ is the longitudinal wavevector associated with the mode. The reflection amplitudes $r_{\mathbf{k}}^{\epsilon}$, here supposed to be the same for both mirrors are causal retarded functions obeying high-frequency transparency.
We now calculate the Casimir energy as a sum over the cavity modes using the plasma model for the mirrors dielectric function $$\varepsilon \lbrack \omega ]=1-\frac{\omega _{\mathrm{p}}^{2}}{\omega ^{2}}$$ with $\omega _{\mathrm{p}}$ the plasma frequency and $\lambda _{\mathrm{p}}=\frac{2\pi c}{\omega _{\mathrm{p}}}$ the plasma wavelength, of the order of 100nm for metals used in experiments [@Lambrecht00]. In this case the zeros of the argument of the integrand in (\[realForce\]) lie on the real axis. In fact, they have to be pushed slightly below this axis by introducing a vanishing dissipation parameter in order to avoid any ambiguity in expression (2) [@GenetPRA03]. We may then rewrite ([realForce]{}) as a sum over the solutions $\left[ \omega _{\mathbf{k}}^{\epsilon}\right] _{m}$ of the equation labelled by an integer index $m$ $$r_{\mathbf{k}}^{\epsilon}[\omega ]^{2}e^{2ik_{z}L}=1. \label{solModes}$$ Simple algebraic manipulations exploiting residues theorem and complex integration techniques [@Schram73] then lead to the Casimir energy expressed as sums over these modes $$\begin{aligned}
E &=& \sum_{\epsilon,\mathbf{k}}\left[\sum_{m}^{\prime}\frac{\hbar \left[
\omega _{\mathbf{k}}^{\epsilon}\right]_{m}}{2}\right]_{L\rightarrow\infty}^{L}.\label{diff}\end{aligned}$$The prime in the sum over $m$ signifies as usually that the term $m=0$ has to be multiplied by 1/2. The sum over the modes is to be understood as a regularized quantity as it involves infinite quantities. This result is well known for perfect mirrors and is not changed by the choice of the plasma model for the mirrors reflection coefficients. The upper expression contains as limiting cases at large distances the Casimir expression with perfect mirrors and at short distances the expression in terms of surface plasmon resonances. For arbitrary distances, photonic modes as well as plasmonic modes are important.
We will now discuss the structure of TE and TM modes inside the cavity formed by the two mirrors. The different modes have been obtained by writing explicitly all solutions of (\[solModes\]), using the standard expressions for the reflection coefficients. Figure \[modes2\] shows the phase shift acquired by the TE modes through the influence of imperfect reflection. They are represented through their longitudinal wavevector as a function of $kL$. The TE polarization admits only photonic modes which can be written under the standard form $k_{z}L=m\pi -\delta $, where the integer $m=1,2\ldots \infty $ is the order of the cavity mode and $\delta $ the phase shift of the mode on a mirror.
![Mode plot of the first photonic TE modes ($m=1,2,...8$) with the plasma model for $ck=0.5\omega_\text{p}$. Modes are presented through their longitudinal wavevector as a function of $kL/\pi$. The dotted lines correspond to the cavity modes with perfect mirrors.[]{data-label="modes2"}](fig1.eps){height="5.5cm"}
Perfect mirrors lead to cavity modes plotted as dotted lines corresponding to $\delta ^{\mathrm{TE}}=0$. With the plasma model, the photonic modes are displaced compared to the perfect cavity modes as a direct consequence of the phase shift $\delta $ acquired by vacuum fields upon reflection. The limit of perfect reflection corresponds to the large distances limit. The high frequency transparency of metallic mirrors imposes an upper bound to their longitudinal wavevector $ck_{z}<\omega _{\mathrm{p}}$, where all photonic modes coincide.
For TM polarization, similar photonic modes are obtained labelled also by a positive integer $m$. They are accompanied by two additional modes, which we label $\left[ \omega _{\mathbf{k}}^{\mathrm{pl}}\right] _{\pm }$ as they tend to the frequencies of surface plasmon modes [@Barton79] in the limit of small distances. These plasmonic modes are shown as solid black lines in Figure \[modes\], while photonic modes correspond to gray lines. In order to make the plasmonic modes with their imaginary wavevector visible, the modes are now represented through their frequency as a function of $kL$. Plasmonic and photonic modes lie respectively in the sector $\omega < ck$ and $\omega > ck$.
![Mode plot of the two plasmonic modes $\omega_\bk^-$ and $\omega_\bk^+$ (black) in the sector $\omega < ck$ and of photonic modes (gray) in the sector $\omega > ck$ for $ck=0.5\omega_\text{p}$. Modes are presented through their frequency as a function of $kL/\pi$.[]{data-label="modes"}](fig2.eps){height="5cm"}
In the limit of infinite mirrors separation, the plasmonic modes are given by the usual dispersion relation for the surface plasmons in a metallic bulk [@Barton79] $$\left[ \omega _{\mathbf{k}}^{\rm pl}\right]_{\pm} \underrightarrow{L\rightarrow\infty}\quad \frac{\omega_{\mathrm p}^2+2|\mathbf{k}|^2-\sqrt{\omega_{\mathrm p}^4+4|\mathbf{k}|^4}}{2}. \label{wsp}$$ For the photonic modes the phase shift $\delta$ tends towards zero for infinite distances where they obey the dispersion relation for perfect mirrors $\left[ \omega _{\mathbf{k}}^{\epsilon}\right]_{m}=\sqrt{\left\vert \mathbf{k}\right\vert ^{2}+k_{z}^{2}}$ with the longitudinal wavevector $k_z=m\pi/L$. For $L\rightarrow \infty$, the sum over $m$ in (\[diff\]) becomes a continuous integral and the mode contribution of photonic modes corresponds to the one of free field vacuum which is substracted from the contribution at finite distances.
Let us now discuss in more detail the behavior of the two plasmonic modes. $\omega_\bk^-$ is restricted to the plasmonic mode sector, while $\omega_\bk^+$ lies in the plasmonic mode sector for large distances, but crosses the barrier $\omega = ck$ and dies in the photonic mode sector for $kL/\pi \rightarrow 0$. In the present calculation, the whole mode was attributed to the plasmonic mode contribution as its frequency tends to the surface plasmon contribution at short distances. The qualitative results do not change if the part of the mode lying in the photonic modes sector is attributed to the photonic modes contribution.
Obviously, when decreasing the distance $L$ the plasmonic mode $\omega_\bk^+$ acquires a phase shift with the same sign as the TM photonic modes below the plasma frequency. Its frequency at short distances is always larger than the one in the large distance limit. In contrast, the frequency of $\omega_\bk^-$ is decreased at short distances compared to long distances. When now performing the difference (\[diff\]) of the contributions at finite and infinite distances, the Casimir energy contribution turns out to be negative for photonic modes, as the mode contribution in free vacuum ($L\rightarrow\infty)$ exceeds the one inside the cavity, in accordance with an attractive force. It is also negative for the plasmonic mode $\omega_\bk^-$. However, the difference is positive for the plasmonic mode $\omega_\bk^+$. An immediate consequence is that the contribution of $\omega_\bk^+$ to the Casimir energy is repulsive.
To asses quantitatively the effect of the plasmonic modes to the Casimir energy, we have computed separately the energies associated with photonic modes $\left[ \omega _{\mathbf{k}}^{p}\right]
_{m}$ and plasmonic modes $\left[ \omega _{\mathbf{k}}^{\mathrm{pl}}\right]
_{\pm }$. All energies in the following will be presented as a reduction factor $\eta$ [@Lambrecht00] $$E=\eta E_{\text{Cas}}. \label{realEnergy}$$ As the ideal Casimir energy is negative corresponding to attraction, positive and negative reduction factors mean respectively attractive or repulsive interaction. The reduction factor due to imperfect reflection described with the plasma model is shown as a solid line in Figure \[beha\] as a function of the ratio $L/\lambda _{\mathrm{p}}$. We also introduce reduction factors corresponding to contributions of the different modes to the Casimir energy $$\eta _{\text{ph}}=E_{\text{ph}}/E_{\text{Cas}} \quad \quad \eta _{\text{pl}}=E_{\text{pl}}/E_{\text{Cas}}.
\notag$$ Their sum corresponds to the whole Casimir energy $\eta =\eta _{\text{ph}}+\eta _{\text{pl}}$.
![Contributions to Casimir energy normalized to (1) of photonic modes (dotted) and plasmonic modes (dashed) to the total Casimir energy (solid line) as functions of $L/\protect\lambda _{\mathrm{p}}$. The inlet shows the separate contributions of $\omega_\bk^-$ and $\omega_\bk^+$.[]{data-label="beha"}](fig3.eps){width="8.5cm"}
The contribution $\eta _{\text{pl}}$ of plasmonic modes (dashed line) dominates at short distances $L\ll \lambda _{\mathrm{p}}$, which confirms the interpretation of the Casimir effect as resulting in this regime from the Coulomb interaction of surface plasmons. There, a simple expression may be given for the reduction factor [@GenetPRA00; @GenetAFLB03] $$\eta \underset{L\ll \lambda _{\mathrm{p}}}{\simeq }\frac{3\alpha }{2}\frac{L}{\lambda _{\mathrm{p}}}\quad ,\quad \alpha \simeq 1.193.
\label{shortSeparations}$$The power law dependence of $E$ then goes from $L^{-3}$ at large distances to $L^{-2}\lambda _{\mathrm{p}}^{-1}$ at short distances [@Lifshitz56]. The contribution of photonic modes $\eta _{\text{ph}}$ scales as $\left( L/\lambda _{\mathrm{p}}\right) ^{4}$ and its contribution may be neglected at the 1% level up to $L/\lambda _{\mathrm{p}}\sim 0.2$. At larger distances, $\eta _{\text{ph}}$ increases while $\eta _{\text{pl}}$ becomes negative at a distance of the order $\lambda _{\mathrm{p}}/4\pi$. This clearly comes from the behavior of $\omega_\bk^+$, shown in the inlet, which gives a repulsive contribution at all distances. For example, the photonic and plasmonic contribution to the Casimir energy at $\lambda_\text{p}/L\sim 1$ are both about 36 times larger than the total Casimir energy between metallic mirrors. They are of opposite sign while the photonic contribution slightly dominates. For large separations $L/\lambda _{\mathrm{p}}\gg 1$, $\eta _{\text{ph}}$ tends to $+\infty $ while $\eta _{\text{pl}}$ tends to $-\infty $. The sum of the two contributions reproduces the known value for $\eta $, which is positive and increasing over all separations going from (\[shortSeparations\]) to unity for large distances, where the Casimir formula (\[CasimirForce\]) is recovered. This feature results from a compensation between the large positive value of $\eta _{\text{ph}}$ and the large negative value of $\eta _{\text{pl}}$. More precise asymptotic laws for the two contributions are $$\eta _{\text{ph}}-1\underset{L\gg \lambda _{\mathrm{p}}}{\simeq }-\eta _{\text{pl}}\ \underset{L\gg \lambda _{\mathrm{p}}}{\simeq }\beta \sqrt{\frac{L}{\lambda _{\mathrm{p}}}}\quad ,\quad \beta \simeq 74.58.$$ The behavior of the whole reduction factor is also recovered $
\eta \underset{L\gg \lambda _{\mathrm{p}}}{\simeq }1-2\lambda _{\mathrm{p}
}/\left( \pi L\right) $.
These results clearly show the crucial importance of the surface plasmon contribution, not only for short distances where it dominates the Casimir effect but also for long distances. For metallic mirrors the existence of surface plasmons are not an additional correction to the Casimir effect, but inherent to it. A single plasmonic mode $\omega_\bk^+$ ensures consistency with the Casimir energy between metallic mirrors at intermediate distances and with the Casimir formula (1) for perfect mirrors. If we had calculated the Casimir effect by accounting only for the photonic modes, we would have found a result much too large. The photonic modes and one of the plasmonic modes are displaced by the phase shifts which induce a systematical deviation towards a larger magnitude of Casimir energy. The discrepancy which would be obtained in this manner is only cured by the contribution of the $\omega_\bk^+$ plasmonic mode. The whole Casimir energy turns out to be the result of a fine balance between the large attractive photonic contribution and the large repulsive plasmonic contribution. As already known from discussions of arbitrary dielectric mirrors [@GenetPRA03], the outcome of this balance keeps the sign of a binding energy. However, this result relies heavily on the symmetry of the Casimir geometry with two plane mirrors. One might thus hope changing this behavior by enhancing the contribution of plasmonic modes, by changing the geometry, using for example the hole arrays used to enhance the transmission of light through metallic structures [@holearrays] or nanostructured metallic surfaces. This could then play a role in micro-electro-mechanical systems (MEMS) in which the Casimir force is known to have a great influence [@mems].
Many thanks are due to S. Reynaud, C. Genet, M.-T. Jaekel and P.A. Maia Neto for discussions.F.I. thanks the Foundation Angelo della Riccia for financial support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The impossibility of perfect cloning and state estimation are two fundamental results in Quantum Mechanics. It has been conjectured that quantum cloning becomes equivalent to state estimation in the asymptotic regime where the number of clones tends to infinity. We prove this conjecture using two known results of Quantum Information Theory: the monogamy of quantum correlations and the properties of entanglement breaking channels.'
author:
- Joonwoo Bae and Antonio Acín
title: Asymptotic quantum cloning is state estimation
---
The impossibility of perfect state estimation is a major consequence of the nonorthogonality of quantum states: *the state of a single quantum system cannot be perfectly measured*. In other words, a measurement on a system in order to acquire information on its quantum state perturbs the system itself. The full reconstruction of the state is only possible by computing statistical averages of different observables on a large number of identically prepared systems. Thus, any measurement at the single-copy level only provides partial information.
The fact that state estimation is in general imperfect leads in a natural way to the problem of building *optimal measurements*. Being a perfect reconstruction impossible, it is relevant to find the measurement strategy that maximizes the gain of information about the unknown state. A standard approach to this problem in Quantum Information Theory (QIT) is to quantify the quality of a measurement by means of the so-called *fidelity* [@MP]. This quantity is defined as follows. Consider the situation in which a quantum state $\ket{\psi}$ is chosen from the ensemble $\{p_i,\ket{\psi_i}\}$, i.e. $\ket{\psi}$ can be equal to $\ket{\psi_i}$ with probability $p_i$. A measurement, defined by $N_M$ positive operators, $M_j\geq 0$, summing up to the identity, $\sum_j M_j=\one$, is applied on this unknown state. For each obtained outcome $j$, a guess $\ket{\phi_j}$ for the input state is made. The overlap between the guessed state and the input state, $|\braket{\psi_i}{\phi_j}|^2$, quantifies the quality of the estimation process. The averaged fidelity of the measurement then reads $$\label{avfidmeas}
\bar F_M=\sum_{i,j}
p_i\, \tr(M_j\proj{\psi_i})\,|\braket{\psi_i}{\phi_j}|^2 .$$ A measurement is optimal according to the fidelity criterion when it provides the largest possible value of $\bar F_M$, denoted in what follows by $F_M$.
The No-cloning theorem [@WZ], one of the cornerstones of QIT [@reviews], represents another known consequence of the nonorthogonality of quantum states. It proves that *given a quantum system in an unknown state $\ket{\psi}$, it is impossible to design a device producing two identical copies, $\ket{\psi}\ket{\psi}$*. Indeed, two nonorthogonal quantum states suffice to prove the no-cloning theorem.
As it happens for state estimation, the impossibility of perfect cloning leads to the characterization of *optimal cloning machines* [@BH]. In this case, one looks for the quantum map $\L$ that, given a state $\ket{\psi}$ chosen from an ensemble $\{p_i,\ket{\psi_i}\}$ in $\compl^d$, produces a state $\L(\psi)=\rho_{C_1\ldots C_N}$ in $(\compl^d)^{\otimes N}$, such that each individual clone $\rho_{C_k}=\tr_{\bar
k}(\rho_{C_1\ldots C_N})$ resembles as much as possible the input state, where $\tr_{\bar k}$ denotes the trace with respect to all the systems $C_1,\ldots,C_N$ but $C_k$. The average fidelity of the cloning process is then $$\label{avfidcl}
\bar F_C(N)=\sum_{i,k}
p_i\, \frac{1}{N}\bra{\psi_i}\tr_{\bar k}\L(\psi_i)\ket{\psi_i} .$$ The goal of the optimal machine is to maximize this quantity, this optimal value being denoted by $F_C(N)$.
One can easily realize that the no-cloning theorem and the impossibility of perfect state estimation are closely related. On the one hand, if perfect state estimation was possible, one could use it to prepare any number of clones of a given state, just by measurement and preparation. On the other hand, if perfect cloning was possible, one could perfectly estimate the unknown state of a quantum system by preparing infinite clones of it and then measuring them. Beyond these qualitative arguments, the connection between state estimation and cloning was strengthened in [@GM; @BEM]. The results of these works suggested that asymptotic cloning, i.e. the optimal cloning process when $N\to\infty$, is equivalent to state estimation, in the sense that $$\label{conj}
F_C=F_C(N\to\infty)=F_M .$$ Later, this equality was rigorously shown to hold in the cases of (i) universal cloning [@KW], where the initial ensemble consists of an arbitrary pure state in $\compl^d$, chosen with uniform probability, and (ii) phase covariant qubit cloning [@BCDM], where the initial ensemble corresponds to a state in $\compl^2$ lying on one of the equators of the Bloch sphere. Since then, the validity of this equality for any ensemble has been conjectured and, indeed, has been identified as one of the open problems in QIT [@web].
In this work, we show that the fidelities of optimal asymptotic cloning and of state estimation are indeed equal for any initial ensemble of pure states. Actually, we prove that *asymptotic cloning does effectively correspond to state estimation*, from which the equality of the two fidelities trivially follows. The proof of this equivalence is based on two known results of QIT: the monogamy of quantum correlations and the properties of the so-called entanglement breaking channels (EBC).
It is easy to prove that $F_M\leq F_C$. Indeed, given the initial state $\ket{\psi}$, a possible asymptotic cloning map, not necessarily optimal, consists of first applying state estimation and then preparing infinite copies of the guessed state. It is sometimes said that the opposite has to be true since “asymptotic cloning cannot represent a way of circumventing optimal state estimation". As already mentioned in [@web], this reasoning is too naive, since it neglects the role correlations play in state estimation. For instance, take the simplest case of universal cloning of a qubit, i.e. a state in $\compl^2$ isotropically distributed over the Bloch sphere. The optimal cloning machines produces $N$ approximate clones pointing in the same direction in the Bloch sphere as the input state, but with a shrunk Bloch vector [@KW]. If the output of the asymptotic cloning machine was in a product form, it would be possible to perfectly estimate the direction of the local Bloch vector, whatever the shrinking was. Then, a perfect estimation of the initial state would be possible. And of course, after the perfect estimation one could prepare an infinite number of perfect clones! This simple reasoning shows that the correlations between the clones play an important role in the discussion. Actually, it has recently been shown that the correlations present in the output of the universal cloning machine are the worst for the estimation of the reduced density matrix [@rafael].
As announced, the proof of the conjecture is based on two known results of QIT: the monogamy of entanglement and the properties of EBC. For the sake of completeness, we state here these results, without proof.
Quantum correlations, or entanglement, represent a monogamous resource, in the sense that they cannot be arbitrarily shared. One of the strongest results in this direction was obtained by Werner in 1989 [@Werner]. There, it was shown that the only states that can be arbitrarily shared are the separable ones. Recall that a bipartite quantum state $\rho_{AC}$ in $\compl^d\otimes\compl^d$ is said to be $N$-shareable when it is possible to find a quantum state $\rho_{AC_1\ldots C_N}$ in $\compl^d\otimes(\compl^d)^{\otimes N}$ such that $\rho_{AC_k}=\tr_{\bar k}\rho_{AC_1\ldots C_N}=\rho_{AC},\,\forall
k$. The state $\rho_{AC_1\ldots C_N}$ is then said to be an $N$-extension of $\rho_{AC}$. The initial correlations between subsystems $A$ and $C$ are now shared between $A$ and each of the $N$ subsystems $C_i$, see Fig. \[entshar\]. It is straightforward to see that $$\rho_{AC_1\ldots C_N}=\sum_i q_i\proj{\alpha_i}\otimes
\proj{\gamma_i}^{\otimes N}$$ gives a valid $N$-extension of a separable state $\rho_{AC}^s=\sum_i q_i\proj{\alpha_i}\otimes\proj{\gamma_i}$ for all $N$. As proven by Werner, if the state is entangled, there exists a finite $N$ where no valid extension can be found.
![The state $\rho_{AC}$ is said to be $N$-shareable when there exists a global state $\rho_{AC_1\ldots C_N}$ such that the local state shared between $A$ and $C_i$ is equal to $\rho_{AC}$, for all $i$.[]{data-label="entshar"}](rhoAcccc "fig:"){width="8cm"}\
The second ingredient needed in what follows are the properties of EBC. A channel $\Upsilon$ is said to be entanglement breaking when it cannot be used to distribute entanglement. In Ref. [@HSR] it was proven that the following three statements are equivalent: (1) $\Upsilon$ is entanglement breaking, (2) $\Upsilon$ can be written in the form $\Upsilon(\rho)=\sum_j\tr(M_j\rho)\rho_j$, where $\rho_j$ are quantum states and $\{M_j\}$ defines a measurement and (3) $(\one\otimes\Upsilon)\ket{\Phi^+}$ is a separable state, where $\ket{\Phi^+}=\sum_i \ket{ii}/\sqrt d$ is a maximally entangled state in $\compl^d\otimes\compl^d$. The equivalence of (1) and (2) simply means that any EBC can be understood as the measurement of the input state, $\rho$, followed by the preparation of a new state $\rho_j$ depending on the obtained outcome. The equivalence of (1) and (3) reflects that the intuitive strategy for entanglement distribution where half of a maximally entangled state is sent through the channel is enough to detect if $\Upsilon$ is entanglement breaking.
After collecting all these results, we are now ready to prove the following
[**Theorem:**]{} Asymptotic cloning corresponds to state estimation. Thus, $F_M=F_C$ for any ensemble of states.
[*Proof:*]{} The idea of the proof is to characterize the quantum maps $\L$ associated to asymptotic cloning machines. First of all, note that, for any number of clones, we can restrict our considerations to symmetric cloning machines, $\L^s$, where the clones are all in the same state. Indeed, given a machine where this is not the case, one can construct a symmetric machine achieving the same fidelity $F_C(N)$, just by making a convex combination of all the permutations of the $N$ clones [@note]. Now, denote by $\L^c$ the effective cloning map consisting of, first, the application of a symmetric machine $\L^s$ and then tracing all but one of the clones, say the first one. The cloning problem can be rephrased as, see Eq. (\[avfidcl\]), $$\label{asclon}
\max_{\L^c}\sum_ip_i\bra{\psi_i}\L^c(\psi_i)\ket{\psi_i} .$$ Note that this maximization runs over all channels that can be written as $\L^c=\tr_{\bar 1}\L^s$. For instance, the identity map, where $\psi\to\psi,\,\forall\psi$, does not satisfy this constraint.
If the $N$-cloning map is applied to half of a maximally entangled state, the resulting state, $$\rho_{AC_1\ldots C_N}=(\one_A\otimes\L^s_B)\ket{\Phi^+}_{AB},$$ is such that, for all $i$, $$\label{rhoac}
\rho_{AC_i}=(\one\otimes\L^c)\ket{\Phi^+}=\rho_{AC}.$$ That is, the output of the $N$-cloning machine acting on half of a maximally entangled state is a valid $N$-extension of $\rho_{AC}$. When taking the limit of an infinite number of clones, and because of the monogamy of entanglement, this implies that $\rho_{AC}$ has to be separable and, thus, $\L^c$ is entanglement breaking (\[rhoac\]). Since any EBC can be seen as measurement followed by state preparation, asymptotic cloning (\[asclon\]) can be written as [@note2] $$\max_{\{M_j,\phi_j\}}\sum_{i,j}
p_i\, \tr(M_j\proj{\psi_i})\,|\braket{\psi_i}{\phi_j}|^2 ,$$ which defines the optimal state estimation problem. Therefore, $F_M=F_C$ for any ensemble of states. $\Box$
The same argument applies to the case in which $L$ copies of the initial state $\ket{\psi}$ are given. The measurement and cloning fidelities now read, see Eqs. (\[avfidmeas\]) and (\[avfidcl\]), $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\bar F_M(L) &=& \sum_{i,j}
p_i\, \tr(M_j\proj{\psi_i}^{\otimes L})\,|
\braket{\psi_i}{\phi_j}|^2 \nonumber\\
\bar F_C(N,L) &=& \sum_{i,k} p_i\, \frac{1}{N}\bra{\psi_i}
\tr_{\bar k}\L(\psi_i^{\otimes L})\ket{\psi_i} .\end{aligned}$$ Using the same ideas as in the previous Theorem, it is straightforward to prove that $$\label{eqfidL}
F_M(L)=F_C(N\to\infty,L),$$ where $F_M(L)$ and $F_C(N,L)$ denote the optimal values of $\bar
F_M(L)$ and $\bar F_C(N,L)$, as above.
One can also extend this result to asymmetric scenarios. An asymmetric cloning machine [@asclon], given an initial input state $\ket{\psi}$, produces $N_A$ clones of fidelity $F_C(N_A)$ and $N_B$ clones of fidelity $F_C(N_B)$. The machine is optimal when it gives the largest $F_C(N_B)$ for fixed $F_C(N_A)$. In the case of measurement, we are thinking of measurement strategies where the goal is to obtain information on an unknown state introducing the minimal disturbance. As above, we consider that a guess for the input state is done depending on the measurement outcome. The information vs disturbance trade-off can again be expressed in terms of fidelities [@banaszek]: the information gain is given by the overlap, $G$, between the initial and the guessed state, while the disturbance is quantified by the overlap, $F$, between the state after the measurement and the initial state. A measurement is optimal when for fixed disturbance, $F$, it provides the largest value of $G$. The optimal trade-off between $F$ and $G$ has been derived in [@banaszek] for the case in which the input ensemble consists of any pure state in $\compl^d$ with uniform probability.
As it happens for the symmetric case, a connection between this state estimation problem and asymmetric cloning machines can be expected when $N_A=1$ and $N_B\to\infty$. Indeed, the previous measurement strategy gives a possible realization of this asymptotic and asymmetric cloning machine, not necessarily optimal. Actually, when the input state is any pure state, with uniform probability, the optimal measurement strategy of [@banaszek] turns out to saturate the optimal cloning $1\to
N_A+N_B$ fidelities of [@IACFFG], with $N_A=1$ and $N_B\to\infty$. Now, the equality between the measurement and asymptotic cloning fidelities in the asymmetric scenario for any ensemble of input states can be proven using the same arguments as above: one has to symmetrize the $N_B$ clones and then use the connection with entanglement shareability and EBC when $N_B\to\infty$.
From a more speculative point of view, there exist several works relating the impossibility of perfect cloning to the no-signaling principle, namely the impossibility of having faster-than-light communication (see for instance [@gisin]). Actually, a no-cloning theorem can be derived just from the no-signaling principle, without invoking any additional quantum feature [@ncsign]. In view of the strong connection between cloning and state estimation, it would be interesting to study whether a similar link between the no-signaling principle and the impossibility of perfect state estimation could also be established.
To conclude, this work proves the long-standing conjecture on the equivalence between asymptotic cloning and state estimation. It represents the strongest link between two fundamental no-go theorems of Quantum Mechanics, namely the impossibilities of perfect cloning and state estimation.
We thank Emili Bagan, John Calsamiglia, Sofyan Iblisdir and Ramon Muñoz-Tapia for discussion. This work is supported by the Spanish MCyT, under “Ramón y Cajal" grant, and the Generalitat de Catalunya, 2006FIR-000082 grant.
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Two independent reviews on the no-cloning theorem have recently appeared: V. Scarani, S. Iblisdir, N. Gisin and A. Acín, Rev. Mod. Phys. [**77**]{}, 1225 (2005); N. J. Cerf and J. Fiurášek, quant-ph/0512172.
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See problem 22 in http://www.imaph.tu-bs.de/qi/problems/.
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Notice that this does not mean that the output of the cloning machine lives in the symmetric subspace.
We can already restrict the guessed states to be pure, without any loss of optimality.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^\omega$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a Hölder continuity class $C^{0,\alpha}$. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.'
address:
- 'Andreas Kriegl: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria'
- 'Peter W. Michor: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria'
- 'Armin Rainer: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4'
author:
- 'Andreas Kriegl, Peter W. Michor, and Armin Rainer'
title: 'Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators '
---
[Theorem]{} Let $t\mapsto A(t)$ for $t\in T$ be a parameterized family of unbounded operators in a Hilbert space $H$ with common domain of definition and with compact resolvent.
If $t\in T=\mathbb R$ and all $A(t)$ are self-adjoint then the following holds:
1. If $A(t)$ is real analytic in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ may be parameterized real analytically in $t$.
2. If $A(t)$ is quasianalytic of class $C^Q$ in $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ may be parameterized $C^Q$ in $t$.
3. If $A(t)$ is non-quasianalytic of class $C^L$ in $t\in \mathbb R$ and if no two unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized $C^L$ in $t$.
4. If $A(t)$ is $C^\infty$ in $t\in \mathbb R$ and if no two unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized $C^\infty$ in $t$.
5. If $A(t)$ is $C^\infty$ in $t\in \mathbb R$, then the eigenvalues of $A(t)$ may be parameterized twice differentiably in $t$.
6. If $A(t)$ is $C^{1,\alpha}$ in $t\in \mathbb R$ for some $\alpha>0$, then the eigenvalues of $A(t)$ may be parameterized in a $C^1$ way in $t$.
If $t\in T=\mathbb R$ and all $A(t)$ are normal then the following holds:
1. If $A(t)$ is real analytic in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized real analytically in $s$ near $s=0$.
2. If $A(t)$ is $C^Q$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^Q$ in $s$ near $s=0$.
3. If $A(t)$ is $C^L$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ at which no two of the unequal continuously arranged eigenvalues (see [@Kato76 II.5.2]) meet of infinite order, there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^L$ in $s$ near $s=0$.
4. If $A(t)$ is $C^\infty$ in $t\in \mathbb R$, then for each $t_0\in \mathbb R$ and for each eigenvalue $\lambda$ of $A(t_0)$ at which no two of the unequal continuously arranged eigenvalues (see [@Kato76 II.5.2]) meet of infinite order, there exists $N\in \mathbb N$ such that the eigenvalues near $\lambda$ of $A(t_0\pm s^N)$ and their eigenvectors can be parameterized $C^\infty$ in $s$ near $s=0$.
5. If $A(t)$ is $C^\infty$ in $t\in \mathbb R$ and no two of the unequal continuously parameterized eigenvalues meet of infinite order at any $t\in \mathbb R$, then the eigenvalues and the eigenvectors of $A(t)$ can be parameterized by absolutely continuous functions, locally in $t$.
If $t\in T=\mathbb R^n$ and all $A(t)$ are normal then the following holds:
1. If $A(t)$ is $C^\omega$ or $C^Q$ in $t\in \mathbb R^n$, then for each $t_0\in \mathbb R^n$ and for each eigenvalue $\lambda$ of $A(t_0)$, there exist a finite covering $\{\pi_k : U_k \to W\}$ of a neighborhood $W$ of $t_0$, where each $\pi_k$ is a composite of finitely many mappings each of which is either a local blow-up along a $C^\omega$ or $C^Q$ submanifold or a local power substitution, such that the eigenvalues and the eigenvectors of $A(\pi_k(s))$ can be chosen $C^\omega$ or $C^Q$ in $s$. If $A$ is self-adjoint, then we do not need power substitutions.
2. If $A(t)$ is $C^\omega$ or $C^Q$ in $t\in \mathbb R^n$, then the eigenvalues and their eigenvectors of $A(t)$ can be parameterized by functions which are special functions of bounded variation (SBV), see [@AmbrosioDeGiorgi88] or [@AFP00], locally in $t$.
If $t\in T\subseteq E$, a $c^\infty$-open subset in an infinite dimensional convenient vector space then the following holds:
1. For $0<\alpha\le 1$, if $A(t)$ is $C^{0,\alpha}$ (Hölder continuous of exponent $\alpha$) in $t\in T$ and all $A(t)$ are self-adjoint, then the eigenvalues of $A(t)$ may be parameterized in a $C^{0,\alpha}$ way in $t$.
2. For $0<\alpha\le 1$, if $A(t)$ is $C^{0,\alpha}$ (Hölder continuous of exponent $\alpha$) in $t\in T$ and all $A(t)$ are normal, then we have: For each $t_0\in T$ and each eigenvalue $z_0$ of $A(t_0)$ consider a simple closed $C^1$-curve $\gamma$ in the resolvent set of $A(t_0)$ enclosing only $z_0$ among all eigenvalues of $A(t_0)$. Then for $t$ near $t_0$ in the $c^\infty$-topology on $T$, no eigenvalue of $A(t)$ lies on $\gamma$. Let $\lambda(t)=(\lambda_1(t),\dots,\lambda_N(t))$ be the $N$-tuple of all eigenvalues (repeated according to their multiplicity) of $A(t)$ inside of $\gamma$. Then $t\mapsto \lambda(t)$ is $C^{0,\alpha}$ for $t$ near $t_0$ with respect to the non-separating metric $$d(\lambda,\mu) = \min_{\sigma\in\mathcal S_N} \max_{1\le i\le N} |\lambda_i - \mu_{\sigma(i)}|$$ on the space of $N$-tuples.
Part [(A)]{} is due to Rellich [@Rellich42V] in 1942, see also [@Baumgaertel72] and [@Kato76 VII, 3.9]. Part [(D)]{} has been proved in [@AKLM98 7.8], see also [@KM97 50.16], in 1997, which contains also a different proof of [(A)]{}. [(E)]{} and [(F)]{} have been proved in [@KM03] in 2003. (G) was proved in [@RainerAC 7.1]; it can be proved as [(H)]{} with some obvious changes, but it is not a special case since $C^\omega$ does not correspond to a sequence which is an $\mathcal L$-intersection (see [@KMRq]). [(J)]{} and [(K)]{} were proved in [@RainerAC 7.1]. [(N)]{} was proved in [@KMR].
The purpose of this paper is to prove the remaining parts , , , , , , and .
Definitions and remarks {#definitions-and-remarks .unnumbered}
------------------------
Let $M=(M_k)_{k \in \mathbb{N}=\mathbb{N}_{\ge0}}$ be an increasing sequence ($M_{k+1}\ge M_k$) of positive real numbers with $M_0=1$. Let $U \subseteq \mathbb{R}^n$ be open. We denote by $C^M(U)$ the set of all $f \in C^\infty(U)$ such that, for each compact $K \subseteq U$, there exist positive constants $C$ and $\rho$ such that $$|\partial^\alpha f(x)| \le C \, \rho^{|\alpha|} \, |\alpha|! \, M_{|\alpha|}
\quad\text{ for all }\alpha \in \mathbb{N}^n\text{ and }x \in K.$$ The set $C^M(U)$ is a *Denjoy–Carleman class* of functions on $U$. If $M_k=1$, for all $k$, then $C^M(U)$ coincides with the ring $C^\omega(U)$ of real analytic functions on $U$. In general, $C^\omega(U) \subseteq C^M(U) \subseteq C^\infty(U)$.
Here $Q=(Q_k)_{k\in\mathbb N}$ is a sequence as above which is quasianalytic, log-convex, and which is also an $\mathcal L$-intersection, see [@KMRq] or [@KMRc] and references therein. Moreover, $L=(L_k)_{k\in\mathbb N}$ is a sequence as above which is non-quasianalytic and log-convex.
That $A(t)$ is a real analytic, $C^M$ (where $M$ is either $Q$ or $L$), $C^\infty$, or $C^{k,\alpha}$ family of unbounded operators means the following: There is a dense subspace $V$ of the Hilbert space $H$ such that $V$ is the domain of definition of each $A(t)$, and such that $A(t)^*=A(t)$ in the self-adjoint case, or $A(t)$ has closed graph and $A(t)A(t)^*=A(t)^*A(t)$ wherever defined in the normal case. Moreover, we require that $t\mapsto \langle A(t)u,v\rangle$ is of the respective differentiability class for each $u\in V$ and $v\in H$. >From now on we treat only $C^M=C^\omega$, $C^M$ for $M=Q$, $M=L$, and $C^M=C^{0,\alpha}$.
This implies that $t\mapsto A(t)u$ is of the same class $C^M(E,H)$ (where $E$ is either $\mathbb R$ or $\mathbb R^n$) or is in $C^{0,\alpha}(E,H)$ (if $E$ is a convenient vector space) for each $u\in V$ by [@KM97 2.14.4, 10.3] for $C^\omega$, by [@KMRc 3.1, 3.3, 3.5] for $M=L$, by [@KMRq 1.10, 2.1, 2.3] for $M=Q$, and by [@KM97 2.3], [@FK88 2.6.2] or [@Faure91 4.14.4] for $C^{0,\alpha}$ because $C^{0,\alpha}$ can be described by boundedness conditions only and for these the uniform boundedness principle is valid.
A sequence of functions $\lambda_i$ is said to [*parameterize the eigenvalues, if for each $z\in \mathbb C$ the cardinality $|\{i: \lambda_i(t)=z\}|$ equals the multiplicity of $z$ as eigenvalue of $A(t)$.*]{}
Let $X$ be a $C^\omega$ or $C^Q$ manifold. A *local blow-up $\Phi$* over an open subset $U$ of $X$ means the composition $\Phi = \iota \circ \varphi$ of a blow-up $\varphi : U' \to U$ with center a $C^\omega$ or $C^Q$ submanifold and of the inclusion $\iota : U \to X$. A *local power substitution* is a mapping $\Psi: V \to X$ of the form $\Psi = \iota \circ \psi$, where $\iota : W \to X$ is the inclusion of a coordinate chart $W$ of $X$ and $\psi : V \to W$ is given by $$(y_1,\ldots,y_q) = ((-1)^{\epsilon_1} x_1^{\gamma_1},\ldots,(-1)^{\epsilon_q} x_q^{\gamma_q}),$$ for some $\gamma=(\gamma_1,\ldots,\gamma_q) \in (\mathbb{N}_{>0})^q$ and all $\epsilon = (\epsilon_1,\ldots,\epsilon_q) \in \{0,1\}^q$, where $y_1,\ldots,y_q$ denote the coordinates of $W$ (and $q = \dim X$).
This paper became possible only after some of the results of [@KMRc] and [@KMRq] were proved, in particular the uniform boundedness principles. The wish to prove the results of this paper was the main motivation for us to work on [@KMRc] and [@KMRq].
Applications {#applications .unnumbered}
-------------
Let $X$ be a compact $C^Q$ manifold and let $t\mapsto g_t$ be a $C^Q$-curve of $C^Q$ Riemannian metrics on $X$. Then we get the corresponding $C^Q$ curve $t\mapsto \Delta(g_t)$ of Laplace-Beltrami operators on $L^2(X)$. By theorem (B) the eigenvalues and eigenvectors can be arranged $C^Q$. Question: Are the eigenfunctions then also $C^Q$?
Let $\Omega$ be a bounded region in $\mathbb R^n$ with $C^Q$ boundary, and let $H(t)=-\Delta + V(t)$ be a $C^Q$-curve of Schrödinger operators with varying $C^Q$ potential and Dirichlet boundary conditions. Then the eigenvalues and eigenvectors can be arranged $C^Q$. Question: Are the eigenvectors viewed as eigenfunctions then also in $C^Q(\Omega\times \mathbb R)$?
Example {#example .unnumbered}
--------
This is an elaboration of [@AKLM98 7.4] and [@KM03 Example]. Let $S(2)$ be the vector space of all symmetric real $(2\times 2)$-matrices. We use the $C^L$-curve lemma [@KMRc 3.6] or [@KMRq 2.5]: [*There exists a converging sequence of reals $t_n$ with the following property: Let $A_n, B_n\in S(2)$ be any sequences which converge fast to 0, i.e., for each $k\in \mathbb N$ the sequences $n^kA_n$ and $n^kB_n$ are bounded in $S(2)$. Then there exists a curve $A\in C^L(\mathbb R,S(2))$ such that $A(t_n+s)=A_n+sB_n$ for $|s|\le \frac1{n^2}$, for all $n$.*]{}
We use it for $$\begin{aligned}
A_n := \frac 1{2^{n^2}}
\begin{pmatrix}
1 & 0 \\
0 & -1\\
\end{pmatrix},\quad
B_n := \frac 1{2^{n^2}\,s_n}
\begin{pmatrix}
0 & 1 \\
1 & 0\\
\end{pmatrix},\quad
\text{ where } s_n := 2^{n-n^2}\le \frac 1{n^2}.\end{aligned}$$ The eigenvalues of $A_n+tB_n$ and their derivatives are $$\lambda_n(t) = \pm\frac 1{2^{n^2}} \sqrt{1+(\tfrac t{s_n})^2},\quad
\lambda_n'(t) = \pm\frac {2^{n^2-2n}t}{\sqrt{1+(\frac t{s_n})^2}}.$$ Then $$\begin{aligned}
\frac{\lambda'(t_n+s_n)-\lambda'(t_n)}{s_n^\alpha} &=
\frac{\lambda_n'(s_n)-\lambda_n'(0)}{s_n^\alpha}
=\pm\frac {2^{n^2-2n}s_n}{s_n^\alpha\sqrt{2}}\\
&=\pm\frac{2^{n(\alpha(n-1)-1)}}{\sqrt{2}} \to \infty \text{ for }\alpha>0.\end{aligned}$$ So condition (in , , , , and ) that no two unequal continuously parameterized eigenvalues meet of infinite order cannot be dropped. By [@AKLM98 2.1], we may always find a twice differentiable square root of a non-negative smooth function, so that the eigenvalues $\lambda$ are functions which are twice differentiable but not $C^{1,\alpha}$ for any $\alpha>0$.
Note that the normed eigenvectors cannot be chosen continuously in this example (see also example [@Rellich37I §2]). Namely, we have $$A(t_n)=A_n=\frac1{2^{n^2}}\begin{pmatrix} 1 & 0 \\ 0 &-1 \end{pmatrix},\qquad
A(t_n+s_n)=A_n +s_n\,B_n=\frac1{2^{n^2}}\begin{pmatrix} 1 & 1 \\ 1 &-1 \end{pmatrix}.$$
[Resolvent Lemma]{} Let $C^M$ be any of $C^\omega$, $C^Q$, $C^L$, $C^\infty$, or $C^{0,\alpha}$, and let $A(t)$ be normal. If $A$ is $C^M$ then the resolvent $(t,z)\mapsto (A(t)-z)^{-1}\in L(H,H)$ is $C^M$ on its natural domain, the global resolvent set $$\{(t,z)\in T\times\mathbb C: (A(t)-z):V\to H \text{ is invertible}\}$$ which is open (and even connected).
[**Proof**]{} By definition the function $t\mapsto \langle A(t)v,u \rangle$ is of class $C^M$ for each $v\in V$ and $u\in H$. We may conclude that the mapping $t\mapsto A(t)v$ is of class $C^M$ into $H$ as follows: For $C^M=C^\infty$ we use [@KM97 2.14.4]. For $C^M=C^\omega$ we use in addition [@KM97 10.3]. For $C^M=C^Q$ or $C^M=C^L$ we use [@KMRq 2.1] and/or [@KMRc 3.3] where we replace $\mathbb R$ by $\mathbb R^n$. For $C^M=C^{0,\alpha}$ we use [@KM97 2.3], [@FK88 2.6.2], or [@Faure91 4.1.14] because $C^{0,\alpha}$ can be described by boundedness conditions only and for these the uniform boundedness principle is valid.
For each $t$ consider the norm $\|u\|_t^2:=\|u\|^2+\|A(t)u\|^2$ on $V$. Since $A(t)$ is closed, $(V,\|\quad\|_t)$ is again a Hilbert space with inner product $\langle u,v\rangle_t:=\langle u,v\rangle+\langle A(t)u,A(t)v\rangle$.
[*[(1)]{} Claim (see [@AKLM98 in the proof of 7.8], [@KM97 in the proof of 50.16], or [@KM03 Claim 1]). All these norms $\|\quad\|_t$ on $V$ are equivalent, locally uniformly in $t$. We then equip $V$ with one of the equivalent Hilbert norms, say $\|\quad\|_0$.*]{}
We reduce this to $C^{0,\alpha}$. Namely, note first that $A(t):(V,\|\quad\|_s)\to H$ is bounded since the graph of $A(t)$ is closed in $H\times H$, contained in $V\times H$ and thus also closed in $(V,\|\quad\|_s)\times H$. For fixed $u,v\in V$, the function $t\mapsto \langle u,v\rangle_t=\langle u,v \rangle+\langle A(t)u,A(t)v
\rangle$ is $C^{0,\alpha}$ since $t\mapsto A(t)u$ is it. By the multilinear uniform boundedness principle ([@KM97 5.18] or [@FK88 3.7.4]) the mapping $t\mapsto \langle \quad,\quad\rangle_t$ is $C^{0,\alpha}$ into the space of bounded sesquilinear forms on $(V,\|\quad\|_s)$ for each fixed $s$. Thus the inverse image of $\langle \quad,\quad \rangle_s + \frac12(\text{unit ball})$ in $L(\overline{(V,\|\quad\|_s)} \oplus (V,\|\quad\|_s);\mathbb C)$ is a $c^\infty$-open neighborhood $U$ of $s$ in $T$. Thus $\sqrt{1/2}\|u\|_s\le \|u\|_t\le \sqrt{3/2}\|u\|_s$ for all $t\in U$, i.e. all Hilbert norms $\|\quad\|_t$ are locally uniformly equivalent, and claim [(1)]{} follows.
By the linear uniform boundedness theorem we see that $t\mapsto A(t)$ is in $C^M(T, L(V,H))$ as follows (here it suffices to use a set of linear functionals which together recognize bounded sets instead of the whole dual): For $C^M=C^\infty$ we use [@KM97 1.7 and 2.14.3]. For $C^M=C^\omega$ we use in addition [@KM97 9.4]. For $C^M=C^Q$ or $C^M=C^L$ we use [@KMRq 2.2 and 2.3] and/or [@KMRc 3.5] where we replace $\mathbb R$ by $\mathbb R^n$. For $C^M=C^{0,\alpha}$ see above.
If for some $(t,z)\in T\times\mathbb C$ the bounded operator $A(t)-z:V\to H$ is invertible, then this is true locally with respect to the $c^\infty$-topology on the product which is the product topology by [@KM97 4.16], and $(t,z)\mapsto (A(t)-z)^{-1}:H\to V$ is $C^M$, by the chain rule, since inversion is real analytic on the Banach space $L(V,H)$.
Note that $(A(t)-z)^{-1}:H\to H$ is a compact operator for some (equivalently any) $(t,z)$ if and only if the inclusion $i:V\to H$ is compact, since $i=(A(t)-z)^{-1}\circ(A(t)-z): V\to H\to H$.
[Polynomial proposition]{} Let $P$ be a curve of polynomials $$P(t)(x)=x^n-a_1(t)x^{n-1}+\dots+(-1)^na_n(t),\quad t\in \mathbb R.$$
1. If $P$ is hyperbolic (all roots real) and if the coefficient functions $a_i$ are all $C^Q$ then there exist $C^Q$ functions $\lambda_i$ which parameterize all roots.
2. If $P$ is hyperbolic (all roots real), if the coefficient functions $a_i$ are $C^L$ and no two of the different roots meet of infinite order, then there exist $C^L$ functions $\lambda_i$ which parameterize all roots.
3. If the coefficient functions $a_i$ are $C^Q$, then for each $t_0$ there exists $N\in \mathbb N$ such that the roots of $s\mapsto P(t_0\pm s^{N})$ can be parameterized $C^Q$ in $s$ for $s$ near 0.
4. If the coefficient functions $a_i$ are $C^L$ and no two of the different roots meet of infinite order, then for each $t_0$ there exists $N\in \mathbb N$ such that the roots of $s\mapsto P(t_0\pm s^{N})$ can be parameterized $C^L$ in $s$ for $s$ near 0.
All $C^Q$ or $C^L$ solutions differ by permutations.
The proof of parts and is exactly as in [@AKLM98] where the corresponding results were proven for $C^\infty$ instead of $C^L$, and for $C^\omega$ instead of $C^Q$. For this we need only the following properties of $C^Q$ and $C^L$:
- They allow for the implicit function theorem (for [@AKLM98 3.3]).
- They contain $C^\omega$ and are closed under composition (for [@AKLM98 3.4]).
- They are derivation closed (for [@AKLM98 3.7]).
Part is also in [@CC04 7.6] which follows [@AKLM98]. It also follows from the multidimensional version [@RainerQA 6.10] since blow-ups in dimension 1 are trivial. The proofs of parts and are exactly as in [@RainerAC 3.2] where the corresponding result was proven for $C^\omega$ instead of $C^Q$, and for $C^\infty$ instead of $C^L$, if none of the different roots meet of infinite order. For these we need the properties of $C^Q$ and $C^L$ listed above.
[Matrix proposition]{} Let $A(t)$ for $t\in T$ be a family of $(N\times N)$-matrices.
1. If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^Q$-curve of Hermitian matrices, then the eigenvalues and the eigenvectors can be chosen $C^Q$.
2. If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^L$-curve of Hermitian matrices such that no two eigenvalues meet of infinite order, then the eigenvalues and the eigenvectors can be chosen $C^L$.
3. If $T=\mathbb R\ni t\mapsto A(t)$ is a $C^L$-curve of normal matrices such that no two eigenvalues meet of infinite order, then for each $t_0$ there exists $N_1\in \mathbb N$ such that the eigenvalues and eigenvectors of $s\mapsto A(t_0\pm s^{N_1})$ can be parameterized $C^L$ in $s$ for $s$ near 0.
4. Let $T\subseteq\mathbb R^n$ be open and let $T\ni t\mapsto A(t)$ be a $C^\omega$ or $C^Q$-mapping of normal matrices. Let $K \subseteq T$ be compact. Then there exist a neighborhood $W$ of $K$, and a finite covering $\{\pi_k : U_k \to W\}$ of $W$, where each $\pi_k$ is a composite of finitely many mappings each of which is either a local blow-up along a $C^\omega$ or $C^Q$ submanifold or a local power substitution, such that the eigenvalues and the eigenvectors of $A(\pi_k(s))$ can be chosen $C^\omega$ or $C^Q$ in $s$. Consequently, the eigenvalues and eigenvectors of $A(t)$ are locally special functions of bounded variation (SBV). If $A$ is a family of Hermitian matrices, then we do not need power substitutions.
The proof of the matrix proposition in case and is exactly as in [@AKLM98 7.6], using the polynomial proposition and properties of $C^Q$ and $C^L$. Item is exactly as in [@RainerAC 6.2], using the polynomial proposition and properties of $C^L$. Item is proved in [@RainerQA 9.1 and 9.6], see also [@KurdykaPaunescu08].
[**Proof of the theorem**]{} We have to prove parts , , , , , , and . So let $C^M$ be any of $C^\omega$, $C^Q$, $C^L$, or $C^{0,\alpha}$, and let $A(t)$ be normal. Let $z$ be an eigenvalue of $A(t_0)$ of multiplicity $N$. We choose a simple closed $C^1$ curve $\gamma$ in the resolvent set of $A(t_0)$ for fixed $t_0$ enclosing only $z$ among all eigenvalues of $A(t_0)$. Since the global resolvent set is open, see the resolvent lemma, no eigenvalue of $A(t)$ lies on $\gamma$, for $t$ near $t_0$. By the resolvent lemma, $A: T\to L((V,\|\quad\|_0),H)$ is $C^M$, thus also $$t\mapsto -\frac1{2\pi i}\int_\gamma (A(t)-z)^{-1}\;dz =: P(t,\gamma) = P(t)$$ is a $C^M$ mapping. Each $P(t)$ is a projection, namely onto the direct sum of all eigenspaces corresponding to eigenvalues of $A(t)$ in the interior of $\gamma$, with finite rank. Thus the rank must be constant: It is easy to see that the (finite) rank cannot fall locally, and it cannot increase, since the distance in $L(H,H)$ of $P(t)$ to the subset of operators of rank $\le N=\operatorname{rank}(P(t_0))$ is continuous in $t$ and is either 0 or 1.
So for $t$ in a neighborhood $U$ of $t_0$ there are equally many eigenvalues in the interior of $\gamma$, and we may call them $\lambda_i(t)$ for $1\le i\le N$ (repeated with multiplicity).
Now we consider the family of $N$-dimensional complex vector spaces $t\mapsto P(t)(H)\subseteq H$, for $t\in U$. They form a $C^M$ Hermitian vector subbundle over $U$ of $U\times H\to U$: For given $t$, choose $v_1,\dots v_N\in H$ such that the $P(t)v_i$ are linearly independent and thus span $P(t)H$. This remains true locally in $t$. Now we use the Gram Schmidt orthonormalization procedure (which is $C^\omega$) for the $P(t)v_i$ to obtain a local orthonormal $C^M$ frame of the bundle.
Now $A(t)$ maps $P(t)H$ to itself; in a $C^M$ local frame it is given by a normal $(N\times N)$-matrix parameterized $C^M$ by $t\in U$.
Now all local assertions of the theorem follow:
- Use the matrix proposition, part .
- Use the matrix proposition, part .
- Use the matrix proposition, part , and note that in dimension 1 blowups are trivial.
- Use the matrix proposition, part .
- Use the matrix proposition, part , for $\mathbb R^n$.
- We use the following
Finally, it remains to extend the local choices to global ones for the cases and only. There $t\mapsto A(t)$ is $C^Q$ or $C^L$, respectively, which imply both $C^\infty$, and no two different eigenvalues meet of infinite order. So we may apply [@AKLM98 7.8] (in fact we need only the end of the proof) to conclude that the eigenvalues can be chosen $C^\infty$ on $T=\mathbb R$, uniquely up to a global permutation. By the local result above they are then $C^Q$ or $C^L$. The same proof then gives us, for each eigenvalue $\lambda_i:T \to \mathbb R$ with generic multiplicity $N$, a unique $N$-dimensional smooth vector subbundle of $\mathbb R\times H$ whose fiber over $t$ consists of eigenvectors for the eigenvalue $\lambda_i(t)$. In fact this vector bundle is $C^Q$ or $C^L$ by the local result above, namely the matrix proposition, part or , respectively.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Boosting algorithms have been widely used to tackle a plethora of problems. In the last few years, a lot of approaches have been proposed to provide standard AdaBoost with cost-sensitive capabilities, each with a different focus. However, for the researcher, these algorithms shape a tangled set with diffuse differences and properties, lacking a unifying analysis to jointly compare, classify, evaluate and discuss those approaches on a common basis. In this series of two papers we aim to revisit the various proposals, both from theoretical (Part I) and practical (Part II) perspectives, in order to analyze their specific properties and behavior, with the final goal of identifying the algorithm providing the best and soundest results.'
address: 'Signal Theory and Communications Department, University of Vigo, Maxwell Street, 36310, Vigo, Spain'
author:
- 'Iago Landesa-Vázquez, José Luis Alba-Castro'
---
AdaBoost ,Classification ,Cost ,Asymmetry ,Boosting
Introduction {#sec:Intro1}
============
The classical approach to solve a classification problem is based on the use of a single expert that must be able to build a solution classifier. However, in the last few decades, a new classification paradigm, based on the combination of several experts in a distributed decision process, has arisen and attracted the attention of the Machine Learning community. The success of this paradigm relies on several theoretical, practical and even biological reasons (such as generalization properties, complexity, data handling, data source fusion, etc.) making these *Ensemble Classifiers* [@Polikar06] preferable to classical ones in many scenarios.
One of the milestones on the history of ensemble methods was the work published by Robert E. Schapire in 1990 [@Schapire90], in which the author proves the equivalence between *weak* learners, algorithms able to generate classifiers performing only slightly better than random guessing, and *strong* learners, those generating classifiers which are correct in all but an arbitrarily small fraction of the instances. This new model of learnability, in which weak learners can be *boosted* to achieve strong performance when they are properly combined, paved the way to one of the most prominent families of algorithms within the ensemble classifiers paradigm: *boosting*.
In 1997, Yoav Freund and Robert E. Schapire [@FreundSchapire97] proposed a more general boosting algorithm called AdaBoost (from Adaptive Boosting). Unlike previous approaches, AdaBoost does not require any prior knowledge on weak hypothesis space, and it iteratively adjusts to weak hypothesis that become part of the ensemble. Apart from theoretical guarantees and practical advantages over its predecessors, early experiments on AdaBoost also showed a surprising resistance to overfitting. As a consequence of all these qualities, AdaBoost has received an attention “rarely matched in computational intelligence” [@Polikar06] being an active research topic in the fields of machine learning, pattern recognition and computer vision [@Schapire98; @SchapireSinger99; @Opitz99; @Friedman00; @MeaseWyner08a; @ViolaJones04; @MasnadiVasconcelos11; @LandesaAlba12] till present.
Throughout this time, several studies have been conducted to analyze AdaBoost from different points of view, relating the algorithm with different theories: margin theory [@Schapire98], entropy [@Kivinen99], game theory [@FreundSchapire96], statistics [@Friedman00], etc. In the same way, numerous AdaBoost and boosting variants have been proposed for the two-class and multiclass problems: Real AdaBoost [@SchapireSinger99; @Friedman00], LogitBoost [@Friedman00], Gentle AdaBoost [@Friedman00], AsymBoost [@ViolaJones02], AdaCost [@Fan99], AdaBoost.M1 [@FreundSchapire96b], AdaBoost.M2 [@FreundSchapire96b], AdaBoost.MH [@SchapireSinger99], AdaBoost.MO [@SchapireSinger99], AdaBoost.MR [@SchapireSinger99], JointBoosting [@Torralba04], AdaBoost.ECC [@GuruswamiSahai99] etc.
Among the different kinds of classification problems, one common subset is that of tasks with clearly different costs depending on each possible decision, or scenarios with very unbalanced class priors in which one class is extremely more frequent or easier to sample than the other one. In such *cost-sensitive* or *asymmetric* conditions (disaster prediction, fraud detection, medical diagnosis, object detection, etc.) classifiers must be able to focus their attention in the rare/most valuable class. Many works in the literature have been devoted to cost-sensitive learning [@Elkan01; @Provost97; @Weiss03], including a significant set of proposals on how to provide AdaBoost with asymmetric properties (e.g. [@Fan99; @Ting00; @ViolaJones04; @ViolaJones02; @Sun07; @MasnadiVasconcelos07; @MasnadiVasconcelos11; @LandesaAlba12; @LandesaAlba13]). The link between AdaBoost and Cost-Sensitive learning has special interest since AdaBoost is the learning algorithm inside the widespread Viola-Jones object detector framework [@ViolaJones04], a seminal work in computer vision dealing with a markedly asymmetric problem and a enormous number of weak classifiers (the order of hundred of thousands).
The different AdaBoost asymmetric variants proposed in the literature are very heterogeneous, and their related works are focused on emphasizing the possible advantages of each respective method, rather than building a common framework to jointly classify, analyze and discuss the different approaches. The final result is that, for the researcher, these algorithms shape a confusing set with no clear theoretical properties to rule their application in practical problems.
In this series of two papers we try to classify, analyze, compare and discuss the different proposals on Cost-Sensitive AdaBoost algorithms, in order to gain a unifying perspective. Our final goal is finding a definitive scheme to directly translate any cost-sensitive learning problem to the AdaBoost framework and shedding light on which algorithm can ensure the best performance.
The current article is focused on the theoretical part of our work and it is organized as follows: next section focuses on standard AdaBoost and its related theoretical framework, Section \[sec:CSvar\] is devoted to cluster and explain, in an homogeneous notational framework, the different cost-sensitive AdaBoost variants proposed in the literature, and in Section \[sec:Discuss\] we analyze in depth those algorithms with a fully theoretical derivation scheme. Finally, we present the preliminary conclusions (Section \[sec:Conclusions1\]) that will be culminated in the accompanying paper “” covering the experimental part of our work.
AdaBoost {#sec:AdaBoost}
========
Let us define $\mathbf{X}$ as the random process from which our observations $\mathbf{x}=\left(x_{1},\ldots,x_{N}\right)^{T}$ are sampled, and $Y$ the random variable governing the related labels $y \in \{-1,1\}$. In this scenario, a *detector* $H(\mathbf{x})$ (we will also refer to it as *classifier* or *hypothesis*) is a function trying to guess the label $y$ of a given sample $\mathbf{x}$, and it can be defined in terms of a more generic function $f\left(\mathbf{x}\right) \in \mathbb{R}$ which we will call *predictor*.
$$\label{pred_eqn}
H(\mathbf{x})=\mathrm{sign}\left[f\left(\mathbf{x}\right)\right]$$
Suppose we have a training set of $n$ examples $\mathbf{x}_i$ with its respective labels $y_i$, a weight distribution $D(i)$ over them and a *weak learner* able to select, according to labels and weights, the best detector $h(\mathbf{x})$ from a predefined collection of weak classifiers. In this scenario, the role of AdaBoost is to compute a goodness measure $\alpha$ depending on the performance obtained by the selected weak classifier, and to update, accordingly, the weight distribution to emphasize misclassified training examples. Then, with a different weight distribution, the weak learner can make a new hypothesis selection and the process restarts. By iteratively repeating this scheme (, , , ) with $t$ indexing the number of learning rounds, AdaBoost obtains an ensemble of weak classifiers with respective goodness parameters $\alpha_{t}$.
$$\label{alphat_eqn}
%\begin{split}
\alpha_{t}=
%\frac{1}{2} \ln \left(\frac{1+\sum D_{t}(i) y_{i} h_{t}(x_{i})}{1-\sum D_{t}(i) y_{i} h_{t}(x_{i})}\right)
\frac{1}{2} \log \left(\frac{1+r_{t}}{1-r_{t}}\right)
%=\frac{1}{2} \ln \left( \frac{1-\epsilon_{t}}{\epsilon_{t}}\right)
%\end{split}$$
$$\label{rt_eqn}
r_{t}=
\sum_{i=1}^{n}D_{t}(i) y_{i} h_{t} (\mathbf{x}_{i})$$
$$\label{weight_rule_eqn}
D_{t+1}(i)=
\frac{D_{t}(i)\exp\left(-\alpha_{t} y_{i} h_{t}(\mathbf{x}_{i})\right)}{Z_{t}}$$
$$\label{zt_eqn}
Z_{t}=
\sum_{i=1}^{n} D_{t}(i) \exp\left(-\alpha_{t} y_{i} h_{t}(\mathbf{x}_{i})\right)$$
Weak hypothesis searching in AdaBoost is guided to maximize goodness $\alpha_{t}$ of each selected classifier, which is equivalent to maximize, at each iteration, weighted correlation $r_{t}$ () between labels $y_{i}$ and predictions $h_{t}$. This iterative searching process can continue until a predefined number $T$ of training rounds have been completed or some performance goal is reached. The final AdaBoost *strong detector* $H(\mathbf{x})$ is defined () in terms of a boosted *predictor* $f(\mathbf{x})$ built as an ensemble of the selected weak classifiers weighted by their respective goodness parameters $\alpha_{t}$.
$$\label{stradb_eqn}
H(\mathbf{x})=\mathrm{sign}\left(f(\mathbf{x})\right)=\mathrm{sign}\left(\sum_{t=1}^{T}\alpha_{t}h_{t}(\mathbf{x})\right)$$
Error Bound Minimization {#subsec:GeneralizedVersion}
------------------------
Robert E. Schapire and Yoram Singer proposed [@SchapireSinger99], from the original derivation of AdaBoost, a generalised and simplified analysis that models the algorithm as an additive (round-by-round) minimization process of an exponential bound on the strong classifier training error ($E_{T}$). This bounding process is explained in equation[^1] () from which all AdaBoost equations we have presented, weight update rule included, can be derived [@LandesaAlba12].
$$\label{bound_ineq_eqn}
{\begin{array}{c}
\underbrace{\strut H(\mathbf{x}_{i})\neq y_{i} \:\Rightarrow\: y_{i} f(\mathbf{x}_{i}) \leq 0 \:\Rightarrow\: \exp\left(-y_{i} f(\mathbf{x}_{i})\right) \geq 1}\\
\Downarrow\\
E_{T}= \sum_{i=1}^{n} D_{1}(i) \llbracket H(\mathbf{x}_{i}) \neq y_{i}\rrbracket \leq \sum_{i=1}^{n} D_{1}(i) \exp \left( -y_{i} f(\mathbf{x}_{i}) \right)
\end{array}
}$$
After (), the final bound of the training error obtained by AdaBoost can be expressed as (), and the additive minimization of the exponential bound $\tilde{E}_{T}$ can be seen as finding, in each round, the weak hypothesis $h_{t}$ that maximizes $r(t)$, the weighted correlation between labels $(y_{i})$ and predictions $(h_{t})$.
$$\label{et_bound_eqn}
E_{T}\leq
\prod_{t=1}^{T}Z_{t}\leq
\prod_{t=1}^{T}\sqrt{1-{r_{t}}^2}=\tilde{E}_{T}$$
When weak hypothesis are binary, $h_{i}\in\{-1,+1\}$, the last inequality on () becomes an equality, and parameter $\alpha_{t}$ can be directly rewritten () in terms of the weighted error $\epsilon_{t}$ of the current weak classifier. As can be seen, the minimization process turns out to be equivalent to simply selecting the weak classifier with less weighted error.
$$\label{round_err_eqn}
\epsilon_{t}=
\sum_{i=1}^{n} D_{t}(i) \llbracket h_{t}(\mathbf{x}_{i}) \neq y_{i}\rrbracket=
\sum_{\textrm{err}}D_{t}(i)$$
$$\label{alphat2_eqn}
%\begin{split}
\alpha_{t}=
\frac{1}{2} \log \left( \frac{1-\epsilon_{t}}{\epsilon_{t}}\right)
%\end{split}$$
In line with other works, for the sake of simplicity and clarity, we will focus our analysis on this *Discrete* version of AdaBoost using binary weak classifiers, which does not prevent our conclusions from being extended to other variations of the algorithm. Also, trying to define an homogeneous notational framework for our work, we have unified the different notations found in the literature to that used by Schapire and Singer [@SchapireSinger99]. A summary of AdaBoost can be found on Algorithm \[adb\_algorithm\] (all the algorithms discussed in this paper are detailed, with homogeneous notation, in Appendix \[app:algorithms\]).
Statistical View of Boosting {#subsec:Statistical View}
----------------------------
One of the milestones in boosting research and the foundation of many variations of AdaBoost is the highly-cited contribution by Jerome H. Friedman et al. [@Friedman00] in which a statistical reinterpretation of boosting is given. Following the exponential criterion seen in the last subsection, Friedman et al. showed that AdaBoost can be motivated as an iterative algorithm building an additive logistic regression model $f(\mathbf{x})$ that minimizes the expectation of the exponential loss, $J(f(\mathbf{x}))$:
$$\label{loss_eqn}
J(f(\mathbf{x}))=\E\left[\exp(-yf(\mathbf{x}))\right]$$
This defined loss is effectively minimized at $$\label{regress_eqn}
f(\mathbf{x})=\frac{1}{2}\log\left(\frac{\Prob(y=1|\mathbf{x})}{\Prob(y=-1|\mathbf{x})}\right)$$ so a direct connection between boosting and additive logistic regression models is drawn. According to this statistical perspective, AdaBoost predictions can be seen as estimations of the posterior class probabilities, which has served as basis to develop many extensions and variants of the algorithm (among them, the Cost-Sensitive Boosting scheme [@MasnadiVasconcelos07; @MasnadiVasconcelos11]).
It is important to mention that, despite the huge and unquestionable value of the statistical view, some enriching controversy, revealed by empirical evidences [@MeaseWyner08a; @Bennet08; @MeaseWyner08b], has arisen about inconsistencies of this interpretation.
Cost-Sensitive Variants of AdaBoost {#sec:CSvar}
===================================
Cost-sensitive classification problems can be fully portrayed by a cost matrix [@Elkan01] whose components map the loss of each possible result. For two-class problems there are four kinds of results: true positives, true negatives, false positives and false negatives; so the cost matrix $\mathbf{C}$ can be defined as follows:
$$\label{cost_matrix}
\begin{tabular} {c c c c c c l l l}
& & \multicolumn{2}{c}{Actual} & & & \\
& & Negative & Positive & & &\\
\multirow{2}{*}{$\mathbf{C}=$} & \multirow{2}{*}{{\Huge(}} & $c_{nn}$ & $c_{np}$ & \multirow{2}{*}{{\Huge)}} & Negative & \multirow{2}{*}{Classified}\\
& & $c_{pn}$ & $c_{pp}$ & & Positive & &
\end{tabular}$$
The optimal decision for a given cost matrix will not change if all its coefficients are added a constant, or if they are multiplied by a constant positive factor. As a result, a cost matrix for two-class classification problems only has two degrees of freedom and can be parametrized by only two coefficients: false negatives normalized cost ($\overline{c}_{np}$) and true positives normalized cost ($\overline{c}_{pp}$):
$$\label{simple_C}
\mathbf{C}=\left(
\begin{array} {c c}
0 & \overline{c}_{np}\\
1 & \overline{c}_{pp}\\
\end{array}
\right)$$
In the most common case correct decisions have null related costs ($c_{nn}=c_{pp}=0$), so $\mathbf{C}$ has eventually only one degree of freedom: the ratio between cost of errors on positives ($c_{np}$) and cost of errors on negatives ($c_{pn}$). In the literature and most practical problems, cost requirements are usually specified by these two error parameters, which, for simplicity, we will denote as $C_P$ and $C_N$ respectively.
$$\label{simple_C2}
\mathbf{C}=\left(
\begin{array} {c c}
0 & C_P/C_N\\
1 & 0\\
\end{array}
\right)
\rightarrow \left(
\begin{array} {c c}
0 & C_P\\
C_N & 0\\
\end{array}
\right)$$
The coefficients of a cost matrix may not be constant in general. While constant coefficients model a scenario where all the examples of each class have the same cost (class-level asymmetry), variable coefficients mean that examples belonging to the same class can have different costs (example-level asymmetry). Whatever the scenario, it is also important to notice that, for “reasonableness” [@Elkan01], correct predictions in a cost matrix should have lower associated costs than mistaken ones ($c_{nn}<c_{np}$ and $c_{pp}<c_{pn}$).
Bearing in mind that class-level asymmetry is the most common for detection problems, and that example-level asymmetry can be modeled by a class-level asymmetry scheme with a resampled training dataset, for our analysis we have homogenized the different Asymmetric AdaBoost approaches to the class-level scheme. Thus, we will follow a prototypical cost-sensitive detection statement specified by two constant coefficients $C_P$ and $C_N$, that can be alternatively described by the “normalized cost asymmetry” of the problem $\gamma \in (0,1)$:
$$\gamma=\frac{C_{P}}{C_{P}+C_{N}}$$
Despite the widespread use of these particularizations, in Appendix \[app:cost\_scen\] we will extend our conclusions to example-level asymmetry and also cases in which correct classification costs are nonzero.
It is also important to emphasize that this work is focused on AdaBoost and its cost-sensitive variants, a realm of methods in the literature that are based on a exponential loss minimization criterion, analogous to that giving rise to the original algorithm (as we have seen in Sections \[subsec:GeneralizedVersion\] and \[subsec:Statistical View\] from different points of view) . Other boosting algorithms based on other kinds of losses beyond the exponential paradigm, like the binomial log-likelihood [@Friedman00] or the p-norm loss [@LozanoAbe08], are outside the scope of the current study.
Classification {#subsec:Classification}
--------------
In order to give a clear overview of the cost-sensitive variants of AdaBoost proposed in the literature, we suggest an analytical classification scheme to cluster them into three categories according to the way asymmetry is reached: *A posteriori*, *Heuristic* and *Theoretical*.
### A Posteriori {#subsec:APosteriori}
The seminal face/object detector framework by Paul Viola and Michael J. Jones [@ViolaJones04] uses a validation set to modify, after training, the threshold of the original (cost-insensitive) AdaBoost strong classifier. The goal is to adjust the balance between false positive and detection rates, building, that way, a cost-sensitive boosted classifier:
$$\label{vjmod_eqn}
\tilde{H}(\mathbf{x})=\mathrm{sign}\left(f(\mathbf{x})-\phi\right)=\mathrm{sign}\left({\sum_{t=1}^{T}\alpha_{t}h_{t}(\mathbf{x})}-\phi\right)$$
Besides the great success of the detection framework, the authors themselves acknowledge that neither this a posteriori cost-sensitive tuning ensures that the selected weak classifiers are optimal for the asymmetric goal [@ViolaJones02], nor their modifications preserve the original AdaBoost training and generalization guarantees [@ViolaJones04].
An useful insight on this can be drawn from the analysis by Masnadi-Shirazi and Vasconcelos [@MasnadiVasconcelos11]. According to the Bayes Decision Rule, the optimal predictor $f^{*}(\mathbf{x})$ can be expressed in terms of the optimal predictor for a cost-insensitive scenario $f^{*}_{0}(\mathbf{x})$ and a threshold $\phi$ depending on costs.
$$\label{pred_thresh}
f^{*}\left(\mathbf{x}\right)=\log\left(\frac{\Prob_{Y|\mathbf{X}}\left(1|\mathbf{x}\right)C_{P}}{\Prob_{Y|\mathbf{X}}\left(-1|\mathbf{x}\right)C_{N}}\right)=\log\left(\frac{\Prob_{Y|\mathbf{X}}\left(1|\mathbf{x}\right)}{\Prob_{Y|\mathbf{X}}\left(-1|\mathbf{x}\right)}\right)-\log\left(\frac{C_N}{C_P}\right)=f^*_0(\mathbf{x})-\phi$$
As a consequence, for any cost requirements, the optimal cost-sensitive *detector* $H^{*}(\mathbf{x})$ can also be expressed as a threshold on the cost-insensitive optimal *predictor* $f^{*}_{0}(\mathbf{x})$.
$$\label{optimal_detector}
H^*\left(\mathbf{x}\right)=\mathrm{sign}\left[f^*(\mathbf{x})\right]=\mathrm{sign}\left[f^*_0(\mathbf{x})-\phi\right]$$
In practical terms, however, learning algorithms do not have access to the exact probability distributions and they must approximate this optimal detector rule. Thus, AdaBoost can be seen as an algorithm obtaining an approximation ($\hat{H}_0(\mathbf{x})$) to the optimal cost-insensitive *detector*, built by means of an estimation ($\hat{f}_0(\mathbf{x})$) of the cost-insensitive *predictor* ().
$$\label{adaboost_estimation}
\hat{H}_0(\mathbf{x})=\mathrm{sign}\left[\hat{f}_0(\mathbf{x})\right]=\mathrm{sign}\left({\sum_{t=1}^{T}\alpha_{t}h_{t}(\mathbf{x})}\right) \approx H^*_0(\mathbf{x})$$
By definition, the purpose of AdaBoost is to obtain a *detector* as close as possible to the optimal one, and this optimality is ensured if the learned *predictor* satisfies two necessary and suficient conditions:
$$\label{costinsens_cond}
{\begin{array}{c}
\hat{H}_0(\mathbf{x})=H_0^*(\mathbf{x}) \\
\Updownarrow \\
\begin{cases}
\hat{f}_0\left(\mathbf{x}\right)=f_0^*(\mathbf{x})=0 & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x})=\Prob_{Y|\mathbf{X}}(-1|\mathbf{x})\\
\mathrm{sign}\left[\hat{f}_0\left(\mathbf{x}\right)\right]=\mathrm{sign}\left[f_0^*(\mathbf{x})\right] & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x}) \neq \Prob_{Y|\mathbf{X}}(-1|\mathbf{x})
\end{cases}
\end{array}
}$$
As can be seen, in order to reach optimal *detection* the predictor learned by AdaBoost should match the optimal predictor in the boundary region, but only its sign elsewhere. Analogously, optimal detection for the cost-sensitive case, would be ensured by two equivalent conditions:
$$\label{costsens_cond}
{\begin{array}{c}
\hat{H}(\mathbf{x})=H^*(\mathbf{x}) \\
\Updownarrow \\
\begin{cases}
\hat{f}\left(\mathbf{x}\right)=f^*(\mathbf{x})=0 & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x})C_P=\Prob_{Y|\mathbf{X}}(-1|\mathbf{x})C_N\\
\mathrm{sign}\left[\hat{f}\left(\mathbf{x}\right)\right]=\mathrm{sign}\left[f^*(\mathbf{x})\right] & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x})C_P \neq \Prob_{Y|\mathbf{X}}(-1|\mathbf{x})C_N
\end{cases}
\end{array}
}$$
Thus, optimality conditions required by the *a posteriori* modification of the AdaBoost threshold would be as follows:
$$\label{threshmod_cond}
{\begin{array}{c}
\hat{H}(\mathbf{x})=H^*(\mathbf{x}) \\
\Updownarrow \\
\begin{cases}
\hat{f}_0\left(\mathbf{x}\right)=f_0^*(\mathbf{x})=\phi & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x})C_P=\Prob_{Y|\mathbf{X}}(-1|\mathbf{x})C_N\\
\mathrm{sign}\left[\hat{f}_0\left(\mathbf{x}\right)-\phi\right]=\mathrm{sign}\left[f_0^*(\mathbf{x})-\phi\right] & \text{if } \Prob_{Y|\mathbf{X}}(1|\mathbf{x})C_P \neq \Prob_{Y|\mathbf{X}}(-1|\mathbf{x})C_N
\end{cases}
\end{array}
}$$
Bearing in mind that AdaBoost predictor $\hat{f}_0(\mathbf{x})$ is geared to satisfy (), the optimality conditions for *threshold modification* are not necessarily fulfilled. The only way to meet these requirements for any cost would be that the predictor obtained by AdaBoost matched the optimal one along the whole space, which is an obviously stronger condition than actually required (). Moreover, recalling the exponential bounding equation in which AdaBoost is based (), we can see that, once the sign of the obtained predictor matches the right label, the error bound is further minimized for increasing absolute values of the estimated predictor, no matter how close they are (or not) to the optimal predictor value.
As a consequence, there are no guarantees that a threshold change on the classical AdaBoost predictor will give us a cost-sensitive detector oriented to be optimal. Nonetheless, this non-optimality has not prevented that asymmetric detectors obtained by the Viola-Jones framework have been very successfully used for object detection.
### Heuristic {#subsec:Heuristic}
Most of the proposed cost-sensitive variations of AdaBoost [@Fan99; @Ting00; @ViolaJones02; @Sun07] try to deal with asymmetry through direct manipulations of the weight update rule (), but they are not full reformulations of AdaBoost for cost-sensitive scenarios. Masnadi-Shirazi and Vasconcelos pointed out that this kind of manipulations “provide no guarantees of asymptotic convergence to a good cost-sensitive decision rule” [@MasnadiVasconcelos11], considering those algorithms as “heuristic” modifications of AdaBoost [@MasnadiVasconcelos07; @MasnadiVasconcelos11].
Although these proposals have, in greater or lesser extent, some theoretical basis, for the sake of clarity and distinctiveness in our analysis, we will maintain the term *heuristic*, as used in [@MasnadiVasconcelos07; @MasnadiVasconcelos11], to label this group of approaches based on the arbitrary modification of the weight update rule, as opposed to the full *theoretical* derivations we will delve into in the next subsection.
### AsymBoost {#subsubsec:AsymBoost .unnumbered}
Assuming the non-optimality of the strong classifier threshold adjustment procedure in their object detector framework (Section \[subsec:APosteriori\]), Paul Viola and Michael J. Jones proposed a different scheme, coined as AsymBoost [@ViolaJones02], trying to optimize AdaBoost for cost-sensitive classification problems.
Discarding the asymmetric weight initialization to be “naive” and only “somewhat effective” due to “AdaBoost’s balanced reweighting scheme” (we will discuss on this point in Section \[subsec:weight\]), AsymBoost proposes to distributedly emphasize weights by an asymmetric modulation before each round. In practical terms, the only change is multiplying weights $D(i)$ by a constant factor $(C_P/C_N)^{y_i/2T}$ before every learning step of a $T$-round process. As a consequence, the overall asymmetric factor seen by positive elements across the whole process is $C_P/C_N$ times the factor seen by negatives.
$$\label{asb_equation}
D(i)_{t+1} = \frac{D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)\left(\frac{C_P}{C_N}\right)^{\frac{y_{i}}{2T}}}{\sum_{i=1}^{n}D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)\left(\frac{C_P}{C_N}\right)^{\frac{y_{i}}{2T}}}$$
AsymBoost, that reduces to AdaBoost when costs are uniform, is detailed in Algorithm \[asb\_algorithm\] (Appendix \[app:algorithms\]).
Though the global AsymBoost procedure seems to be theoretically sound, the *equitable* asymmetry sharing among a *fixed* number of rounds entails significant problems: Why such a rigid equitable sharing procedure should be optimal inserted in an adaptive framework such as AdaBoost? Why should we have to know in advance the number of training rounds while standard AdaBoost does not require that? Note that standard AdaBoost allows flexible performance tests to decide when to stop training, since any change in the total number of rounds is directly performed by training new additional rounds or trimming the current ensemble. However, a change in the size of the final ensemble (number of rounds) would strictly require Asymboost to re-train the whole classifier with a new asymmetry distribution.
### AdaCost {#adacost .unnumbered}
Wei Fan et al. proposed [@Fan99] a cost-sensitive variation of AdaBoost called AdaCost. The idea behind AdaCost is to modify the weight update rule, so examples with higher costs have sharper increases of their weights after misclassification but lighter decreases when are succesfully classified. This scheme is essentially addresed by introducing a misclassification adjustment function $\beta(i)$ into the weight update rule ().
$$\label{adc_weights}
D_{t+1}(i) = \frac{D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\beta(i)\right)}{\sum_{i=1}^{n}D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\beta(i)\right)}$$
The misclassification adjustment function must depend on the cost ($C(i)$) of each example/class and the success/fail of its classification. As a result, $\beta(i)$ is imposed to be non-decreasing respect to $C(i)$ when classification fails, and non-increasing when classification succeeds. This opens the door to a huge amount of functions satisfying such requirements, from which authors chose the next:
$$\beta(i)=
\left\{
\begin{array}{ll}
0.5 \left(1-C(i)\right) & \mbox{$\text{if } h_{f}(\mathbf{x}_{i}) = y_{i}$},\\
0.5 \left(1+C(i)\right) & \mbox{$\text{if } h_{f}(\mathbf{x}_{i}) \neq y_{i}$}.
\end{array} \right.$$
As can be seen, AdaCost does not match with AdaBoost for uniform costs and also applies a cost-dependent weight pre-emphasis (see Algorithm \[adc\_algorithm\]).
### CSB0, CSB1 and CSB2 {#csb0-csb1-and-csb2 .unnumbered}
Following the same idea of modifying the weight update rule, the CSB (acronym from Cost-Sensitive Boosting) family of algorithms [@Ting98; @Ting00] propose three different updating schemes depending on which parameters are involved, resulting in CSB0, CSB1 and CSB2 algorithms (see respective Algorithms \[csb0\_algorithm\], \[csb1\_algorithm\] and \[csb2\_algorithm\]). These rules are complemented, for all the three alternatives, by an asymmetric weight initialization and a minimum expected cost criterion for strong classification replacing the usual weighted voting scheme:
$$H(\mathbf{x})=\mathrm{sign}\left(\sum_{t=1}^{T}\alpha_{t}h_{t}(\mathbf{x}) \left( C_P \llbracket h_{t}(\mathbf{x})=+1 \rrbracket + C_N \llbracket h_{t}(\mathbf{x})=-1 \rrbracket \right)\right)$$
This new voting rule gives emphasis, in run time, to weak hypothesis deciding in favor of the costly class. Of the three alternatives, only the last one, CSB2, is reduced to standard AdaBoost when costs are equal.
### AdaC1, AdaC2 and AdaC3 {#adac1-adac2-and-adac3 .unnumbered}
Defining new ways to modify the weight update rule, Yanmin Sun et al. [@Sun05; @Sun07] proposed another family of asymmetric AdaBoost alternatives called AdaC1, AdaC2 and AdaC3. These variants couple the cost factor in different parts of the update equation: inside the exponent (AdaC1), outside the exponent (AdaC2) and both (AdaC3):
$$\label{ac1_update}
D_{t+1}(i) = \frac{D_{t}(i)\exp\left(-\alpha_{t}c_{i}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}{\sum_{i=1}^{n}D_{t}(i)\exp\left(-\alpha_{t}c_{i}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}$$
$$\label{ac2_update}
D_{t+1}(i) = \frac{c_{i}D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}{\sum_{i=1}^{n}c_{i}D_{t}(i)\exp\left(-\alpha_{t}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}$$
$$\label{ac3_update}
D_{t+1}(i) = \frac{c_{i}D_{t}(i)\exp\left(-\alpha_{t}c_{i}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}{\sum_{i=1}^{n}c_{i}D_{t}(i)\exp\left(-\alpha_{t}c_{i}y_{i}h_{t}\left(\mathbf{x}_i\right)\right)}$$
As a difference from previous approaches, these changes in the weight update are also propagated to the way goodness parameter $\alpha_t$ is defined and, as a consequence, have influence on how the weak classifier error is computed (see Algorithms \[ac1\_algorithm\], \[ac2\_algorithm\], \[ac3\_algorithm\]). All these variants reduce to AdaBoost when the cost function $C(i)$ is 1 for all examples.
### Theoretical {#subsec:Theoretical}
The methods in the previous subsection have one key point in common: the starting point of their derivations is an arbitrary modification of the weight update rule. However, as can be easily shown following the work by Schapire and Singer [@SchapireSinger99], weight update in standard AdaBoost is actually a *consequence* of the error minimization procedure () and not an arbitrary starting point of it. Thus, the way to reach theoretically sound cost-sensitive boosting algorithms should be to walk the path in the opposite direction: designing a new asymmetric derivation scheme to obtain a new full formulation (that may include a new weight update rule), instead of partially adapting previous equations.
There are three alternatives in the literature that follow different theoretically sound derivation schemes reaching cost-sensitive variants of AdaBoost: Cost-Sensitive AdaBoost [@MasnadiVasconcelos07; @MasnadiVasconcelos11], AdaBoostDB [@LandesaAlba13] and Cost-Generalized AdaBoost [@LandesaAlba12].
### Cost-Sensitive AdaBoost {#cost-sensitive-adaboost .unnumbered}
The Cost-Sensitive Boosting framework proposed by Hamed Masnadi-Shirazi and Nuno Vasconcelos [@MasnadiVasconcelos07; @MasnadiVasconcelos11] has its roots in the Statistical View of Boosting [@Friedman00], by adapting the standard loss in equation () with asymmetric exponential arguments for each class component.
$$\label{csloss_eqn}
J(f(\mathbf{x}))=\E\left[\llbracket y=1 \rrbracket \mathrm{e}^{-C_{P}f(\mathbf{x_i})} + \llbracket y=-1 \rrbracket \mathrm{e}^{C_{N}f(\mathbf{x_i})}\right]$$
This asymmetric loss is theoretically minimized by the asymmetric logistic transform of $\Prob\left(y=1|\mathbf{x}\right)$ (see Section \[subsec:Statistical View\]), which should ensure cost-sensitive optimality.
$$\label{stat_sol_asym}
\begin{split}
f(x)=\frac{1}{C_{P}+C_{N}}\log\frac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}
\end{split}$$
The empirical minimization of the asymmetric loss proposed by Masnadi-Shirazi and Vasconcelos follows a gradient descent scheme on the space of boosted (combined and modulated) binary weak classifiers, resulting in the Cost-Sensitive Adaboost algorithm shown in Algorithm \[csa\_algorithm\]. As can be seen, the final solution involves hyperbolic functions and scalar search procedures, being extremely more complex and computing demanding than the original AdaBoost.
### AdaBoostDB {#adaboostdb .unnumbered}
Following the generalizad analysis of AdaBoost [@SchapireSinger99] instead of the Statistical View of Boosting, a different approach to provide AdaBoost with Cost-Sensitive properties through a fully theoretical derivation procedure is presented in [@LandesaAlba13]. This algorithm, coined as AdaBoostDB (from Double Base), is based on the use of different exponential bases $\beta_P$ and $\beta_N$ for each class error component, thus defining a class-dependent error bound to minimize.
$$\label{exp_bound_eqn_asym}
%\begin{split}
E_{T} \leq \tilde{E}_T = \sum_{i=1}^{m} D_{1}(i){\beta_P}^{ -y_{i} f(\mathbf{x_{i}})} + \sum_{i=m+1}^{n} D_{1}(i){\beta_N}^{ -y_{i} f(\mathbf{x_{i}})}
%\end{split}$$
On the one hand, the derivation scheme followed and the polynomial model used to address the problem, enable a different and extremely efficient formulation, able to achieve over 99% training time saving with respect to Cost-Sensitive AdaBoost (see Algorithm \[abdb\_algorithm\]). On the other hand, this class-dependent error is fully equivalent to the cost-sensitive loss () defined for Cost-Sensitive Boosting, so both minimizations will converge to the same solution and ensure the same formal guarantees.
As a result, AdaBoostDB is a much more efficient framework to reach the same solution as Cost-Sensitive Boosting (except for numerical errors related to the different models adopted, hyperbolyc vs. polynomial). However, despite its large improvement in training complexity and performance, AdaBoostDB is still much more complex than standard AdaBoost.
### Cost-Generalized AdaBoost {#cost-generalized-adaboost .unnumbered}
The Asymmetric AdaBoost problem is addressed in [@LandesaAlba12] from a different theoretical perspective, realizing that one kind of modification have systematically been either overlooked or undervalued in the related literature: weight initialization.
Even though some preliminary studies by Freund and Schapire [@FreundSchapire97], creators of AdaBoost, left the initial weight distribution free to be controlled by the learner, AdaBoost is “de facto” defined, almost everywhere in the literature (e.g. [@Schapire98; @SchapireSinger99; @Fan99; @FreundSchapire99; @Ting00; @Friedman00; @Polikar06; @Sun07; @Polikar07; @MasnadiVasconcelos11]), with a fixed initial uniform weight distribution. From there, some asymmetric boosting algorithms (like AdaCost or CSB) use cost-sensitive initialization as a lateral or secondary strategy respect to their proposed weight update rules, while others (like AsymBoost or Cost-Sensitive Boosting), immediately discard asymmetric weight initialization to be “naive” and ineffective, arguing that the first boosting round would absorb the full introduced asymmetry and the rest of the process would keep entirely symmetric.
In [@LandesaAlba12], following a different insight to analyze AdaBoost and obtaining a novel error bound interpretation, asymmetric weight initialization is shown to be an effective way to reach cost-sensitiveness, and, as occurs with everything related to boosting, it is achieved in an additive round-by-round (asymptotic) way. All, with the added advantage that weight initialization is the only needed change to gain asymmetry with regard to standard AdaBoost (even weight update rule is unchanged). Hence, for whatever desired asymmetry, both complexity and formal guarantees of the original AdaBoost remain intact.
In this work, we will refer to the algorithm underlying this perspective as Cost-Generalized AdaBoost (see Algorithm \[adbg\_algorithm\]).
Theoretical Algorithms: Analysis and Discussion {#sec:Discuss}
===============================================
Though in the experimental part of our work (see the accompanying paper [@LandesaAlba??b]) we will show comparative results of all the alternatives presented in the previous section, at this point we will focus our attention on the three proposals with a fully theoretical derivation scheme: Cost-Sensitive AdaBoost [@MasnadiVasconcelos07; @MasnadiVasconcelos11], AdaBoostDB [@LandesaAlba13] and Cost-Generalized AdaBoost [@LandesaAlba12]. The first important aspect we should notice is that these three proposals can be effectively analyzed as if they were only two, since Cost-Sensitive AdaBoost and AdaBoostDB, despite following different perspectives and obtaining markedly different algorithms, share an equivalent theoretical root and drive to the same solution [@LandesaAlba13]. As a consequence, if not otherwise specified, in this section we will refer to one or another interchangeably, giving priority to the name Cost-Sensitive AdaBoost due to its chronological precedence.
The Question of Weight Initialization {#subsec:weight}
-------------------------------------
As commented in Section \[subsec:Theoretical\], despite some initial studies pointing to free initial weight distributions [@FreundSchapire97] or works proposing cost-proportional weighting as an effective way to transform generic cost-insensitive learning algorithms into cost-sensitive ones [@Zadrozny03], subsequent works on boosting have insisted on two recurrent ideas: On the one hand, uniform distribution has been assumed as the “de facto” standard for weight initialization when defining AdaBoost (e.g. [@Schapire98; @SchapireSinger99; @Fan99; @FreundSchapire99; @Ting00; @Friedman00; @Polikar06; @Sun07; @Polikar07; @MasnadiVasconcelos11]); on the other hand, asymmetric weight initialization has been systematically rejected as a valid method to achieve cost-sensitive boosted classifiers, arguing that it is insufficient [@Fan99; @Ting00] or ineffective [@ViolaJones02; @MasnadiVasconcelos07; @MasnadiVasconcelos11].
However, in [@LandesaAlba12], AdaBoost is demonstrated to have inherent and sound cost-sensitive properties embedded in the way the weight distribution is initialized. In fact, the method we are referring to as Cost-Generalized AdaBoost, was not even originally proposed as a new algorithm: “it is just AdaBoost” [@LandesaAlba12] with appropriate initial weights. Such an analysis, supported by a novel class-conditional interpretation of AdaBoost, is, thus, in clear contradiction to the supposed ineffectiveness of cost-sensitive weight initialization underlying previous works.
In order to definitely clarify this contradiction, we will connect both perspectives by demonstrating the validity of asymmetric weight initialization in the same scenarios and lines of reasoning that have been previously used in the literature to decline its use.
### The Supposed Symmetry {#subsubsec:supposed_symmetry}
Masnadi-Shirazi and Vasconcelos [@MasnadiVasconcelos11] when explaining their Cost-Sensitive Boosting framework, immediately discard the unbalanced weight initialization (calling it “naive implementation”) with the argument that iterative weight update in AdaBoost “quickly destroys the initial asymmetry” obtaining a “predictor” which “is usually not different from that produced with symmetric initial conditions”. Though their statement is not explicitly supported for any further test or bibliographic reference, it seems to be extracted from the work by Viola and Jones [@ViolaJones02] in which AsymBoost is presented. In that work, the initial weight modification technique is rejected arguing that “the first classifier selected absorbs the entire effect of the initial asymmetric weights”, and assuming the rest of the process as “entirely symmetric”. It is because of this seeming problem that AsymBoost was designed for distributing an equitable asymmetry among a fixed number of rounds.
The cost-sensitive analysis by Viola and Jones [@ViolaJones02] is illustrated by a four-round boosted classifier graphic representation that supports their conclusions against asymmetric weight initialization. However, this example can be misleading: what would happen if boosting were run for more than those four rounds? An answer can be found in Figure \[counter\_example\_1\_fig\], where we have reproduced and extended that illustrative experiment.
![Synthetic counterexample to the example by Viola and Jones [@ViolaJones02], with costs $C_P=4$ and $C_N=1$, and the same polarity as the original:(a) training set with the first four weak classifiers superimposed; (b) weak classifiers after 50 training rounds; (c) Global error evolution through 50 training rounds. Weak classifiers are stumps in the linear 2D space. Positive examples are marked as ‘$+$’, ‘$\circ$’ are the negative ones, and ‘1’ denotes the first selected weak classifier. Positives are the costly class.[]{data-label="counter_example_1_fig"}](Fig01.pdf){width="0.8\columnwidth"}
Strictly following Viola and Jones [@ViolaJones02], after Figure \[counter\_example\_1\_fig\]a we could reach the seeming conclusion that, once an initial asymmetric weak classifier has been selected, the selection of the remaining weak classifiers is not guided by an asymmetric goal. However, as showed by Schapire and Singer [@SchapireSinger99], AdaBoost is an additive minimization process and, as such, it has an *asymptotic* behavior, a kind of behavior that can not be properly judged by stopping after only a few training rounds. Running the algorithm for many more rounds in the same example (see Figure \[counter\_example\_1\_fig\]b), we appreciate that many other subsequent selected classifiers are, at least, as asymmetric as the first one.
The class-conditional interpretation of AdaBoost in [@LandesaAlba12] shows that the asymmetry encoded by the initial weight distribution is actually translated to a cost-sensitive global error (a weighted error), and what AdaBoost is actually minimizing is a bound on that global error. Thus, instead of inspecting the individual asymmetry of each single hypothesis, the cost-sensitive behavior of AdaBoost should be evaluated, for correctness, in terms of the *cumulative contribution* of *all* the selected weak classifiers giving rise to the strong one. Figure \[counter\_example\_1\_fig\]c shows how, even in a scenario like the one proposed by Viola and Jones [@ViolaJones02], the classifier obtained by AdaBoost after an asymmetric weight initialization follows a real cost-sensitive iterative profile.
Moreover, postulates by Viola and Jones [@ViolaJones02] and Masnadi-Shirazi and Vasconcelos [@MasnadiVasconcelos11] can also be refuted by simply inverting labels on the same set (see Figure \[counter\_example\_2\_fig\]). As can be seen, no weak classifier is able to satisfy, by itself, the requirements of that “supposed” initial round absorbing the full asymmetry of the problem. However, even in such an unfavorable scenario, the desired asymmetry is effectively achieved, from cost-proportionate initial weights, after a (boosted) round-by-round cumulative process.
![Synthetic counterexample to the example by Viola and Jones [@ViolaJones02], with costs $C_P=4$ and $C_N=1$, and with opposite polarity to the original:(a) training set with the first four weak classifiers superimposed;(b) weak classifiers after 50 training rounds; (c) Global error evolution through 50 training rounds. Weak classifiers are stumps in the linear 2D space. Positive examples are marked as ‘$+$’, ‘$\circ$’ are the negative ones, and ‘1’ denotes the first selected weak classifier. Positives are the costly class.[]{data-label="counter_example_2_fig"}](Fig02.pdf){width="0.8\columnwidth"}
Further comments on these experiments can be found in Appendix \[app:comments\_figures\].
### Weight Initialization inside the Cost-Sensitive Boosting Framework {#subsubsec:weight_cost_sensitive}
Cost-Sensitive AdaBoost [@MasnadiVasconcelos11] is an algorithm that, despite having a rigorous theoretical derivation, is built upon the belief that cost-sensitive initial weighting is not a valid method to achieve asymmetric boosted classifiers. However, as we have already mentioned, the theoretical analysis in [@LandesaAlba12] refutes that supposed invalidity. A clarifying experiment at this point is to introduce asymmetric weight initialization inside the Cost-Sensitive AdaBoost theoretical framework, to assess the theoretical validity of the former with the tools used by the latter.
Based on the Statistical View of Boosting [@Friedman00], the cost-sensitive expected loss (i.e. the risk function) proposed by Masnadi-Shirazi and Vasconcelos to derive Cost-Sensitive AdaBoost, consists on two class-dependent exponential components with asymmetry embedded in its exponents:
$$\label{csloss_eqn1}
J_{CSA}(f(\mathbf{x}))=\E\left[\llbracket y=1 \rrbracket \exp(-C_{P}f(\mathbf{x})) + \llbracket y=-1 \rrbracket \exp(C_{N}f(\mathbf{x}))\right]$$
Following the proof derivation scheme in [@MasnadiVasconcelos11], if the derivatives of this loss are set to zero, we will obtain the function of minimum expected loss (minimum risk) conditioned on $\mathbf{x}$ for Cost-Sensitive AdaBoost, that, as can be seen, is based on the asymmetric logistic transform of $\Prob(y=1|\mathbf{x})$.
$$\label{statistical_deriv_cda}
{\begin{array}{c}
J_{CSA}(f(\mathbf{x}))=\Prob\left(y=1|\mathbf{x}\right)\exp(-C_{P}f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)\exp(C_{N}f(\mathbf{x}))\\
\Downarrow\\
\dfrac{\partial J_{CSA}(f(\mathbf{x}))}{\partial f(\mathbf{x})}= -C_{P}\Prob\left(y=1|\mathbf{x}\right)\exp(-C_{P}f(\mathbf{x}))+C_{N}\Prob\left(y=-1|\mathbf{x}\right)\exp(C_{N}f(\mathbf{x}))=0\\
\Downarrow\\
\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}=\exp((C_{P}+C_{N})f(\mathbf{x}))\\
\Downarrow\\
f_{CGA}(\mathbf{x})=\dfrac{1}{C_{P}+C_{N}}\log\left(\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}\right)
\end{array}
}$$
Now, let us suppose that the two cost parameters $C_P$ and $C_N$, rather than in the exponents, are incorporated as direct modulators of the exponentials (). This procedure is equivalent to model the initial weight distribution by means of two uniform *class-conditional* distributions, respectively modulated by $C_{P}/\left(C_{P}+C_{N}\right)$ and $C_{N}/\left(C_{P}+C_{N}\right)$, i.e. an asymmetric weight initialization as the one proposed giving rise to Cost-Generalized AdaBoost.
$$\label{csloss_exp}
J_{CGA}(f(\mathbf{x}))=\E\left[\llbracket y=1 \rrbracket C_{P} \exp(-f(\mathbf{x})) + \llbracket y=-1 \rrbracket C_{N} \exp(f(\mathbf{x}))\right]$$
If we repeat the above derivation scheme on this new loss, we will find the function of minimum expected loss (minimum risk) conditioned on $\mathbf{x}$ for Cost-Generalized AdaBoost:
$$\label{statistical_deriv_cga}
{\begin{array}{c}
J_{CGA}(f(\mathbf{x}))=\Prob\left(y=1|\mathbf{x}\right)C_{P}\exp(-f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)C_{N}\exp(f(\mathbf{x}))\\
\Downarrow\\
\dfrac{\partial J_{CGA}(f(\mathbf{x}))}{\partial f(\mathbf{x})}= -\Prob\left(y=1|\mathbf{x}\right)C_{P}\exp(-f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)C_{N}\exp(f(\mathbf{x}))=0\\
\Downarrow\\
\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}=\exp(2f(\mathbf{x}))\\
\Downarrow\\
f_{CGA}(\mathbf{x})=\dfrac{1}{2}\log\left(\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}\right)
\end{array}
}$$
As can be seen, the obtained minimizer is also based on the asymmetric logistic transform of $\Prob(y=1|\mathbf{x})$, showing us that, even from the Cost-Sensitive AdaBoost derivation perspective, there is no reason to discard asymmetric weight initialization as a valid approach to build cost-sensitive boosted classifiers[^2].
Comparative Analysis of the Theoretical Approaches {#subsec:algorithms_cmp}
--------------------------------------------------
As we have seen, among the three asymmetric AdaBoost algorithms with a full theoretical derivation, two of them (Cost-Sensitive AdaBoost and AdaBoostDB) drive to the same solution, while the other one (Cost-Generalized AdaBoost) has been shown to guarantee the same theoretical validity than its counterparts. At this point, we may wonder if Cost-Generalized AdaBoost is also obtaining the same solution as Cost-Sensitive AdaBoost/AdaBoostDB. As we will see in the experimental part of our work (in the second part of this series of two papers [@LandesaAlba??b]) the answer to this question is “no”: classifiers obtained by Cost-Sensitive AdaBoost and Cost-Generalized AdaBoost in the same scenarios are markedly different. In this section, from a theoretical perspective, we will analyze the differences between the two algorithms, with the aim of achieving the intrinsic distinctivenesses of their respective classifiers.
### Error Bound Minimization {#subsubsec:error_bound_cmp}
As commented in Section \[sec:CSvar\], the most common detection problem can be parametrized by the next cost matrix:
$$\mathbf{C}
=\left(
\begin{array} {c c}
c_{nn} & c_{np}\\
c_{pn} & c_{pp}\\
\end{array}
\right)
=\left(
\begin{array} {c c}
0 & C_P\\
C_N & 0\\
\end{array}
\right)$$
We will start our comparative analysis by following the error bound minimization perspective originally proposed by Schapire and Singer [@SchapireSinger99], also used in the derivation of Cost-Generalized AdaBoost and AdaBoostDB. From that point of view, classical AdaBoost, with its initial uniform weight distribution, is an algorithm driven to minimize an exponential bound ($\tilde{E}_T$) on the training error ($E_T$) (), as illustrated in Figure \[std\_adb\_bound\_fig\]. In that figure, the horizontal axis ($y_if(\mathbf{x}_i)$) represents the *performance score* of a classification, whose sign indicates the success (if $y_if(\mathbf{x}_i)>0$) or failure (if $y_if(\mathbf{x}_i)<0$) of the decision, and whose magnitude indicates the confidence expected by the classifier on its decision. The exponential bound is decreasing for increasing performance scores, so the classical AdaBoost minimization process is aimed to maximize correct classifications and their margin (distance to the boundary), in a scenario where all the training examples follow a common cost scheme.
$$\label{std_adb_bound}
E_{T}= \sum_{i=1}^{n} \frac{1}{n} \llbracket H(\mathbf{x}_{i}) \neq y_{i}\rrbracket \leq \sum_{i=1}^{n} \frac{1}{n} \exp \left( -y_{i} f(\mathbf{x}_{i}) \right) = \tilde{E}_{T}$$
![Training error bound of AdaBoost. The loss (y-axis) associated to each decision has an exponential dependency on the performance score of the strong classifier (x-axis).[]{data-label="std_adb_bound_fig"}](Fig03.pdf){width="7.5cm"}
Cost-Sensitive AdaBoost and AdaBoostDB, assumming that the training set is divided into two significant subsets (positives and negatives), define two different exponential bounds ($\tilde{E}_{TP}$ and $\tilde{E}_{TN}$) with different associated costs ($C_P$ and $C_N$) over each subset. These costs are inserted as exponent modulators into each class-dependent exponential bound (), reaching a cost-sensitive behavior that can be graphically interpreted as shown in Figure \[csb\_bound\_fig\]. The goal is, again, to maximize correct classifications and their margin, but this time in a scenario where positives and negatives have different associated losses.
$$\label{csb_bound}
\begin{split}
E_{T} &= \sum_{i=1}^{n} \frac{1}{n} \llbracket H(\mathbf{x}_{i}) \neq y_{i}\rrbracket\\
& \leq \sum_{i=1}^{m} \frac{1}{n} \exp \left( -C_{P}y_{i} f(\mathbf{x}_{i}) \right) + \sum_{i=m}^{n} \frac{1}{n} \exp \left( -C_{N}y_{i} f(\mathbf{x}_{i}) \right)\\
& =\tilde{E}_{TP} + \tilde{E}_{TN} = \tilde{E}_{T}
\end{split}$$
![Training error bound of Cost-Sensitive AdaBoost and AdaBoostDB for $C_P=2$ and $C_N=1$. Loss has a class-dependent definition and is composed of two different exponential functions.[]{data-label="csb_bound_fig"}](Fig04.pdf){width="7.5cm"}
As can be seen, asymmetric modifications in Cost-Sensitive AdaBoost (and AdaBoostDB) are based on new bounds for the training error, while the error definition itself remains unchanged from original (cost-insensitive) AdaBoost.
Cost-Generalized AdaBoost, on the other hand, is based on redefining the training error and then applying the standard exponential bounding process. To achieve this, training error in positives (${E}_{TP}$) and in negatives (${E}_{TN}$) are computed separately, and then are modulated by its respective normalized costs. The resulting class-dependent weighted error components (${E}_{TP}'$ and ${E}_{TN}'$) jointly define the *cost-sensitive global training error* ($E_{T}'$). The same way as in standard AdaBoost, each of these weighted error components can be exponentially bounded ($\tilde{E}_{TP}$ and $\tilde{E}_{TN}$), and the combination of the two resulting *class-dependent* bounds will define a cost-sensitive global bound ($\tilde{E}_{T}$) (), that is the function being minimized by Cost-Generalized AdaBoost. The scenario is graphically depicted in Figure \[cgb\_bound\_fig\].
$$\label{cgb_bound}
\begin{split}
E_{T}' &= E_{TP}' + E_{TN}' = \frac{C_{P}}{C_{P}+C_{N}} E_{TP} + \frac{C_{N}}{C_{P}+C_{N}} E_{TN} \\
& = \frac{C_{P}}{C_{P}+C_{N}} \sum_{i=1}^{m} \frac{1}{m} \llbracket H(\mathbf{x}_{i}) \neq y_{i}\rrbracket + \frac{C_{N}}{C_{P}+C_{N}} \sum_{i=m+1}^{n} \frac{1}{n-m} \llbracket H(\mathbf{x}_{i}) \neq y_{i}\rrbracket \\
& \leq \frac{C_{P}}{C_{P}+C_{N}} \sum_{i=1}^{m} \frac{1}{m} \exp \left( -y_{i} f(\mathbf{x}_{i}) \right) + \frac{C_{N}}{C_{P}+C_{N}} \sum_{i=m}^{n} \frac{1}{n-m} \exp \left( -y_{i} f(\mathbf{x}_{i}) \right)\\
& =\tilde{E}_{TP} + \tilde{E}_{TN} = \tilde{E}_{T}
\end{split}$$
![Training error bound of Cost-Generalized AdaBoost for $C_P=2$ and $C_N=1$. Loss keeps again an exponential dependency, but now modulated by a class-dependent behavior.[]{data-label="cgb_bound_fig"}](Fig05.pdf){width="7.5cm"}
It is important to notice that, by definition, all these algorithms have the goal of obtaining the best possible classifier able to deal with the problem in a cost-sensitive sense, and that the bounding loss functions $\tilde{E}_{T}$ are a mere mathematical tool to make the minimization problem tractable. Thus, from a formal point of view, the direct definition of a cost-sensitive error to be subsequently bounded, as proposed by Cost-Generalized AdaBoost, seems to be more suitable than using the standard cost-insensitive error and manipulate its bound to be asymmetric, as suggested by Cost-Sensitive AdaBoost or AdaBoostDB.
Figure \[preval\_fig\] illustrates the prevalence of the class-dependent error bounds of the two algorithms, assuming, without loss of generality, that positives have a greater cost than negatives $C_P>C_N$ (the opposite case can be modeled by a simple label swap). As can be seen, in Cost-Generalized AdaBoost (Figure \[preval\_fig\]a) the loss associated to positives is always greater than the loss associated to negatives, and the ratio between the two class-dependent losses remains constant along the performance scores. However, in Cost-Sensitive AdaBoost (Figure \[preval\_fig\]b), the ratio between losses varies according to the score, to the extent that class prevalence is inverted depending on which side of the success boundary ($y_if(\mathbf{x}_i)=0$) we are.
![Class prevalence of error bounds for Cost-Generalized AdaBoost (a) and Cost-Sensitive AdaBoost (b) ($C_P=2$, $C_N=1$).[]{data-label="preval_fig"}](Fig06.pdf){width="0.9\columnwidth"}
The iterative learning process behind AdaBoost builds a predictor function $f(\mathbf{x}_i)$ aimed to progressively (round by round) minimize the respective loss function over the training dataset. In terms of classification, this means that AdaBoost classifiers are trained not only to maximize the accuracy of the classifier over the training set, but also to maximize the margin of its decisions. So, once one training example is correctly classified, the tendency of the learner will be to continue increasing the confidence of its prediction ($\mathrm{abs}(f(\mathbf{x}_i))$) to move it away from the decision boundary ($f(\mathbf{x}_i)=0$). For Cost-Generalized AdaBoost, this means that any positive training example will always be more costly (and in the same ratio) than any negative example with its same performance score, whatever this score is. However, in the case of Cost-Sensitive AdaBoost, prevalence ratio varies exponentially with performance scores. So, when scores are positive, negative training examples become the prevalent ones.
Bearing in mind that the performance score of any training example, at any iteration of the learning process, is determined by the evaluation over the example of the boosted predictor learned so far, and that the weight of this example for the next learning round will depend on the value of the related bounding loss for that particular score, we can draw the two following consequences:
- In Cost-Generalized AdaBoost positives will always be the costly class, and the same cost asymmetry is preserved throughout the whole learning process.
- In Cost-Sensitive AdaBoost cost asymmetry changes. While the classifier is wrong, positives are the costly class (learning is positive-driven), but when classification is correct, negatives are prevalent (learning is negative-driven). The more accurate the classifier obtained is, the more costly will be negatives over positives in subsequent training rounds.
In terms of training error, these differences seem to be anecdotal, since the change of class prevalence occurs once the classifier succeeds for each example. However, what is really relevant, is the effect in terms of generalization error: when the classifier works on unseen instances it will make mistakes and it is essential, from a cost-sensitive perspective, to characterize which class is the most prone to errors and to what extent.
As the iterative training process progresses, the performance scores associated to the training examples tend to increase, and their respective losses tend to decrease moving along the Y axis on Figures \[csb\_bound\_fig\] and \[cgb\_bound\_fig\], so, the more rounds we train, the more on the right of these figures we will be. In the case of Cost-Sensitive AdaBoost this trend will increasingly emphasize negatives at the expense of positives, while Cost-Generalized AdaBoost keeps the ratio between classes intact throughout the whole learning process. Thus, due to its changing emphasis, Cost-Sensitive AdaBoost may run the risk of obtaining classifiers in which the supposed costly class is the most prone to errors: just the opposite of what was originally intended!
In the companion paper of the series [@LandesaAlba??b] we will see empirical evidences confirming this *asymmetry swapping* behavior that, by definition, is expected to be more noticeable the closer the system is to overfitting, but that may have an implicit detrimental effect on the performance reached by all classifiers trained by Cost-Sensitive AdaBoost.
### Statistical View of Boosting {#subsubsec:statistical_cmp}
Instead of the exponential error bound minimization perspective that originally gave rise to AdaBoost (and that also is the derivation core of Cost-Generalized AdaBoost and AdaBoostDB) we will now adopt a different point of view: the Statistical View of Boosting [@Friedman00], the other major analytical framework to interpret and derive AdaBoost that, in addition, is the foundation of Cost-Sensitive AdaBoost.
As we have seen in Section \[subsec:Statistical View\], from the Statistical View of Boosting perspective, AdaBoost can be interpreted as an algorithm that iteratively builds an additive regression model based on the following loss function:
$$\label{loss_ab_eqn}
l_{AB}(f(\mathbf{x}),y)=\exp\left(-yf(\mathbf{x})\right)$$
From that loss, an associated risk function $J_{AB}(f(\mathbf{x}))$ (the expected loss) is defined:
$$\label{risk_ab_eqn}
\begin{split}
J_{AB}(f(\mathbf{x}))&=\E\left[l_{AB}(f(\mathbf{x}),y)\right]\\
&=\Prob\left(y=1|\mathbf{x}\right)\exp(-f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)\exp(f(\mathbf{x}))
\end{split}$$
If we minimize that risk we will obtain the optimal predictor $f_{AB}(\mathbf{x})$, that turns out to be the symmetric logistic transform of $\Prob\left(y=1|\mathbf{x}\right)$.
$$\label{min_ab_eqn}
f_{AB}(\mathbf{x})=\dfrac{1}{2}\log\dfrac{\Prob\left(y=1|\mathbf{x}\right)}{\Prob\left(y=-1|\mathbf{x}\right)}$$
AdaBoost is geared to approximate, in an additive way, that optimal predictor without embedded costs. Thus, the obtained model will be cost-insensitive, only depending on the likelihood of each class (see Figure \[stat\_cmp\_ab\_fig\]).
![Risk minimizing function (optimal predictor) for AdaBoost ($f_{AB}(\mathbf{x})$). It only depends on the likelihood of each class.[]{data-label="stat_cmp_ab_fig"}](Fig095a.pdf){width="7.5cm"}
In the case of Cost-Generalized AdaBoost, from this same perspective, we will have a loss function in which costs are included as modulators of the exponentials.
$$\label{loss_cga_eqn}
l_{CGA}(f(\mathbf{x}),y)=\llbracket y=1 \rrbracket C_{P} \exp(-f(\mathbf{x})) + \llbracket y=-1 \rrbracket C_{N} \exp(f(\mathbf{x}))$$
Thus, as explained in Section \[subsubsec:weight\_cost\_sensitive\], the respective risk function $J_{CGA}(f(\mathbf{x}))$ and its minimizer $f_{CGA}(\mathbf{x})$ will be the following ones:
$$\label{risk_cga_eqn}
\begin{split}
J_{CGA}(f(\mathbf{x}))&=\E\left[l_{CGA}(f(\mathbf{x}),y)\right]\\
&=\Prob\left(y=1|\mathbf{x}\right)C_{P}\exp(-f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)C_{N}\exp(f(\mathbf{x}))
\end{split}$$
$$\label{min_cga_eqn}
f_{CGA}(\mathbf{x})=\dfrac{1}{2}\log\left(\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}\right)$$
As can be seen, now we have a cost-sensitive risk function with a cost-sensitive minimizer gearing to an optimal predictor $f_{CGA}(\mathbf{x})$ based on the asymmetric logistic transform of $\Prob\left(y=1|\mathbf{x}\right)$. Thus, in contrast to AdaBoost, the model pursued by Cost-Generalized AdaBoost will not exclusively depend on the likelihood of each class, but also on the related costs.
![Risk minimizing function (optimal predictor) for Cost-Generalized AdaBoost ($f_{CGA}(\mathbf{x})$). It depends on the likelihood of each class and on the related costs, having a homogeneous and continuous cost-sensitive behavior for whatever likelihood.[]{data-label="stat_cmp_cga_fig"}](Fig095b.pdf){width="7.5cm"}
On the other hand, the loss function of Cost-Sensitive AdaBoost embeds the costs inside the exponents $$\label{loss_csa_eqn}
l_{CSA}(f(\mathbf{x}),y)=\llbracket y=1 \rrbracket \exp(-C_{P} f(\mathbf{x})) + \llbracket y=-1 \rrbracket \exp(C_{N}f(\mathbf{x}))$$ so the risk function and its associated minimizer will be as follows (see Section \[subsubsec:weight\_cost\_sensitive\]): $$\label{risk_csa_eqn}
\begin{split}
J_{CSA}(f(\mathbf{x}))&=\E\left[l_{CSA}(f(\mathbf{x}),y)\right]\\
&=\Prob\left(y=1|\mathbf{x}\right)\exp(-C_{P}f(\mathbf{x}))+\Prob\left(y=-1|\mathbf{x}\right)\exp(C_{N}f(\mathbf{x}))
\end{split}$$ $$\label{min_csa_eqn}
f_{CSA}(\mathbf{x})=\dfrac{1}{C_{P}+C_{N}}\log\left(\dfrac{C_{P}\Prob\left(y=1|\mathbf{x}\right)}{C_{N}\Prob\left(y=-1|\mathbf{x}\right)}\right)$$
Then, Cost-Sensitive AdaBoost is also aimed to fit a model based on the asymmetric logistic transform of $\Prob\left(y=1|\mathbf{x}\right)$, depending both on the likelihood of each class as well as on the related costs (see Figure \[stat\_cmp\_csa\_fig\]).
![Risk minimizing function (optimal predictor) for Cost-Sensitive AdaBoost ($f_{CSA}(\mathbf{x})$). It depends on the likelihood of each class and on the related costs, but in this case the cost-sensitive behavior is not homogeneous with respect to likelihood (solutions for different costs cross each other depending on $\Prob\left(y=1|\mathbf{x}\right)$).[]{data-label="stat_cmp_csa_fig"}](Fig095c.pdf){width="7.5cm"}
Notwithstanding, the optimal predictors guiding Cost-Sensitive AdaBoost and Cost-Generalized AdaBoost, despite being both cost-sensitive, have different equations. Such differences become apparent in their graphic representations (see Figures \[stat\_cmp\_cga\_fig\] and \[stat\_cmp\_csa\_fig\]).
To delve into the consequences of these differences, we will analyze the optimal predictors of Cost-Generalized AdaBoost and Cost-Sensitive AdaBoost as functions depending on two magnitudes: likelihood and cost asymmetry [^3]. In Figure \[stat\_cmp\_cgacsamap\_fig\] we have represented the outputs of the optimal predictors as colormaps (we have used isolines for the sake of clarity) onto the plane defined by the likelihood and the cost asymmetry. As can be seen, the optimal predictor of Cost-Generalized AdaBoost (Figure \[stat\_cmp\_cgacsamap\_fig\]a) obtains higher predictor values for increasing $\Prob\left(y=1|\mathbf{x}\right)$ and increasing $C_P$ (vice versa for negatives). However, that is not the case for Cost-Sensitive AdaBoost (Figure \[stat\_cmp\_cgacsamap\_fig\]b) where, for a given likelihood, we can find lower predictor outputs for increasing positive costs (and vice versa for negatives). This inhomogeneous behavior can explain the *asymmetry swapping* effect we have commented in Section \[subsubsec:error\_bound\_cmp\], and to which we will come back in the companion paper of the series [@LandesaAlba??b] when analyzing the experimental behavior of Cost-Sensitive AdaBoost.
[ \[training\_set\_nonover\_fig\] ![Isolines of the optimal predictors for Cost-Generalized AdaBoost (a), and Cost-Sensitive AdaBoost (b), with respect to the likelihood ($\Prob\left(y=1|\mathbf{x}\right)$) and the normalized cost asymmetry ($\gamma=C_P/(C_P+C_N)$).[]{data-label="stat_cmp_cgacsamap_fig"}](Fig095d1.pdf "fig:"){width="5.5cm"} ]{} [ \[training\_classifiers\_nonover\_fig\] ![Isolines of the optimal predictors for Cost-Generalized AdaBoost (a), and Cost-Sensitive AdaBoost (b), with respect to the likelihood ($\Prob\left(y=1|\mathbf{x}\right)$) and the normalized cost asymmetry ($\gamma=C_P/(C_P+C_N)$).[]{data-label="stat_cmp_cgacsamap_fig"}](Fig095d2.pdf "fig:"){width="5.5cm"} ]{}
Summary and Conclusions {#sec:Conclusions1}
=======================
In this first paper of the series we have introduced our working scenario, presenting the algorithms under study (AdaBoost with threshold modification [@ViolaJones04]; AsymBoost [@ViolaJones02]; AdaCost [@Fan99]; CSB0, CSB1 and CSB2 [@Ting98; @Ting00]; AdaC1, AdaC2 and AdaC3 [@Sun05; @Sun07]; Cost-Sensitive AdaBoost [@MasnadiVasconcelos07; @MasnadiVasconcelos11]; AdaBoostDB [@LandesaAlba13]; and Cost-Generalized AdaBoost [@LandesaAlba12]) in a homogeneous notational framework and proposing a clustering scheme for them based on the way asymmetry is inserted in the learning process: *theoretically*, *heuristically* or *a posteriori*. Then, for those algorithms with a fully theoretical derivation, we performed a thorough theoretical analysis and discussion, adopting the different perspectives that have been used to explain and derive the related approaches in the literature (Error Bound Minimization perspective [@SchapireSinger99] and Statistical View of Boosting [@Friedman00]).
The presented analysis clearly shows that the asymmetric weight initialization mechanism used by Cost-Generalized AdaBoost, from whatever point of view, is definitely a valid mechanism to build theoretically sound cost-sensitive boosted classifiers, despite having being recurrently overlooked or rejected in many previous works (e.g. [@Fan99; @Ting00; @ViolaJones02; @MasnadiVasconcelos07; @MasnadiVasconcelos11]). In addition, and besides being the simplest algorithm, Cost-Generalized AdaBoost exhibits the most consistent error bound definition and it is able to preserve the class-dependent loss ratio regardless of the training round whereas Cost-Sensitive AdaBoost and AdaBoostDB, the other theoretical alternatives, may end up emphasizing the least costly class.
After this purely theoretical study, an empirical analysis of the different approaches, also including the non-fully-theoretical methods (a posteriori and heuristic), is needed to reach global conclusions and culminate the analysis we have started in this paper. Such experimental part can be found in the next article of the series: “” [@LandesaAlba??b].
[^1]: Notation: $\llbracket a \rrbracket$ is $1$ when $a$ is true and $0$ otherwise.
[^2]: As analyzed in Appendix \[app:weight\], the way asymmetry is applied across the different boosting variants covered by the Cost-Sensitive Boosting framework [@MasnadiVasconcelos11] is not homogeneus either. In fact, despite having discarded cost-proportionate weight initialization as a valid method, one of the algorithms (Cost-Sensitive LogitBoost) proposed in the same work is actually based on that strategy.
[^3]: In the case of Cost-Sensitive AdaBoost (and AdaBoostDB) we can actually distinguish three different involved magnitudes (likelihood, cost of positives and cost of negatives), since the optimal predictor changes when costs are multiplied by a positive factor. This behavior (that does not happen for Cost-Generalized AdaBoost) violates the rules of the cost matrix [@Elkan01] explained at the beginning of Section \[sec:CSvar\]. In order to tackle this problem for our analysis, we have restricted the possible costs to combinations $(C_P, C_N)$ in which one of the coefficients is always 1, and the other one is $\geq1$. This decision allows us to homogeneously interpret the scenarios in which negatives are the costliest class as label inversions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'While sorting is an important procedure in computer science, the `argsort` operator - which takes as input a vector and returns its sorting permutation - has a discrete image and thus zero gradients almost everywhere. This prohibits end-to-end, gradient-based learning of models that rely on the `argsort` operator. A natural way to overcome this problem is to replace the `argsort` operator with a continuous relaxation. Recent work has shown a number of ways to do this, but the relaxations proposed so far are computationally complex. In this work we propose a simple continuous relaxation for the `argsort` operator which has the following qualities: it can be implemented in three lines of code, achieves state-of-the-art performance, is easy to reason about mathematically - substantially simplifying proofs - and is faster than competing approaches. We open source the code to reproduce all of the experiments and results.'
bibliography:
- 'references.bib'
---
Introduction
============
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Gradient-based optimization lies at the core of the success of deep learning. However, many common operators have discrete images and thus exhibit zero gradients almost everywhere, making them unsuitable for gradient-based optimization. Examples include the *Heaviside step* function, the `argmax` operator, and - the central object of this work - the `argsort` operator. To overcome this limitation, continuous relaxations for these operators can be constructed. For example, the sigmoid function serves as a continuous relaxation for the *Heaviside step* function, and the `softmax` operator serves as a continuous relaxation for the `argmax`. These continuous relaxations have the crucial property that they provide meaningful gradients that can drive optimization. Because of this, operators such as the `softmax` are ubiquitous in deep learning. In this work we are concerned with continuous relaxations for the `argsort` operator.
Formally, we define the `argsort` operator as the mapping $\texttt{argsort} : \mathbb R^n \rightarrow \mathcal S_n$ from $n$-dimensional real vectors $s \in \mathbb R^n$ to the set of permutations over $n$ elements $\mathcal S_n \subseteq \{1,2,\dots,n\}^n$, where $\texttt{argsort}(s)$ returns the permutation that sorts $s$ in decreasing order[^1]. For example, for the input vector $s = [9,1,5,2]^T$ we have $\texttt{argsort}(s) = [1, 3, 4, 2]^T$. If we let $\mathcal P_n \subseteq \{0, 1\}^{n \times n} \subset \mathbb R^{n \times n}$ denote the set of permutation matrices of dimension $n$, following [@2018_grover] we can define, for a permutation $\pi \in \mathcal S_n$, the permutation matrix $P_\pi \in \mathcal P_n$ as: $$P_\pi[i, j] =
\begin{cases}
1\text{ if }j = \pi_i,\\
0\text{ otherwise}
\end{cases}$$ This is simply the one-hot representation of $\pi$. Note that with these definitions, the mapping $\texttt{sort} : \mathbb R^n \rightarrow \mathbb R^n$ that sorts $s$ in decreasing order is $\texttt{sort}(s) = P_{\texttt{argsort}(s)}s$. Also, if we let $\bar{1}_n = [1,2,\dots,n]^T$, then the `argsort` operator can be recovered from $P_{\texttt{argsort}(\cdot)} : \mathbb R^n \rightarrow \mathcal P_n$ by a simple matrix multiplication via $$\texttt{argsort}(s) = P_{\texttt{argsort}(s)}\bar{1}_n$$ Because of this reduction from the `argsort` operator to the $P_{\texttt{argsort}(\cdot)}$ operator, in this work, as in previous works [@2018_mena; @2018_grover; @2019_cuturi], our strategy to derive a continuous relaxation for the `argsort` operator is to instead derive a continuous relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator. This is analogous to the way that the `softmax` operator relaxes the `argmax` operator.
The main contribution of this paper is the proposal of a family of simple continuous relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator, which we call $\SimpleSort{}$, and define as follows: $$\label{eq:simplesort}
\SimpleSort{}^d_{\tau}(s) = \texttt{softmax}\left(\frac{-d\left(\texttt{sort}(s) \mathds 1^T, \mathds 1 s^T\right)}{\tau}\right)$$ where the softmax operator is applied row-wise, $d$ is any differentiable almost everywhere, semi–metric function of $\mathbb R$ applied pointwise, and $\tau$ is a temperature parameter that controls the degree of the approximation. In simple words: *the $r$-th row of the operator is the `softmax` of the negative distances to the $r$-th largest element*.
Throughout this work we will predominantly use $d(x, y) = |x - y|$ (the $L_1$ norm), but our theoretical results hold for any such $d$, making the approach flexible. The operator is trivial to implement in automatic differentiation libraries such as TensorFlow [@2016_abadi] and PyTorch [@2017_paszke], and we show that:
- achieves state-of-the-art performance on multiple benchmarks that involve reordering elements.
- shares the same desirable properties as the `NeuralSort` operator [@2018_grover]. Namely, it is row-stochastic, converges to $P_{\texttt{argsort}(\cdot)}$ in the limit of the temperature, and can be projected onto a permutation matrix using the row-wise `argmax` operation. However, is significantly easier to reason about mathematically, which substantially simplifies the proof of these properties.
- The operator is faster than the `NeuralSort` operator of [@2018_grover], the fastest competing approach, and empirically just as easy to optimize in terms of the number of gradient steps required for the training objective to converge.
Therefore, the operator advances the state of the art in differentiable sorting by significantly simplifying previous approaches. To better illustrate the usefulness of the mapping defined by , we show in Figure \[fig:simplesort\_plots\] the result of applying the operator to soft-sort a sequence of vectors $v_i$ $(1\leq i \leq n)$ according to the order given by respective scores $s_i \in \mathbb R$. Soft-sorting the $v_i$ is achieved by multiplying to the left by $\SimpleSort(s)$.
The code and experiments are available at <https://github.com/sprillo/softsort>
Related Work
============
Relaxed operators for sorting procedures were first proposed in the context of Learning to Rank with the end goal of enabling direct optimization of Information Retrieval (IR) metrics. Many IR metrics, such as the Normalized Discounted Cumulative Gain (NDCG) [@2002_kalervo], are defined in terms of *ranks*. Formally, the `rank` operator is defined as the function $\texttt{rank} : \mathbb R^n \rightarrow \mathcal S_n$ that returns the inverse of the `argsort` permutation: $\texttt{rank}(s) = \texttt{argsort}(s)^{-1}$, or equivalently $P_{\texttt{rank}(s)} = P_{\texttt{argsort}(s)}^T$. The $\texttt{rank}$ operator is thus intimately related to the $\texttt{argsort}$ operator; in fact, a relaxation for the $P_{\texttt{rank}(\cdot)}$ operator can be obtained by transposing a relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator, and vice-versa; this duality is apparent in the work of [@2019_cuturi].
We begin by discussing previous work on relaxed `rank` operators in section \[sec:relaxed\_rank\_operators\]. Next, we discuss more recent work, which deals with relaxations for the $P_{\texttt{argsort}(\cdot)}$ operator.
Relaxed Rank Operators {#sec:relaxed_rank_operators}
----------------------
The first work to propose a relaxed `rank` operator is that of [@2008_taylor]. The authors introduce the relaxation $\texttt{SoftRank}_\tau : \mathbb R^n \rightarrow \mathbb R^n$ given by $\texttt{SoftRank}_\tau(s) = \mathbb E[\texttt{rank}(s + z)]$ where $z \sim \mathcal N_n(0, \tau I_n)$, and show that this relaxation, as well as its gradients, can be computed exactly in time $\mathcal O(n^3)$. Note that as $\tau \rightarrow 0$, $\texttt{SoftRank}_\tau(s) \rightarrow \texttt{rank}(s)$[^2]. This relaxation is used in turn to define a surrogate for NDCG which can be optimized directly.
In [@2010_qin], another relaxation for the `rank` operator $\widehat{\pi}_\tau : \mathbb R^n \rightarrow \mathbb R^n$ is proposed, defined as: $$\label{eqn:2010_qin}
\widehat{\pi}_\tau(s)[i] = 1 + \sum_{j \neq i} \sigma\left(\frac{s_i - s_j}{\tau}\right)$$ where $\sigma(x) = (1 + \exp\{-x\})^{-1}$ is the sigmoid function. Again, $\widehat{\pi}_\tau(s) \rightarrow \texttt{rank}(s)$ as $\tau \rightarrow 0$. This operator can be computed in time $\mathcal{O}(n^2)$, which is faster than the $\mathcal O(n^3)$ approach of [@2008_taylor]. Note that the above two approaches cannot be used to define a relaxation for the `argsort` operator. Indeed, $\texttt{SoftRank}_\tau(s)$ and $\widehat{\pi}_\tau(s)$ are not relaxations for the $P_{\texttt{rank}(\cdot)}$ operator. Instead, they directly relax the $\texttt{rank}$ operator, and there is no easy way to obtain a relaxed `argsort` or $P_{\texttt{argsort}(\cdot)}$ operator from them.
Sorting via Bipartite Maximum Matchings
---------------------------------------
The work of [@2018_mena] draws an analogy between the `argmax` operator and the `argsort` operator by means of bipartite maximum matchings: the `argmax` operator applied to an $n$-dimensional vector $s$ can be viewed as a maximum matching on a bipartite graph with $n$ vertices in one component and $1$ vertex in the other component, the edge weights equal to the given vector $s$; similarly, a permutation matrix can be seen as a maximum matching on a bipartite graph between two groups of $n$ vertices with edge weights given by a matrix $X \in \mathbb R^{n \times n}$. This induces a mapping $M$ (for ‘matching’) from the set of square matrices $X \in \mathbb R^{n \times n}$ to the set $\mathcal P_n$. Note that this mapping has arity $M : \mathbb R^{n \times n} \rightarrow \mathcal P_n$, unlike the $P_{\texttt{argsort}(\cdot)}$ operator which has arity $P_{\texttt{argsort}(\cdot)} : \mathbb R^n \rightarrow \mathcal P_n$.
Like the $P_{\texttt{argsort}(\cdot)}$ operator, the $M$ operator has discrete image $\mathcal P_n$, so to enable end-to-end gradient-based optimization, [@2018_mena] propose to relax the matching operator $M(X)$ by means of the Sinkhorn operator $S(X / \tau)$; $\tau$ is a temperature parameter that controls the degree of the approximation; as $\tau \rightarrow 0$ they show that $S(X / \tau) \rightarrow M(X)$. The Sinkhorn operator $S$ maps the square matrix $X / \tau$ to the Birkhoff polytope $\mathcal B_n$, which is defined as the set of doubly stochastic matrices (i.e. rows and columns summing to $1$).
The computational complexity of the [@2018_mena] approach to differentiable sorting is thus $\mathcal O(Ln^2)$ where $L$ is the number of Sinkhorn iterates used to approximate $S(X / \tau)$; the authors use $L = 20$ for all experiments.
Sorting via Optimal Transport
-----------------------------
The recent work of [@2019_cuturi] also makes use of the Sinkhorn operator to derive a continuous relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator. This time, the authors are motivated by the observation that a sorting permutation for $s \in \mathbb R^n$ can be recovered from an optimal transport plan between two discrete measures defined on the real line, one of which is supported on the elements of $s$ and the other of which is supported on arbitrary values $y_1 < \dots < y_n$. Indeed, the optimal transport plan between the probability measures $\frac{1}{n} \sum_{i = 1}^n \delta_{s_i}$ and $\frac{1}{n} \sum_{i = 1}^n \delta_{y_i}$ (where $\delta_x$ is the Dirac delta at $x$) is given by the matrix $P_{\texttt{argsort}(s)}^T$.
Notably, a variant of the optimal transport problem with entropy regularization yields instead a continuous relaxation $P^\epsilon_{\texttt{argsort}(s)}$ mapping $s$ to the Birkhoff polytope $\mathcal B_n$; $\epsilon$ plays a role similar to the temperature in [@2018_mena], with $P^\epsilon_{\texttt{argsort}(s)} \rightarrow P_{\texttt{argsort}(s)}$ as $\epsilon \rightarrow 0$. This relaxation can be computed via Sinkhorn iterates, and enables the authors to relax $P_{\texttt{argsort}(\cdot)}$ by means of $P^\epsilon_{\texttt{argsort}(s)}$. Gradients can be computed by backpropagating through the Sinkhorn operator as in [@2018_mena].
The computational complexity of this approach is again $\mathcal O(Ln^2)$. However, the authors show that a generalization of their method can be used to compute relaxed quantiles in time $\mathcal O(Ln)$, which is interesting in its own right.
Sorting via a sum-of-top-k elements identity
--------------------------------------------
Finally, a more computationally efficient approach to differentiable sorting is proposed in [@2018_grover]. The authors rely on an identity that expresses the sum of the top $k$ elements of a vector $s \in \mathbb R^n$ as a symmetric function of $s_1,\dots,s_n$ that only involves `max` and `min` operations [@2003_ogryczak Lemma 1]. Based on this identity, and denoting by $A_s$ the matrix of *absolute* pairwise differences of elements of $s$, namely $A_s[i, j] = |s_i - s_j|$, the authors prove the identity: $$\label{eq:2018_grover_eq4}
P_{\texttt{argsort}(s)}[i, j] =
\begin{cases}
1\text{ if }j = \texttt{argmax}(c_i),\\
0\text{ otherwise}
\end{cases}$$ where $c_i = (n + 1 - 2i)s - A_s \mathds 1$, and $\mathds 1$ denotes the column vector of all ones.
Therefore, by replacing the $\texttt{argmax}$ operator in Eq. \[eq:2018\_grover\_eq4\] by a row-wise `softmax`, the authors arrive at the following continuous relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator, which they call `NeuralSort`: $$\label{eq:2018_grover_eq5}
\texttt{NeuralSort}_\tau(s)[i, :] = \texttt{softmax}\left(\frac{c_i}{\tau}\right)$$ Again, $\tau > 0$ is a temperature parameter that controls the degree of the approximation; as $\tau \rightarrow 0$ they show that $\texttt{NeuralSort}_\tau(s) \rightarrow P_{\texttt{argsort}(s)}$. Notably, the relaxation proposed by [@2018_grover] can be computed in time $\mathcal O(n^2)$, making it much faster than the competing approaches of [@2018_mena; @2019_cuturi].
: A simple relaxed sorting operator
===================================
$$\texttt{NeuralSort}_\tau(s) = g_\tau
\begin{pmatrix}
0 & s_2 - s_1 & 3s_3 - s_1 - 2s_2 & 5s_4 - s_1 - 2s_2 - 2s_3 \\
s_2 - s_1 & 0 & s_3 - s_2 & 3s_4 - s_2 - 2s_3 \\
2s_2 + s_3 - 3s_1 & s_3 - s_2 & 0 & s_4 - s_3 \\
2s_2 + 2s_3 + s_4 - 5s_1 & 2s_3 + s_4 - 3s_2 & s_4 - s_3 & 0 \\
\end{pmatrix}$$
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$$\SimpleSort{}^{|\cdot|}_{\tau}(s) = g_\tau
\begin{pmatrix}
0 & s_2 - s_1 & s_3 - s_1 & s_4 - s_1 \\
s_2 - s_1 & 0 & s_3 - s_2 & s_4 - s_2 \\
s_3 - s_1 & s_3 - s_2 & 0 & s_4 - s_3 \\
s_4 - s_1 & s_4 - s_2 & s_4 - s_3 & 0 \\
\end{pmatrix}$$
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In this paper we propose , a simple continuous relaxation for the $P_{\texttt{argsort}(\cdot)}$ operator. We define as follows: $$\label{eq:simplesort2}
\SimpleSort{}^d_{\tau}(s) = \texttt{softmax}\left(\frac{-d\left(\texttt{sort}(s) \mathds 1^T, \mathds 1 s^T\right)}{\tau}\right)$$ where $\tau > 0$ is a temperature parameter that controls the degree of the approximation and $d$ is semi–metric function applied pointwise that is differentiable almost everywhere. Recall that a semi–metric has all the properties of a metric except the triangle inequality, which is not required. Examples of semi–metrics in $\mathbb R$ include any positive power of the absolute value. The operator has similar desirable properties to those of the `NeuralSort` operator, while being significantly simpler. Here we state and prove these properties. We start with the definition of *Unimodal Row Stochastic Matrices* [@2018_grover], which summarizes the properties of our relaxed operator:
(Unimodal Row Stochastic Matrices). An $n \times n$ matrix is Unimodal Row Stochastic (URS) if it satisfies the following conditions:
1. **Non-negativity:** $U[i, j] \ge 0\quad \forall i,j \in \{1,2,\dots,n\}$.
2. **Row Affinity:** $\sum_{j = 1}^n U[i, j] = 1\quad \forall i \in \{1,2,\dots,n\}$.
3. **Argmax Permutation:** Let $u$ denote a vector of size $n$ such that $u_i = \arg\max_j U[i, j]\quad \forall i \in \{1,2,\dots,n\}$. Then, $u \in \mathcal S_n$, i.e., it is a valid permutation.
While `NeuralSort` and `SoftSort` yield URS matrices (we will prove this shortly), the approaches of [@2018_mena; @2019_cuturi] yield bistochastic matrices. It is natural to ask whether URS matrices should be preferred over bistochastic matrices for relaxing the $P_{\texttt{argsort}(\cdot)}$ operator. Note that URS matrices are not comparable to bistochastic matrices: they drop the column-stochasticity condition, but require that each row have a distinct `argmax`, which is not true of bistochastic matrices. This means that URS matrices can be trivially projected onto the probability simplex, which is useful for e.g. straight-through gradient optimization, or whenever hard permutation matrices are required, such as at test time. The one property URS matrices lack is column-stochasticity, but this is not central to soft sorting. Instead, this property arises in the work of [@2018_mena] because their goal is to relax the bipartite matching operator (rather than the `argsort` operator), and in this context bistochastic matrices are the natural choice. Similarly, [@2019_cuturi] yields bistochastic matrices because they are the solutions to optimal transport problems (this does, however, allow them to simultaneously relax the `argsort` and `rank` operators). Since our only goal (as in the `NeuralSort` paper) is to relax the `argsort` operator, column-stochasticity can be dropped, and URS matrices are the more natural choice.
Now on to the main Theorem, which shows that has the same desirable properties as `NeuralSort`. These are [@2018_grover Theorem 4]:
\[main\_theorem\] The operator has the following properties:
1. Unimodality: $\forall \tau > 0$, $\SimpleSort{}^d_\tau(s)$ is a unimodal row stochastic matrix. Further, let $u$ denote the permutation obtained by applying $\texttt{argmax}$ row-wise to $\SimpleSort{}^d_\tau(s)$. Then, $u = \texttt{argsort}(s)$.
2. Limiting behavior: If no elements of $s$ coincide, then: $$\lim_{\tau \rightarrow 0^+} \SimpleSort{}^d_\tau(s) = P_{\texttt{argsort}(s)}$$ In particular, this limit holds almost surely if the entries of $s$ are drawn from a distribution that is absolutely continuous w.r.t. the Lebesgue measure on $\mathbb R$.
**Proof**.
1. Non-negativity and row affinity follow from the fact that every row in $\SimpleSort{}^d_\tau(s)$ is the result of a `softmax` operation. For the third property, we use that `softmax` preserves maximums and that $d(\cdot, x)$ has a unique minimum at $x$ for every $x\in\mathbb R$. Formally, let $\texttt{sort}(s) = [s_\ord{1}, \dots, s_\ord{n}]^T$, i.e. $s_\ord{1} \ge \dots \ge s_\ord{n}$ are the decreasing order statistics of $s$. Then: $$\begin{aligned}
u_i &= \argmax_j \SimpleSort{}^d_\tau(s)[i, j] \\
&= \argmax_j \left(\texttt{softmax}(-d(s_\ord{i}, s_j) / \tau)\right) \\
&= \argmin_j \left(d(s_\ord{i}, s_j)\right) \\
&= \texttt{argsort}(s)[i]\end{aligned}$$ as desired.
2. It suffices to show that the $i$-th row of $\SimpleSort{}^d_{\tau}(s)$ converges to the one-hot representation $h$ of $\texttt{argsort}(s)[i]$. But by part 1, the $i$-th row of $\SimpleSort{}^d_{\tau}(s)$ is the softmax of $v / \tau$ where $v$ is a vector whose unique argmax is $\texttt{argsort}(s)[i]$. Since it is a well-known property of the softmax that $\lim_{\tau \rightarrow 0^+} \texttt{softmax}(v / \tau) = h$ [@1994_elfadel], we are done.
Note that the proof of unimodality of the operator is straightforward, unlike the proof for the `NeuralSort` operator, which requires proving a more technical Lemma and Corollary [@2018_grover Lemma 2, Corollary 3].
The row-stochastic property can be loosely interpreted as follows: row $r$ of and `NeuralSort` encodes a distribution over the value of the rank $r$ element, more precisely, the probability of it being equal to $s_j$ for each $j$. In particular, note that the first row of the $\SimpleSort{}^{|\cdot|}$ operator is precisely the `softmax` of the input vector. In general, the $r$-th row of the $\SimpleSort{}^d$ operator is the `softmax` of the negative distances to the $r$-th largest element.
Finally, regarding the choice of $d$ in , even though a large family of semi–metrics could be considered, in this work we experimented with the absolute value as well as the square distance and found the absolute value to be marginally better during experimentation. With this in consideration, in what follows we fix $d = |\cdot|$ the absolute value function, unless stated otherwise. We leave for future work learning the metric $d$ or exploring a larger family of such functions.
Comparing to `NeuralSort`
=========================
def neural_sort(s, tau):
n = tf.shape(s)[1]
one = tf.ones((n, 1), dtype = tf.float32)
A_s = tf.abs(s - tf.transpose(s, perm=[0, 2, 1]))
B = tf.matmul(A_s, tf.matmul(one, tf.transpose(one)))
scaling = tf.cast(n + 1 - 2 * (tf.range(n) + 1), dtype = tf.float32)
C = tf.matmul(s, tf.expand_dims(scaling, 0))
P_max = tf.transpose(C-B, perm=[0, 2, 1])
P_hat = tf.nn.softmax(P_max / tau, -1)
return P_hat
def soft_sort(s, tau):
s_sorted = tf.sort(s, direction='DESCENDING', axis=1)
pairwise_distances = -tf.abs(tf.transpose(s, perm=[0, 2, 1]) - s_sorted)
P_hat = tf.nn.softmax(pairwise_distances / tau, -1)
return P_hat
-0.2in
Mathematical Simplicity {#sec:mathematical_simplicity}
-----------------------
The difference between and `NeuralSort` becomes apparent once we write down what the actual operators look like; the equations defining them (Eq. \[eq:2018\_grover\_eq5\], Eq. \[eq:simplesort2\]) are compact but do not offer much insight. Note that even though the work of [@2018_grover] shows that the `NeuralSort` operator has the desirable properties of Theorem \[main\_theorem\], the paper never gives a concrete example of what the operator instantiates to in practice, which keeps some of its complexity hidden.
Let $g_\tau : \mathbb R^{n \times n} \rightarrow \mathbb R^{n \times n}$ be the function defined as $g_\tau(X) = \texttt{softmax}(X / \tau)$, where the `softmax` is applied row-wise. Suppose that $n = 4$ and that $s$ is sorted in decreasing order $s_1 \ge s_2 \ge s_3 \ge s_4$. Then the `NeuralSort` operator is given in Figure \[eq:neuralsort\_logits\] and the $\SimpleSort{}$ operator is given in Figure \[eq:simplesort\_logits\]. Note that the diagonal of the logit matrix has been $0$-centered by subtracting a constant value from each row; this does not change the `softmax` and simplifies the expressions. The $\SimpleSort{}$ operator is straightforward, with the $i,j$-th entry of the logit matrix given by $-|s_i - s_j|$. In contrast, the $i,j$-th entry of the `NeuralSort` operator depends on all intermediate values $s_i,s_{i+1},\dots,s_j$. This is a consequence of the coupling between the `NeuralSort` operator and the complex identity used to derive it. As we show in this paper, this complexity can be completely avoided, and results in further benefits beyond aesthetic simplicity such as flexibility, speed and mathematical simplicity.
Note that for an arbitrary region of space other than $s_1 \ge s_2 \ge s_3 \ge s_4$, the `NeuralSort` and $\SimpleSort{}$ operators look just like Figures \[eq:neuralsort\_logits\] and \[eq:simplesort\_logits\] respectively except for relabelling of the $s_i$ and column permutations. Indeed, we have:
\[prop:invariance\] For both $f = \SimpleSort^d_\tau$ and $f = \texttt{NeuralSort}_\tau$, the following identity holds: $$f(s) = f(\texttt{sort}(s))P_{\texttt{argsort}(s)}$$
We defer the proof to appendix \[sec:proof\_of\_invariance\]. This proposition is interesting because it implies that the behaviour of the and `NeuralSort` operators can be completely characterized by their functional form on the region of space where $s_1 \ge s_2 \ge \dots \ge s_n$. Indeed, for any other value of $s$, we can compute the value of $\SimpleSort(s)$ or $\texttt{NeuralSort}(s)$ by first sorting $s$, then applying $\SimpleSort$ or $\texttt{NeuralSort}$, and finally sorting the columns of the resulting matrix with the inverse permutation that sorts $s$. In particular, to our point, the proposition shows that Figures \[eq:neuralsort\_logits\] and \[eq:simplesort\_logits\] are valid for *all* $s$ up to relabeling of the $s_i$ (by $s_\ord{i}$) and column permutations (by the inverse permutation that sorts $s$).
To further our comparison, in appendix \[sec:size\] we show how the $\SimpleSort{}$ and `NeuralSort` operators differ in terms of the *size* of their matrix entries.
Code Simplicity
---------------
In Figures \[code:neuralsort\] and \[code:simplesort\] we show TensorFlow implementations of the `NeuralSort` and operators respectively. has a simpler implementation than `NeuralSort`, which we shall see is reflected in its faster speed. (See section \[speed-comparison\])
Note that our implementation of is based directly off Eq. \[eq:simplesort2\], and we rely on the `sort` operator. We would like to remark that there is nothing wrong with using the `sort` operator in a stochastic computation graph. Indeed, the `sort` function is continuous, almost everywhere differentiable (with non-zero gradients) and piecewise linear, just like the `max`, `min` or `ReLU` functions.
Finally, the unimodality property (Theorem \[main\_theorem\]) implies that any algorithm that builds a relaxed permutation matrix can be used to construct the true discrete permutation matrix. This means that any relaxed sorting algorithm (in particular, `NeuralSort`) is lower bounded by the complexity of sorting, which justifies relying on sorting as a subroutine. As we show later, is faster than `NeuralSort`. Also, we believe that this modular approach is a net positive since sorting in CPU and GPU is a well studied problem [@2017_singh] and any underlying improvements will benefit ’s speed as well. For instance, the current implementation in TensorFlow relies on radix sort and heap sort depending on input size.
Experiments
===========
\[tab:run\_sort\]
\[tab:run\_median\]
Algorithm $n = 5$ $n = 9$ $n = 15$
------------------------------- ----------------------------------- ----------------------------------- -----------------------------------
Deterministic NeuralSort $\textbf{21.52}\ (\textbf{0.97})$ $\textbf{15.00}\ (\textbf{0.97})$ $18.81\ (\textbf{0.95})$
Stochastic NeuralSort $24.78\ (\textbf{0.97})$ $17.79\ (0.96)$ $18.10\ (0.94)$
Deterministic SoftSort (Ours) $23.44\ (\textbf{0.97})$ $19.26\ (0.96)$ $\textbf{15.54}\ (\textbf{0.95})$
Stochastic SoftSort (Ours) $26.17\ (\textbf{0.97})$ $19.06\ (0.96)$ $20.65\ (0.94)$
\[tab:knn\]
Algorithm MNIST Fashion-MNIST CIFAR-10
------------------------------------- ----------- --------------- ------------
kNN+Pixels 97.2% 85.8% 35.4%
kNN+PCA 97.6% 85.9% 40.9%
kNN+AE 97.6% 87.5% 44.2%
kNN + Deterministic NeuralSort **99.5%** 93.5% 90.7%
kNN + Stochastic NeuralSort 99.4% 93.4% 89.5%
kNN + Deterministic SoftSort (Ours) 99.37% 93.60 % **92.03**%
kNN + Stochastic SoftSort (Ours) 99.42% **93.67**% 90.64%
CNN (w/o kNN) 99.4% 93.4% **95.1%**
We first compare to `NeuralSort` on the benchmarks from the `NeuralSort` paper [@2018_grover], using the code kindly open-sourced by the authors. We show that performs comparably to `NeuralSort`. Then, we devise a specific experiment to benchmark the speeds of the and `NeuralSort` operators in isolation, and show that is faster than `NeuralSort` while taking the same number of gradient steps to converge. This makes not only the simplest, but also the fastest relaxed sorting operator proposed to date.
Models
------
For both and `NeuralSort` we consider their deterministic and stochastic variants as in [@2018_grover]. The deterministic operators are those given by equations \[eq:2018\_grover\_eq5\] and \[eq:simplesort2\]. The stochastic variants are Plackett-Luce distributions reparameterized via Gumbel distributions [@2018_grover Section 4.1], where the $P_{\texttt{argsort}(\cdot)}$ operator that is applied to the samples is relaxed by means of the or `NeuralSort` operator; this is analogous to the Gumbel-Softmax trick where the `argmax` operator that is applied to the samples is relaxed via the `softmax` operator [@2017_jang; @2017_maddison].
Sorting Handwritten Numbers
---------------------------
The *large-MNIST* dataset of handwritten *numbers* is formed by concatenating $4$ randomly selected MNIST *digits*. In this task, a neural network is presented with a sequence of $n$ large-MNIST numbers and must learn the permutation that sorts these numbers. Supervision is provided only in the form of the ground-truth permutation. Performance on the task is measured by:
1. The proportion of *permutations* that are perfectly recovered.
2. The proportion of *permutation elements* that are correctly recovered.
Note that the first metric is always less than or equal to the second metric. We use the setup in [@2019_cuturi] to be able to compare against their Optimal-Transport-based method. They use $100$ epochs to train all models.
The results for the first metric are shown in Table \[tab:run\_sort\]. We report the mean and standard deviation over 10 runs. We see that and `NeuralSort` perform identically for all values of $n$. Moreover, our results for `NeuralSort` are better than those reported in [@2019_cuturi], to the extent that `NeuralSort` and outperform the method of [@2019_cuturi] for $n = 9, 15$, unlike reported in said paper. We found that the hyperparameter values reported in [@2018_grover] and used by [@2019_cuturi] for `NeuralSort` are far from ideal: [@2018_grover] reports using a learning rate of $10^{-4}$ and temperature values from the set $\{1,2,4,8,16\}$. However, a higher learning rate dramatically improves `NeuralSort`’s results, and higher temperatures also help. Concretely, we used a learning rate of $0.005$ for all the and `NeuralSort` models, and a value of $\tau = 1024$ for $n = 3,5,7$ and $\tau = 128$ for $n = 9, 15$. The results for the second metric are reported in appendix \[app:run\_sort\_aux\]. In the appendix we also include learning curves for and `NeuralSort`, which show that they converge at the same speed.
Quantile Regression
-------------------
As in the sorting task, a neural network is presented with a sequence of $n$ large-MNIST numbers. The task is to predict the median element from the sequence, and this is the only available form of supervision. Performance on the task is measured by mean squared error and Spearman’s rank correlation. We used $100$ iterations to train all models.
The results are shown in Table \[tab:run\_median\]. We used a learning rate of $10^{-3}$ for all models - again, higher than that reported in [@2018_grover] - and grid-searched the temperature on the set $\{128, 256, 512, 1024, 2048, 4096\}$ - again, higher than that reported in [@2018_grover]. We see that and `NeuralSort` perform similarly, with `NeuralSort` sometimes outperforming and vice versa. The results for `NeuralSort` are also much better than those reported in [@2018_grover], which we attribute to the better choice of hyperparameters, concretely, the higher learning rate. In the appendix we also include learning curves for and `NeuralSort`, which show that they converge at the same speed.
Differentiable kNN
------------------
In this setup, we explored using the operator to learn a differentiable $k$-nearest neighbours (kNN) classifier that is able to learn a representation function, used to measure the distance between the candidates.
In a supervised training framework, we have a dataset that consists of pairs $(x, y)$ of a datapoint and a label. We are interested in learning a map $\Phi$ to embed every $x$ such that we can use a kNN classifier to identify the class of $\hat{x}$ by looking at the class of its closest neighbours according to the distance $\|\Phi(x) - \Phi(\hat{x})\|$. Such a classifier would be valuable by virtue of being interpretable and robust to both noise and unseen classes.
This is achieved by constructing episodes during training that consist of one pair $\hat{x}, \hat{y}$ and $n$ candidate pairs $(x_i, y_i)$ for $i = 1\dots n$, arranged in two column vectors $X$ and $Y$. The probability $P(\hat{y}|\hat{x},X,Y)$ of class $\hat{y}$ under a kNN classifier is the average of the first $k$ entries in the vector $$P_{\texttt{argsort}(-\|\Phi(X) - \Phi(\hat{x})\|^2)} \mathbb{I}_{Y = \hat{y}}$$ where $\|\Phi(X) - \Phi(\hat{x})\|^2$ is the vector of squared distances from the candidate points and $\mathbb{I}_{Y = \hat{y}}$ is the binary vector indicating which indexes have class $\hat{y}$. Thus, if we replace $P_\texttt{argsort}$ by the operator we obtain a differentiable relaxation $\widehat{P}(\hat{y}|\hat{x},X,Y)$. To compute the loss we follow [@2018_grover] and use $-\widehat{P}(\hat{y}|\hat{x},X,Y)$. We also experimented with the cross entropy loss, but the performance went down for both methods.
When $k=1$, our method is closely related to *Matching Networks* [@2016_vinyals]. This follows from the following result: (See proof in Appendix \[sec:proof\_of\_dknn\])
\[dknn\] Let $k=1$ and $\widehat{P}$ be the differentiable kNN operator using $\SimpleSort{}^{|\cdot|}_2$. If we choose the embedding function $\Phi$ to be of norm $1$, then $$\widehat{P}(\hat{y}|\hat{x},X,Y) =
{\sum_{i:y_i = \hat{y}} e^{\Phi(\hat{x})\cdot\Phi(x_i)}} \bigg/
{\sum_{i=1\dots n} e^{\Phi(\hat{x})\cdot\Phi(x_i)}}$$
This suggests that our method is a generalization of *Matching Networks*, since in our experiments larger values of $k$ yielded better results consistently and we expect a kNN classifier to be more robust in a setup with noisy labels. However, *Matching Networks* use contextual embedding functions, and different networks for $\hat{x}$ and the candidates $x_i$, both of which could be incorporated to our setup. A more comprehensive study comparing both methods on a few shot classification dataset such as *Omniglot* [@2011_lake] is left for future work.
We applied this method to three benchmark datasets: MNIST, Fashion MNIST and CIFAR-10 with canonical splits. As baselines, we compare against `NeuralSort` as well as other kNN models with fixed representations coming from raw pixels, a PCA feature extractor and an auto-encoder. All the results are based on the ones reported in [@2018_grover]. We also included for comparison a standard classifier using a convolutional network.
Results are shown in Table \[tab:knn\]. In every case, we achieve comparable accuracies with `NeuralSort` implementation, either slightly outperforming or underperforming `NeuralSort`. See hyperparameters used in appendix \[sec:experimental\_details\].
Speed Comparison
----------------
We set up an experiment to compare the speed of the and `NeuralSort` operators. We are interested in exercising both their forward and backward calls. To this end, we set up a dummy learning task where the goal is to perturb an $n$-dimensional input vector $\theta$ to make it become sorted. We scale $\theta$ to $[0, 1]$ and feed it through the or `NeuralSort` operator to obtain $\widehat{P}(\theta)$, and place a loss on $\widehat{P}(\theta)$ that encourages it to become equal to the identity matrix, and thus encourages the input to become sorted.
Concretely, we place the cross-entropy loss between the true permutation matrix and the predicted soft URS matrix: $$L(\widehat{P}) = - \frac{1}{n}\sum_{i = 1}^n \log \widehat{P}[i, i]$$ This encourages the probability mass from each row of $\widehat{P}$ to concentrate on the diagonal, which drives $\theta$ to sort itself. Note that this is a trivial task, since for example a pointwise ranking loss $\frac{1}{n} \sum_{i = 1}^n (\theta_i + i)^2$ [@2008_taylor Section 2.2] leads the input to become sorted too, without any need for the or `NeuralSort` operators. However, this task is a reasonable benchmark to measure the speed of the two operators in a realistic learning setting.
We benchmark $n$ in the range $100$ to $4000$, and batch $20$ inputs $\theta$ together to exercise batching. Thus the input is a parameter tensor of shape $20 \times n$. Models are trained for $100$ epochs, which we verified is enough for the parameter vectors to become perfectly sorted by the end of training (i.e., to succeed at the task).
In Figure \[fig:benchmark\_tensorflow\] we show the results for the TensorFlow implementations of `NeuralSort` and given in Figures \[code:neuralsort\] and \[code:simplesort\] respectively. We see that on both CPU and GPU, is faster than `NeuralSort`. For $N = 4000$, is about 6 times faster than the `NeuralSort` implementation of [@2018_grover] on both CPU and GPU. We tried to speed up the `NeuralSort` implementation of [@2018_grover], and although we were able to improve it, `NeuralSort` was still slower than , concretely: $80\%$ slower on CPU and $40\%$ slower on GPU. Details of our improvements to the speed of the `NeuralSort` operator are provided in appendix \[appendix:neuralsort\_performance\_improvement\].
The performance results for PyTorch are provided in the appendix and are similar to the TensorFlow results. In the appendix we also show that the learning curves of with $d = |\cdot|^2$ and `NeuralSort` are almost identical; interestingly, we found that using $d = |\cdot|$ converges more slowly on this synthetic task.
We also investigated if relying on a sorting routine could cause slower run times in worst-case scenarios. When using sequences sorted in the opposite order we did not note any significant slowdowns. Furthermore, in applications where this could be a concern, the effect can be avoided entirely by shuffling the inputs before applying our operator.
As a final note, given that the cost of sorting is sub-quadratic, and most of the computation is payed when building and applying the $n \times n$ matrix, we also think that our algorithm could be made faster asymptotically by constructing sparse versions of the operator. For applications like differentiable nearest neighbors, evidence suggests than processing longer sequences yields better results, which motivates improvements in the asymptotic complexity. We leave this topic for future work.
Conclusion
==========
We have introduced , a simple continuous relaxation for the `argsort` operator. The $r$-th row of the operator is simply the `softmax` of the negative distances to the $r$-th largest element.
has similar properties to those of the `NeuralSort` operator of [@2018_grover]. However, due to its simplicity, is trivial to implement, more modular, faster than `NeuralSort`, and proofs regarding the operator are effortless. We also empirically find that it is just as easy to optimize.
Fundamentally, advances the state of the art in differentiable sorting by significantly simplifying previous approaches. Our code and experiments can be found at <https://github.com/sprillo/softsort>.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Assistant Professor Jordan Boyd-Graber from University of Maryland and Visting Researcher at Google, and Thomas Müller from Google Language Research at Zurich for their feedback and comments on earlier versions of the manuscript. We would also like to thank the anonymous reviewers for their feedback that helped improve this work.
[^1]: This is called the `sort` operator in [@2018_grover]. We adopt the more conventional naming.
[^2]: Except when there are ties, which we assume is not the case. Ties are merely a technical nuisance and do not represent a problem for any of the methods (ours or other’s) discussed in this paper.
| {
"pile_set_name": "ArXiv"
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---
abstract: |
Theoretical arguments for a new higher-color quark sector, based on Pomeron physics in QCD, are briefly described. The electroweak symmetry-breaking, Strong CP conservation, and electroweak scale CP violation, that is naturally produced by this sector is also outlined.
A further consequence is that above the electroweak scale there will be a radical change in the strong interaction. Electroweak states, in particular multiple $W$’s and $Z$’s, and new, semi-stable, very massive , baryons, will be commonly produced. The possible correlation of expected phenomena with a wide range of observed Cosmic Ray effects at and above the primary spectrum knee is described.
Related phenomena that might be seen in the highest energy hard scattering events at the Fermilab Tevatron, some of which could be confused with top production, are also briefly discussed.
---
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[**NEW STRONG INTERACTIONS ABOVE THE ELECTROWEAK SCALE**]{}
Alan R. White[^1]\
High Energy Physics Division\
Argonne National Laboratory\
Argonne, IL 60439\
Invited talk presented at the 8th International Symposium on Very High Energy Cosmic Ray Interactions, July 1994, Waseda University, Tokyo, Japan.
The physics of very high energy Cosmic Rays, as seen in Mountain Emulsion Chambers and Extensive Air Showers, is predominantly that of strong interaction fragmentation and diffraction. It is well known in the Cosmic Ray community that a significant number of effects now suggest the existence of new strong interaction physics at energies around $10^{16}$ eV or higher. The most radical proposal being[@nik; @dy] that the famous “knee”, in the induced incoming energy spectrum, around this value is actually evidence for new physics rather than a discontinous change in the primary spectrum. The corresponding center of mass threshold for hadron-hadron scattering is $\sqrt{s} \sim 3-5$ TeV. This implies that the new physical processes can probably not be seen directly at the Fermilab Tevatron but, as I shall discuss, the physics involved might be glimpsed in the highest energy virtual processes .
I have studied the QCD Pomeron responsible for diffraction for many years, and for some time have advocated the theoretical necessity for a new, [**higher-color**]{}, quark sector. From my analysis[@arw1] this new sector is directly required for the consistency of the Pomeron with both confinement and perturbative QCD at high-energy. Remarkably such a sector can replace the unaesthetic Higgs sector of the Standard Model (in partial analogy with technicolor) and provide an essentially complete mechanism for mass generation in the electroweak sector. This links the strong and electroweak interactions in a direct manner and, in particular, implies that the electroweak scale is explained as a second QCD scale. As I have studied this possibility more and more seriously, I have gradually realized that other deep puzzles of the Standard Model may also be resolved. For example, the problems of Strong CP Conservation and CP Violation at the Electroweak Scale.
In this talk I will first explain qualitatively why the new quark sector is required in QCD and also describe the dynamical electroweak symmetry-breaking and CP properties that result. I will then spend the remainder of the talk elaborating on the essential feature for this conference. That is, not very far [**above the electroweak scale there will be a radical change in the strong interaction**]{}. Electroweak states, in particular multiple $W's$ and $Z's$, and new semi-stable baryons, will be commonly produced. We anticipate that diffractive production of the new states will be a major (if not the major) effect. Clearly this will dramatically change the nature of Cosmic Ray showers and the states they produce above such energies. Although very difficult to predict in any detail, I will suggest that the new phenomena to be expected have the right characteristics to explain, qualitatively, a wide range of observed Cosmic Ray effects, including the following.
- [Strong attenuation of family production, as observed in emulsion chambers, together with a sharp change in the electromagnetic and hadronic energy spectra.]{}
- [Small $X_{max}$ for high-energy air showers with $E^0 \sim 10^{17}$ eV together with a fast rise of $X_{max}$ as the energy increases.]{}
- [Shorter “hadronic” interaction length in emulsion and lead chambers.]{}
- [Anomalous penetration in the atmosphere and in detectors, often involving the production of intense “halos”.]{}
- [Coplanarity of multi-halos.]{}
- [Large $p_{\perp}$ production of “Centauros” - with low electromagnetic energy, and “Chirons” - with apparent anomalously low $p_{\perp}$ in secondary showers.]{}
- [Excess of (underground) muon pairs with large separation.]{}
- [Large zenith angle excess of high-energy air showers and azimuthal asymmetry in $\gamma$ and hadron family production.]{}
In general the situation seems very interesting and a reasonable case can be made that the type of modification of the strong interaction that I am arguing for is actually being observed in the highest energy Cosmic Rays. Of course, many of the above Cosmic Ray effects suffer from low statistics and it will remain essential that they be observed in accelerator experiments if they are to be confirmed and studied. The LHC will cover most of the relevant energy range but it is a decade away from operating. As I discuss at the end of the talk, it is also possible that the accumulating number of very high energy hard scattering events at the Fermilab Tevatron (mostly involving photon and weak vector boson states) could be a glimpse of the physics involved. Indeed [**this physics might well be closely correlated with top production**]{} and there could be confusion, experimentally, between new and expected production processes.
In pre-QCD days the Pomeron was a phenomenological object - a Regge pole which (essentially) was thought to reflect the low $k_{\perp}$ multiperipheral production of multiple pions (with, say, $<n> ~ \sim 10-20$). As illustrated in Fig. 1, $$\eqalign{
\sigma_T = \Sigma \int d\Omega_n |A_n|^2 ~\sim~ s^{\alpha_{\spom}(0) - 1}}
\auto$$ so that $\sigma_T \sim C$ requires $\alpha_{\spom}(0) = 1$. [**Pomeron**]{} (reggeon) [**Field Theory**]{} had both a phenomenological and, via reggeon unitarity, a theoretical basis[@arw2] as an effective field theory accounting for all the additional diffractive and absorptive effects that unitarity requires must accompany multiperipheral pion production. High-mass diffraction determines the magnitude of the triple Pomeron coupling. Multi-Pomeron diagrams can be thought of as representing multiplicity fluctuations. That is, an N-Pomeron state appearing on some part of the rapidity axis is associated with a multiplicity fluctuation of N times the average multiplicity in that rapidity interval. The origin of some reggeon graphs is illustrated in Fig. 2.
The renormalization group can be applied to Pomeron Field Theory and, with a triple Pomeron coupling, there is a [**Critical Pomeron**]{} solution[@mpt] for $\alpha_{\spom}(0) = 1$. This gives $\sigma_T ~\sim ~ (ln s)^{\eta}$ where $\eta$ is an anomalous dimension. The Critical Pomeron is the only known theoretical description of rising total cross-sections which satisfies [**all $s$ and $t$ - channel unitarity constraints.**]{}
Within Pomeron Field Theory we can study what happens if we initially violate unitarity by setting $\alpha_{\spom}(0) > 1$. The result is a new [**Super-Critical Pomeron**]{} phase[@arw2] in which there is a Pomeron condensate - giving rise to the vacuum production of Pomerons! At first sight, it is difficult to understand how the vacuum production of Pomerons could possibly have a physical interpretation. However, a detailed study of the structure of the induced graphs,the transverse momentum singularities that appear, and the resultant reggeon unitarity properties, leads to the following remarkable result.
Vacuum production of Pomerons directly produces additional particle pole factors in reggeon graphs that correspond to [**vector reggeon intermediate states**]{}. Consequently, the phase-transition when $\alpha_{\spom}(0) > 1$ involves the conversion of the divergences in rapidity (due to $\alpha_{\spom}(0) > 1$) into particle divergences in transverse momentum. In effect, if we try to make the total cross-section increase more rapidly than allowed by unitarity, a [**vector particle V is “deconfined”**]{} and appears in the theory coupling pair-wise to the Pomeron. The vector reggeon trajectory $\alpha_V(t)$ is exchange-degenerate with the Pomeron trajectory and, away from the critical point, the physical intercepts satisfy $\alpha_{\spom}(0) = \alpha_V(0) < 1$. As a result, in both the Sub-Critical and Super-Critical phases, we have $\sigma_T \to~ 0$ asymptotically. This implies that [**to obtain a rising cross-section the Pomeron must be Critical**]{}.
The Super-Critical Pomeron was discovered independently of QCD but a fundamental question is, of course, what is the physical interpretation (if any) of the Pomeron phase transition in QCD? Or, equivalently, can the vector particle V be related to a (deconfined) gluon?
In the multiperipheral model it is clear[@gt] that adding more hadron states increases $\alpha_{\spom}(0)$. Therefore, [**if**]{} there is a Regge pole Pomeron in QCD, we anticipate that adding quarks moves the Pomeron closer to criticality and that [**the Critical Pomeron might be related to QCD with the “maximum” number of quarks.**]{} All quarks become (close to) massless in the asymptotic Regge limit, and so we are led to ask - can the physics of QCD with a large number of massless quarks be related to the Pomeron phase-transition?
Several properties of (massless) QCD suggest that the Critical Pomeron does indeed occur when the number of flavors $N_f$ is a “maximum”. Since these can be described without introducing the technology of Reggeon Field Theory we briefly describe them here. $N_f = N_f^{max} = 16$ is the maximum value before asymptotic freedom is lost and (presumably) gluons are deconfined. (The deconfinement implies, of course, that there is a phase transition but can we identify the Critical Pomeron transition?) Consider first the behavior of the $\beta$-function as a function of $N_f$. It is very likely[@bz; @arw1] that at $N_f = N_f^{max}$ (and probably only at this value) the $\beta$-function develops an infra-red fixed-point, as illustrated in Fig. 3.
The scaling properties, and in particular, the variety of anomalous dimensions that develop at this fixed-point clearly could be directly related to the scaling properties of the Critical Pomeron. Indeed, it can be shown[@arw1] that in the limit of zero meson mass (which we assume to correspond to the limit of zero quark mass in QCD), the Critical Pomeron forward diffraction peak behaves as illustrated in Fig. 4. That is the diffraction pattern collapses into a simple peak that has[@arw1] the character of a massless vector singularity with an anomalous dimension, suggesting directly a relationship with a massless, fixed-point, vector theory.
Also, when $N_f = N_f^{max}$ (and [**only**]{} at this value), the first term in the $\beta$-function is small enough[@gw] that adding a triplet (“Higgs”) scalar to the theory retains asymptotic freedom of both the gauge coupling and the scalar coupling. This implies that for this “maximum” number of quarks, the (dynamical?) Higgs mechanism can break the gauge symmetry, from SU(3) to SU(2), without destroying the short-distance properties of the theory. Consequently [**a vector reggeon V, i.e. a massive gluon reggeon , can enter the theory smoothly**]{} (with no $k_{\perp}$ cut-off in particular) [**only when when $N_f = N_f^{max}$**]{}. Since the entry of V into the theory characterises the Super-Critical Pomeron we have another independent argument that $N_f = N_f^{max}$ is the critical point.
Of course, we expect confinement to be crucial for the emergence of a multiperipheral like Pomeron related to pion production etc.. The study of confinement in the Regge limit is a major topic. After an elaborate technical analysis utilising reggeon diagrams, it can be shown[@arw1] that within QCD
- [**quarks play a crucial role in the simultaneous emergence of confinement and a Regge pole Pomeron in the small $k_{\perp}$ high-energy regime**]{}
- [**the energy-dependence of small $k_{\perp}$ physics (i.e. $\alpha_{\spom}(0)$ )is strongly dependent on both $N_f$ and the $k_{\perp}$ cut-off and the Critical Pomeron occurs without a cut-off only when $N_f =
N_f^{max}$.**]{}
To remove the $k_{\perp}$ cut-off and obtain a smooth matching with QCD perturbation theory at large $k_{\perp}$, we must have a cross-section that does not go to zero asymptotically (i.e. $\alpha_{\spom}(0) = 1$), since this is what the perturbation expansion gives. Therefore, for confinement and QCD perturbation theory to coexist in the high-energy region, we must obtain the Critical Pomeron as the large $k_{\perp}$ cut-off is removed. Since this requires $N_f = N_f^{max}$, [**a further quark sector must exist!!**]{}
It is very important, however, that (assuming six flavors are known to exist)
- [**the further quark sector need not be 10 more flavors of color triplet quarks. $N_f = N_f^{max}$ is also produced by an additional flavor doublet of color sextet quarks.**]{}
From the perspective of QCD Pomeron physics, it is a remarkable coincidence that two flavors of color sextet quarks can provide[@wjm] a natural form of dynamical symmetry-breaking for the electroweak interaction which meshes perfectly with the observed experimental features. Indeed this provides a self-contained motivation for introducing the higher color quark sector which we can briefly outline as follows.
Consider adding to the Standard Model (with no scalar Higgs sector), a massless flavor doublet $(u_6,d_6)$ of color sextet quarks with the usual quark quantum numbers, except that the role of quarks and antiquarks is interchanged. For the $SU(2)\otimes U(1)$ anomaly to be cancelled there must also be other fermions with electroweak quantum numbers added to the theory[@wjm; @kk], but we shall not consider this here except to note that this could be the color octet leptoquarks discussed below. We consider first the QCD interaction of the massless sextet quark sector. There is a $U(2)\otimes
U(2)$ chiral flavor symmetry. QCD chiral dynamics will break the axial symmetries spontaneously and produce four massless pseudoscalar mesons (Goldstone bosons), which we denote as $\pi^+_6,\;\pi^-_6,\;\pi^0_6$ and $\eta_6$, in analogy with the usual notation for mesons composed of $u$ and $d$ color triplet quarks.
As long as all quarks are massless, QCD is necessarily $CP$ conserving in both the sextet and triplet quark sectors. Therefore, in the massless theory we can, in analogy with the familiar treatment of flavor isospin in the triplet quark sector, define sextet quark vector and axial-vector currents $V^{\tau}_{\mu}$ and $A^{\tau}_{\mu}$ which are “isotriplets” under the unbroken $SU(2)$ vector flavor symmetry and singlet currents $v_{\mu}$, $a_{\mu}$. The pseudoscalar mesons couple “longitudinally” to the axial currents, that is $$\eqalign{ <0|A^\tau_{\mu}|\pi^{\tau}_6(q)>~\sim F_{\pi_6}q_{\mu}~~,
{}~~~~~~<0|a_{\mu}|\eta_6(q)>~\sim F_{\eta_6}q_{\mu}}
\auto$$ while the vector currents remain conserved. (Note that $a_{\mu}$ should actually contain a small admixture of the triplet quark flavor singlet axial current if it is to generate the U(1) symmetry orthogonal to that broken by the QCD $U(1)$ anomaly).
We consider next the coupling of the electroweak gauge fields to the sextet quark sector. The massless $SU(2)$ gauge fields $W^{\tau}_{\mu}$ couple to the isotriplet sextet quark currents in the standard manner, that is $$\eqalign{{\cal L}_I=gW^{\tau\mu}\Bigl(V^{\tau}_{\mu}-A^{\tau}_{\mu}\Bigr)}
\auto$$ It follows from (2) and (3) that the $\pi^+_6,\;\pi^-_6$ and $\pi^0_6$ are “eaten” by the $SU(2)$ gauge bosons and (after the hypercharge interaction is included) respectively become the third components of the $W^+,\;W^-$ and $Z^0$. Consequently, QCD chiral symmetry breaking generates masses for the $W^+,\;W^-$ and $Z^0$ with $M_W\sim g\;F_{\pi_6}$ where $F_{\pi_6}$ is [*a QCD scale*]{}. We anticipate that the relative scales of triplet and sextet chiral symmetry breaking are determined by the “Casimir Scaling” rule[@wjm], i.e. if $C_6$ and $C_3$ are sextet and triplet Casimirs respectively, then $$\eqalign{C_6\alpha_s(F^2_{\pi_6})~\sim~C_3\alpha_s(F^2_{\pi})}
\auto$$ which is consistent with $F_{\pi_6}\sim 250$ GeV!
We conclude that a sextet sector of $QCD$ produces a special version of “technicolor” symmetry breaking in which [**the electroweak scale is naturally explained as a second $QCD$ scale**]{}. Also since we are completely restricted to a flavor doublet the form of the symmetry-breaking is automatically equivalent to that of an $SU(2)$ Higgs sector and so $$\eqalign{\rho=~(M^2_W/M^2_Zcos^2\theta_W)~=~1}
\auto$$ as required by experiment.
Therefore introducing a sextet quark sector not only produces a matching of the asymptotic freedom and confinement properties of $QCD$ via the Critical Pomeron, but also gives a natural solution to the major problem of today’s Standard Model i.e. the nature of electroweak symmetry breaking. [**The sextet sector may**]{}, as we now discuss, [**also be deeply tied up[@kkw; @arw3] with the issue of Strong $CP$ conservation.**]{}
The $\eta_6$ is not involved in generating mass for the electroweak gauge bosons, but instead remains as a Goldstone boson associated with a $U(1)$ axial chiral symmetry. It is therefore an axion[@wil] in the original sense of the Peccei-Quinn mechanism[@pq] and it remains massless until triplet quark masses are added to the theory. In the present context, this involves the addition of triplet/sextet four-fermion couplings (that should ultimately be traceable to a larger unifying gauge group), which, when combined with the sextet quark condensate, provide triplet quark masses. That $CP$ remains conserved by QCD triplet quark interactions then follows from the original Peccei-Quinn argument utilising the sextet axial $U(1)$ symmetry.
At this stage another very important property of QCD with $N_f = N_f^{max}$ is crucial. It seems that renormalon singularities are completely absent in the Borel plane[@arw1; @cjm]. This implies that perturbation theory is much more convergent and that [**instanton interactions are both infra-red finite and provide all the non-perturbative physics**]{} of the theory. Instanton interactions are therefore well-defined at the lowest infra-red scale of the theory, i.e. the electroweak scale. Combining this with the extremely slow evolution of the gauge coupling, the instanton interactions are then enhanced by integration over an extremely wide size range (for the instanton involved). Consequently the $\eta_6$ can aquire a large, i.e. electroweak scale, mass as a result of electroweak scale (and higher) color instanton interactions[@arw3] and, unlike a conventional Peccei-Quinn axion, is certainly not ruled out experimentally. Indeed it may even have been seen[@kkw]. Clearly all of the particular properties of QCD with color sextet quarks play an intrinsic role in this very special resolution of the Strong CP problem.
A rather complicated set of fermion vertices is actually generated by the electroweak scale instanton interactions. Because of the distinct Casimirs involved, the singlet current $$\eqalign{J^0_{\mu}~=~a^6_{\mu}-5a^3_{\mu}}
\auto$$ is conserved in the presence of instantons (6 and 3 now denote sextet and triplet currents respectively). Consequently the minimum instanton interaction involves one quark/antiquark pair of each triplet flavor and five pairs of each sextet flavor. Combining this interaction with the existence of both sextet and triplet chiral condensates (and, also, four-fermion vertices coupling triplets and sextets) a wide assortment of fermion vertices is produced. As we discuss further in the next Section, we expect that these vertices will play a major role in strong interactions above the electroweak scale.
Finally we note that [**the sextet sector may also be responsible for $CP$ violation at the weak scale**]{}. Because the [**sextet sector has no axion**]{} the QCD interactions at this scale will naturally be [**“Strong $CP$-violating”. The familiar triplet quark hadrons will contain a small admixture of sextet quark states - which could provide their $CP$ violating interactions**]{}.
Before we go on to the the new strong interactions and their consequences for Cosmic Ray physics, we would like to emphasize the (unconventional) implication of the foregoing arguments. Namely that understanding the intricacies of the strong interaction may actually provide answers to remaining problems of the weaker interactions. Or equivalently
- [**the QCD Pomeron may be the Key to Many of the Remaining Puzzles of the Standard Model**]{}
Above the sextet chiral scale, that is the electroweak scale, the sextet sector will be a major part of the QCD interaction. $QCD_{max}$ (that is QCD with $N_f = N_f^{max}$ - via the triplet and sextet sectors) is a very different gauge theory to those conventionally studied. The gauge coupling is relatively small and effectively does not run. While the sextet sector can, presumably, be integrated out to give conventional QCD at low energies, at high energy we can expect very different behavior. Some “non-perturbative” physics will perhaps be understood via conventional non-perturbative QCD ideas in terms of sextet flux tubes etc.. However, many non-perturbative effects will surely be directly dependent on the multitude of higher-order (instanton) fermion interactions involving sextet quarks. (We have emphasized that these interactions are enhanced by the gauge coupling not running). As illustrated in Fig. 5, these interactions will generate high-order vertices coupling $W$’s, $Z$’s, and $\eta_6$’s, with $${\openup3\jot
\eqalign{ \Gamma_{mW,~nZ,~r\eta_6}~~
&\sim ~~~{F_{\pi_6}~(momentum scale)^2 \over \VEV{q_6\bar{q}_6}} ~~~
\Gamma_{(m-1)W,~nZ,~r\eta_6}\cr
&\sim ~~~{F_{\pi_6}~(momentum scale)^2 \over \VEV{q_6\bar{q}_6}} ~~~
\Gamma_{mW,~(n-1)Z,~r\eta_6}\cr
&\sim ~~~{F_{\pi_6}~(momentum scale)^2 \over \VEV{q_6\bar{q}_6}} ~~~
\Gamma_{mW,~nZ,~(r-1)\eta_6}\cr}}
\auto$$ where $\VEV{q_6\bar{q}_6}$ is the sextet condensate and $(\VEV{q_6\bar{q}_6})^{{1 \over 3}}~\sim~ F_{\pi_6}~\sim$ 250 GeV.
The $W^{+,-}$, $Z^0$ and $\eta_6$ are the “PIONS” of the sextet sector and, as we have just described, they will be multiply produced, via a “hard” interaction at the electroweak scale. Since the mass and decay properties of the $\eta_6$ are not well understood[@kkw] and it has, of course, not yet been discovered, we will concentrate mainly on multiple $W$ and $Z$ production.
Because of the Casimir effect, we anticipate that sextet states will have a stronger coupling to gluons, and hence to the Pomeron, than does the triplet sector. Therefore
- [**sextet states will have larger hadronic cross-sections than triplet states (i.e. conventional hadrons)**]{}
My work[@arw1] on high-energy hadrons interacting via the Pomeron can be heuristically understood if we visualize a hadron as a conventional bag containing quarks but with the surface containing a “topological condensate” due to instanton interactions and expanding as illustrated in Fig. 6.
- Heuristic picture of a high-energy hadron and the Pomeron as perturbative gluon exchange in a topological condensate background.
In first approximation, the Pomeron can then, as illustrated, be thought of as one gluon exchange within the overlapping topological gauge fields of the scattering hadrons. (Note that this automatically gives the “additive quark model” result that the Pomeron couples directly to a single quark in a hadron).
The topological gauge fields of the hadrons will also, via instanton interactions, be responsible for multiple $W$ and $Z$ production accompanying the perturbative gluon interaction. Therefore we conclude that a major component of the new strong interactions above the electroweak scale will be
- [**diffractive production, with very high transverse momentum, of states containing large numbers of $W$’s and $Z$’s.**]{}
This will be one major ingredient of our discussion of Cosmic Ray effects.
Next we note that the sextet quark sector will produce new BARYONS of the form $$\eqalign{~\bar{q}_6qq~,~q_6\bar{q}\bar{q}~,~\bar{q}_6\bar{q}_6q~,
{}~q_6q_6\bar{q}~,~\bar{q}_6\bar{q}_6\bar{q}_6~,~
q_6q_6q_6~,}
\auto$$ There will also be VECTOR MESONS of the form $$\eqalign{q_6\bar{q}_6}
\auto$$
The sextet quark constituent mass is presumably of the same order of magnitude or a little larger than the chiral scale and so for definiteness we will take it to be $\sim 400$ GeV. Clearly the lightest new states will be the BARYONS containing just one sextet quark. Their mass will be very close to the sextet mass i.e. $\sim 400$ GeV and since, within $QCD_{max}$, sextet and triplet baryon numbers are separately conserved, they will be very stable. We refer to BARYONS containing two (triplet) quarks as $P$’s and those containing two antiquarks as $\bar{P}$’s. The VECTOR MESONS will decay into the PIONS of the theory and so will give resonance production of $W$’s, $Z$’s and $\eta_6$’s at the TeV scale. The higher mass BARYONS will presumably decay into $P$’s and $\bar{P}$’s (together with appropriate combinations of normal hadrons).
For the next Section it will be crucial that [**the $P$’s and $\bar{P}$’s are sufficiently stable that sometimes (but not always) they survive a trip (with collisions) from near the top of the atmosphere down to mountain-top detectors.**]{}
If the $P$’s and $\bar{P}$’s are to decay, there must be a further (unifying) interaction coupling the two distinct quark sectors. At first sight this could be a high mass (GUT) gauge boson. But the absence of proton decay probably makes it very difficult to construct such a theory consistently if the BARYONS are to decay much faster than protons! An alternative[@kk] is that within the unified theory, there are further color octet quarks ($q_8$) (these could be “leptoquarks”) that enter at a mass scale just a few orders of magnitude above the electroweak scale. At this scale the unified theory can be asymptotically free even though the QCD subsector will not be. If the unified theory is chiral then in general two-fermion condensates are not gauge invariant. Gauge-invariant condensates must contain at least four fermion fields and so it is natural[@kk] to expect that, at the new high mass scale, condensates of the form $$\eqalign{ \VEV{q_8q_8q_6q_3}}
\auto$$ will exist. QCD instanton interactions, at this scale, will then produce the appropriate sextet quark decays. We therefore assume that
- [**$P$’s and $\bar{P}$’s are “semistable” with a decay rate determined by a mass scale much larger than the electroweak scale. We anticipate that their decay modes will include states containing multiple $W$’s and $Z$’s.**]{}
Clearly BARYONS can be pair-produced diffractively by the Pomeron (in particular, via an instanton interaction), also VECTOR MESONS can be produced with an accompanying $W$’s and/or $Z$’s. From the experimental evidence on the diffractive production of strange baryons[@R608] illustrated in Fig. 7, we can assume that this production process will have some important properties, which we can explain as follows.
Because the Pomeron couples predominantly to a single quark in a proton, two constituent quarks persist in the forward direction of the initial proton (with around 90% probability) during any diffractive excitation process. If a new forward going baryon is to be formed then this is achieved by the vacuum production of additional quark-antiquark pairs in the center of mass of the scattered quark and the forward going diquark system. (This process can be an instanton interaction). There are two consequences of this production mechanism which will carry over directly into the diffractive production of BARYONS.
Firstly, because only a single quark can be replaced in the fast proton if a BARYON-ANTIBARYON pair is produced
- [**there is a charge bias in the production of the forward produced BARYON - it is necessarily positively charged or neutral. Correspondingly the charge of the ANTIBARYON state produced away from the forward direction is either negative or neutral.**]{}
Secondly, if all vacuum pairs involved are produced (almost) at rest in the center of mass of the scattered quark and diquark system then, as is illustrated by the data for diffractive production of $\Lambda^0\bar{\Lambda}^0$ pairs shown in Fig. 7,
- [**the full diffractively produced state is approximately coplanar - it lies in the plane formed by the momenta of the forward going fast BARYON and the (ANTI-)BARYON with the smallest forward momentum.**]{}
With all of the properties highlighted in this Section in hand we can now try to explain at least part of the wide range of Cosmic Ray exotica.
In this Section I will go through the phenomena I listed in the Introduction, giving a brief summary of the experimental results and then describing their interpretation in terms of the physics of the last Section. Since I am not an expert I may well have misunderstood some of the phenomena involved. If so I apologize to the authors involved.
[**Strong attenuation of family production, as observed in emulsion chambers, together with a sharp change in the electromagnetic and hadronic energy spectra.**]{}
Such effects have been seen by the Chacaltaya and Pamir collaborations and more distinctively in the highest energy results of the HADRON experiment at Tien-Shan. Figs. 8(a) and 8(b) show that the combination of low family flux and small energy spectrum indices for constituent showers in the Chacaltaya/Pamir data[@cp] is not fit by any of the conventional models. As shown in Fig. 8(c), the discrepancy is less if a heavy nuclei primary composition is assumed.
Fig. 9(a) shows a possibly related effect in the data[@nik] from the HADRON experiment. The $\gamma$ spectrum of shower cores scales up to a certain energy and then softens as the energy increases. As is shown, the softening could be reproduced by a heavy primary composition but the overall intensity would be much too high. Fig. 9(b) shows that the assumption of a heavy primary composition is inconsistent with the muon multiplicity distribution obtained at Tien-Shan.
The change of the $E_{\gamma}$ spectrum suggests the existence of a physical threshold around the knee energy. It also implies that, for some fraction of the Chacaltaya/Pamir events, the primary energy may be higher than given by conventional physics models. Therefore new physics above the threshold may be involved in these events also.
My explanation of these phenomena is close to that already suggested by those working on the HADRON experiment[@nik]. At high enough energies, production of the heavy, semistable, $P$’s and $\bar{P}$’s will be a significant part of the diffractive and fragmentation cross-sections. Since these BARYONS are semi-stable they will propagate for large distances within the shower, sometimes reaching the detector. The evolution of that part of the shower energy not in the heavy BARYONS wil be normal but will clearly produce far fewer gamma and hadron families. This effect is labeled “fragmentation region disappearance" by Nikolsky[@nik] who argues that the particles involved should have a mass $\geq$ 400 GeV. This explanation achieves the same effective reduction of primary energy as heavy nuclei primaries would do, but without the high multiplicity muon production that is not seen in Fig. 9(b).
A heavy primary composition for energies around $10^{16}$ GeV is also incompatible with the Soudan 1 underground muon multiplicity[@soud]. Recent MACRO data[@mac] shown in Fig. 10(a) leads to a similar conclusion. As is shown in Fig. 10(b), the high multiplicity tail of this distribution is determined by the highest primary energies, whatever (conventional physics) composition model is utilised. [**The absence of a large number of high multiplicity muon events is clearly a major problem for any explanation of very high energy Cosmic Ray phenomena that apppeals to a large heavy nuclei composition.**]{}
[**Small $X_{max}$ for high-energy air showers with $E^0 \sim 10^{17}$ eV together with a fast rise of $X_{max}$ as the energy increases.**]{}
This is the Fly’s Eye result[@fly]. As illustrated in Fig. 11, results for the lowest energies i.e. $E^0 \sim 10^{17}$ eV, give a sufficiently low average value for $X_{max}$ that a very strong heavy nuclei composition has to be used to fit the data with conventional physics models. However, as the energy rises this average increases too fast and the distribution changes too much for a single composition model to fit the data. It is necessary to vary the composition with energy as illustrated. The initial heavy nuclei composition is again at variance with the lack of high-multiplicity underground muon events mentioned above.
My explanation here is, in part, the same as for the previous effect. At the lower end of the energy range the production of the heavy, semistable, BARYONS will reduce the development of the shower and the consequent average $X_{max}$ in the same manner as the heavy nuclei composition. However, as we get to energies high compared even to the sextet scale we can expect that, in analogy with the triplet sector, high multiplicity PION states will be the dominant sextet states produced. That is the production of $W$’s, $Z$’s and $\eta_6$’s will dominate. Since these states are all unstable the showers will develop more like normal proton showers. This could produce naturally the required energy dependence of the $X_{max}$ distribution without any dramatic change in composition.
[**Shorter “hadronic” interaction length in emulsion and lead chambers.**]{}
The results of the Chacaltaya/Pamir collaboration[@cp1] are shown in Fig. 12. Both in the Chacaltaya emulsion chambers and in the Pamir lead chambers there is a pronounced decrease in the hadronic interaction length in the highest energy showers.
I attribute this in part to the higher hadronic cross-section of those sextet BARYONS that reach the detector. Also multiple $W$, $Z$ and $\eta_6$ intermediate states will produce major decay modes involving heavy flavors, leptons, and photons, which may be partly responsible for the effect.
[**Anomalous penetration in the atmosphere and in detectors, involving the production of intense “halos” in the highest energy showers.**]{}
Examples of events which have extreme penetration in lead[@cp1] and in emulsion chambers[@cp] are shown in Fig. 13. Some of them continue producing new showers down to an extraordinary depth. At the highest energies the cores of such showers contain very intense halos recorded on X-ray films.
These effects have to be produced by BARYONS that enter the detector. Multi-halo events should presumably be interpreted as involving multi-BARYON states, although the initial production of very energetic $W$’s and $Z$’s could also be involved. In the very highest energy events there could even be VECTOR MESON resonances.
[**Coplanarity of multi-halos.**]{}
The coplanarity of multi-halos in very high energy showers is a striking phenomenon having an established statistical significance. Results from the Pamir collaboration[@pam] are illustrated in Fig. 14. The X-rays for individual events are shown as well as the energy-dependence of the alignment. A table also illustrates how conventional models fail to produce the alignment. The experimenters emphasize that the total cross-section for halo events is far too large for them to originate from minijet configurations[@hal].
My description of the alignment phenomenon in diffractive production of BARYONS provides a direct explanation of this phenomenon. It is the same as is seen in the diffractive production of strange baryons at the ISR!
[**Large $p_{\perp}$ production of “Centauros” - with low electromagnetic energy, and “Chirons” - with apparent anomalously low $p_{\perp}$ in secondary showers.**]{}
Familiar plots of hadronic versus electromagnetic energy for Chacaltaya/Pamir[@cp] data and the comparison with simulations are shown in Fig. 15. Centauro events represent the extreme of a general phenomenon that less electromagnetic energy is produced than in normal pion production events. The overall $p_{\perp}$ involved is apparently large but from their narrowness, the $p_{\perp}$ in secondary showers appears to be anomalously small. The general class of events with these $p_{\perp}$ properties are referred to as Chirons.
As we have said, BARYONS will generally be produced in the initial atmospheric collision of the Cosmic Ray primary. We can assume they will sometimes decay directly just above or in the detector. Often they will undergo secondary collisions and then decay similarly. The collisions and (or) the decay will involve very high initial $p_{\perp}$ and can take place sufficiently close to the detector that secondary $p_{\perp}$ within produced showers is normal even though they appear anomalously narrow. Since multiple $W$, $Z$ and $\eta_6$ intermediate states will again be involved, we can anticipate that in general the production of heavy flavors, taus and muons, will produce final states that will be interpreted as anomalously “hadron-rich” in the detector.
[**Excess of (underground) muon pairs with large separation.**]{}
The underground muon experiments also measure the distribution of the distance separation of muon pairs. The MACRO distributions[@mac1] are shown in Fig. 16. There is an apparent excess at large distances which would not be expected from conventional physics models. Not surprisingly, I would like to interpret the excess as evidence that $Z^0$’s are being directly produced, with a hadronic cross-section, in high-energy Cosmic Rays. Potentially $Z^0$ events could be explicitly identified. This could provide, strong, direct evidence for new physics such as I am proposing.
[**Large zenith angle excess of high-energy air showers and azimuthal asymmetry in $\gamma$ and hadron family production.**]{}
Finally I come to some further results from the HADRON experiment. Fig. 17 shows[@dy] the zenith angle dependence for high-energy showers at two energies. The straight lines are conventional physics simulations at the two energies. There is a clear excess at large zenith angle at the highest energy. Also shown, in Fig. 17(b), is the zenith angle dependence of a break in the general size (energy) spectra of the showers. It is interesting that the break is essentially independent of the zenith angle and is located at the energy of the knee, in the conventionally induced primary energy spectrum. Since showers at different angles degrade differently in the atmosphere this, in itself, suggests that there is some physics effect in the break which is not simply related to the primary spectrum. Indeed it clearly leads to the suggestion[@nik; @dy], referred to in the Introduction, that there is a “new physics” effect involved in the knee, and not just a simple change in spectrum.
In Fig. 18 we show the most exotic (and, if it should be confirmed, perhaps the most exciting) result from the HADRON experiment. A striking asymmetry in the azimuthal angular dependence[@che] of the large zenith angle showers is shown. This could perhaps be explained[@che] as due to the earth’s magnetic field if massive, [**negatively charged**]{}, particles are preferentially responsible for the showers.
I interpret these last results as evidence that in initial atmospheric collisions secondary, semi-stable, very energetic, particles are produced at varying (relatively) large angle, which are then interpreted as (separated) large zenith angle primary showers. If such secondary particles are responsible for the excess, this could explain why the spectrum break is at the same shower size independently of the zenith angle. My proposal is, of course, that the secondary particles are BARYONS. According to our diffractive production argument above, the larger angle BARYONS contributing to the larger zenith angle showers will be [**preferentially negatively charged.**]{}
It seems possible therefore that [**the azimuthal asymmetry could be explained by the charge asymmetry of the larger angle versus forward angle diffractive production of BARYON pairs**]{} described in the last Section.
If the new physics seen in Cosmic Rays were simply diffractive production of $W$ and $Z$ pairs (or perhaps $\eta_6$ pairs) then we might estimate the effective threshold to be, say, $x = (1 - M^2/s) \geq 0.96$ i.e. $$\eqalign{ \sqrt{s}~~\geq~~5~M ~~&\sim~~5~\times~160~~ GeV\cr
&\sim ~~800~ GeV}$$ and so to be visible at the Fermilab Tevatron. Indeed, it remains possible that $W$ pairs are produced, in some number, relatively far forward since this would be impossible to determine with the present detectors.
From our discussion in previous sections it is clear that the more distinctive effects involve at least BARYON pair production. The corresponding diffractive threshold would then be roughly $$\eqalign{ \sqrt{s}~~\geq~~5~M ~~&\sim~~5~\times~800~~ GeV\cr
&\sim ~~4~ TeV}$$ which is consistent with the Cosmic Ray effects. Of course, we can also expect some effects of the new sector to show up at energies well below the diffractive threshold.
Indeed we might expect the new quark sector to first show up in the highest transverse energy (but very rare) hard scattering events. Instanton interactions will provide transitions from the (light) triplet quark sector, to the sextet sector. Amongst the simplest possible states that can be produced are $W^+W^-$, $Z^0Z^0$, $\eta_6\eta_6$, $Z^0\gamma$ and $\gamma\gamma$. There were indications from UA1[@UA1] that the hard scattering cross-section for $W$ pairs is indeed anomalously large. The events at CERN were detected in the $WW \to $ leptons + 2 jet channel, which, of course, has a relatively large branching ratio. However, at the Tevatron this channel may be obscured since the background from conventional QCD processes is much larger than at the CERN collider. Nevertheless an excess of very high energy hard scattering events may be accumulating at the Tevatron (including $Z^0\gamma$ and $\gamma\gamma$ events). Although these events are not yet statistically significant, they may become so as the experiments continue to take data.
Enhanced electroweak scale instanton interactions can provide transitions from the familiar (light) triplet quark sector, not only to the sextet sector we have been discussing, but also to states that are a combination of sextet and (preferentially) heavy triplet quarks, and even to purely triplet heavy quark states. This implies that the top quark (with a mass of $\sim$ 170 Gev according to recent CDF results[@cdf]) may have a larger production cross-section than standard perturbative estimates would give. Additional states that can be produced include $$\eqalign{ W^+W^-~+ ~b\bar{b}~,~~~~Z^0Z^0~+~b\bar{b}~, ~~~...}$$ The first state can clearly be directly confused with top production. Indeed in the CDF analysis[@cdf] searching for candidate top events, a few events have been found which are candidates to be identified with the second final state. In many respects, these events strongly resemble those identified as top events, but they should not be present at all according to the Standard Model. This clearly suggests that some of the candidate top events might in fact be direct $WW + b\bar{b}$ events. CDF also has a clear $WZ$ event which has a very low probability to occur, according to the Standard Model. An instanton interaction has to conserve charge in the sextet sector but could produce a $WWZ$ state, with one $W$ in a region of phase space where it escapes detection.
We conclude that [**a glimpse of sextet quark physics at the Tevatron collider may have already been provided**]{}. As data is accumulated it should become clear whether this is indeed the case. If it is, [**there will be a lot more than top quark production that provides “new physics” in the highest-energy hard scattering events.**]{}
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[**Figure Captions**]{}
- Multiperipheral pion production generating the Regge pole Pomeron.
- Varying multiplicity densities on the rapidity axis generate higher-order Pomeron diagrams.
- Evolution of the $\beta$-function with $N_f$ - (a) $N_f \sim 5,6$ (b) $N_f \sim 14,15$ (c) $N_f = 16$
- The Critical Pomeron asymptotic diffraction peak appproaches a simple peak if quark masses are sent to zero.
- Instanton interactions combined with condensates generate multiple vertices for $W$’s, $Z$’s and $\eta_6$’s,
- Coplanar diffractive production of strange baryons in experiment R608 at the ISR.
- Relation between the the family flux and power indices of the energy spectrum $\beta$ (a) for single core showers and (b) for shower clusters, in the energy range of 10-50 TeV, for the joint (J), Pamir (P) and Chacaltaya (C) chambers, compared to simulation models. (b) The single-core comparison when a heavy primary composition is used in the models.
- \(a) Energy spectra of $\gamma$-quanta and electrons in EAS cores with varying primary energies. The wide shaded strip is a simulation with primary protons, the narrow strip is with a heavy nuclei composition. (b) The experimental muon multiplicity distribution compared to simulations with 1) heavy nuclei primary composition 2) an increasing inelasticity coefficient and 3) the energy of the fragmentation region is lost from the hadron-electron cascades.
- \(a) The underground muon multiplicity distribution results from the MACRO experiment. (b) The relationship between muon multiplicities and primary energy in conventional physics simulation models.
- Average $X_{max}$ as a function of primary energy. Black dots : data. Open squares : simulation with a proton dominant primary composition. Open circles : simulation with a dominant heavy nuclei composition. Diamonds : simulation with an energy-dependent composition.
- \(a) Distribution of shower starting position for Chiron-type families observed by Chacaltaya two-storey chambers. The dotted line represents exponential decrease with the geometrical attenuation mean free path. (b) The $\Delta$T distribution of 170 hadrons with $E_{\gamma} \geq 10$ TeV in 16 high energy famlies with $E_{family} \geq 700$ TeV observed in the thick lead chambers of the Pamir experiment. (c) For comparison the $\Delta$T distribution of all hadrons in the Pamir lead chamber.
- \(a) Shower transitions with spot darkness plotted against depth for two events in the Pamir lead chambers (b) Examples of strongly penetrating, small spread, shower clusters in Chacaltaya chambers.
- a\) Darkness contours on X-ray films for 6 Pamir multicore halo events. b) The energy dependence of the percentage of four-core events satisfying $\lambda_4 \geq 0.8$ where $\lambda_4$ is a suitably defined alignment parameter. c) Table comparing alignment percentages for simulations and experimental data.
- \(a) The upper graph is a scatter plot of the number of hadrons, $N_h (E_h^{(\gamma)} \geq 4$ TeV) in a family and the fraction of the family energy carried by hadrons $Q_h ~(\equiv \sum E_h^{(\gamma)} /
\sum E_{(\gamma)} + E_h^{(\gamma)})$. Closed circles are for 173 families in the joint chambers and 135 families in the Pamir chambers, and open circles are for 121 families in the Chacaltaya chambers. The lower graph is a simulation. The primary composition assumed is - solid black dot = proton, open dot = alpha, diamond = CNO, x = heavy, + = iron. (b) Is the same as (a) but with the families selected to have lateral spread $\VEV{E^*R^*} < 300$ GeV.m.
- The MACRO distance separation for muon pairs compared with simulations (a) for muon pairs within all muon events (b) for dimuon events only.
[^1]: Work supported by the U.S. Department of Energy, Division of High Energy Physics, ContractW-31-109-ENG-38
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Consider a family of Boolean models, indexed by integers $n \ge 1$, where the $n$-th model features a Poisson point process in $\R^n$ of intensity $e^{n \rho_n}$ with $\rho_n \to \rho$ as $n \to \infty$, and balls of independent and identically distributed radii distributed like $\bar X_n \sqrt{n}$, with $\bar X_n$ satisfying a large deviations principle. It is shown that there exist three deterministic thresholds: $\tau_d$ the degree threshold; $\tau_p$ the percolation threshold; and $\tau_v$ the volume fraction threshold; such that asymptotically as $n$ tends to infinity, in a sense made precise in the paper: (i) for $\rho < \tau_d$, almost every point is isolated, namely its ball intersects no other ball; (ii) for $\tau_d< \rho< \tau_p$, almost every ball intersects an infinite number of balls and nevertheless there is no percolation; (iii) for $\tau_p< \rho< \tau_v$, the volume fraction is 0 and nevertheless percolation occurs; (iv) for $\tau_d< \rho< \tau_v$, almost every ball intersects an infinite number of balls and nevertheless the volume fraction is 0; (v) for $\rho > \tau_v$, the whole space covered. The analysis of this asymptotic regime is motivated by related problems in information theory, and may be of interest in other applications of stochastic geometry.'
author:
- |
Venkat Anantharam\
[*EECS Department*]{}\
[*University of California, Berkeley*]{}\
[email protected]\
- |
François Baccelli\
[*Department of Mathematics and ECE Department*]{}\
[*University of Texas, Austin*]{}\
and\
[*INRIA/ENS*]{}, [*Paris, France*]{}\
[email protected] [*or*]{} [email protected]
title: |
The Boolean Model in the Shannon Regime:\
Three Thresholds and Related Asymptotics
---
Introduction
============
The Boolean model was considered in high dimensions in a few papers, both within the framework of stochastic geometry [@Gouere; @PenroseCP] and within the framework of information theory [@AB]. The present paper discusses three thresholds and some asymptotics related to these thresholds in a setting analogous to that in [@AB], which is that where the radii of the balls in the Boolean model scale with the dimension $n$ of the ambient space like ${\bar X_n} \sqrt{n}$, where $({\bar X_n}, n \ge 1)$ is a sequence of random variables. In this paper, we assume that this sequence of random variables satisfies a large deviations principle (LDP).
The first threshold is the volume fraction threshold, which is the threshold at which the probability of coverage of the origin by the Boolean model switches from asymptotically vanishing to asymptotically approaching $1$ as the dimension $n$ tends to $\infty$. The second one is the percolation threshold; it was first studied in detail in [@PenroseCP] in the particular case where $\bar X_n$ is a constant. The case with random $\bar X_n$ was also discussed in [@Gouere]. This is the threshold at which the probability of percolation in the Boolean model switches from asymptotically approaching $0$ to being asymptotically nonzero as $n \to \infty$. The last is the degree threshold. This is the threshold at which the mean number of grains of the Palm version of the Boolean model that intersect the grain of the origin switches from asymptotically being finite to asymptotically approaching $\infty$ as $n \to \infty$. It is not hard to see that these three thresholds are decreasing in the order in which they were presented.
The main new contributions of the present paper are (a) representations of these three thresholds in terms of optimization problems based on the rate function of the LDP and (b) explicit asymptotics for various rates of convergence in the neighborhood of these thresholds.
Setup {#s.setup}
=====
In each dimension $n \ge 1$ we have a homogeneous Poisson process of intensity $e^{n \rho_n}$ (i.e. normalized logarithmic intensity $\rho_n$). Assume that $\rho_n \to \rho$ as $n \to \infty$. Note that $\rho$ is a real number (which can be negative). It is called the [*asymptotic normalized logarithmic intensity*]{} of this sequence of Poisson processes. We will assume that the processes are defined on a single probability space $(\Omega,{\cal F},\PP)$, although the coupling between the different dimensions is not relevant for the issues we consider. We will denote by $\PP^0_n$ the Palm probability of the Poisson point process in dimension $n$. See [@DVJ2 Chapter 13] for the definition of and basic facts about Palm probabilities.
To each point $T^{(k)}_n$, $k \ge 1$, of the Poisson process in dimension $n$ (enumerated in some way) we associate a mark $X^{(k)}_n\in \R^+$. The $X^{(k)}_n$, $k \ge 1$, are assumed to be independent and identically distributed (i.i.d.) and independent of the points. For each dimension $n$, let $X^{(k)}_n \stackrel{d}{=} \bar{X}_n$ for all $k$. Let $R^*_n$ denote $\EE[ \bar{X}_n]$, and let $R^* := \lim_{n \to \infty} R^*_n$, where the last limit is assumed to exist. We assume that $0 < R^* < \infty$. We assume that the sequence $(\bar{X}_n, n \ge 1)$ satisfies an LDP, with good and convex rate function $I(\cdot)$ [@DZ], e.g. by assuming the Gärtner-Ellis conditions. We also assume that the following condition holds: $$\begin{aligned}
\limsup_{n\to \infty} \EE [ (\bar X_n)^{\gamma n}]^{\frac 1 n} < \infty
\quad \mbox{for some $\gamma>1$}.\label{cond1}
$$ By the deterministic setting we mean that $\bar{X}_n$ is deterministic and equal to $R^*_n$ for each $n \ge 1$, with $R^*_n \to R^*$ as $n \to \infty$. The deterministic setting is a special case of the general setting, but we will separately highlight the results in this case, since it is of particular interest.
To the marked point process in dimension $n$, we associate a Boolean model where the grain of point $T^{(k)}_n$ is a closed ball of radius $X^{(k)}_n \sqrt{n}$. Let $$\label{eq:defbool}
{\cal C}_n := \bigcup_k B(T^{(k)}_n, X^{(k)}_n \sqrt{n})$$ denote this Boolean model, with $B(t,r)$ denoting the closed ball of center $t\in \R^n$ and radius $r\ge 0$. Here, and in the rest of the paper, $:=$ denotes equality by definition.
From Slivnyak’s theorem [@DVJ2 Chapter 13], the Palm version of the process in each dimension $n$ (i.e. its law under $\PP^0_n$) is equivalent in law to the superposition of a stationary version of the process and a process with a single point at the origin carrying a ball with radius having law $\bar{X}_n \sqrt{n}$, and independent of the stationary version (which is called the reduced process of the Palm version).
Our motivations for the analysis of this setting came from related problems in information theory that we studied recently [@AB]. More specifically, in the study of error probabilities for coding over an additive white Gaussian noise channel [@Gallager Section 7.4], it is natural to consider a sequence of Poisson processes, one in each dimension $n \ge 1$, with well defined asymptotic logarithmic intensity, as was done in [@AB], motivated by the ideas in [@P94]. The error exponent questions studied in [@AB] are related to [^1] the consideration of a Boolean model where the grains associated with the individual points are defined in terms of additive white Gaussian noise: for all $n \ge 1$ and $k \ge 1$, let $W_n^{(i,k)}$, $n\ge i\ge 1$, denote an i.i.d. sequence of Gaussian random variables, all centered and of variance $\sigma^2 > 0$. Let $W_n^{(k)}$ denote the $n$-dimensional vector with coordinates $W_n^{(i,k)}$, $n\ge i\ge 1$. Then $T_n^{(k)}+W_n^{(k)}$ belongs to the closed ball of center $T_n^{(k)}$ and radius $X_n^{(k)} \sqrt{n},$ with $$X_n^{(k)} := \left(\frac 1 n
\sum_{i=1}^n \left(W_n^{(i,k)}\right)^2\right)^{\frac 1 2}$$ satisfying an LDP and all the assumptions listed above. For the error exponent problem what is of interest is not this Boolean model, but the related Boolean model where the grain associated to each point is not the random ball described above, but rather an associated [*typicality region*]{}, which in this case we may define as the set $$\{ T_n^{(k)} + v ~:~ v \in \mathbb{R}^n,~
\| v \|_2 \le \sigma \sqrt{n} + \alpha_n \}~,$$ where $\| v \|_2$ denotes the usual Euclidean length of $v$ and $0 < \alpha_n = O(\sqrt{n})$ are chosen such that [^2] $$\begin{aligned}
&& ~ \frac{\alpha_n}{\sqrt{n}} \to 0 \mbox{ as $n \to \infty$};\\
&&P( \| W_n^{(k)} \| \le
\sigma \sqrt{n} + \alpha_n ) \to 1 \mbox{ as $n \to \infty$
(for each $1 \le k \le n$, of course)};\\
&&
\frac{1}{n} \log \mbox{Vol} \{ v \in \mathbb{R}^n ~:~
\| v \|_2 \le \sigma \sqrt{n} + \alpha_n \} \to \frac{1}{2} \log (2 \pi e \sigma^2)
\mbox{ as $n \to \infty$}~.\end{aligned}$$ This fits within the class of deterministic Boolean models considered in this paper. Thus, having carried out the analysis in [@AB], it was natural for us to become curious about the asymptotic properties in dimension of the sequence of Boolean models with the grains being balls whose radii obey a large deviations principle in the sense described above, and the current paper may be viewed as a start in that direction. In particular, it is to be hoped that this particular asymptotic regime, which is so natural from an information theoretic viewpoint, will also be of value in the applications of stochastic geometry in other domains of science and engineering.
The paper is structured as follows. We start with a summary of results and heuristic explanations in Section \[sec:res\]. We then give proofs in Section \[sec:proofs\]. For smoothness of exposition, we first discuss the volume fraction threshold, then the degree threshold, and finally the percolation threshold in each of these sections. Some concluding remarks, making connections between the issues addressed here and the information theoretic concerns of [@AB], are made in Section \[s.concluding\], where in particular the instantiation of our general results in the case of Gaussian grains is worked out in detail.
Results {#sec:res}
=======
Volume Fraction Threshold
-------------------------
Consider the stationary version of the marked Poisson process in each dimension. We are interested in the asymptotic behavior of the probability with which the origin is covered, namely $ \PP( {\underline{0}}\in \mathcal{C}_n)$. By stationarity, for any Borel set of $\R^n$, this probability is also the mean fraction of the volume of the Borel set which is covered by the Boolean model. We claim that there is a number $\tau_v$, called the [*volume fraction threshold*]{}, such that for $\rho < \tau_v$ this probability asymptotically approaches $0$ as $n$ tends to infinity, while for $\rho > \tau_v$ it asymptotically approaches $1$. The value of $\tau_v$ depends on the large deviations rate function $I(\cdot)$ associated to the sequence of distributions of the radii of the marks.
The idea of the proof is based on the fact that most of the volume of a ball is at the boundary. Hence for all $R>0$, the mean number of points at distance roughly $R \sqrt{n}$ from the origin grows like $$e^{n \rho_n} e^{ \frac{n}{2} \log ( 2 \pi e) + o(n)} e^{n \log R}~.$$
Each such point covers the origin with probability $\PP(\bar{X}_n \ge R)$. For $R < R^*$ this probability is asymptotically $1$. For $R > R^*$ this probability decays like $e^{- n I(R) + o(n)}$, where $I(\cdot)$ denotes the rate function for the convergence $\bar{X}_n \stackrel{\PP}{\to} R^*$.
Let ${\underline{0}}$ denote the origin in $\mathbb{R}^n$. We should therefore have $$\lim_{n \to \infty} \frac{1}{n} \log \PP( {\underline{0}}\in \mathcal{C}_n) =
\rho + \frac{1}{2} \log ( 2 \pi e) + \sup_{R \ge R^*} (\log R - I(R))~,$$ as long as $$\rho + \frac{1}{2} \log ( 2 \pi e) + \sup_{R \ge R^*} (\log R - I(R)) < 0~,$$ where we used the fact that $I(R^*)=0$ which implies that $\log(R)\le \log(R^*)-I(R^*)$ for $R\le R^*$.
Also $$\lim_{n \to \infty} \PP( {\underline{0}}\in \mathcal{C}_n) = 1~,$$ if $$\rho + \frac{1}{2} \log ( 2 \pi e) + \sup_{R \ge R^*} (\log R - I(R)) > 0~.$$
This gives a heuristic explanation of the value of the threshold in the following theorem:
\[thm:vf\] Under the foregoing assumptions, the volume fraction threshold is equal to $$\label{Thresh.Vol}
\tau_v = - \frac{1}{2} \log ( 2 \pi e) + \inf_{R \ge R^*} (I(R) - \log R)~.$$ More precisely, for $\rho < \tau_v$, as $n$ tends to infinity, the volume fraction in dimension $n$, namely $\PP( {\underline{0}}\in \mathcal{C}_n)$, tends to 0 exponentially fast with $$\label{eq:vfexp}
\lim_{n\to \infty} \frac 1 n
\log(\PP( {\underline{0}}\in \mathcal{C}_n)) =\rho-\tau_v,$$ whereas for $\rho > \tau_v$, it tends to 1 with $$\label{Limit.Above}
\lim_{n \to \infty} \frac{1}{n}
\log(- \log \PP( {\underline{0}}\notin \mathcal{C}_n))
=
\rho - \tau_v~.$$
Note that $$\label{eq:dettauv}
\tau_v \le - \frac{1}{2} \log ( 2 \pi e) - \log R^*.$$ In the case of deterministic radii, i.e. when $\bar{X}_n$ equals the deterministic value $R^*_n$ for each $n \ge 1$, with $R^*_n \to R^*$ as $n \to \infty$, we have equality in eqn. (\[eq:dettauv\]). The R.H.S. of eqn. (\[eq:dettauv\]) is identical to what is called the Poltyrev threshold in [@AB], where it in effect arose in the context of the Boolean models with Gaussian grains truncated to their typicality regions, as described at the end of Section \[s.setup\]. In Section \[s.concluding\] we will discuss in more depth this connection between the questions addressed in this paper and the information theoretic questions studied in [@AB]. This threshold can also be described as follows: the volume of the $n$-ball of random radius ${\bar X_n} \sqrt{n}$ scales like $e^{nV + o(n)}$ as $n$ tends to infinity, for some constant $V$; we have $\tau_v=- V$ or equivalently the critical density $e^{n\tau_v + o(n)}$ scales like the inverse of the volume of this $n$-ball.
Degree Threshold
-----------------
We are interested in the number $D_n$ of points other than ${\underline{0}}$ whose ball intersects the ball of the point at the origin under $\PP^0_n$.
We claim that there is a number $\tau_d$, that we will call the [ *degree threshold*]{}, such that if $\rho < \tau_d$, then $\EE^0_n [D_n]$ asymptotically goes to $0$ when $n$ tends to infinity, while for $\rho > \tau_d$ it asymptotically goes to $\infty$.
We argue as follows: condition on the radius of the ball of the point at the origin, call it $s \sqrt{n}$. Every point that lands in the ball of radius $s \sqrt{n}$ will have its ball meeting the ball of the origin. The number of such points grows like $$e^{n \rho_n} e^{\frac{n}{2} \log (2 \pi e) + o(n)} e^{n \log s}~.$$ Next consider points at a distance roughly $R \sqrt{n}$ from the (point at the) origin, with $R>s$. The number of such points grows like $$e^{n \rho_n} e^{\frac{n}{2} \log (2 \pi e) + o(n)} e^{n \log R}~.$$ Each such point has its ball intersecting the ball of the point at the origin with probability asymptotically equal to $1$ if $R - s < R^*$ and with probability decaying like $e^{ - n I(R - s) + o(n)}$ if $R - s > R^*$. The number of points meeting the ball of the origin, conditioned on this ball having radius $s \sqrt{n}$, therefore grows like $$\label{eq:meanasy}
e^{n (\rho + \frac{1}{2} \log (2 \pi e) + \sup_{R \ge s + R^*} (\log R - I(R - s))) + o(n)}~.$$ The probability that the ball of the origin has radius roughly $s \sqrt{n}$ decays like $e^{ - n I(s) + o(n)}~.$ Thus, the overall growth rate of the number of points whose ball meets the ball of the origin is $$\begin{aligned}
\label{Boolean}
&&~\sup_{s > 0} \left( - I(s) +
\rho + \frac{1}{2} \log (2 \pi e) + \sup_{R \ge s + R^*} (\log R - I(R - s)) \right)
\nonumber \\
&&~~~ = \rho + \frac{1}{2} \log (2 \pi e) + \sup_{s > 0} \sup_{R \ge s + R^*}
\left( - I(s) + \log R - I(R - s)\right)
\nonumber \\
&&~~~ = \rho + \frac{1}{2} \log (2 \pi e) + \sup_{R > R^*} \left( \log R +
\sup_{0 < s \le R - R^*} \left( - I(s) - I(R - s) \right) \right)
\nonumber \\
&&~~~ \stackrel{(a)}{=} \rho + \frac{1}{2} \log (2 \pi e) +
\max \left(
\sup_{R^* \le R < 2R^*} \left( \log R - I(R - R^*) \right)\right.,
\nonumber \\
& & \hspace{7cm}
\left. \sup_{R \ge 2R^*} \left( \log R - 2 I(\frac{R}{2}) \right) \right)~,
\nonumber\\
&&~~~ = \rho + \frac{1}{2} \log (2 \pi e) + \sup_{R \ge 2 R^*}
\left( \log R - 2 I(\frac{R}{2}) \right)~,\end{aligned}$$ where in step (a) we have used the convexity of the rate function $I(\cdot)$ and the fact that $I(R^*) = 0$, and in the last step we have observed that the maximum in the first of the terms in the overall maximum occurs at $R =2R^*$. This gives intuition for the value of the threshold in the following theorem:
\[thm:deg\] Under the conditions of Theorem \[thm:vf\], the degree threshold is $$\label{Thresh.Boolean.Old}
\tau_d = - \frac{1}{2} \log (2 \pi e) + \inf_{R \ge 2 R^*}
\left( 2 I(\frac{R}{2}) - \log R \right)~.$$ That is, for $\rho < \tau_d$, as $n$ tends to infinity, in dimension $n$, $\EE_n^0 [D_n]$ tends to 0 exponentially fast, whereas for $\rho > \tau_d$ it tends to infinity exponentially fast. In both cases, $$\label{eq:degexp}
\lim_{n\to \infty} \frac 1 n
\log (\EE_n^0 [D_n]) =
\rho- \tau_d~.$$
It is sometimes more convenient to write the degree threshold as $$\label{Thresh.Boolean}
\tau_d = - \frac{1}{2} \log (2 \pi e) + \inf_{R \ge R^*}
\left( 2 I(R) - \log (2 R) \right)~.$$ Note that $$\tau_d \le - \frac{1}{2} \log (2 \pi e) -\log (2 R^*)$$ and that the R.H.S. of the last inequality is the degree threshold in the case of deterministic radii [@PenroseCP]. In the general case, the degree threshold can be described as follows: let $\bar X_n'$ be an independent random variable with the same law as $\bar X_n$. The volume of the $n$-ball of random radius $(\bar X_n+\bar X_n') \sqrt{n}$ scales like $e^{nV + o(n)}$ as $n$ tends to infinity for some constant $V$; we have $\tau_d=- V$ or equivalently the critical density $e^{n\tau_d + o(n)}$ scales like the inverse of the volume of this $n$-ball.
Percolation Threshold
---------------------
Consider the Palm version of the process in dimension $n$. Consider the connected component of $\mathcal{C}_n$ that contains the origin, called the [*cluster of the origin*]{}, and denote the set of points of the underlying Poisson process that lie in this connected component by $K_n$. The [*percolation probability*]{} in dimension $n$ is denoted by $$\theta_n := \PP^0_n( |K_n| = \infty)~,$$ with $|A|$ the cardinality of set $A$. This is one of the standard definitions for percolation probability in continuum percolation theory, see [@MeesterRoyBook Section 1.4].
We are interested in the asymptotics of the percolation probability as $n \to \infty$. We claim that there is a number $\tau_p$, called the [*percolation threshold*]{}, such that for $\rho < \tau_p$ we have $\theta_n \to 0$ as $n \to \infty$, while for $\rho > \tau_p$ we have $\liminf_n \theta_n > 0$.
In the case of deterministic radii, the percolation and the degree thresholds coincide, i.e. $ \tau_p = \tau_d~$.
To see that $\tau_p \ge \tau_d$, note that if $\rho < \tau_d$ then $\EE^0_n[D_n] \to 0$ as $n \to \infty$ from Theorem \[thm:deg\]. It follows that $\PP^0_n(D_n = 0) \to 1$ as $n \to \infty$. Hence $\PP^0_n( |K_n| = 1) \to 1$ as $n \to \infty$, from which it follows that $\theta_n \to 0$ as $n \to \infty$. This means $\rho < \tau_p$. This argument actually works in the general case, i.e. it does not require the assumption of deterministic radii.
To see that $\tau_p \le \tau_d$ in the case of deterministic radii, we need to prove that if $\rho > \tau_d$ then $\liminf_n \theta_n > 0$. To this end, let us recall the main result of [@PenroseCP]. In our notation, in [@PenroseCP] Penrose considers the sequence of Poisson Boolean models with deterministic radii $R^*_n = R^*$ for each $n \ge 1$, and with normalized logarithmic intensities $\rho^y_n$ defined via $$e^{n \rho^y_n} \frac{(\pi n)^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}
(2 R^*)^n = y~,~~\mbox{ for all $n \ge 1$}~,$$ where $y > 0$ is a fixed real number. Let $\theta^y_n$ denote the percolation probability in dimension $n$ with these choices. The main result [@PenroseCP Theorem 1] is that $\lim_{n \to \infty} \theta^y_n$ exists and equals the survival probability of a branching process with offspring distribution Poisson with mean $y$, and started with a single individual. In particular, this means that if $y > 1$, then $\liminf_{n \to \infty} \theta^y_n > 0$.
In our scenario with deterministic radii, the degree threshold (see eqn. (\[Thresh.Boolean\])) is $$\tau_d = - \frac{1}{2} \log (2 \pi e) - \log (2 R^*)~,~~\mbox{ (deterministic radii)}~.$$ It suffices to observe that if $\rho > \tau_d$, then $$\lim_{n \to \infty} e^{n \rho_n} \frac{(\pi n)^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}
(2 R^*_n)^n = \infty~.$$ That $\liminf_{n \to \infty} \theta_n > 0$ then follows from the result of [@PenroseCP] cited above.
The main result on the case with random radii is:
\[thm:perc\] The percolation threshold is given by the formula $$\label{Thresh.Perc}
\tau_p = - \frac{1}{2} \log (2 \pi e) +
\inf_{R \ge R^*} \left( I(R) - \log(R + R^*) \right)~.$$ That is, for $\rho<\tau_p$, $\theta_n\to 0$ when $n$ tends to infinity, whereas for $\rho>\tau_p$ we have $\liminf_n \theta_n >0$.
Note that $$\tau_p \le - \frac{1}{2} \log (2 \pi e) - \log (2 R^*)~.$$ In the case of deterministic radii the minimum in the expression for the percolation threshold in eqn. (\[Thresh.Perc\]) is achieved at $R = R^*$ and so we have $$\tau_p = - \frac{1}{2} \log (2 \pi e) - \log (2 R^*)~,~~\mbox{ (deterministic radii)}~.$$ This also equals the value of the degree threshold in the case of deterministic radii. The volume of the $n$-ball of random radius $(\bar X_n+R^*) \sqrt{n}$ scales like $e^{nV + o(n)}$ as $n$ tends to infinity, for some constant $V$; we have $\tau_p=- V$ or equivalently the critical density $e^{n\tau_p + o(n)}$ scales like the inverse of the volume of this $n$-ball. The intuition for this result is that what matters for percolation is the mean number of balls that intersect a ball with typical radius (namely roughly $R^*\sqrt{n}$): if $\rho<\tau_p$, then on an event whose probability tends to 1 as $n$ tends to infinity, namely the event that the ball of the point at ${\underline{0}}$ has a radius in the interval $(R^*\sqrt{n}-\alpha_n, R^*\sqrt{n}+\alpha_n)$ for appropriate $0 < \alpha_n = O(\sqrt{n})$, no other ball intersects the latter ball asymptotically (because $\rho<\tau_p$) and hence there is no percolation. Conversely, for $\rho>\tau_p$, when the ball of ${\underline{0}}$ is at typicality, i.e. its radius lies in an interval of the kind defined above, we can consider a thinned version of the reduced process where we only retain points whose balls have radii that are at least above a threshold slightly less than the value of $R$ achieving the infimum in the definition of $\tau_p$ (assume for the moment that this infimum is achieved), and we will still have that the mean number of balls intersecting the ball of the origin tends to infinity like $e^{n\delta}$ with some $\delta >0$. Since these balls themselves have radius at least as big as the typical ball of the unconditional distribution, this scenario propagates via a supercritical branching process, implying asymptotic percolation. [^3]
Ordering of the Thresholds {#ss.ordering}
--------------------------
\[thm:order\] Under the foregoing assumptions, $$\label{Relations.Betwn.Threshs}
\tau_d \le \tau_p \le \tau_v~.$$
#### Remark
The ordering relation of the last theorem is not limited to the Poisson case. The family of Boolean models considered here can naturally be extended to a family of particle processes [@SW], where the $n$-th particle process features a stationary and ergodic point process $\mu_n$ in $\R^n$ with normalized logarithmic intensity $\rho_n$ such that $\rho_n \to \rho$ as $n \to \infty$, and i.i.d. marks satisfying the same independence and LDP assumptions as above. This family of particle processes will be said to admit a volume fraction threshold $\tau_v$ if the associated ${\cal C}_n$, still defined by eqn. (\[eq:defbool\]), is such that $\PP( {\underline{0}}\in \mathcal{C}_n)$ asymptotically approaches $0$ as $n$ tends to infinity for $\rho < \tau_v$, while for $\rho > \tau_v$ it asymptotically approaches $1$. Similarly, it will be said to admit a degree threshold $\tau_d$ if the Palm expectation of $D_n$ tends to 0 as $n$ tends to infinity for $\rho < \tau_d$, while for $\rho > \tau_d$ it tends to $\infty$. The definition of the percolation threshold can also be extended verbatim. Assuming that these three thresholds exist, then they must satisfy eqn. (\[Relations.Betwn.Threshs\]). This follows from first principles. If the volume fraction asymptotically tends to 1, then percolation must hold asymptotically; hence $\tau_p\le \tau_v$. If the mean number of balls that intersect the ball of the origin tends to 0, then percolation cannot hold asymptotically; hence $\tau_d\le \tau_p$.\
Returning to the Poisson case, to better understand the thresholds, we first need to recall some facts from basic convex analysis [@Rockafellar]. Since it is a good convex rate function, $I(\cdot)$ is [*proper*]{}, as defined in [@Rockafellar pg. 24]. Further, since it is lower semicontinuous, its epigraph is closed [@Rockafellar Thm. 7.1], so $I(\cdot)$ is [*closed*]{} in the sense of [@Rockafellar pg. 52]. Recall that the domain of $I(\cdot)$, defined as the set of $R$ for which $I(R)$ is finite, is an interval, which is nonempty because $I(R^*) = 0$. Since $I(\cdot)$ is closed and proper, the right and left derivatives, $I^\prime_+(\cdot)$ and $I^\prime_-(\cdot)$ respectively, are well-defined as functions on $\mathbb{R}$ (both defined to be $+\infty$ to the right of the domain of $I(\cdot)$ and to be $-\infty$ to the left of the domain of $I(\cdot)$). These are nondecreasing functions, each of which is finite on the interior of the domain of $I(\cdot)$, and satisfy ([@Rockafellar Thm. 24.1]): $$I^\prime_+(z_1) \le I^\prime_-(x) \le I^\prime_+(x) \le I^\prime_-(z_2)~,~~
\mbox{ if $z_1 < x < z_2$}~,$$ and, for all $x \in \mathbb{R}$, $$\lim_{z \uparrow x} I^\prime_-(z) = \lim_{z \uparrow x} I^\prime_+(z)
= I^\prime_-(x) \mbox{ and }
\lim_{z \downarrow x} I^\prime_-(z) = \lim_{z \downarrow x} I^\prime_+(z)
= I^\prime_+(x)~.$$ We further note that $0 \in [I^\prime_-(R^*), I^\prime_+(R^*)]$, since $I(\cdot)$ is a nonnegative function with $I(R^*) = 0$.
This means we can define the following radii:
- $R_v \ge R^*$ as a value of $R$ satisfying $$\frac{1}{R_v} \in [I^\prime_-(R_v), I^\prime_+(R_v)]~.$$ Such $R_v$ achieves the infimum in eqn. (\[Thresh.Vol\]) for the volume fraction threshold. Further, since $R \mapsto \frac{1}{R}$ is strictly decreasing and decreases to $0$ as $R \to \infty$, it follows that $R_v$ is uniquely defined and finite.
- $R_d \ge R^*$, as a value of $R$ satisfying $$\frac{1}{2 R_d} \in [I^\prime_-(R_d), I^\prime_+(R_d)]~.$$ Such $R_d$ achieves the minimum in eqn. (\[Thresh.Boolean\]) for the degree threshold. Further, since $R \mapsto \frac{1}{2 R}$ is strictly decreasing and decreases to $0$ as $R \to \infty$, it follows that $R_d$ is uniquely defined and finite.
- $R_p \ge R^*$, as a value of $R$ satisfying $$\frac{1}{R_p + R^*} \in [I^\prime_-(R_p), I^\prime_+(R_p)]~.$$ Such $R_p$ achieves the minimum in eqn. (\[Thresh.Perc\]) for the percolation threshold. Further, since $R \mapsto \frac{1}{R + R^*}$ is strictly decreasing and decreases to $0$ as $R \to \infty$, it follows that $R_p$ is uniquely defined and finite.
\[thm:order2\] With the foregoing definitions, we have $$\begin{aligned}
\label{eq:total-order} R^*\le R_d \le R_p \le R_v \le R_p + R^* \le 2 R_d~.\end{aligned}$$
Proofs {#sec:proofs}
======
Proof of Theorem \[thm:vf\]
---------------------------
Below we will use the [*directed random geometric graph*]{} built as follows: its vertices are the nodes of the point process and there is an edge from $T_n^{(k)}$ to $T_n^{(l)}$, $l\ne k$ if $T_n^{(l)}\in B(T_n^{(k)}, X_n^{(k)} \sqrt{n})$.
Let $d^-_n$ denote the in-degree of the node at the origin in this random directed graph under $\PP^0$, namely the number of points whose ball contains the origin. Let $d_n^+$ denote the out-degree of the origin, namely the number of points which fall in the ball of the point at the origin. From the mass transport principle [@Last], or by straightforward elementary arguments based on an ergodic theorem for spatial averages, we have $$\EE_n^0 [d^+_n] = \EE_n^0 [d^-_n].$$
Now, we have $$\begin{aligned}
\EE_n^0[d_n^+] &\stackrel{(a)}{=}&
e^{n \rho_n} \EE[ \frac{(\pi n)^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}
\bar{X}_n^n]\\
&=& e^{n \rho_n} \EE[e^{\frac{n}{2} \log ( 2 \pi e) + o(n)} e^{n \log \bar{X}_n}]\\
&=& e^{n (\rho + \frac{1}{2} \log (2 \pi e)) + o(n)} \EE[ e^{n \log \bar{X}_n}]~.\end{aligned}$$
Here step (a) follows from Slivnyak’s theorem. Then we have the following result.
$$\label{Asymp.Count}
\frac{1}{n} \log \EE_n^0[ d_n^-] \to
\rho + \frac{1}{2} \log ( 2 \pi e) + \sup_{R \ge R^*} (\log R - I(R))~,~~\mbox{ as $n \to \infty$}~.$$
From what precedes, $$\begin{aligned}
\EE^0_n[d_n^-] =
e^{n (\rho +\frac{n}{2} \log ( 2 \pi e)) + o(n)}
\EE[e^{n \log \bar X_n}]~.\end{aligned}$$ It follows from Assumption (\[cond1\]) and from Varadhan’s lemma [@DZ] that $$\lim_{n\to \infty} \frac 1 n
\EE[e^{n \log \bar X_n}]
= \sup_{R \ge R^*} (\log R - I(R)) ~,$$ where we have used the observation that $\log R - I(R) \le \log R^*$ for $0 \le R \le R^*$. This completes the proof.
Now, from the independent thinning theorem [@DVJ2 Exercise 11.3.1], the distribution of $d_n^-$ is Poisson. Thus $$\label{Prob.From.Count}
\PP( {\underline{0}}\in \mathcal{C}_n) = 1 - \exp( - \EE^0_n[d_n^-])\le
\EE^0_n[d_n^-]~.$$ For $\rho <\tau_v$, we see from eqn. (\[Asymp.Count\]) that $$\EE[ d_n^-] \to 0 \mbox{ as $n \to \infty$}~,$$ which implies that $$\PP( {\underline{0}}\in \mathcal{C}_n) \to 0 \mbox{ as $n \to \infty$}.$$ In addition, for all $\alpha < 1$ we have $$1 - \alpha x \ge \exp( -x) \ge 1 - x \mbox{ for all sufficiently small $x > 0$}~.$$ Thus, from eqn. (\[Prob.From.Count\]) we get that, for all $\alpha < 1$ and $n$ sufficiently large $$\frac{1}{n} \log \EE^0_n[ d_n^-] \ge
\frac{1}{n} \log \PP( {\underline{0}}\in \mathcal{C}_n) \ge \frac{1}{n} \log \EE^0_n[ d_n^-]
+ \frac{1}{n} \log \alpha~.$$ Thus we have $$\label{Limit.Below}
\lim_{n \to \infty} \frac{1}{n} \log P( {\underline{0}}\in \mathcal{C}_n)
= \rho + \frac{1}{2} \log ( 2 \pi e) + \sup_{R \ge R^*} (\log R - I(R))~,$$ which concludes the proof of eqn. (\[eq:vfexp\]).
Suppose now that $\rho > \tau_v$ Since $$\label{Prob.From.Count.2}
\PP( {\underline{0}}\notin \mathcal{C}_n) = \exp( - \EE^0_n[d_n^-])~,$$ we then immediately get eqn. (\[Limit.Above\]).
Volume Fraction Threshold for Deterministic Radii
-------------------------------------------------
The proof above also works for the case of deterministic radii (equal to $R^*_n \sqrt{n}$ in dimension $n$ with $R^*_n \to R^*$ as $n \to \infty$). The only change needed is to replace $\EE[ e^{n \log \bar{X}_n}]$ by $e^{n \log R^*_n}$. Then eqn. (\[Asymp.Count\]) is replaced by $$\frac{1}{n} \log \EE^0_n[ d_n^-] \to
\rho + \frac{1}{2} \log ( 2 \pi e) + \log R^*~,$$ and we learn that if $$\rho < - \frac{1}{2} \log ( 2 \pi e) - \log R^*$$ then $$\lim_{n \to \infty} \frac{1}{n} \log \PP( {\underline{0}}\in \mathcal{C}_n)
= \rho + \frac{1}{2} \log ( 2 \pi e) + \log R^*~,$$ which replaces eqn. (\[Limit.Below\]), while if $$\rho > - \frac{1}{2} \log ( 2 \pi e) - \log R^*$$ then $$\lim_{n \to \infty} \frac{1}{n} \log (-\log \PP( {\underline{0}}\notin \mathcal{C}_n))
= \rho + \frac{1}{2} \log ( 2 \pi e) + \log R^*~,$$ which replaces eqn. (\[Limit.Above\]).
Proof of Theorem \[thm:deg\] {#ss.Boolean.Sketch.RR}
----------------------------
In each dimension $n$, consider the stationary version of the process. Given $s > 0$, let $N_n({\underline{0}}, s)$ denote the number of points whose balls intersect the ball of radius $s \sqrt{n}$ centered at the origin ${\underline{0}}$ in $\mathbb{R}^n$. Then $$\begin{aligned}
\EE[N_n({\underline{0}},s)] &\stackrel{(a)}{=}&
e^{n \rho_n} \EE[ \frac{(\pi n)^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}
(\bar{X}_n + s)^n]\\
&=& e^{n \rho_n} \EE[e^{\frac{n}{2} \log ( 2 \pi e) + o(n)} e^{n \log (\bar{X}_n + s)}]\\
&=& e^{n (\rho
+ \frac{1}{2} \log (2 \pi e)) + o(n)} \EE[ e^{n \log (\bar{X}_n + s)}]~.\end{aligned}$$ Step (a) again follows from Slivnyak’s theorem and the mass transport principle applied to the directed graph with an edge from $T_n^{(k)}$ to $T_n^{(l)}$, $l\ne k$ if $T_n^{(l)}\in B(T_n^{(k)},(X_n^{(k)}+s) \sqrt{n})$.
Consider now the Palm version of the point process. Recall that, from Slivnyak’s theorem, ignoring the point at ${\underline{0}}$ and its mark leaves behind a stationary version of the marked point process, which is called the reduced process. We therefore have, $$\EE^0_n[D_n] = \EE[ \EE[ N_n({\underline{0}}, S) |S]]~,$$ where $S \stackrel{d}{=} \bar{X}_n$ and is independent of the reduced process.
Thus $$\EE^0_n[D_n] = e^{n (\rho
+ \frac{1}{2} \log (2 \pi e)) + o(n)} \EE[ e^{n \log (\bar{X}_n + \bar{X}^\prime_n)}]~,$$ with $\bar{X}_n$ and $\bar{X}^\prime_n$ being i.i.d. We now use the fact that $\bar{X}_n + \bar{X}^\prime_n$ satisfies an LDP with good convex rate function $2I(\frac u 2)$ to derive (\[Thresh.Boolean.Old\]) from Varadhan’s lemma. For this, we have to check that if $\bar X'_n$ is a variable with the same law as $\bar X_n$ and such that $\bar X'_n$ and $\bar X_n$ are independent, then $$\begin{aligned}
\limsup_{n\to \infty} \EE [ (\bar X_n+ \bar X'_n)^{\gamma n}]^{\frac 1 n} < \infty
\quad \mbox{for some $\gamma>1$}.
\label{cond2}\end{aligned}$$ But this can be obtained from the following convexity argument $$\begin{aligned}
\EE [ (\bar X_n+ \bar X'_n)^{\gamma n}] & \le &
2^{\gamma n} \EE [(\bar X_n)^{\gamma n}]~,\end{aligned}$$ which implies that $$\begin{aligned}
\EE [ (\bar X_n+ \bar X'_n)^{\gamma n}]^{\frac 1 n} & \le &
2^{\gamma } \EE [(\bar X_n)^{\gamma n}]^{\frac 1 n}.\end{aligned}$$ Hence eqn. (\[cond2\]) follows from eqn. (\[cond1\]).
This completes the proof of the results on the degree threshold.
Degree Threshold for Deterministic Radii
-----------------------------------------
The proof given above also works for the case of deterministic radii. The changes needed are analogous to those that were needed in the case of the volume fraction threshold.
Proof of Theorem \[thm:perc\]
-----------------------------
We need to prove two things: (1) if $\rho < \tau_p$ then $\lim_{n \to \infty} \theta_n = 0$ and (2) if $\rho > \tau_p$, then $\liminf_{n \to \infty} \theta_n > 0$. To prove (1) we follow the lines of the proof in Section \[ss.Boolean.Sketch.RR\]. Rather than considering $\EE^0_n[D_n]$, we consider $\PP^0_n(D_n > 0)$. As in Section \[ss.Boolean.Sketch.RR\], we write $$\PP^0_n(D_n > 0) = \EE^0_n [ \PP_n^0( D_n > 0 | S)]~,$$ where $S \sqrt{n}$ now refers to the radius of the ball of the point at the origin. Now, if $$\rho < - \frac{1}{2} \log ( 2 \pi e) + \inf_{R \ge R^*}
\left( I(R) - \log(R + R^*) \right)~,$$ then, because $R_p$, as defined in Section \[ss.ordering\] is finite, for sufficiently small $\epsilon > 0$ we also have $$\rho < - \frac{1}{2} \log ( 2 \pi e) + \inf_{R \ge R^*}
\left( I(R) - \log(R + R^* + \epsilon) \right)~.$$ On the event $\{ S \le R^* + \epsilon^\prime \}$, with $\epsilon > \epsilon^\prime > 0$, we have $$\begin{aligned}
&&~ \lim_n \frac 1 n \log \left(\EE_n^0[D_n | S]\right)
\nonumber\\
&&~~~= \rho + \frac{1}{2} \log (2 \pi e)
+ \sup_{R \ge R^* + S} \left( \log R - I( R - S) \right)\nonumber \\
&&~~~ = \rho + \frac{1}{2} \log (2 \pi e)
+ \sup_{\tilde{R} \ge R^*} \left( \log( \tilde{R} + S) - I(\tilde{R})
\right) \nonumber
\\
&&~~~ \le \inf_{R \ge R^*}
\left( I(R) - \log(R + R^* + \epsilon) \right) +
\sup_{\tilde{R} \ge R^*} \left( \log( \tilde{R} + S) - I(\tilde{R})
\right) \nonumber
\\
&&~~~ = \sup_{\tilde{R} \ge R^*} \left( \log( \tilde{R} + S) - I(\tilde{R}) \right)
- \sup_{\tilde{R} \ge R^*} \left( \log(\tilde{R} + R^* +\epsilon) - I(\tilde{R}) \right) \nonumber\\
&&~~~ < 0~.
\label{eq:argab}\end{aligned}$$ We also have $$\begin{aligned}
\PP^0_n( D_n > 0) & = & \PP^0_n( D_n > 0 , S > R^* + \epsilon^\prime )\\
& & + \EE^0_n[ \PP^0_n(D_n > 0 | S) 1( S \le R^* + \epsilon^\prime)]\\
& \le & \PP (S > R^* + \epsilon^\prime )
+ \EE^0_n[ \EE^0_n[D_n | S] 1( S \le R^* + \epsilon^\prime)]~.
$$ In the last expression, the first term has probability asymptotically approaching $0$ as $n \to \infty$. From eqn. (\[eq:argab\]), for all $s$ in the integration interval, the integrand in the second term tends pointwise to 0 as $n \to \infty$. From this and dominated convergence, we conclude that $\PP^0_n( D_n > 0 ) \to 0$ as $n \to \infty$, which proves (1).
We now prove (2), assuming that $$\rho > - \frac{1}{2} \log ( 2 \pi e) + \inf_{R \ge R^*}
\left( I(R) - \log(R + R^* + \epsilon) \right)~.$$ Let $R_p$, as defined in Section \[ss.ordering\], achieve the minimum in the definition of $\tau_p$ in eqn. (\[Thresh.Perc\]). We need to distinguish between the two cases $R_p = R^*$ and $R_p > R^*$.
Consider first the case $R_p = R^*$. Then $$\rho = - \frac{1}{2} \log ( 2 \pi e) - \log(2 R^*) + \delta~,$$ for some $\delta > 0$. This means that we can choose $\gamma > 0$ such that $$\rho > - \frac{1}{2} \log ( 2 \pi e) - \log(2 (R^* - \gamma)) + \frac{\delta}{2}~.$$ For each dimension $n$ we consider the thinned version of the reduced process, where we only retain the points whose balls have radius at least $(R^* - \gamma) \sqrt{n}$. We also consider only the event that the ball of the point at the origin has radius at least $(R^* - \gamma) \sqrt{n}$. Let $\tilde{\theta}_n$ denote the probability of percolation from the origin via its ball and through the balls of the thinned reduced point process, on the event that the ball of the origin has radius at least $(R^* - \gamma) \sqrt{n}$. Since $\theta_n \ge \tilde{\theta}_n$, if we can show that $\liminf_{n \to \infty} \tilde{\theta}_n > 0$, then we will be done.
Let us now show that
- the probability that the ball of ${\underline{0}}$ has radius at least $(R^*-\gamma)\sqrt{n}$ tends to 1 as $n$ tends to infinity;
- the probability that the number $J_n$ of balls of the thinned point process intersecting the ball of ${\underline{0}}$ (with radius at least $(R^*-\gamma)\sqrt{n}$) is positive tends to 1 as $n$ tends to infinity;
- the Boolean model with deterministic radii $(R^* - \gamma) \sqrt{n}$ for the points of the thinned reduced process percolates, i.e. the associated percolation probability has a positive liminf.
Property (a) is immediate. For proving (b), we show that $\EE[J_n]$ tends to infinity with $n$, which will complete the proof since $J_n$ is Poisson. The probability that the $\bar X_n$ exceeds $(R^* - \gamma)\sqrt{n}$ is asymptotically $1$. Hence, by arguments similar to those used earlier, $$\lim_n \frac 1 n \EE[J_n]
= \rho +\frac 1 2 \log(2\pi e) +\ln(2 (R^*-\gamma)) > \frac{\delta}{2}~,$$ which completes the proof of (b). For proving (c), we use the results in [@PenroseCP]. The percolation threshold of the Boolean models with deterministic radii $(R^* - \gamma)\sqrt{n}$ is $- \frac{1}{2} \log ( 2 \pi e) - \log( 2(R^* - \gamma))$. Since the normalized logarithmic intensity of the thinned reduced process is still asymptotically $\rho$, and since this exceeds $- \frac{1}{2} \log ( 2 \pi e) - \log( 2(R^* - \gamma))$, the proof of (c) follows from [@PenroseCP].
The proof of the desired result, in the case $R_p = R^*$, now follows immediately from (a), (b), and (c).
We next turn to the case $R_p > R^*$. Note that in this case we must have $I(R_p) < \infty$. Since $$\rho = - \frac{1}{2} \log ( 2 \pi e) + I(R_p) - \log(R_p + R^*) + \delta~,$$ for some $\delta > 0$, we can choose $\gamma > 0$ such that $R^* < R_p - \gamma < R_p$ (which implies that $I(R_p - \gamma) < \infty$), and such that $$\rho > - \frac{1}{2} \log ( 2 \pi e) + I(R_p - \gamma) - \log(R_p + R^* - 2 \gamma)
+ \frac{\delta}{2}~.$$ For each dimension $n$ we consider the thinned version of the reduced process, where we only retain the points whose balls have radius at least $(R_p - \gamma) \sqrt{n}$. We also consider only the event that the ball of the point at the origin has radius at least $(R^* - \gamma) \sqrt{n}$. Let $\tilde{\theta}_n$ denote the probability of percolation from the origin via its ball and through the balls of the thinned reduced point process, on the event that the ball of the origin has radius at least $(R^* - \gamma) \sqrt{n}$. Since $\theta_n \ge \tilde{\theta}_n$, if we can show that $\liminf_{n \to \infty} \tilde{\theta}_n > 0$, then we will be done.
Let us now show that
- the probability that the ball of ${\underline{0}}$ has radius at least $(R^*-\gamma)\sqrt{n}$ tends to 1 as $n$ tends to infinity;
- the probability that the number $J_n$ of balls of the thinned point process intersecting the ball of ${\underline{0}}$ (with radius at least $(R^*-\gamma)\sqrt{n}$) is positive tends to 1 as $n$ tends to infinity;
- the Boolean model with deterministic radii $(R_p - \gamma) \sqrt{n}$ for the points of the thinned reduced process percolates, i.e. the associated percolation probability has a positive liminf.
Property (a) is immediate. For proving (b), we show that $\EE[J_n]$ tends to infinity with $n$, which will complete the proof since $J_n$ is Poisson. The probability that the $\bar X_n$ exceeds $(R_p - \epsilon)\sqrt{n}$ scales like $e^{-nI(R_p - \gamma) + o(n)}$. Hence, by arguments similar to those used earlier, $$\lim_n \frac 1 n \EE[J_n]
= \rho +\frac 1 2 \log(2\pi e) - I(R_p - \gamma) + \ln(R_p + R^*- 2 \gamma)) >
\frac{\delta}{2}~,$$ which completes the proof of (b). For proving (c), we use the results in [@PenroseCP]. The percolation threshold of the Boolean models with deterministic radii $(R_p - \gamma)\sqrt{n}$ is $- \frac{1}{2} \log ( 2 \pi e) - \log( 2(R_p - \gamma))$. Since the normalized logarithmic intensity of the thinned reduced process is asymptotically $\rho - I(R_p - \gamma)$, and since this exceeds $- \frac{1}{2} \log ( 2 \pi e) - \log( R_p + R^* - 2 \gamma))$ (because $R_p > R^*$), the proof of (c) follows from [@PenroseCP].
The proof of the desired result, in the case $R_p > R^*$, now follows immediately from (a), (b), and (c). This also completes the overall proof.
Percolation Threshold for Deterministic Radii
---------------------------------------------
The proof of the percolation threshold in the case of deterministic radii (i.e. when the radii in dimension $n$ equal a constant $R^*_n \sqrt{n}$, with $R_n^* \to R^*$ as $n \to \infty$) can be completed in a much simpler way than the general proof. Since $\tau_p = \tau_d$ in the case of deterministic radii, the absence of percolation when $\rho < \tau_p$ is an immediate consequence of the result proved earlier that $\PP(D_n > 0) \to 0$ as $n \to \infty$ when $\rho < \tau_d$. For the case $\rho > \tau_p$ the proof can be done in a way exactly as the general case where $R_p =R^*$ was handled above.
Proof of Theorem \[thm:order\]
------------------------------
We first prove that $\tau_d\le \tau_p$. By the convexity of the rate function $I(\cdot)$, and because $I(R^*) = 0$, we have, for all $R\ge R^*$, $$2 I(\frac{R + R^*}{2}) \le I(R)~.$$ Hence, for all $R\ge R^*$, $$2 I(\frac{R + R^*}{2}) - \log(2 (\frac{R+R^*}{2}))
\le I(R) - \log(R + R^*)~.$$ This establishes the result. The fact that $\tau_p\le \tau_v$ immediately follows from $$\inf_{R \ge R^*} \left( I(R) - \log(R + R^*) \right) \le
\inf_{R \ge R^*} \left( I(R) - \log(R) \right)~.$$
Proof of Theorem \[thm:order2\]
-------------------------------
The fact that $R^*\le R_d$ is immediate from the definition of $R_d$.
We first prove that $R_d \le R_p$. Recall that $R_d$ is uniquely defined by $$\frac{1}{2 R_d}\in
[I^\prime_-(R_d), I^\prime_+(R_d)]~,$$ and $R_p$ is uniquely defined by $$\frac{1}{R_p + R^*}\in
[I^\prime_-(R_p), I^\prime_+(R_p)]~.$$ Now $$R_d > R_p \Longrightarrow 2 R_d > R_p + R^* \Longrightarrow
\frac{1}{2R_d} < \frac{1}{R_p + R^*} \Longrightarrow R_d < R_p~.$$ This contradiction implies that $R_d \le R_p$.
We next prove that $R_p \le R_v$. Here we also need to recall that $R_v$ is uniquely defined by $$\frac{1}{R_v} \in [I^\prime_-(R_v), I^\prime_+(R_v)]~.$$ Now $$R_p > R_v \Longrightarrow R_p + R^* > R_v \Longrightarrow
\frac{1}{R_p + R^*} < \frac{1}{R_v} \Longrightarrow R_p < R_v~.$$ This contradiction proves that $R_p \le R_v$.
We next prove that $R_v \le R_p + R^*$. For this, it suffices to observe the contradiction $$R_v > R_p + R^* \Longrightarrow \frac{1}{R_v} < \frac{1}{R_p+R^*}
\Longrightarrow R_v < R_p \Longrightarrow R_v < R_p + R^*~.$$
We finally prove that $R_p + R^* \le 2 R_d$. For this we observe the contradiction $$R_p + R^* > 2 R_d \Longrightarrow \frac{1}{R_p + R^*} < \frac{1}{2 R_d}
\Longrightarrow R_p < R_d \Longrightarrow R_p + R^* < 2 R_d~.$$
This completes the proof.
Concluding Remarks {#s.concluding}
==================
Connections with Error Exponents
--------------------------------
In this subsection we make some remarks about the connections between the concerns of this paper and the problem of error exponents in channel coding over the additive white Gaussian noise channel [@Gallager Section 7.4], as discussed in [@AB] in the Poltyrev regime. For all $n \ge 1$ and $k \ge 1$, let $W_n^{(i,k)}$, $n\ge i\ge 1$, denote an i.i.d. sequence of Gaussian random variables, all centered and of variance $\sigma^2$. Let $W_n^{(k)}$ denote the $n$-dimensional vector with coordinates $W_n^{(i,k)}$, $n\ge i\ge 1$. Then $T_n^{(k)}+W_n^{(k)}$ belongs to the closed ball of center $T_n^{(k)}$ and radius $X_n^{(k)} \sqrt{n},$ with $$\label{eq:GGrain}
X_n^{(k)} := \left(\frac 1 n
\sum_{i=1}^n \left(W_n^{(i,k)}\right)^2\right)^{\frac 1 2} \stackrel{d}{=} \bar{X}_n~,$$ where $\bar{X}_n$ denotes a random variable having the distribution of the normalized radius random variables in the preceding equation. One can check that $(\bar{X}_n, n \ge 1)$ satisfy an LDP and all the assumptions listed above. For each $\sigma^2 > 0$, we call such a family of Boolean models (parametrized by $n \ge 1$, as usual) the case with [*Gaussian grains*]{}.
For Shannon’s channel coding problem in the Poltyrev regime, as considered in [@AB], the focus is on the probability of error. As a result, one only wants to associate those points in Euclidean space that have a high probability of being of the type $T_n^{(k)} + W_n^{(k)}$ to the point $T_n^{(k)}$. Therefore one considers, instead of the Boolean model where a Gaussian grain is associated to each point, another Boolean model where this Gaussian grain is replaced by an associated [*typicality region*]{}, namely the set $$\{ T_n^{(k)} + v ~:~ v \in \mathbb{R}^n,~
\| v \|_2 \le \sigma \sqrt{n} + \alpha_n \}~,$$ where $\| v \|_2$ denotes the usual Euclidean length of $v$ and where $0 < \alpha_n = O(\sqrt{n})$ are chosen such that $$\begin{aligned}
&& ~ \frac{\alpha_n}{\sqrt{n}} \to 0 \mbox{ as $n \to \infty$};\\
&&P(\| W_n^{(k)} \| \le
\sigma \sqrt{n} + \alpha_n ) \to 1 \mbox{ as $n \to \infty$
(for each $1 \le k \le n$)};\\
&&
\frac{1}{n} \log \mbox{Vol} \{ v \in \mathbb{R}^n ~:~
\| v \|_2 \le \sigma \sqrt{n} + \alpha_n \} \to \frac{1}{2} \log (2 \pi e \sigma^2)
\mbox{ as $n \to \infty$}~.\end{aligned}$$ This now gives rise to a family of deterministic Boolean models which will be referred to as the [*truncated Gaussian grain*]{} models below. The [*Poltyrev capacity*]{} is the threshold for the asymptotic logarithmic intensity of such a family of Boolean models. This threshold is the asymptotic logarithmic intensity up to which it is possible to make such an association with asymptotically vanishing probability of error. It is also the threshold up to which there is a vanishingly small probability that a point in Euclidean space is covered by multiple truncated Gaussian grains, which directly corresponds to what is called the volume fraction threshold in the present paper.
It turns out that the volume fraction threshold is smaller for Gaussian grains than for truncated Gaussian grains, even though the normalized radii of the grains in the two models have the same asymptotic limit $R^*$. To see this, consider Gaussian grains with per-coordinate variance $\sigma^2$, as above. Then $\bar{X}_n^2$ is distributed as the average of $n$ i.i.d. squared Gaussian random variables of mean $0$ and variance $\sigma^2$, so we have $\bar{X}_n^2 \stackrel{\PP}{\to} \sigma^2$ as $n \to \infty$, which implies $\bar{X}_n \stackrel{\PP}{\to} \sigma$ as $n \to \infty$. This means $E[ \bar{X}_n]$ (which is bounded above by $( E[ (\bar{X}_n)^2])^{\frac{1}{2}}
= \sigma$) converges to $\sigma$ as $n \to \infty$. This means $R^* = \sigma$. The volume fraction threshold for deterministic grains with radius $R^* \sqrt{n}$ in dimension $n$ is then given by the R.H.S. of eqn. (\[eq:dettauv\]). The exponent of the growth rate in $n$ of the volume of each Gaussian grain is strictly bigger than this. That it is at least as big follows immediately from the convexity of the function $R \mapsto R^n$, defined for $R \ge 0$. To see the strict inequality, first note that the density of the radius of the Gaussian grain in dimension $n$ can be written as $g_n^\sigma(r)$, $r \ge 0$, where $g_n^\sigma(r) = \frac{1}{\sigma} g_n^1(\frac{r}{\sigma})$, with $$g_1^\sigma(r) := \frac{n r^{n-1} e^{ - \frac{r^2}{2}}}
{2^{\frac{n}{2}} \Gamma(\frac{n}{2} + 1)}~,~~r \ge 0~,$$ where $\Gamma(\cdot)$ denotes the standard Euler gamma function. To figure out the asymptotic growth rate of the expected volume of a grain, we need to compute $$\lim_{n \to \infty} \frac{1}{n} \log \int_0^\infty V_n(1) r^n g_n^\sigma(r) dr~,$$ where $V_n(1) := \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}$ denotes the volume of the ball of unit radius in $\mathbb{R}^n$.
It is convenient to reparametrize $r$ as $v \sigma \sqrt{n}$, giving $$g_n^\sigma(v \sigma \sqrt{n}) = e^{- n( \frac{v^2}{2} - \frac{1}{2} - \log(v) + o(1))}~.$$ Thus $$\begin{aligned}
&~& \lim_{n \to \infty} \frac{1}{n} \log \int_0^\infty
\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)} r^n g_n^\sigma(r) dr\\
~~~~~~~~~~~~~~~~ &=& \lim_{n \to \infty} \frac{1}{n} \log
\left( (2 \pi e \sigma)^{\frac{n}{2}} \int_0^\infty e^{ n (\log v + o(1))}
e^{- n( \frac{v^2}{2} - \frac{1}{2} - \log(v) + o(1))} dv \right)\\
~~~~~~~~~~~~~~~~ &\stackrel{(a)}{=}& \frac{1}{2} \log ( 2 \pi e \sigma^2) + \frac{1}{2} ( \log 4 - 1)\\
~~~~~~~~~~~~~~~~ &>& \frac{1}{2} \log ( 2 \pi e \sigma^2)~,\end{aligned}$$ where step (a) comes from Laplace’s principle that the asymptotics is controlled by the exponential term in the integrand with the largest exponent. Laplace’s principle as just applied is only a heuristic, of course, but this calculation makes the point that the volume fraction threshold for Gaussian grains is strictly smaller than the volume fraction threshold for the truncated Gaussian grains (i.e. the Poltyrev threshold).
Thresholds in the Gaussian Grain Case
-------------------------------------
It is interesting to consider the case of Gaussian grains in detail as an illustration of the general results in this paper, and we turn to this next, giving, in the process, a rigorous derivation of the volume fraction threshold for this case as discussed in the preceding subsection.
We first need to determine the large deviations rate function for the sequence $(\bar{X}_n, n \ge 1)$, with each $\bar{X}_n$ defined as in eqn. (\[eq:GGrain\]). Here we think of $\sigma^2 > 0$ as being fixed. It is easy to do this by first observing that $(\bar{X}^2_n, n \ge 1)$ satisfies the large deviations principle with rate function $J(\cdot)$ given by $$J(z) = \begin{cases}
\frac{z}{2 \sigma^2} - \frac{1}{2} - \frac{1}{2} \log \frac{z}{\sigma^2} &
\mbox{ if $z > 0$}\\
\infty & \mbox{ otherwise}
\end{cases}~,$$ which follows from the fact that if $Z_n$ is a Gaussian random variable with mean zero and variance $\sigma^2$ then $$\log E[ e^{\theta Z_n^2}] =
\begin{cases}
-\frac{1}{2} \log (1 - 2 \theta \sigma^2) & \mbox{ if $\theta < \frac{1}{2 \sigma^2}$} \\
\infty & \mbox{ otherwise}~.
\end{cases}$$ The contraction principle [@DZ Thm. 4.2.1] then gives the rate function of the sequence $(\bar{X}_n, n \ge 1)$ as being $I(\cdot)$, where $$I(R) = \begin{cases}
\frac{R^2}{2 \sigma^2} - \frac{1}{2} - \frac{1}{2} \log \frac{R^2}{\sigma^2}
& \mbox{ if $R > 0$}\\
\infty & \mbox{ otherwise}~.
\end{cases}$$ Another way to see this is to note that the convex conjugate dual of this function is the function $$\Lambda(\theta) = \frac{\theta \sigma}{2}
\left( \frac{\theta \sigma + \sqrt{ \theta^2 \sigma^2 + 4}}{2} \right)
+ \log \left( \frac{\theta \sigma + \sqrt{ \theta^2 \sigma^2 + 4}}{2} \right)~,$$ and to check that $$\begin{aligned}
\Lambda(\theta) &=& \lim_{n \to \infty} \frac{1}{n} \log E[ e^{n \theta \bar{X}_n}]\\
&=& \lim_{n \to \infty} \frac{1}{n} \log
\int_0^\infty e^{\sqrt{n} \theta r} g_n^\sigma(r) dr\\
&=& \lim_{n \to \infty} \frac{1}{n} \log
\int_0^\infty e^{ n \left( \theta v \sigma - \frac{v^2}{2} + \frac{1}{2} + \log v + o(1)
\right)} dv~,\end{aligned}$$ by the use of Laplace’s principle in the last expression on the right hand side.
The volume fraction threshold for the case of Gaussian grains (with $\sigma^2 > 0$ being fixed) it then given by finding the solution $R _v \ge \sigma$ to the equality $$\frac{R}{\sigma^2} - \frac{1}{R} = \frac{1}{R}~.$$ There is a unique solution to this equation, namely $R_v = \sigma \sqrt{2}$, and this turns out to satisfy $R_v \ge \sigma$, as it should. Here $\sigma$ is playing the role of $R^*$ in the general theory. Substituting back into the formula $$\tau_v = - \frac{1}{2} \log( 2 \pi e) + I(R_v) - \log(R_v)$$ for the volume fraction threshold gives $$\tau_v = - \frac{1}{2} \log(2 \pi e \sigma^2) - \frac{1}{2} ( \log 4 - 1)~.$$ This is the announced rigorous derivation of the formula that was found above by the heuristic application of Laplace’s principle.
The degree threshold is given by finding the solution $R _d \ge \sigma$ to the equality $$\frac{R}{\sigma^2} - \frac{1}{R} = \frac{1}{2 R}~.$$ There is a unique solution to this equation, namely $R_d = \sigma \sqrt{\frac{3}{2}}$, and this turns out to satisfy $R_d \ge \sigma$, as it should. Substituting back into the formula $$\tau_d = - \frac{1}{2} \log( 2 \pi e) + 2 I(R_d) - \log(2 R_d)$$ for the degree threshold gives $$\tau_d = - \frac{1}{2} \log(2 \pi e \sigma^2) - \frac{1}{2} ( \log \frac{27}{2} - 1)~.$$
The percolation threshold is given by finding the solution $R _p \ge \sigma$ to the equality $$\frac{R}{\sigma^2} - \frac{1}{R} = \frac{1}{R + \sigma}~.$$ There is a unique solution to this equation, namely $R_p = \sigma c$, where $c$ is the unique root of the equation $$c^3 + c^2 - 2 c -1 = 0,$$ which satisfies $c \ge 0$. In fact, this root satisfies $c > 1$. [^4] Substituting back into the formula $$\tau_p = - \frac{1}{2} \log( 2 \pi e) + I(R_p) - \log(R_p + \sigma)$$ for the percolation threshold gives $$\tau_p = - \frac{1}{2} \log(2 \pi e \sigma^2)
- \frac{1}{2} ( \log (c^2 (1 + c)^2) - c^2 + 1)~.$$
Numerical evaluation of $c$ gives $1.2469796 < c < 1.2469797$. This approximation suffices to verify that $$\log(\frac{27}{2}) - 1 > \log (c^2 (1 + c)^2) - c^2 + 1 > \log(4) - 1~,$$ which confirms that $
\tau_d \le \tau_p \le \tau_v
$ in the case of Gaussian grains, as required by Theorem \[thm:order\]. This approximation also suffices to verify that $$1 < \sqrt{\frac{3}{2}} < c < \sqrt{2} < 1 + c < \sqrt{6}~,$$ which confirms that $$R^*\le R_d \le R_p \le R_v \le R_p + R^* \le 2 R_d~,$$ in the case of Gaussian grains, as required by Theorem \[thm:order2\].
Returning to the Boolean model with truncated Gaussian grains discussed in the last subsection, since, for every $\sigma^2 > 0$, this family of Boolean models is a deterministic model with $R^* = \sigma$, the rate function for this model satisfies $$I(\sigma) = 0,\mbox{ and } I(R) = \infty \mbox{ for all $R \neq \sigma$}~.$$ Thus, in this case the deterministic threshold equals the percolation threshold, and both are $\log 2$ below the volume fraction threshold. It is interesting to note, as observed in [@P94] and [@AB], that this threshold also has a meaning; it is the threshold for the asymptotic logarithmic intensity up to which the truncated Gaussian grain of any given point of the Poisson process is so small that with probability asymptotically equal to $1$ it does not meet the grain of any other point of the Poisson process. This feature, which relates to a study of pairwise conflict in decoding between two codewords, is central to Gallager’s analysis of error exponents in the power constrained channels that are of interest to engineers; for more details see Section 7.4 of [@Gallager] and in particular the study there of what is called Gallager’s $E_0$ function.
Acknowledgements {#acknowledgements .unnumbered}
================
The research of the first author was supported by the ARO MURI grant W911NF- 08-1-0233, Tools for the Analysis and Design of Complex Multi-Scale Networks, the NSF grants CNS-0910702 and ECCS-1343398, and the NSF Science & Technology Center grant CCF-0939370, Science of Information. This work of the second author was supported by an award from the Simons Foundation (\# 197982 to The University of Texas at Austin).
[99]{} V. Anantharam and F. Baccelli, Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements, to appear in [*Advances in Applied Probability*]{}. A. Dembo and O. Zeitouni, [*Large Deviation Techniques and Applications*]{}, Jones and Bartlett, Boston, 1993. D. J. Daley and D. Vere-Jones, [*An Introduction to the Theory of Point Processes*]{}, Vol. 2, Second Edition, Springer, 2008. R. G. Gallager, [*Information Theory and Reliable Communications*]{}, John Wiley & Sons, 1968. J.-B. Gouéré and R. Marchand, Continuum percolation in high dimensions, http://arxiv.org/abs/1108.6133, 2011. G. Last and H. Thorisson, Invariant transports of stationary random measures and mass-stationarity, The Annals of Probability, Pages 790–813, Vol. 37, Number 2, 2009. R. Meester and R. Roy, [*Continuum Percolation*]{}. Cambridge University Press, 1996. M. D. Penrose, Continuum Percolation and Euclidean Minimal Spanning Trees in High Dimensions. [*The Annals of Applied Probability*]{}, Vol. 6, No. 2, pp. 528 -544, 1996. G. Poltyrev, “On Coding Without Restrictions for the AWGN Channel". [*IEEE Trans. on Inform. Theory*]{}, Vol. 40, No. 2, pp. 409-417, Mar. 1994. R. T. Rockafellar, [*Convex Analysis*]{}, Princeton University Press, 1972. R. Schneider and W. Weil, [*Stochastic and Integral Geometry*]{}, Springer Verlag, 2008. S.R.S. Varadhan, [*Large Deviations and Applications*]{}, SIAM, 1984.
[^1]: The scenario considered in [@AB] goes beyond additive white Gaussian noise to consider a setting where the additive noise comes from sections of a stationary and ergodic process. The Boolean models that arise in the more general case involve grains, defined by the typicality sets of the noise process, that are not necessarily spherically symmetric. Even more generally, in [@AB] the underlying point process in each dimension is allowed to be an arbitrary stationary ergodic process (while still requiring the existence of an asymptotic logarithmic intensity).
[^2]: It is straightforward to check that it is possible to choose $(\alpha_n, n \ge 1)$ satisfying these requirements.
[^3]: For technical reasons, the formal proof looks slightly different from this sketch, but this is the basic intuition.
[^4]: That there is a unique such root and that it satisfies $c > 1$ can be verified by noting that the expression on the left hand side of this equation equals $-1$ at $c =0$ and at $c =1$ and goes to $\infty$ as $c \to \infty$ and, further, the derivative in $c$ of the expression is $3 c^2 + 2 c - 2$, which equals $-2$ at $c =0$ and is a convex function.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Matthieu <span style="font-variant:small-caps;">Faitg</span>'
title: '**Projective representations of mapping class groups in combinatorial quantization**'
---
IMAG, Univ Montpellier, CNRS, Montpellier, France.\
E-mail address: `[email protected]`
[2cm]{}[2cm]{} [<span style="font-variant:small-caps;">Abstract</span>. Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev–Grosse–Schomerus and Buffenoir–Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. We construct a projective representation of the mapping class group of $\Sigma_{g,n}$ using $\mathcal{L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods. ]{}
Introduction
============
Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. It is readily seen that $\Sigma_{g,n} \setminus D$ (where $D$ is an open disk) is homeomorphic to the tubular neighborhood of the graph $\Gamma$ whose edges are the generators of the fundamental group of the surface (see Figure \[figureIntro\]); we will denote $\Sigma_{g,n}^{\mathrm{o}} = \Sigma_{g,n} \setminus D$. This particular choice of graph is not a loss of generality.\
Let $G$ be an algebraic Lie group (generally assumed connected and simply-connected, *e.g.* $G = \mathrm{SL}_2(\mathbb{C})$). A lattice gauge field theory on $\Gamma$ is a discretization of the moduli space of flat $G$-connections on $\Sigma_{g,n}^{\mathrm{o}}$. It consists of a set of discrete connections $\mathcal{A} = G^{2g+n}$, a gauge group $\mathcal{G} = G$ and an algebra of functions $\mathbb{C}[\mathcal{A}] = \mathbb{C}[G]^{\otimes 2g+n}$ (see *e.g.* [@W 2.3], [@labourie Chap. 2] for the general definitions). There is also a notion of discrete holonomy defined in a natural way. The gauge group acts on $\mathcal{A}$ (and dually on $\mathbb{C}[\mathcal{A}]$ on the right) by conjugation; the invariant functions are called classical observables.\
Lattice gauge field theory on $\Gamma$ is another description of the character variety of $\Sigma_{g,n}^{\mathrm{o}}$. More precisely, the discrete holonomy is a bijection between the set $\mathcal{A}/\mathcal{G}$ of discrete $G$-connections up to gauge equivalence and $\mathrm{Hom}\!\left(\pi_1(\Sigma_{g,n}^{\mathrm{o}}), G\right)/G$. The space $\mathcal{A}$ is endowed with a Poisson structure defined by Fock and Rosly [@FockRosly]. This Poisson structure descends to $\mathcal{A}/\mathcal{G}$ and moreover, $\mathbb{C}[\mathcal{A}/\mathcal{G}] = \mathbb{C}[\mathcal{A}]^{\mathcal{G}}$ is isomorphic to $\mathbb{C}\!\left[ \mathrm{Hom}\!\left(\pi_1(\Sigma_{g,n}^{\mathrm{o}}), G\right) \right]^G$, namely the space of functions on the character variety. Under this isomorphism, the Fock–Rosly Poisson structure corresponds to that given by the Goldman bracket, or equivalently, by the Atiyah–Bott symplectic form.\
The previous remarks apply to the original surface $\Sigma_{g,n}$ if we consider the subset of discrete flat connections $\mathcal{A}_f$ instead of $\mathcal{A}$. These are the discrete connections whose holonomy along the boundary of the unique face of the graph $\Gamma$ is trivial.
It is worthwhile to describe the algebra of functions $\mathbb{C}[\mathcal{A}]$ in terms of matrix coefficients $\overset{I}{T}{^i_j} \in \mathbb{C}[G]$ (where $I$ is a finite-dimensional $G$-module), since they linearly span $\mathbb{C}[G]$. For instance, we can construct a function $\overset{I}{A}(k){^i_j} \in \mathbb{C}[\mathcal{A}]$ by putting the function $\overset{I}{T}{^i_j}$ over the edge $a_k$ and the trivial function $1$ on the other edges. In particular, we get a matrix $\overset{I}{A}(k)$ with coefficients in $\mathbb{C}[\mathcal{A}]$, see Figure \[figureIntro\]. The coefficients of such matrices span $\mathbb{C}[\mathcal{A}]$ as an algebra. The action of the gauge group is by conjugation, for instance $\overset{I}{A} \cdot g = \overset{I}{g} \overset{I}{A} \overset{I}{g}{^{-1}}$, where $\overset{I}{g} = \overset{I}{T}(g)$ is the representation of $g$ on $I$.
In the works of Alekseev [@alekseev], Alekseev–Grosse–Schomerus [@AGS; @AGS2] and Buffenoir–Roche [@BR; @BR2], the Lie group $G$ is replaced by a quantum group $U_q(\mathfrak{g})$, with $\mathfrak{g} = \mathrm{Lie}(G)$. The notions described above can be generalized in this setting. Then the graph algebra $\mathcal{L}_{g,n}(U_q(\mathfrak{g}))$ is a quantization of the Fock-Rosly Poisson structure on $\mathcal{A}$. It is an associative (non-commutative) deformation of $\mathbb{C}[\mathcal{A}]$, defined by means of equalities involving the matrices $\overset{I}{A}(k), \overset{I}{B}(k), \overset{I}{M}(l)$ and the $R$-matrix of $U_q(\mathfrak{g})$ ($I$ now runs in the set of finite-dimensional $U_q(\mathfrak{g})$-modules). This algebra is endowed with an action of $U_q(\mathfrak{g})$, analogous to the action of the gauge group $G$ on $\mathbb{C}[\mathcal{A}]$. The multiplication in $\mathcal{L}_{g,n}(U_q(\mathfrak{g}))$ is designed so that it is an $U_q(\mathfrak{g})$-module-algebra with respect to this action. In particular, we have a subalgebra of invariant elements $\mathcal{L}^{\mathrm{inv}}_{g,n}(U_q(\mathfrak{g}))$, which is a quantized analogue of the algebra of classical observables of the initial lattice gauge field theory.
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These quantized algebras of functions and their generalizations appear in various works, *e.g.* [@BFKB2; @BNR; @MW; @BZBJ; @AGPS].\
The definition of the algebras $\mathcal{L}_{g,n}(U_q(\mathfrak{g}))$ is purely algebraic and we can replace the quantum group $U_q(\mathfrak{g})$ by any ribbon Hopf algebra $H$. The representation theory of $\mathcal{L}_{g,n}(H)$ and of its subalgebra of invariant elements is investigated in [@alekseev] when $H$ is the quantum group $U_q(\mathfrak{g})$ for $q$ generic and in [@AS] when $H$ is finite-dimensional and semi-simple, or a semisimple truncation of quantum group at a root of unity (the latter being defined in the setting of quasi-Hopf algebras).\
Moreover, in [@AS; @AS2], a projective representation of the mapping class group of $\Sigma_{g,n}$ based on $\mathcal{L}_{g,n}(H)$ is described. This representation is an analogue in the quantized setting of the obvious representation of the mapping class groups on $\mathbb{C}[\mathcal{A}]$ and $\mathbb{C}[\mathcal{A}_f].$\
In this paper, we consider the algebras $\mathcal{L}_{g,n}(H)$ from a purely algebraic viewpoint, under the general assumption that the gauge algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. The algebras $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$, which are the building blocks of the theory (see Definiton \[definitionLgn\]), and the associated projective representation of $\mathrm{SL}_2(\mathbb{Z})$, have already been studied under these assumptions in [@Fai18].\
In section \[sectionLgnAlekseev\], we first quickly recall the definition and main properties of $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$. Then we recall the definition of $\mathcal{L}_{g,n}(H)$, and we show that the Alekseev isomorphism [@alekseev], which is a fundamental tool to construct representations of $\mathcal{L}_{g,n}(H)$, holds under our assumptions. In particular, when $n=0$, the Alekseev isomorphism implies that $\mathcal{L}_{g,0}(H)$ is isomorphic to a matrix algebra (because the Heisenberg double is a matrix algebra, see subsection \[heisenbergDouble\] and ) and that the only indecomposable (and simple) representation of $\mathcal{L}_{g,0}(H)$ is $(H^*)^{\otimes g}$.\
We construct representations of the subalgebras of invariant elements $\mathcal{L}^{\mathrm{inv}}_{g,n}(H)$ in section \[RepInvariants\] with a generalization of the method used in [@alekseev]. More precisely, for each representation $V$ of $\mathcal{L}_{g,n}(H)$ we associate a representation $\mathrm{Inv}(V) \subset V$ of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$, defined by the requirement that the holonomy of a connection along the boundary of the unique face of the graph $\Gamma$ acts trivially on it.\
In section \[sectionRepMCG\], we construct a projective representation of the mapping class groups $\mathrm{MCG}(\Sigma_{g,0}^{\mathrm{o}})$ and $\mathrm{MCG}(\Sigma_{g,0})$ (we discuss the case $n > 0$ in subsection \[CasGeneral\]). The idea of the construction is to associate an automorphism $\widetilde{f}$ of $\mathcal{L}_{g,0}(H)$ to each element $f$ of the mapping class group (Proposition \[liftHumphries\]), called the lift of $f$. To define such a lift, we just replace generators of the fundamental group by matrices of generators of $\mathcal{L}_{g,0}(H)$ (up to some normalization), see and . Since $\mathcal{L}_{g,0}(H)$ is isomorphic to a matrix algebra, this automorphism is inner and we get an element $\widehat{f} \in \mathcal{L}_{g,0}(H)$, unique up to scalar. Then to $f$ we associate the representation of $\widehat{f}$ on $(H^*)^{\otimes g}$ (in the case of $\Sigma_{g,0}^{\mathrm{o}}$) and on $\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$ (in the case of $\Sigma_{g,0}$). This construction was first introduced by Alekseev and Schomerus in [@AS] and [@AS2] in the semi-simple setting. Here we generalize and complete this approach with detailed proofs in the non-semi-simple setting.\
Finally, we give explicit formulas for the representation of the Dehn twists about the curves depicted in Figure \[figureCourbesCanoniques\] (Theorem \[formulesExplicites\]) and in particular this allows us to prove that the representation of the mapping class group described above is equivalent (Theorem \[thmEquivalenceReps\]) to another one constructed by Lyubashenko using categorical techniques based on the coend of a ribbon category $\mathcal{C}$ satisfying some assumptions [@lyu95a; @lyu95b; @lyu96]. For this equivalence we take $\mathcal{C} = \mathrm{mod}_l(H)$, the category of finite-dimensional left modules. For works based on the Lyubashenko representation, see *e.g.* [@FSS1; @FSS2].
Although the two representations are equivalent, the combinatorial quantization provides additional structure and tools. Indeed, it also gives rise to a representation of the quantized version of the classical observables $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$; this is interesting because these quantum observables are related to skein theory [@BFKB; @BFKB2]. Moreover, as a deformation of the algebra of functions on the character variety, combinatorial quantization is a natural and explicit setting to derive mapping class group representations.\
To sum up, the main results of this paper are:
- The construction of a projective representation of $\mathrm{MCG}(\Sigma_{g,0}^{\mathrm{o}})$ and $\mathrm{MCG}(\Sigma_{g,0})$ (Theorem \[thmRepMCG\]),
- Explicit formulas for the representation of the Dehn twists about the curves of Figure \[figureCourbesCanoniques\] (Theorem \[formulesExplicites\]),
- The equivalence with the Lyubashenko representation for $\mathrm{mod}_l(H)$ (Theorem \[thmEquivalenceReps\]).
Let us conclude with a few remarks about our results and further work. First, as already said, all our constructions are explicit; this feature of the theory could be helpful to make computations when one studies the representation of the mapping class group for a given $H$ (see for instance the proof of [@Fai18 Theorem 6.4] for computations in the case of the torus with $H = \overline{U}_q(\mathfrak{sl}(2))$). Second, for $H = \overline{U}_q(\mathfrak{sl}(2))$, our representations of the mapping class group should be associated to logarithmic conformal field theory in arbitrary genus. For the torus $\Sigma_{1,0}$, this is indeed the case: combining the results of [@FGST] and [@Fai18], the projective representation of $\mathrm{SL}_2(\mathbb{Z})$ obtained via the combinatorial quantization is equivalent to that coming from logarithmic conformal field theory. Hence, a natural problem is to study in depth the representation of the mapping class group obtained for $H = \overline{U}_q(\mathfrak{sl}(2))$ (basis of the representation space, explicit formulas for the action on this basis and structure of the representation). Another question is to study the relation between $\mathcal{L}_{g,n}^{\mathrm{inv}}\!\left( \overline{U}_q(\mathfrak{sl}(2)) \right)$ and skein theory (work in progress).
**Acknowledgments.** I am grateful to my advisors, Stéphane Baseilhac and Philippe Roche, for their regular support and their useful remarks. I thank A. Gainutdinov for inviting me to present this work at the algebra seminar of the University of Hamburg.\
**Notations.** If $A$ is a $\mathbb{C}$-algebra, $V$ is a finite-dimensional $A$-module and $x \in A$, we denote by $\overset{V}{x} \in \operatorname{End}_{\mathbb{C}}(V)$ the representation of $x$ on the module $V$. Similarly, if $X \in A^{\otimes n}$ and if $V_1, \ldots , V_n$ are $A$-modules, we denote by $\overset{V_1 \ldots V_n}{X}$ the representation of $X$ on $V_1 \otimes \ldots \otimes V_n$. Here we consider only finite-dimensional representations, hence $H$-module implicitly means finite-dimensional $H$-module.\
Let $\mathrm{Mat}_m(A) = \mathrm{Mat}_m(\mathbb{C}) \otimes A$. Every $M \in \mathrm{Mat}_m(A)$ is written as $M = \sum_{i,j} E^i_j \otimes M^i_j$, where $E^i_j$ is the matrix with $1$ at the intersection of the $i$-th row and the $j$-th column and $0$ elsewhere. If $f : A \to A$ is a morphism, then we define $f(M)$ by $f(M) = \sum_{i,j} E^i_j \otimes f(M^i_j)$. Let moreover $N = \sum_{i,j}E^i_j \otimes N^i_j \in \mathrm{Mat}_n(\mathbb{C}) \otimes A$. We embed $M,N$ in $\mathrm{Mat}_m(\mathbb{C}) \otimes \mathrm{Mat}_n(\mathbb{C}) \otimes A = \mathrm{Mat}_{mn}(A)$ by $$M_1 = \sum_{i,j}E^i_j \otimes \mathbb{I}_n \otimes M^i_j \in \mathrm{Mat}_m(\mathbb{C}) \otimes \mathrm{Mat}_n(\mathbb{C}) \otimes A, \:\:\:\:
N_2 = \sum_{i,j} \mathbb{I}_m \otimes E^i_j \otimes N^i_j \in \mathrm{Mat}_m(\mathbb{C}) \otimes \mathrm{Mat}_n(\mathbb{C}) \otimes A.$$ where $\mathbb{I}_k$ is the identity matrix of size $k$. Then $M_1N_2$ (resp. $N_2M_1$) contains all the possible products of coefficients of $M$ (resp. of $N$) by coefficients of $N$ (resp. of $M$): $(M_1N_2)^{ik}_{j\ell} = M^i_jN^k_{\ell}$ (resp. $(N_2M_1)^{ik}_{j\ell} = N^k_{\ell}M^i_j$).\
In order to simplify notation we use Sweedler’s notation (see [@kassel Not. III.1.6]) without summation sign for coproducts, that is we write $$\Delta(x) = x' \otimes x'', \:\:\: \Delta^{(2)}(x) = (\Delta \otimes \mathrm{id}) \circ \Delta(x) = x' \otimes x'' \otimes x''', \:\:\: \ldots, \:\:\: \Delta^{(n)}(x) = x^{(1)} \otimes \ldots \otimes x^{(n+1)}.$$ We write the universal $R$-matrix as $R = a_i \otimes b_i$ with implicit summation on $i$ and define $R' = b_i \otimes a_i$. We also denote $RR' = X_i \otimes Y_i$, $(RR')^{-1} = \overline{X}_i \otimes \overline{Y}_i$.\
The symbol “?” will mean a variable in functional constructions. For instance if $H$ is a finite-dimensional Hopf algebra and $\varphi, \psi \in H^*$, $a,b \in H$, then for all $x, y \in H$, $\varphi(?a) : x \mapsto \varphi(xa)$, $\varphi(?a) \otimes \psi(b?) : x\otimes y \mapsto \varphi(xa)\psi(by)$ and $\varphi(?a)\psi(b?) : x \mapsto \varphi(x'a)\psi(bx'')$ (thanks to the dual Hopf algebra structure on $H^*$, see below).
Preliminaries
=============
In all this paper, $H$ is a finite-dimensional, factorizable, ribbon Hopf algebra.
Factorizable ribbon Hopf algebras
---------------------------------
We recall basic facts about Hopf algebras. For more details, see [@kassel].\
If $I$ is a (finite-dimensional) $H$-module, we denote by $\overset{I}{T} \in \mathrm{Mat}_{\dim(I)}(H^*)$ the matrix defined by $\overset{I}{T}(x) = \overset{I}{x}$. Since $H$ is finite-dimensional, the coefficients of the matrices $\overset{I}{T}$ span $H^*$ when $I$ runs in the set of $H$-modules. We assume that $H$ is factorizable, which means that the coefficients of the matrices $(\overset{I}{T} \otimes \mathrm{id})(RR')$ span $H$ when $I$ runs in the set of $H$-modules. Let $R^{(+)} = R$, $R^{(-)} = R'^{-1}$, and consider the matrices $\overset{I}{L}{^{(\pm)}} = (\overset{I}{T} \otimes \mathrm{id})(R^{(\pm)})$. Since $H$ is factorizable, the coefficients of the matrices $\overset{I}{L}{^{(+)}}, \overset{I}{L}{^{(-)}}$ generate $H$ as an algebra when $I$ runs in the set of $H$-modules. As a consequence of the properties of the universal $R$-matrix (see [@kassel VIII.2]), we have the following relations: $$\label{propertiesL}
\begin{aligned}
&\overset{I}{L} \,\!^{(\epsilon)}_1\overset{J}{L} \,\!^{(\epsilon)}_2 = \overset{\!\!\!\!\!I\otimes J}{L^{(\epsilon)}_{12}}, \:\:\:\:\: \Delta(\overset{I}{L} \,\!^{(\epsilon)}\,\!^a_b) = \sum_i\overset{I}{L} \,\!^{(\epsilon)}\,\!^i_b \otimes \overset{I}{L} \,\!^{(\epsilon)}\,\!^a_i,\\
&\overset{IJ}{R}\,\!^{(\epsilon)}_{12} \overset{I}{L} \,\!^{(\epsilon)}_1 \overset{J}{L} \,\!^{(\sigma)}_2 = \overset{J}{L} \,\!^{(\sigma)}_2 \overset{I}{L} \,\!^{(\epsilon)}_1 \overset{IJ}{R}\,\!^{(\epsilon)}_{12} \:\:\:\:\, \forall \, \epsilon, \sigma \in \{\pm\}\\
&\overset{IJ}{R}\,\!^{(\epsilon)}_{12} \overset{I}{L} \,\!^{(\sigma)}_1 \overset{J}{L} \,\!^{(\sigma)}_2 = \overset{J}{L} \,\!^{(\sigma)}_2 \overset{I}{L} \,\!^{(\sigma)}_1 \overset{IJ}{R}\,\!^{(\epsilon)}_{12} \:\:\:\: \forall \, \epsilon, \sigma \in \{\pm\}.
\end{aligned}$$ Recall that the Drinfeld element $u$ (see [@kassel VIII.4]) and its inverse are: $$\label{u}
u = S(b_i)a_i = b_iS^{-1}(a_i), \:\:\:\:\: u^{-1} = S^{-2}(b_i)a_i = S^{-1}(b_i)S(a_i) = b_iS^2(a_i).$$ We assume that $H$ contains a ribbon element $v$ (see [@kassel XIV.6]); it satisfies $$\label{ribbon}
v \text{ is central and invertible, } \:\:\:\:\: v^2 = uS(u), \:\:\:\:\: \Delta(v) = (R'R)^{-1} v \otimes v, \:\:\:\:\: S(v) = v.$$ Then $H$ contains a canonical pivotal element $g = uv^{-1}$. It satisfies $\Delta(g) = g \otimes g$ and $S^2(x) = gxg^{-1}$ for all $x \in H$.\
We denote by $\mathcal{O}(H)$ the vector space $H^*$ endowed with the dual Hopf algebra structure, which in terms of matrix coefficients is: $$\label{structureOH}
\overset{I}{T_1}\overset{J}{T_2} = \overset{I \otimes J}{T}\!\!\!_{12}, \:\: 1_{H^*} = \overset{\mathbb{C}}{T}, \:\: \Delta(\overset{I}{T^{\, a}_{\, b}}) = \sum_i\overset{I}{T^{\, a}_{\, i}} \otimes \overset{I}{T^{\, i}_{\, b}}, \:\: \varepsilon(\overset{I}{T}) = I_{\dim(I)}, \:\: S(\overset{I}{T}) = \overset{I}{T}{^{-1}}$$ where $\mathbb{C}$ is the trivial representation, so $\overset{\mathbb{C}}{T} = \varepsilon$, the counit of $H$. In particular, in $\mathcal{O}(H)$ holds the following exchange relation: $$\overset{IJ}{R}_{12} \overset{I}{T}_1 \overset{J}{T}_2 = \overset{J}{T}_2 \overset{I}{T}_1 \overset{IJ}{R}_{12}.$$ Since $H$ is finite-dimensional, it exists right and left integrals $\mu^r, \mu^l \in \mathcal{O}(H)$ defined by $$\forall \, \varphi \in \mathcal{O}(H), \:\:\: \mu^r \varphi = \varepsilon(\varphi)\mu^r, \:\:\: \varphi \mu^l = \varepsilon(\varphi)\mu^l.$$ They are unique up to scalar and we fix $\mu^l = \mu^r \circ S$. Moreover, it holds $$\begin{aligned}
&\forall \, h \in H, \:\:\: \mu^r(h?) \varphi = \mu^r\!\left(h'?\right) \varphi\!\left(S^{-1}(h'')\right), \label{integraleShifte}\\
&\mu^l = \mu^r(g^2 ?), \label{muLmuRgCarre}\\
&\forall\, x,y \in H, \:\:\: \mu^r(xy) = \mu^r(S^2(y)x), \:\:\: \mu^l(xy) = \mu^l(S^{-2}(y)x), \label{quasiCyclic}\\
&\forall\, x,y \in H, \:\:\: \mu^r(gxy) = \mu^r(gyx), \:\:\: \mu^l(g^{-1}xy) = \mu^l(g^{-1}yx) \label{integraleShifteSLF}.\end{aligned}$$ These properties are well-known, for proofs see *e.g.* [@Fai18 Prop. 5.3, Lemma 5.9, Lemma 5.10] and the references therein; is easy, is an obvious consequence of .
Heinsenberg double of $\mathcal{O}(H)$ {#heisenbergDouble}
--------------------------------------
Let $H$ be a Hopf algebra. We recall the definition of the Heisenberg double $\mathcal{H}(\mathcal{O}(H))$ (see *e.g.* [@Mon 4.1.10]). As a vector space, $\mathcal{H}(\mathcal{O}(H)) = \mathcal{O}(H) \otimes H$. We identify $\psi \otimes 1$ with $\psi \in \mathcal{O}(H)$ and $1\otimes h$ with $h \in H$. Then the structure of algebra on $\mathcal{H}(\mathcal{O}(H))$ is defined by the following conditions:
- $\mathcal{O}(H) \otimes 1$ and $1 \otimes H$ are subalgebras of $\mathcal{H}(\mathcal{O}(H))$,
- Under the previous identifications, we have the exchange rule $$\label{relDefHeisenberg}
h\psi = \psi(?h')h''$$ where $\psi(?z) \in \mathcal{O}(H)$ is defined by $\psi(?z)(x) = \psi(xz)$.
In terms of matrices, the exchange relation is $$\label{echangeHeisenberg}
\overset{I}{L}\,\!^{(\pm)}_1 \overset{J}{T}_2 = \overset{J}{T}_2 \overset{I}{L}\,\!^{(\pm)}_1 \overset{IJ}{R}\,\!^{(\pm)}_{12}.$$ There is a faithful representation $\triangleright$ of $\mathcal{H}(\mathcal{O}(H))$ on $\mathcal{O}(H)$ (see [@Mon Lem. 9.4.2]) defined by $$\label{repHO}
\psi \triangleright \varphi = \psi\varphi, \:\:\:\:\: h \triangleright \varphi = \varphi(?h).$$ Hence we have an injective morphism $\rho : \mathcal{H}(\mathcal{O}(H)) \to \operatorname{End}_{\mathbb{C}}(H^*)$; by equality of the dimensions, it follows that $$\mathcal{H}(\mathcal{O}(H)) \cong \operatorname{End}_{\mathbb{C}}(H^*).$$ In particular, the elements of $\mathcal{H}(\mathcal{O}(H))$ can be defined by their action on $\mathcal{O}(H)$ under $\triangleright$. In terms of matrices, the representation $\triangleright$ is $$\label{repTriangleHeisenberg}
\overset{I}{T_1}\triangleright \overset{J}{T}_2 = \overset{I \otimes J}{T}\!\!\!_{12}, \:\:\:\:\:\:\: \overset{I}{L}\,\!^{(\pm)}_1 \triangleright \overset{J}{T}_2 = (\overset{I}{a_i^{(\pm)}})_1 \, b_i^{(\pm)} \triangleright \overset{J}{T}_2 = (\overset{I}{a_i^{(\pm)}})_1 \overset{J}{T}_2 (\overset{I}{b_i^{(\pm)}})_2 = \overset{J}{T}_2 \overset{IJ}{R}\,\!^{(\pm)}_{12}$$ where $R^{(\pm)} = a_i^{(\pm)} \otimes b_i^{(\pm)}$.\
For $h \in H$, let $\widetilde{h}\in \mathcal{H}(\mathcal{O}(H))$ be defined by $$\label{operateursTilde}
\widetilde{h} \triangleright \varphi = \varphi(S^{-1}(h)?).$$ It is easy to see that $$\label{echangeTilde}
\forall\, g \in H, \: \forall \, \psi \in \mathcal{O}(H), \:\:\:\: \widetilde{g}\widetilde{h} = \widetilde{gh}, \:\:\: g\widetilde{h} = \widetilde{h}g, \:\:\: \widetilde{h}\psi = \psi\!\left(S^{-1}(h'')?\right)h'.$$ Applying this to the matrices $\overset{I}{L}{^{(\pm)}}$ of generators of $H$, we define $$\overset{I}{\widetilde{L}}{^{(+)}} = \overset{I}{a_i} \widetilde{b_i}, \:\:\: \overset{I}{\widetilde{L}}{^{(-)}} = \overset{I}{S^{-1}(b_i)} \widetilde{a_i} \in \mathrm{Mat}_{\dim(I)}\!\left(\mathcal{H}(\mathcal{O}(H))\right)$$ or equivalently $\overset{I}{\widetilde{L}}{^{(\pm)}_1} \triangleright \overset{J}{T}_2 = \overset{IJ}{R}\,\!^{(\pm)-1}_{12} \overset{J}{T}_2$. Using the standard properties of the $R$-matrix, it is not difficult to show the following relations: $$\label{LTilde}
\begin{array}{l}
\overset{I}{\widetilde{L}}{^{(\epsilon)}_1} \overset{J}{\widetilde{L}}{^{(\epsilon)}_2} = \overset{I\otimes J}{\widetilde{L}}\!\!{^{(\epsilon)}_{12}}, \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \overset{I}{\widetilde{L}}{^{(\epsilon)}_1}\overset{J}{L}{^{(\sigma)}_2} = \overset{J}{L}{^{(\sigma)}_2}\overset{I}{\widetilde{L}}{^{(\epsilon)}_1}, \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \overset{IJ}{R}{^{(\epsilon)}_{12}} \overset{I}{\widetilde{L}}{^{(\epsilon)}_1}\overset{J}{T}_2 = \overset{J}{T}_2\overset{I}{\widetilde{L}}{^{(\epsilon)}_1},\\
\overset{IJ}{R}\,\!^{(\epsilon)}_{12} \overset{I}{\widetilde{L}} \,\!^{(\epsilon)}_1 \overset{J}{\widetilde{L}} \,\!^{(\sigma)}_2 = \overset{J}{\widetilde{L}} \,\!^{(\sigma)}_2 \overset{I}{\widetilde{L}} \,\!^{(\epsilon)}_1 \overset{IJ}{R}\,\!^{(\epsilon)}_{12} \:\:\:\: \forall\, \epsilon, \sigma \in \{\pm\}, \:\:\:\:\:\:\:\:\: \overset{IJ}{R}\,\!^{(\epsilon)}_{12} \overset{I}{\widetilde{L}} \,\!^{(\sigma)}_1 \overset{J}{\widetilde{L}} \,\!^{(\sigma)}_2 = \overset{J}{\widetilde{L}} \,\!^{(\sigma)}_2 \overset{I}{\widetilde{L}} \,\!^{(\sigma)}_1 \overset{IJ}{R}\,\!^{(\epsilon)}_{12} \:\:\:\: \forall\, \epsilon, \sigma \in \{\pm\}.
\end{array}$$
Definition of $\mathcal{L}_{g,n}(H)$ and the Alekseev isomorphism {#sectionLgnAlekseev}
=================================================================
Recall that $H$ is a finite-dimensional factorizable ribbon Hopf algebra. The algebras $\mathcal{L}_{g,n}(H)$ where introduced by Alekseev for $H = U_q(\mathfrak{g})$, which gave a presentation of them by generators and relations close to . Here we will define $\mathcal{L}_{g,n}(H)$ using the braided tensor product, as in [@AS2]. This has the advantage to show immediately that $\mathcal{L}_{g,n}(H)$ is a $H$-module-algebra and to emphasize the role of the two building blocks of the theory, namely $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$. We quickly recall the main properties of these building blocks, and we refer to [@Fai18] for more details about them under our assumptions on $H$.
Definition and properties of $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$
------------------------------------------------------------------------------
Let $\mathrm{T}(H^*)$ be the tensor algebra associated to $H^*$, which by definition is spanned by all the formal products $\psi_1 \cdots \psi_n$ of elements of $H^*$, modulo the obvious multilinear relations. There is a canonical injection $j : H^* \to \mathrm{T}(H^*)$ and we denote $\overset{I}{M} = j(\overset{I}{T})$.
\[defL01\] The loop algebra $\mathcal{L}_{0,1}(H)$ is the quotient of $\mathrm{T}(H^*)$ by the following fusion relations: $$\overset{I \otimes J}{M}\!_{12} = \overset{I}{M}_1\overset{IJ}{(R')}_{12}\overset{J}{M}_2\overset{IJ}{(R')}{^{-1}_{12}}$$ for all finite-dimensional $H$-modules $I,J$.
See for an explicit description of the product in $\mathcal{L}_{0,1}(H)$. The matrix coefficients $\overset{I}{M}{^i_j}$ for all $I,i,j$ linearly span $\mathcal{L}_{0,1}(H)$. The following exchange relation, called reflection equation, holds in $\mathcal{L}_{0,1}(H)$: $$\overset{IJ}{R}_{12}\overset{I}{M}_1\overset{IJ}{(R')}_{12}\overset{J}{M}_2 = \overset{J}{M}_2\overset{IJ}{R}_{12}\overset{I}{M}_1\overset{IJ}{(R')}_{12}.$$ An important fact is that $\mathcal{L}_{0,1}(H)$ is endowed with a structure of left $\mathcal{O}(H)$-comodule-algebra $\Omega : \mathcal{L}_{0,1}(H) \to \mathcal{O}(H) \otimes \mathcal{L}_{0,1}(H)$ (*i.e.* $\Omega$ is a morphism of algebras, see [@kassel Def. III.7.1]) defined by $$\Omega(\overset{I}{M}{^a_b}) = \overset{I}{T^a_{\,i}}S(\overset{I}{T}\,\!^{j}_{b}) \otimes \overset{I}{M}\,\!^i_{j}.$$ If we view $\mathcal{O}(H)$ and $\mathcal{L}_{0,1}(H)$ as subalgebras of $\mathcal{O}(H) \otimes \mathcal{L}_{0,1}(H)$ in the canonical way, then $\Omega$ is simply written $\Omega(\overset{I}{M}) = \overset{I}{T}\overset{I}{M}S(\overset{I}{T})$. Equivalently, evaluating the coaction $\Omega$ on $H$, $\mathcal{L}_{0,1}(H)$ is endowed with a structure of right $H$-module-algebra (see [@kassel Def. V.6.1] for this notion) defined by $$\label{actionL01}
\overset{I}{M} \cdot h = \overset{I}{h'}\overset{I}{M}\overset{I}{S(h'')}$$ for $h \in H$. Moreover, if we endow $H$ with the right adjoint action defined by $a \cdot h = S(h')ah''$ ($a, h \in H$), then $$\label{isoPsi01}
{ \begin{array}{crll}\Psi_{0,1} :& \mathcal{L}_{0,1}(H) & \rightarrow & H \\ &\overset{I}{M} &\mapsto & (\overset{I}{T} \otimes \mathrm{id})(RR') = \overset{I}{L}{^{(+)}}\overset{I}{L}{^{(-)-1}} \end{array}}$$ is an isomorphism of $H$-module-algebras. In particular, $\mathcal{L}_{0,1}^{\mathrm{inv}}(H) \cong \mathcal{Z}(H)$, where $\mathcal{L}_{0,1}^{\mathrm{inv}}(H)$ is the space of coinvariants (that is, the elements such that $x \cdot h = \varepsilon(h)$ for all $h \in H$ or equivalently $\Omega(x) = 1 \otimes x$). Moreover, the matrices $\overset{I}{M}$ are invertible.\
Now consider the free product $\mathcal{L}_{0,1}(H) \ast \mathcal{L}_{0,1}(H)$. Let $j_1$ (resp. $j_2$) be the canonical algebra embeddings in the first (resp. second) copy of $\mathcal{L}_{0,1}(H)$ in $\mathcal{L}_{0,1}(H) \ast \mathcal{L}_{0,1}(H)$, and define $\overset{I}{A} = j_1(\overset{I}{M})$, $\overset{I}{B} = j_2(\overset{I}{M})$.
The handle algebra $\mathcal{L}_{1,0}(H)$ is the quotient of $\mathcal{L}_{0,1}(H) \ast \mathcal{L}_{0,1}(H)$ by the following exchange relations: $$\overset{IJ}{R}_{12}\overset{I}{B}_1\overset{IJ}{(R')}_{12}\overset{J}{A}_2 = \overset{J}{A}_2\overset{IJ}{R}_{12}\overset{I}{B}_1\overset{IJ}{R}{^{-1}_{12}}$$ for all finite-dimensional $H$-modules $I, J$.
Similarly to $\mathcal{L}_{0,1}(H)$, $\mathcal{L}_{1,0}(H)$ is endowed with a structure of left $\mathcal{O}(H)$-comodule-algebra structure $\Omega : \mathcal{L}_{1,0}(H) \to \mathcal{O}(H) \otimes \mathcal{L}_{1,0}(H)$ defined by $$\Omega(\overset{I}{A}) = \overset{I}{T}\overset{I}{A}S(\overset{I}{T}), \:\:\:\: \Omega(\overset{I}{B}) = \overset{I}{T}\overset{I}{B}S(\overset{I}{T}).$$ As previously, it is equivalent to deal with the right action defined by $\overset{I}{A} \cdot h = \overset{I}{h'}\overset{I}{A}\overset{I}{S(h'')}$, $\overset{I}{B} \cdot h = \overset{I}{h'}\overset{I}{B}\overset{I}{S(h'')}$. The map $$\begin{array}{crll}
\Psi_{1,0} :& \mathcal{L}_{1,0}(H) & \rightarrow & \mathcal{H}(\mathcal{O}(H)) \\
& \overset{I}{A} &\mapsto & \overset{I}{L}\,\!^{(+)}\overset{I}{L}\,\!^{(-)-1}\\
& \overset{I}{B} &\mapsto & \overset{I}{L}\,\!^{(+)}\overset{I}{T}\overset{I}{L}\,\!^{(-)-1}\\
\end{array}$$ is an isomorphism of algebras (see [@Fai18] for a proof). It follows that $\mathcal{L}_{1,0}(H)$ is isomorphic to a matrix algebra, and in particular has trivial center.
Braided tensor product and definition of $\mathcal{L}_{g,n}(H)$ {#defLgn}
---------------------------------------------------------------
Let $\mathrm{mod}_r(H)$ be the category of finite-dimensional right $H$-modules (or, equivalently, of finite-dimensional left $H$-comodules). The braiding in $\mathrm{mod}_r(H)$ is given by: $${ \begin{array}{crll}c_{I,J} :& I \otimes J & \rightarrow & J \otimes I \\ &v \otimes w &\mapsto & w \cdot a_i \otimes v \cdot b_i \end{array}}$$ with $R = a_i \otimes b_i$. Let $(A, m_A, 1_A)$ and $(B, m_B, 1_B)$ be two algebras in $\mathrm{mod}_r(H)$ (that is, $H$-module-algebras), and define: $$\begin{aligned}
&m_{A \widetilde{\otimes} B} = (m_A \otimes m_B) \circ (\mathrm{id}_A \otimes c_{B,A} \otimes \mathrm{id}_B) \: : \: A \otimes B \to A \otimes B,\\
&1_{A \widetilde{\otimes} B} = 1_A \otimes 1_B \: : \: \mathbb{C} \to A \otimes B.\end{aligned}$$ This endows $A \otimes B$ with a structure of algebra in $\mathrm{mod}_r(H)$, denoted $A\,\widetilde{\otimes}\,B$ and called braided tensor product of $A$ and $B$ (see [@majid Lemma 9.2.12]). Note that $\widetilde{\otimes}$ is associative.\
There are two canonical algebra embeddings $j_A, j_B : A, B \hookrightarrow A\,\widetilde{\otimes}\,B$ respectively defined by $j_A(x) = x \otimes 1_B$, $j_B(y) = 1_A \otimes y$. We identify $x \in A$ (resp. $y \in B$) with $j_A(x) \in A\,\widetilde{\otimes}\,B$ (resp. $j_B(y)$). Under these identifications, the multiplication rule in $A\,\widetilde{\otimes}\,B$ is entirely given by: $$\label{braidedProduct}
\forall \, x \in A, \forall \, y \in B, \:\: yx = (x \cdot a_i)(y \cdot b_i) .$$ Since $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$ are algebras in $\mathrm{mod}_r(H)$, we can apply the braided tensor product to them.
\[definitionLgn\] $\mathcal{L}_{g,n}(H)$ is the $H$-module-algebra $\mathcal{L}_{1,0}(H)^{\widetilde{\otimes} g} \, \widetilde{\otimes} \, \mathcal{L}_{0,1}(H)^{\widetilde{\otimes} n}$.
It is useful to keep in mind that the $H$-module-algebra $\mathcal{L}_{g,n}(H)$ is associated with the surface $\Sigma_{g,n}\setminus D$; in order to make this precise we now define the matrices introduced in Figure \[figureIntro\]. There are canonical algebra embeddings $j_i : \mathcal{L}_{1,0}(H) \hookrightarrow \mathcal{L}_{g,n}(H)$ for $1 \leq i \leq g$ and $j_i : \mathcal{L}_{0,1}(H) \hookrightarrow \mathcal{L}_{g,n}(H)$ for $g+1 \leq i \leq g+n$, given by $j_i(x) = 1^{\otimes i-1} \otimes x \otimes 1^{\otimes g+n-i}$. Define $$\overset{I}{A}(i) = j_i(\overset{I}{A}), \:\:\: \overset{I}{B}(i) = j_i(\overset{I}{B}) \: \text{ for } 1 \leq i \leq g \:\: \text{ and } \:\: \overset{I}{M}(i) = j_i(\overset{I}{M}) \: \text{ for } g+1 \leq i \leq g+n.$$ Relation indicates that $\mathcal{L}_{g,n}(H)$ is an exchange algebra. Let us write the exchange relations in a matrix form. Let $\overset{I}{U}$ be $\overset{I}{A}$ or $\overset{I}{B}$ or $\overset{I}{M}$, let $\overset{J}{V}$ be $\overset{J}{A}$ or $\overset{J}{B}$ or $\overset{J}{M}$ and let $i < j$. Then, by definition of the right action and by : $$\begin{aligned}
\overset{J}{V}(j)_2 \overset{I}{U}(i)_1 = (\overset{I}{a'_k})_1 \overset{I}{U}(i)_1 \overset{I}{S(a''_k)}_1 (\overset{J}{b'_k})_2 \overset{J}{V}(j)_2 \overset{J}{S(b''_k)}_2
= (\overset{I}{a_l})_1 \overset{IJ}{R}_{12} \overset{I}{U}(i)_1 \overset{IJ}{R}{^{-1}_{12}}\overset{J}{V}(j)_2 \overset{IJ}{R}_{12} \overset{J}{S(b_l)}_2\end{aligned}$$ where for the second equality we applied properties of the $R$-matrix and obvious commutation relations in $\mathrm{End}_{\mathbb{C}}(I) \otimes \mathrm{End}_{\mathbb{C}}(J) \otimes \mathcal{L}_{g,n}(H)$. Using that $a_m a_l \otimes S(b_l) b_m = 1 \otimes 1$ together with obvious commutation relations, we obtain the desired exchange relation: $$\overset{IJ}{R}_{12} \overset{I}{U}(i)_1 \overset{IJ}{R}{^{-1}_{12}}\overset{J}{V}(j)_2 = \overset{J}{V}(j)_2 \overset{IJ}{R}_{12} \overset{I}{U}(i)_1 \overset{IJ}{R}{^{-1}_{12}}.$$ To sum up, the presentation of $\mathcal{L}_{g,n}(H)$ by generators and relations is: $$\label{PresentationLgn}
\left\{
\begin{aligned}
& \overset{I \otimes J}{U}\!\!(i)_{12} = \overset{I}{U}(i)_1\,(\overset{IJ}{R'})_{12}\,\overset{J}{U}(i)_2\, (\overset{IJ}{R'}){_{12}^{-1}}& &\text{ for } 1 \leq i \leq g+n\\
& \overset{IJ}{R}_{12}\,\overset{I}{U}(i)_1\, \overset{IJ}{R}{^{-1}_{12}} \,\overset{J}{V}(j)_2 = \overset{J}{V}(j)_2\, \overset{IJ}{R}_{12}\,\overset{I}{U}(i)_1\, \overset{IJ}{R}{^{-1}_{12}}& &\text{ for } 1 \leq i < j \leq g+n\\
& \overset{IJ}{R}_{12}\,\overset{I}{B}(i)_1\, (\overset{IJ}{R'})_{12}\,\overset{J}{A}(i)_2 = \overset{J}{A}(i)_2\, \overset{IJ}{R}_{12}\,\overset{I}{B}(i)_1\, \overset{IJ}{R}{_{12}^{-1}} & &\text{ for } 1 \leq i \leq g
\end{aligned}
\right.$$ where $U(i)$ (resp. $V(i)$) is $A(i)$ or $B(i)$ if $1 \leq i \leq g$ and is $M(i)$ if $g+1 \leq i \leq g+n$. Such a presentation was first introduced in [@alekseev] and [@AGS]. Recall that the first line of relations is the $\mathcal{L}_{0,1}(H)$-fusion relation on each loop, the second line is the exchange relation of the braided tensor product and the third line is the $\mathcal{L}_{1,0}(H)$-exchange-relation.\
**Notation.** Let $\overset{I}{N} = \overset{I}{v}{^m} \overset{I}{N}{^{n_1}_1} \ldots \overset{I}{N}{^{n_l}_l} \in \mathrm{Mat}_{\dim(I)}\!\left(\mathcal{L}_{g,n}(H)\right)$, where $m, n_i \in \mathbb{Z}$ and each $N_i$ is one of the $A(j), B(j), M(k)$ for some $j$ or $k$. By definition of the right action on $\mathcal{L}_{g,n}(H)$, we have a morphism of $H$-modules $j_N : \mathcal{L}_{0,1}(H) \to \mathcal{L}_{g,n}(H)$ defined by $j_N(\overset{I}{M}) = \overset{I}{N}$. Let $x \in H \cong \mathcal{L}_{0,1}(H)$, then we denote $x_N = j_N(x)$. The following lemma is an obvious fact.
\[injectionFusion\] If $N$ satisfies the fusion relation of $\mathcal{L}_{0,1}(H)$, $\overset{I \otimes J}{N}\!\!_{12} = \overset{I}{N}(i)_1\,\overset{IJ}{(R')}_{12}\,\overset{J}{N}(i)_2\, \overset{IJ}{(R')}{_{12}^{-1}}$, then $j_N$ is a morphism of $H$-module-algebras: $(xy)_N = x_N y_N$.
See e.g. for an application of this lemma.
The Alekseev isomorphism
------------------------
Consider the tensor product algebra $\mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n}$. We have canonical algebra embeddings $j_i : \mathcal{L}_{1,0}(H) \hookrightarrow \mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n}$ for $1 \leq i \leq g$ and $j_i : \mathcal{L}_{0,1}(H) \hookrightarrow \mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n}$ for $g+1 \leq i \leq g+n$, defined by $j_i(x) = 1^{\otimes i-1} \otimes x \otimes 1^{\otimes g+n-i}$. Define $\overset{I}{\underline{A}}(i) = j_i(\overset{I}{A})$, $\overset{I}{\underline{B}}(i) = j_i(\overset{I}{B})$ for $1 \leq i \leq g$ and $\overset{I}{\underline{M}}(i) = j_i(\overset{I}{M})$ for $g+1 \leq i \leq g+n$. We underline these matrices to avoid confusion with prior matrices having coefficients in $\mathcal{L}_{g,n}(H)$. By definition, the exchange relation between copies in $\mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n}$ is simply $$\overset{I}{\underline{U}}(i)_1\,\overset{J}{\underline{V}}(j)_2 = \overset{J}{\underline{V}}(j)_2\,\overset{I}{\underline{U}}(i)_1$$ where $i \neq j$, $\underline{U}(i), \underline{V}(i)$ is $\underline{A}(i)$ or $\underline{B}(i)$ if $1 \leq i \leq g$ and is $\underline{M}(i)$ if $g+1 \leq i \leq g+n$.\
The next result is due to Alekseev (see [@alekseev]). Consider the matrices $\overset{I}{M}{^{(-)}} = \Psi_{0,1}^{-1}(\overset{I}{L}{^{(-)}})$ and $\overset{I}{C}{^{(-)}} = \Psi_{1,0}^{-1}(\overset{I}{L}{^{(-)}}\overset{I}{\widetilde{L}}{^{(-)}})$. Let $$\label{matricesAlekseev}
\begin{array}{ll}
\overset{I}{\Lambda}_1 = \mathbb{I}_{\dim(I)}, & \:\: \overset{I}{\Lambda}_i = \overset{I}{\underline{C}}{^{(-)}}(1) \ldots \overset{I}{\underline{C}}{^{(-)}}(i-1) \:\: \text{ for } 2 \leq i \leq g+1,\\
\overset{I}{\Gamma}_{g+1} = \mathbb{I}_{\dim(I)}, & \:\: \overset{I}{\Gamma}_i = \overset{I}{\Lambda}_{g+1} \overset{I}{\underline{M}}{^{(-)}}(g+1) \ldots \overset{I}{\underline{M}}{^{(-)}}(i-1) \:\: \text{ for } g+2 \leq i \leq g+n.
\end{array}$$ be matrices with coefficients in $\mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n}$ (with $\mathbb{I}_s$ the identity matrix of size $s$).
\[isoAlekseev\] The map $$\begin{array}{crll}\alpha_{g,n} :& \mathcal{L}_{g,n}(H) = \mathcal{L}_{1,0}(H)^{\widetilde{\otimes} g} \, \widetilde{\otimes} \, \mathcal{L}_{0,1}(H)^{\widetilde{\otimes} n} & \rightarrow & \mathcal{L}_{1,0}(H)^{\otimes g} \otimes \mathcal{L}_{0,1}(H)^{\otimes n} \\
& \overset{I}{A}(i) &\mapsto & \overset{I}{\Lambda}_i\,\overset{I}{\underline{A}}(i)\,\overset{I}{\Lambda}{_i^{-1}} \:\: \text{ for } 1 \leq i \leq g \\
& \overset{I}{B}(i) &\mapsto & \overset{I}{\Lambda}_i\,\overset{I}{\underline{B}}(i)\,\overset{I}{\Lambda}{_i^{-1}} \:\: \text{ for } 1 \leq i \leq g \\
& \overset{I}{M}(i) &\mapsto & \overset{I}{\Gamma}_i\,\overset{I}{\underline{M}}(i)\,\overset{I}{\Gamma}{_i^{-1}} \:\: \text{ for } g+1 \leq i \leq g+n \\
\end{array}$$ is an isomorphism of algebras, which we call the Alekseev isomorphism.
[*Proof:* ]{}In order to show that it is a morphism of algebras, one must check using various exchange relations that the defining relations of $\mathcal{L}_{g,n}(H)$ are preserved under $\alpha_{g,n}$. This is a straightforward but tedious task and we will not give the details. Let us prove that $\alpha_{g,n}$ is bijective. We first show that $\alpha_{g,0}$ is surjective for all $g$ by induction. For $g=1$, $\alpha_{1,0}$ is the identity. For $g \geq 2$, we embed $\mathcal{L}_{g-1,0}(H)$ in $\mathcal{L}_{g,0}(H)$ in an obvious way by $\overset{I}{A}(i) \mapsto \overset{I}{A}(i)$ and $\overset{I}{B}(i) \mapsto \overset{I}{B}(i)$ for $1 \leq i \leq g-1$. Then the restriction of $\alpha_{g,0}$ to $\mathcal{L}_{g-1,0}(H)$ is $\alpha_{g-1,0}$, and by induction we assume that $\alpha_{g-1,0}(\mathcal{L}_{g-1,0}(H)) = \mathcal{L}_{1,0}(H)^{\otimes g-1}$. Since $\overset{I}{\Lambda}_i \in \mathrm{Mat}_{\dim(I)}\!\left( \mathcal{L}_{1,0}(H)^{\otimes i-1} \otimes \mathbb{C}^{\otimes g+1-i} \right)$, there exists matrices $\overset{I}{\mathcal{N}}_i$ $(1 \leq i \leq g)$ such that $\alpha_{g,0}(\overset{I}{\mathcal{N}}_i) = \overset{I}{\Lambda}_i$. Then $\alpha_{g,0}(\overset{I}{\mathcal{N}}{^{-1}_{i}} \overset{I}{U}(i) \overset{I}{\mathcal{N}}_{i}) = \overset{I}{\underline{U}}(i)$, with $U = A$ or $B$ and $\alpha_{g,0}$ is surjective. Similarly, for $g$ fixed and $n \geq 1$, we can embed $\mathcal{L}_{g,n-1}(H)$ into $\mathcal{L}_{g,n}(H)$ and reproduce the same reasoning. Hence $\alpha_{g,n}$ is surjective for all $g,n$. Since the domain and the range of $\alpha_{g,n}$ have the same dimension, it is an isomorphism. [[$\Box$]{} ]{}\
We can now generalize the isomorphisms $\Psi_{0,1}$ and $\Psi_{1,0}$ by $$\label{isoPsi}
\Psi_{g,n} = \left(\Psi_{1,0}^{\otimes g} \otimes \Psi_{0,1}^{\otimes n}\right) \circ \alpha_{g,n} \: : \: \mathcal{L}_{g,n}(H) \overset{\sim}{\rightarrow} \mathcal{H}(\mathcal{O}(H))^{\otimes g} \otimes H^{\otimes n}.$$ In particular $\mathcal{L}_{g,0}(H)$ is a matrix algebra, since $\mathcal{H}(\mathcal{O}(H))$ is.\
Thanks to $\Psi_{g,n}$, the representation theory of $\mathcal{L}_{g,n}(H)$ is entirely determined by the representation theory of $H$. Indeed, the only indecomposable (and simple) representation of $\mathcal{H}(\mathcal{O}(H)) \cong \operatorname{End}_{\mathbb{C}}(H^*)$ is $H^*$, thus it follows that the indecomposable representations of $\mathcal{L}_{g,n}(H)$ are of the form $$(H^*)^{\otimes g} \otimes I_1 \otimes \ldots \otimes I_n$$ where $I_1, \ldots, I_n$ are indecomposable representations of $H$. We will denote the action of $\mathcal{L}_{g,n}(H)$ on $(H^*)^{\otimes g} \otimes I_1 \otimes \ldots \otimes I_n$ by $\triangleright$, namely: $$\label{actionTriangle}
x \triangleright \varphi_1 \otimes \ldots \otimes \varphi_g \otimes v_1 \otimes \ldots \otimes v_n = \Psi_{g,n}(x) \cdot \varphi_1 \otimes \ldots \otimes \varphi_g \otimes v_1 \otimes \ldots \otimes v_n$$ for $x \in \mathcal{L}_{g,n}(H)$, where $\cdot$ is the action component-by-component of $\Psi_{g,n}(x)$ on $(H^*)^{\otimes g} \otimes I_1 \otimes \ldots \otimes I_n$.
Representation of $\mathcal{L}^{\mathrm{inv}}_{g,n}(H)$ {#RepInvariants}
=======================================================
Recall that an element $x \in \mathcal{L}_{g,n}(H)$ is invariant if $x \cdot h = \varepsilon(h)x$ for all $h \in H$, or equivalently, if $\Omega(x) = 1 \otimes x$. In this section we construct representations of the subalgebra of invariants $\mathcal{L}^{\mathrm{inv}}_{g,n}(H)$. For this, we use an idea introduced in [@alekseev] (the matrices $\overset{I}{C}$), but adaptated to our assumptions on $H$.
The matrices $\overset{I}{C}_{g,n}$
-----------------------------------
We first consider the case of $\mathcal{L}_{1,0}(H)$. Let us define matrices $$\label{matriceCL10}
\overset{I}{C} = \overset{I}{v}{^{2}}\overset{I}{B}\overset{I}{A}{^{-1}}\overset{I}{B}{^{-1}}\overset{I}{A}, \:\:\:\:\: \overset{I}{C}{^{(\pm)}} = \Psi_{1,0}^{-1}(\overset{I}{L}{^{(\pm)}}\overset{I}{\widetilde{L}}{^{(\pm)}})$$
\[decGauss\] The following equality holds in $\mathcal{L}_{1,0}(H)$: $$\overset{I}{C} = \overset{I}{C}{^{(+)}}\overset{I}{C}{^{(-)-1}}.$$ Moreover, the matrices $\overset{I}{C}$ satisfy the fusion relation of $\mathcal{L}_{0,1}(H)$: $$\overset{I \otimes J}{C}\!\!_{12} = \overset{I}{C}_1 \, \overset{IJ}{(R')}_{12} \, \overset{J}{C}_2 \, \overset{IJ}{(R')}{^{-1}_{12}}.$$
[*Proof:* ]{}We have $$\Psi_{1,0}\!\left(\overset{I}{v}{^{2}}\overset{I}{B}\overset{I}{A}{^{-1}}\overset{I}{B}{^{-1}}\overset{I}{A}\right) = \overset{I}{L}{^{(+)}} \left(\overset{I}{v}{^{2}} \overset{I}{T}\overset{I}{L}{^{(+)-1}}\overset{I}{L}{^{(-)}} S(\overset{I}{T}) \right)\overset{I}{L}{^{(-)-1}}.$$ Let us simplify the middle term: $$\begin{aligned}
\overset{I}{v}{^{2}} \overset{I}{T} \overset{I}{S(a_i)} \overset{I}{S^{-1}(b_j)} b_i a_j S(\overset{I}{T}) &= \overset{I}{v}{^{2}} \overset{I}{T} \overset{I}{S(a_i)} \overset{I}{S^{-1}(b_j)} \overset{I}{S(a'_j)} \overset{I}{S(b'_i)} S(\overset{I}{T}) b''_i a''_j \\
&= \overset{I}{v}{^{2}}\overset{I}{T} \overset{I}{S(a_ia_k)}\overset{I}{S^{-1}(b_j b_{\ell})} \overset{I}{S(a_j)} \overset{I}{S(b_k)} S(\overset{I}{T}) b_ia_{\ell} = \overset{I}{T} \overset{I}{S(a_i)}\overset{I}{S(b_{\ell})} S(\overset{I}{T}) a_{\ell} b_i.\end{aligned}$$ The first equality is the exchange relation in $\mathcal{H}(\mathcal{O}(H))$ and the second follows from the properties of the $R$-matrix. The third equality is obtained as follows: denoting $m : H \otimes H \to H$ the multiplication, we can write $$\begin{aligned}
&v^2 S(a_ia_k) S^{-1}(b_{\ell}) S^{-1}(b_j) S(a_j) S(b_k) \otimes b_ia_{\ell} = v S(a_ia_k) S^{-1}(b_k b_{\ell}) g^{-1} \otimes b_ia_{\ell}\\
= \:&v \otimes 1 (m \circ (S \otimes S^{-1}) \otimes \mathrm{id})(R_{13} R_{12} R_{32}) g^{-1} \otimes 1 = v \otimes 1 (m \circ (S \otimes S^{-1}) \otimes \mathrm{id})(R_{32} R_{12} R_{13}) g^{-1} \otimes 1\\
= \:&v S(a_k a_i) S^{-1}(b_{\ell} b_k) g^{-1} \otimes a_{\ell} b_i = v S(a_i) S\!\left(S^{-2}(b_k)a_k\right) S^{-1}(b_{\ell})g^{-1} \otimes a_{\ell} b_i = S(a_i) S(b_{\ell}) \otimes a_{\ell}b_i.\end{aligned}$$ We used formula for $u^{-1}$ twice, a Yang-Baxter relation and the standard properties for $g$ and $v$. Now, we have: $$\overset{I}{T}_1 \overset{I}{S(b_{\ell} a_i)}_1 S(\overset{I}{T})_1 a_{\ell} b_i \triangleright \overset{J}{T}_2 = \overset{I}{T}_1 \overset{I}{S(b_{\ell} a_i)}_1 S(\overset{I}{T})_1 \overset{J}{T}_2 \overset{J}{(a_{\ell} b_i)}_2
= \overset{I}{S(b_{\ell} a_i)}_1 \overset{J}{(a_{\ell} b_i)}_2 \overset{J}{T}_2 = \overset{I}{(a_i b_{\ell})}_1 \widetilde{b_i} \widetilde{a_{\ell}} \triangleright \overset{J}{T}_2.$$ For the second equality, we used that for any $h \in H$: $$\langle \overset{I}{S(b_{\ell} a_i)}_1 S(\overset{I}{T})_1 \overset{J}{T}_2 \overset{J}{(a_{\ell} b_i)}_2, h \rangle = \overset{I}{S(h'b_{\ell} a_i)}_1 \overset{J}{(h''a_{\ell} b_i)}_2 = \overset{I}{S(b_{\ell} a_ih')}_1 \overset{J}{(a_{\ell} b_ih'')}_2 = \langle S(\overset{I}{T})_1 \overset{I}{S(b_{\ell} a_i)}_1 \overset{J}{(a_{\ell} b_i)}_2 \overset{J}{T}_2, h \rangle.$$ Since $\triangleright$ is faithful, we finally get $$\overset{I}{v}{^{2}} \overset{I}{T} \overset{I}{S(a_i)} \overset{I}{S^{-1}(b_j)} b_i a_j S(\overset{I}{T}) = \overset{I}{(a_{\ell}b_i)} \widetilde{b_{\ell}} \widetilde{a_i} = \overset{I}{\widetilde{L}}{^{(+)}}\overset{I}{\widetilde{L}}{^{(-)-1}}.$$ Hence $$\Psi_{1,0}(\overset{I}{C}) = \overset{I}{L}{^{(+)}}\overset{I}{\widetilde{L}}{^{(+)}}(\overset{I}{L}{^{(-)}}\overset{I}{\widetilde{L}}{^{(-)}})^{-1}
= \Psi_{1,0}(\overset{I}{C}{^{(+)}}\overset{I}{C}{^{(-)-1}})$$ as desired. To prove the fusion relation, it suffices to consider $\Psi_{1,0}(\overset{I}{C}) = \overset{I}{L}{^{(+)}}\overset{I}{\widetilde{L}}{^{(+)}}\overset{I}{\widetilde{L}}{^{(-)-1}}\overset{I}{L}{^{(-)-1}}$ and to use the exchange relations in . This is a straightforward computation left to the reader. [[$\Box$]{} ]{}\
We now give the general definition. For $i \leq g$, let $\overset{I}{C}(i)$ be the embedding of $\overset{I}{C}$ previously defined on the $i$-th copy of $\mathcal{L}_{1,0}(H)$ in $\mathcal{L}_{g,n}(H)$.
$\overset{I}{C}_{g,n} = \overset{I}{C}(1) \ldots \overset{I}{C}(g) \overset{I}{M}(g+1) \ldots \overset{I}{M}(g+n)$.
Geometrically (see Figure \[figureIntro\]), for each $I$ the matrix $\overset{I}{C}_{g,n}$ corresponds to the holonomy along the boundary of the unique face of the graph $\Gamma$ defined in the Introduction.\
There is a decomposition analogous to Lemma \[decGauss\], which was the case $g=1, n=0$. Indeed, let $$\overset{I}{C}{_{g,n}^{(\pm)}} = \alpha_{g,n}^{-1}\!\left(\overset{I}{\underline{C}}{^{(\pm)}}(1) \ldots \overset{I}{\underline{C}}{^{(\pm)}}(g) \overset{I}{\underline{M}}{^{(\pm)}}(g+1) \ldots \overset{I}{\underline{M}}{^{(\pm)}}(g+n)\right) \in \mathrm{Mat}_{\dim(I)}\!\left(\mathcal{L}_{g,n}(H)\right).$$
The following equality holds in $\mathcal{L}_{g,n}(H)$: $$\overset{I}{C}_{g,n} = \overset{I}{C}{^{(+)}_{g,n}}\,\overset{I}{C}{^{(-)-1}_{g,n}}.$$ Moreover, the matrices $\overset{I}{C}_{g,n}$ satisfy the fusion relation of $\mathcal{L}_{0,1}(H)$: $$(\overset{I \otimes J}{C}\!\!_{g,n})_{12} = (\overset{I}{C}_{g,n})_1 \, \overset{IJ}{(R')}_{12} \, (\overset{J}{C}_{g,n})_2 \, \overset{IJ}{(R'^{-1})}_{12}.$$
[*Proof:* ]{}The first claim is a simple consequence of the definition of $\alpha_{g,n}$ and of Lemma \[decGauss\]. The fusion relation is a consequence of a more general fact which is easy to show, namely: if $i_1 < \ldots < i_k$ and if $\overset{I}{X^1}(i_1), \ldots, \overset{I}{X^k}(i_k)$ are matrices satisfying the fusion relation of $\mathcal{L}_{0,1}(H)$, then their product $\overset{I}{X^1}(i_1) \ldots \overset{I}{X^k}(i_k)$ also satisfies the fusion relation of $\mathcal{L}_{0,1}(H)$. [[$\Box$]{} ]{}
The image of these matrices have simple expressions in $\mathcal{H}(\mathcal{O}(H))^{\otimes g} \otimes H^{\otimes n}$:
\[expressionM\] It holds $$\begin{aligned}
\Psi_{g,n}( \overset{I}{C}{_{g,n}^{(+)}} ) &= \overset{I}{a_i}\:\widetilde{b_i^{(2g -1 + n)}} b_i^{(2g + n)} \otimes \ldots \otimes \widetilde{b_i^{(1+n)}}b_i^{(2+n)} \otimes b_i^{(n)} \otimes \ldots \otimes b^{(1)}_i\\
\Psi_{g,n}( \overset{I}{C}{_{g,n}^{(-)}} ) &= \overset{I}{S^{-1}(b_i)} \: \widetilde{a_i^{(2g -1 + n)}} a_i^{(2g + n)} \otimes \ldots \otimes \widetilde{a_i^{(1+n)}}a_i^{(2+n)} \otimes a_i^{(n)} \otimes \ldots \otimes a^{(1)}_i\\
\Psi_{g,n}( \overset{I}{C}_{g,n} ) &= \overset{I}{X_i}\: \widetilde{Y_i^{(2g -1 + n)}} Y_i^{(2g + n)} \otimes \ldots \otimes \widetilde{Y_i^{(1+n)}}Y_i^{(2+n)} \otimes Y_i^{(n)} \otimes \ldots \otimes Y^{(1)}_i\end{aligned}$$ where $X_i \otimes Y_i = RR'$ and the superscripts mean iterated coproduct.
[*Proof:* ]{}As an immediate consequence of quasitriangularity, we have for all $n \geq 2$ $$(\mathrm{id} \otimes \Delta^{(n-1)})(R) = a_i \otimes b^{(1)}_i \otimes \ldots \otimes b^{(n)}_i = a_{i_1} \ldots a_{i_n} \otimes b_{i_n} \otimes \ldots \otimes b_{i_1}.$$ with implicit summation on $i_1, \ldots, i_n $. It follows that $$\begin{aligned}
\Psi_{g,n}( \overset{I}{C}{_{g,n}^{(+)}} ) &= \overset{I}{L}{^{(+)}}(1)\overset{I}{\widetilde{L}}{^{(+)}}(1) \ldots \overset{I}{L}{^{(+)}}(g)\overset{I}{\widetilde{L}}{^{(+)}}(g) \overset{I}{L}{^{(+)}}(g+1) \ldots \overset{I}{L}{^{(+)}}(g+n)\\
&= \overset{I}{a_{i_1}} \ldots \overset{I}{a_{i_{2g+n}}} \: \widetilde{b_{i_2}} b_{i_1} \otimes \ldots \otimes \widetilde{b_{i_{2g}}} b_{i_{2g-1}} \otimes b_{i_{2g+1}} \otimes \ldots \otimes b_{i_{2g+n}}\\
&= \overset{I}{a_i}\:\widetilde{b_i^{(2g -1 + n)}} b_i^{(2g + n)} \otimes \ldots \otimes \widetilde{b_i^{(1+n)}}b_i^{(2+n)} \otimes b_i^{(n)} \otimes \ldots \otimes b^{(1)}_i\end{aligned}$$ as desired. The second is shown similarly since $R'^{-1}$ is also an universal $R$-matrix. The third is an immediate consequence. [[$\Box$]{} ]{}\
The matrices $\overset{I}{C}_{g,n}$ satisfying the fusion relation of $\mathcal{L}_{0,1}(H)$, we can apply Lemma \[injectionFusion\] and define a representation of $H$ on $V = (H^*)^{\otimes g} \otimes I_1 \otimes \ldots \otimes I_n$ by $$\label{actionHsurFormes}
h \cdot v = h_{C_{g,n}} \triangleright v.$$ Since $H$ is factorizable, each $h \in H$ is a linear combination of coefficients of the matrices $\overset{I}{X}_i Y_i$. Hence, $h_{C_{g,n}}$ is a linear combination of coefficients of the matrices $\overset{I}{C}_{g,n}$. It follows from Lemma \[expressionM\] that this representation is explicitly given by $$\label{actionH}
\begin{split}
&h \cdot \varphi_1 \, \otimes \, \ldots \, \otimes \, \varphi_g \, \otimes v_1 \otimes \, \ldots \, \otimes v_n\\
&= \varphi_1\!\left(S^{-1}\!\left(h^{(2g -1 + n)}\right) ? h^{(2g + n)}\right)\, \otimes \ldots \otimes \, \varphi_g\!\left(S^{-1}\!\left(h^{(1+n)}\right) ? h^{(2+n)}\right)
\otimes \,h^{(n)}v_1 \otimes \ldots \otimes \, h^{(1)}v_n.
\end{split}$$
Determination and representation of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$
-------------------------------------------------------------------------
The matrices $\overset{I}{C}_{g,n}$ introduced above allow one to give a simple characterization of the invariant elements of $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ and to construct representations of them. We begin with a technical lemma.
\[conjugaisonM\] It holds $$(\overset{I}{C}{^{(\pm)}_{g,n}})_1 \, \overset{J}{U}(i)_2 \, (\overset{I}{C}{^{(\pm)}_{g,n}})^{-1}_1 \:\:= \overset{IJ}{R}{^{(\pm)-1}_{12}} \, \overset{J}{U}(i)_2 \, \overset{IJ}{R}{^{(\pm)}_{12}}$$ where $U$ is $A$ or $B$.
[*Proof:* ]{}Applying the isomorphisms $\Psi_{0,1}$ and $\Psi_{1,0}$ and using relations , and , it is easy to show the result for $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$. We get similarly: $$\overset{IJ}{R}{^{(\pm)}_{12}} \, \overset{I}{C}{^{(\pm)}_{1}} \, \overset{J}{C}{^{(-)}_{2}} = \overset{J}{C}{^{(-)}_{2}} \, \overset{I}{C}{^{(\pm)}_{1}} \, \overset{IJ}{R}{^{(\pm)}_{12}},
\:\:\:\: \overset{IJ}{R}{^{(\pm)}_{12}} \, \overset{I}{M}{^{(\pm)}_{1}} \, \overset{J}{M}{^{(-)}_{2}} = \overset{J}{M}{^{(-)}_{2}} \, \overset{I}{M}{^{(\pm)}_{1}} \, \overset{IJ}{R}{^{(\pm)}_{12}}.$$ Using these preliminary facts, we can carry out the general computation. For instance, for $i \leq g$ [$$\begin{aligned}
&\alpha_{g,n}\!\left( (\overset{I}{C}{^{(\pm)}_{g,n}})_1 \, \overset{J}{U}(i)_2 \, (\overset{I}{C}{^{(\pm)}_{g,n}})^{-1}_1\:\, \right) \\
& = \overset{I}{\underline{C}}{^{(\pm)}}(1)_1 \, \ldots \, \overset{I}{\underline{C}}{^{(\pm)}}(i)_1 \, \overset{J}{\underline{C}}{^{(-)}}(1)_2 \, \ldots \, \overset{J}{\underline{C}}{^{(-)}}(i-1)_2 \,
\overset{J}{\underline{U}}(i)_2 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)^{-1}_2 \,
\ldots \, \overset{J}{\underline{C}}{^{(-)}}(1)^{-1}_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(i)^{-1}_1 \, \ldots \, \overset{I}{\underline{C}}{^{(\pm)}}(1)^{-1}_1\\
& = \overset{I}{\underline{C}}{^{(\pm)}}(1)_1 \, \overset{J}{\underline{C}}{^{(-)}}(1)_2 \, \ldots \, \overset{I}{\underline{C}}{^{(\pm)}}(i-1)_1 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)_2 \,
\overset{I}{\underline{C}}{^{(\pm)}}(i)_1 \, \overset{J}{\underline{U}}(i)_2 \,
\overset{I}{\underline{C}}{^{(\pm)}}(i)^{-1}_1 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)^{-1}_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(i-1)^{-1}_1 \, \ldots\\
& \:\:\:\:\:\: \overset{J}{\underline{C}}{^{(-)}}(1)^{-1}_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(1)^{-1}_1\\
& = \overset{I}{\underline{C}}{^{(\pm)}}(1)_1 \, \overset{J}{\underline{C}}{^{(-)}}(1)_2 \, \ldots \overset{I}{\underline{C}}{^{(\pm)}}(i-1)_1 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)_2 \,
\overset{IJ}{R}{^{(\pm)-1}_{12}} \, \overset{J}{\underline{U}}(i)_2 \,
\overset{IJ}{R}{^{(\pm)}_{12}} \, \overset{J}{\underline{C}}{^{(-)}}(i-1)^{-1}_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(i-1)^{-1}_1 \, \ldots \\
& \:\:\:\:\:\: \overset{J}{\underline{C}}{^{(-)}}(1)^{-1}_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(1)^{-1}_1\\
& =\overset{IJ}{R}{^{(\pm)-1}_{12}} \, \overset{J}{\underline{C}}{^{(-)}}(1)_2 \, \overset{I}{\underline{C}}{^{(\pm)}}(1)_1 \, \ldots \overset{J}{\underline{C}}{^{(-)}}(i-1)_2 \,
\overset{I}{\underline{C}}{^{(\pm)}}(i-1)_1 \, \overset{J}{\underline{U}}(i)_2 \,
\overset{I}{\underline{C}}{^{(\pm)}}(i-1)^{-1}_1 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)^{-1}_2 \, \ldots \\
& \:\:\:\:\:\: \overset{I}{\underline{C}}{^{(\pm)}}(1)^{-1}_1 \, \overset{J}{\underline{C}}{^{(-)}}(1)^{-1}_2 \, \overset{IJ}{R}{^{(\pm)}_{12}}\\
& =\overset{IJ}{R}{^{(\pm)-1}_{12}} \, \overset{J}{\underline{C}}{^{(-)}}(1)_2 \, \ldots \overset{J}{\underline{C}}{^{(-)}}(i-1)_2 \,
\overset{J}{\underline{U}}(i)_2 \, \overset{J}{\underline{C}}{^{(-)}}(i-1)^{-1}_2 \, \ldots \overset{J}{\underline{C}}{^{(-)}}(1)^{-1}_2 \, \overset{IJ}{R}{^{(\pm)}_{12}}
= \alpha_{g,n}\!\left( \overset{IJ}{R}{^{(\pm)-1}_{12}} \, \overset{J}{U}(i)_2 \, \overset{IJ}{R}{^{(\pm)}_{12}} \right).\end{aligned}$$]{} The case $i > g$ is treated in a similar way. [[$\Box$]{} ]{}\
For $(V, \triangleright)$ a representation of $\mathcal{L}_{g,n}(H)$, let $$\mathrm{Inv}(V) = \left\{ v \in V \, \left| \, \forall \, I, \: \overset{I}{C}_{g,n} \triangleright v = \mathbb{I}_{\dim(I)}v \right.\right\} = \left\{ v \in V \, \left| \, \forall \, h \in H, \:\: h \cdot v = \varepsilon(h)v \right.\right\}$$ where $\mathbb{I}_k$ is the identity matrix of size $k$, and the action $\cdot$ of $H$ on $V$ is defined in and .
\[thmInv\] 1) An element $x \in \mathcal{L}_{g,n}(H)$ is invariant under the action of $H$ (or equivalently under the coaction $\Omega$ of $\mathcal{O}(H)$) if, and only if, for every $H$-module $I$, $\overset{I}{C}_{g,n}x = x\overset{I}{C}_{g,n}$.\
2) Let $V$ be a representation of $\mathcal{L}_{g,n}(H)$. Then $\mathrm{Inv}(V)$ is stable under the action of invariant elements and thus provides a representation of $\mathcal{L}^{\mathrm{inv}}_{g,n}(H)$.
[*Proof:* ]{}1) The right action of $H$ on $\mathcal{L}_{g,n}(H)$ is by definition $$\overset{I}{U}(i) \cdot h = \overset{I}{h'} \overset{I}{U}(i) \overset{I}{S(h'')}.$$ where $U(i)$ is $A(i)$ or $B(i)$ if $i \leq g$ and is $M(i)$ if $i > g$. Then, denoting $a^{(\pm)}_j \otimes b^{(\pm)}_j = R^{(\pm)}$, $$\begin{split}
\overset{J}{U}(i)_2 \, \cdot \, S^{-1}(\overset{I}{L}{^{(\pm)}})_1 &= \overset{J}{U}(i)_2 \, \cdot \, S^{-1}(b^{(\pm)}_j) \, \overset{I}{(a_j^{(\pm)})}_1 = \overset{J}{S^{-1}(b_j^{(\pm)})}_2 \overset{J}{U}(i)_2 \overset{J}{(b^{(\pm)}_k)}_2 (\overset{I}{a_j^{(\pm)}} \overset{I}{a_k^{(\pm)}})_1 \\ &= \overset{IJ}{R}{^{(\pm)-1}_{12}} \overset{J}{U}(i)_2 \overset{IJ}{R}{^{(\pm)}_{12}} = (\overset{I}{C}{^{(\pm)}_{g,n}})_1 \, \overset{J}{U}(i)_2 \, (\overset{I}{C}{^{(\pm)}_{g,n}})_1^{-1}.
\end{split}$$ where the last equality is Lemma \[conjugaisonM\]. Observe that the matrix $\overset{J}{U}(i)_2 \, \cdot \, S^{-1}(\overset{I}{L}{^{(\pm)}})_1$ contains all the elements obtained by acting by the coefficients of $S^{-1}(\overset{I}{L}{^{(\pm)}})$ on the coefficients of $\overset{J}{U}(i)$. The coefficients of the $S^{-1}(\overset{I}{L}{^{(\pm)}}) = \overset{I}{L}{^{(\pm)-1}}$ generate $H$ as an algebra, hence we deduce that an $x \in \mathcal{L}_{g,n}(H)$ is invariant if, and only if, $\overset{I}{C}{^{(\pm)}_{g,n}}x = x\overset{I}{C}{^{(\pm)}_{g,n}}$. Since $H$ is factorizable, it is generated by the coefficients of the matrices $\overset{I}{X}_i \otimes Y_i$ and we see by Lemma \[expressionM\] that the subalgebra of $\mathcal{L}_{g,n}(H)$ generated by the coefficients of the matrices $\overset{I}{C}{^{(\pm)}_{g,n}}$ equals the subalgebra generated by the coefficients of the matrices $\overset{I}{C}_{g,n}$. Thus $x$ commutes with the $\overset{I}{C}{^{(\pm)}_{g,n}}$ if, and only if, $x$ commutes with the $\overset{I}{C}_{g,n}$.\
2) If $v \in \mathrm{Inv}(V)$ and $x \in \mathcal{L}_{g,n}^{\mathrm{inv}}(H)$, we have $$\overset{I}{C}_{g,n} \triangleright (x \triangleright v) = \overset{I}{C}_{g,n} x \triangleright v = x \overset{I}{C}_{g,n} \triangleright v = \mathbb{I}_{\dim(I)} (x \triangleright v)$$ which shows that $x \triangleright v \in \mathrm{Inv}(V)$. [[$\Box$]{} ]{} By definition, a $H$-connection $\nabla = (h_e)_{e \in E}$ (with $E = \{ b_1, a_1, \ldots, b_g, a_g, m_{g+1}, \ldots, m_{g+n} \}$, the set of edges) is flat if its holonomy along the boundary $c = b_1 a_1^{-1} b_1^{-1} a_1 \ldots b_g a_g^{-1} b_g^{-1} a_g m_{g+1} \ldots m_{g+n}$ of the unique face of the graph $\Gamma$ is trivial: $$\mathrm{Hol}^{\nabla}(C) = h_{b_1} h_{a_1}^{-1} h_{b_1}^{-1} h_{a_1} \ldots h_{b_g} h_{a_g}^{-1} h_{b_g}^{-1} h_{a_g} h_{m_{g+1}} \ldots h_{m_{g+n}} = 1.$$ Hence, the subrepresentation $\mathrm{Inv}(V)$ implements this flatness constraint. This constraint was directly implemented in $\mathcal{L}_{g,n}(H)$ (and not just on representations) in [@AGS2; @AS] by means of characteristic projectors, giving rise to the moduli algebra, a quantum analogue of $\mathbb{C}[\mathcal{A}_f/\mathcal{G}]$ (see Introduction). However, the definition of these projectors requires the $S$-matrix, which has nice properties in the semi-simple case only. In [@AS], the representation space of the mapping class group is the moduli algebra. Here we do not consider the moduli algebra; in particular we will not need to generalize these projectors to construct the projective representation of the mapping class group.
Projective representations of mapping class groups
==================================================
Let $\Sigma_{g,n}$ be the compact orientable surface of genus $g$ with $n$ open disks removed. For simplicity we consider the case of $\Sigma_g$ ($n = 0$). The particular features in this case are that the presentation of the mapping class group is easier and that the associated algebra $\mathcal{L}_{g,0}(H) \cong \mathcal{H}(\mathcal{O}(H))^{\otimes g}$ is isomorphic to a matrix algebra.\
We will discuss the case of $n>0$ in subsection \[CasGeneral\].
Mapping class group of $\Sigma_g$ {#sectionRepMCG}
---------------------------------
Let $D \subset \Sigma_g$ be an embedded open disk and define $\Sigma_g^{\mathrm{o}} = \Sigma_g \setminus D$. We put a basepoint on the boundary circle $c = \partial(\Sigma_g^{\mathrm{o}})$ and we take the curves $a_i, b_i$ ($1 \leq i \leq g$) represented in Figure \[figureSurface\] as generators for the free group $\pi_1(\Sigma_g^{\mathrm{o}})$.
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(6.481430737229788,12.826529098498881) node [$1$]{}; (10.0304404379994802,12.837681256432324) node [$i$]{}; (13.007378022759817,12.837681256432324) node [$i\!+\!1$]{}; (15.513570399115622,12.826529098498881) node [$g$]{}; (17.191817204379586,12.199771390729145) circle (3pt); (0,0) ++(0 pt,3pt) – ++(2.598076211353316pt,-4.5pt)–++(-5.196152422706632pt,0 pt) – ++(2.598076211353316pt,4.5pt); (0,0) ++(0 pt,3pt) – ++(2.598076211353316pt,-4.5pt)–++(-5.196152422706632pt,0 pt) – ++(2.598076211353316pt,4.5pt); (13.9,12.9) circle (1pt); (14.6,12.9) circle (1pt); (14.25,12.9) circle (1pt); (7.599886378457062,12.92127233955011) circle (1pt); (8.79930680637218,12.91904706973951) circle (1pt); (8.19959659241462,12.92015970464481) circle (1pt);
With these generators, $$c = b_1 a_1^{-1} b_1^{-1} a_1 \ldots b_g a_g^{-1} b_g^{-1} a_g.$$ Recall from the Introduction the embedded oriented graph $\Gamma = \left(\{\bullet\}, \{b_1, a_1, \ldots, b_g, a_g\}\right)$ with vertex $\bullet$ and edges the generators of the fundamental group. It is readily seen that $\Sigma_g^{\mathrm{o}}$ is homeomorphic to the thickening of $\Gamma$ represented in Figure \[figureSurfaceRuban\].
plot\[domain=0:3.141592653589793,variable=\]([1\*1.7075037526311627\*cos(r)+0\*1.7075037526311627\*sin(r)]{},[0\*1.7075037526311627\*cos(r)+1\*1.7075037526311627\*sin(r)]{}); plot\[domain=0:3.141592653589793,variable=\]([1\*1.309575195744216\*cos(r)+0\*1.309575195744216\*sin(r)]{},[0\*1.309575195744216\*cos(r)+1\*1.309575195744216\*sin(r)]{}); plot\[domain=0:1.8118338027760237,variable=\]([1\*1.4030639367644646\*cos(r)+0\*1.4030639367644646\*sin(r)]{},[0\*1.4030639367644646\*cos(r)+1\*1.4030639367644646\*sin(r)]{}); plot\[domain=0:1.9894438844649243,variable=\]([1\*1.776642756206547\*cos(r)+0\*1.776642756206547\*sin(r)]{},[0\*1.776642756206547\*cos(r)+1\*1.776642756206547\*sin(r)]{}); plot\[domain=2.2704846801922587:3.141592653589793,variable=\]([1\*1.3953711574291767\*cos(r)+0\*1.3953711574291767\*sin(r)]{},[0\*1.3953711574291767\*cos(r)+1\*1.3953711574291767\*sin(r)]{}); plot\[domain=2.3877511379579315:3.141592653589793,variable=\]([1\*1.789237366642512\*cos(r)+0\*1.789237366642512\*sin(r)]{},[0\*1.789237366642512\*cos(r)+1\*1.789237366642512\*sin(r)]{}); (5,6)– (5,5); (5,5)– (10.794708954246598,5.00203777401749); (5,6)– (5.409328425798604,6); (5.786212515300553,6)– (6.8080282517522,6); (7.200815462007147,6)– (8.405362906788985,6); (8.82433593106093,6)– (10,6); (10.373578819442082,6)– (10.80786927095924,6); plot\[domain=0:3.141592653589793,variable=\]([1\*1.7075037526311627\*cos(r)+0\*1.7075037526311627\*sin(r)]{},[0\*1.7075037526311627\*cos(r)+1\*1.7075037526311627\*sin(r)]{}); plot\[domain=0:3.141592653589793,variable=\]([1\*1.309575195744216\*cos(r)+0\*1.309575195744216\*sin(r)]{},[0\*1.309575195744216\*cos(r)+1\*1.309575195744216\*sin(r)]{}); plot\[domain=0:1.8118338027760237,variable=\]([1\*1.4030639367644646\*cos(r)+0\*1.4030639367644646\*sin(r)]{},[0\*1.4030639367644646\*cos(r)+1\*1.4030639367644646\*sin(r)]{}); plot\[domain=0:1.9894438844649243,variable=\]([1\*1.776642756206547\*cos(r)+0\*1.776642756206547\*sin(r)]{},[0\*1.776642756206547\*cos(r)+1\*1.776642756206547\*sin(r)]{}); plot\[domain=2.2704846801922587:3.141592653589793,variable=\]([1\*1.3953711574291767\*cos(r)+0\*1.3953711574291767\*sin(r)]{},[0\*1.3953711574291767\*cos(r)+1\*1.3953711574291767\*sin(r)]{}); plot\[domain=2.3877511379579315:3.141592653589793,variable=\]([1\*1.789237366642512\*cos(r)+0\*1.789237366642512\*sin(r)]{},[0\*1.789237366642512\*cos(r)+1\*1.789237366642512\*sin(r)]{}); plot\[domain=3.729595257137362:4.701699287460754,variable=\]([1\*2.2187359355090455\*cos(r)+0\*2.2187359355090455\*sin(r)]{},[0\*2.2187359355090455\*cos(r)+1\*2.2187359355090455\*sin(r)]{}); plot\[domain=0:3.141592653589793,variable=\]([1\*1.4991378524730479\*cos(r)+0\*1.4991378524730479\*sin(r)]{},[0\*1.4991378524730479\*cos(r)+1\*1.4991378524730479\*sin(r)]{}); plot\[domain=3.327560085751379:3.7729570768128284,variable=\]([1\*2.455301608888626\*cos(r)+0\*2.455301608888626\*sin(r)]{},[0\*2.455301608888626\*cos(r)+1\*2.455301608888626\*sin(r)]{}); plot\[domain=2.484254690839589:3.0336235085501446,variable=\]([1\*2.3423271124786536\*cos(r)+0\*2.3423271124786536\*sin(r)]{},[0\*2.3423271124786536\*cos(r)+1\*2.3423271124786536\*sin(r)]{}); plot\[domain=4.936459140732812:5.903246179167029,variable=\]([1\*1.6651881573061873\*cos(r)+0\*1.6651881573061873\*sin(r)]{},[0\*1.6651881573061873\*cos(r)+1\*1.6651881573061873\*sin(r)]{}); plot\[domain=0.006881023462768673:1.8643557340986399,variable=\]([1\*1.6100153249433073\*cos(r)+0\*1.6100153249433073\*sin(r)]{},[0\*1.6100153249433073\*cos(r)+1\*1.6100153249433073\*sin(r)]{}); plot\[domain=4.785288706191459:5.326972569147958,variable=\]([1\*5.513782387193336\*cos(r)+0\*5.513782387193336\*sin(r)]{},[0\*5.513782387193336\*cos(r)+1\*5.513782387193336\*sin(r)]{}); (11.7,4.99)– (17.494708954246594,4.99203777401749); (11.7,5.99)– (12.1093284257986,5.99); (12.486212515300549,5.99)– (13.508028251752195,5.99); (13.900815462007142,5.99)– (15.105362906788981,5.99); (15.524335931060925,5.99)– (16.7,5.99); (17.073578819442076,5.99)– (17.507869270959233,5.99); plot\[domain=0:3.141592653589793,variable=\]([1\*1.7075037526311627\*cos(r)+0\*1.7075037526311627\*sin(r)]{},[0\*1.7075037526311627\*cos(r)+1\*1.7075037526311627\*sin(r)]{}); plot\[domain=0:3.141592653589793,variable=\]([1\*1.309575195744216\*cos(r)+0\*1.309575195744216\*sin(r)]{},[0\*1.309575195744216\*cos(r)+1\*1.309575195744216\*sin(r)]{}); plot\[domain=0:1.8118338027760237,variable=\]([1\*1.4030639367644646\*cos(r)+0\*1.4030639367644646\*sin(r)]{},[0\*1.4030639367644646\*cos(r)+1\*1.4030639367644646\*sin(r)]{}); plot\[domain=0:1.9894438844649243,variable=\]([1\*1.776642756206547\*cos(r)+0\*1.776642756206547\*sin(r)]{},[0\*1.776642756206547\*cos(r)+1\*1.776642756206547\*sin(r)]{}); plot\[domain=2.2704846801922587:3.141592653589793,variable=\]([1\*1.3953711574291767\*cos(r)+0\*1.3953711574291767\*sin(r)]{},[0\*1.3953711574291767\*cos(r)+1\*1.3953711574291767\*sin(r)]{}); plot\[domain=2.3877511379579315:3.141592653589793,variable=\]([1\*1.789237366642512\*cos(r)+0\*1.789237366642512\*sin(r)]{},[0\*1.789237366642512\*cos(r)+1\*1.789237366642512\*sin(r)]{}); (18.4,4.98)– (24.19470895424659,4.98203777401749); (18.4,5.98)– (18.809328425798597,5.98); (19.186212515300546,5.98)– (20.20802825175219,5.98); (20.600815462007137,5.98)– (21.80536290678898,5.98); (22.22433593106092,5.98)– (23.4,5.98); (23.773578819442072,5.98)– (24.20786927095923,5.98); (24.20786927095923,5.98)– (24.19470895424659,4.98203777401749); (13.406499243040994,5.405300111476876) node [$b_i$]{}; (14.146452087701703,5.480060431237907) node [$a_i$]{}; (0,0) ++(0 pt,3pt) – ++(2.598076211353316pt,-4.5pt)–++(-5.196152422706632pt,0 pt) – ++(2.598076211353316pt,4.5pt); (0,0) ++(0 pt,3pt) – ++(2.598076211353316pt,-4.5pt)–++(-5.196152422706632pt,0 pt) – ++(2.598076211353316pt,4.5pt); (14.126199467643557,4.998184518464418) circle (3.5pt); (17.75,5.5) circle (1pt); (18.25,5.5) circle (1pt); (18,5.5) circle (1pt); (11,5.5) circle (1pt); (11.5,5.5) circle (1pt); (11.25,5.5) circle (1pt);
Simple closed curves on a surface will simply be called circles. Elements of $\pi_1(\Sigma_g^{\mathrm{o}})$ will be called loops. We consider circles up to free homotopy. In particular, if $\gamma \in \pi_1(\Sigma_g^{\mathrm{o}})$, we denote by $[\gamma]$ the free homotopy class of $\gamma$. For $\alpha$ a circle, we denote by $\tau_{\alpha}$ the Dehn twist about it. If $\gamma \in \pi_1(\Sigma_g^{\mathrm{o}})$, then $\tau_{\gamma}$ is a shortand for $\tau_{[\gamma]}$, thus defined as follows: consider a circle $\gamma'$ freely homotopic to $\gamma$ and which does not intersect the boundary circle $c$; then $\tau_{\gamma} = \tau_{\gamma'}$.\
If $S$ is a compact oriented surface, we denote by $\mathrm{MCG}(S)$ its mapping class group, that is the group of isotopy classes of orientation preserving homeomorphisms of $S$ which fix the boundary pointwise.\
There exists presentations of $\mathrm{MCG}(\Sigma_g)$ and $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ due to Wajnryb [@wajnryb] (also see [@FM Sect. 5.2.1]). Let $$\begin{aligned}
& d_1 = a_1, \:\:\: d_i = a_{i-1} b_{i} a_{i}^{-1} b_{i}^{-1} \:\:\: \text{ for } 2 \leq i \leq g, \\
& e_1 = a_1, \:\:\: e_i = b_1 a_1^{-1} b_1^{-1} a_1 \ldots b_{i-1} a_{i-1}^{-1} b_{i-1}^{-1} a_{i-1} b_i a_i^{-1} b_i^{-1} \:\:\: \text{ for } 2 \leq i \leq g.\end{aligned}$$ The correspondence of notations with [@FM Figure 5.7] is $c_0 = [e_2]$, $c_{2j} = [b_j]$, $c_{2j-1} = [d_j]$. The free homotopy class of these loops are depicted in Figure \[figureCourbesCanoniques\] below.
plot\[domain=3.421902658602023:6.006573738596067,variable=\]([1\*0.51666717817069\*cos(r)+0\*0.51666717817069\*sin(r)]{},[0\*0.51666717817069\*cos(r)+1\*0.51666717817069\*sin(r)]{}); plot\[domain=0.8615978203133784:2.312375095503757,variable=\]([1\*0.6611530960524286\*cos(r)+0\*0.6611530960524286\*sin(r)]{},[0\*0.6611530960524286\*cos(r)+1\*0.6611530960524286\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=0.8615978203133784:2.312375095503757,variable=\]([1\*0.6611530960524286\*cos(r)+0\*0.6611530960524286\*sin(r)]{},[0\*0.6611530960524286\*cos(r)+1\*0.6611530960524286\*sin(r)]{}); plot\[domain=3.421902658602023:6.006573738596067,variable=\]([1\*0.51666717817069\*cos(r)+0\*0.51666717817069\*sin(r)]{},[0\*0.51666717817069\*cos(r)+1\*0.51666717817069\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=0.9265211537579431:2.19965921018045,variable=\]([1\*1.1485995723359301\*cos(r)+0\*1.1485995723359301\*sin(r)]{},[0\*1.1485995723359301\*cos(r)+1\*1.1485995723359301\*sin(r)]{}); plot\[domain=3.953952961918977:5.455765981919788,variable=\]([1\*1.0009623273059072\*cos(r)+0\*1.0009623273059072\*sin(r)]{},[0\*1.0009623273059072\*cos(r)+1\*1.0009623273059072\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=0.8615978203133784:2.312375095503757,variable=\]([1\*0.6611530960524286\*cos(r)+0\*0.6611530960524286\*sin(r)]{},[0\*0.6611530960524286\*cos(r)+1\*0.6611530960524286\*sin(r)]{}); plot\[domain=3.421902658602023:6.006573738596067,variable=\]([1\*0.51666717817069\*cos(r)+0\*0.51666717817069\*sin(r)]{},[0\*0.51666717817069\*cos(r)+1\*0.51666717817069\*sin(r)]{}); plot\[domain=0.9265211537579431:2.19965921018045,variable=\]([1\*1.1485995723359301\*cos(r)+0\*1.1485995723359301\*sin(r)]{},[0\*1.1485995723359301\*cos(r)+1\*1.1485995723359301\*sin(r)]{}); plot\[domain=3.953952961918977:5.455765981919788,variable=\]([1\*1.0009623273059072\*cos(r)+0\*1.0009623273059072\*sin(r)]{},[0\*1.0009623273059072\*cos(r)+1\*1.0009623273059072\*sin(r)]{}); plot\[domain=0.8615978203133784:2.312375095503757,variable=\]([1\*0.6611530960524286\*cos(r)+0\*0.6611530960524286\*sin(r)]{},[0\*0.6611530960524286\*cos(r)+1\*0.6611530960524286\*sin(r)]{}); plot\[domain=3.421902658602023:6.006573738596067,variable=\]([1\*0.51666717817069\*cos(r)+0\*0.51666717817069\*sin(r)]{},[0\*0.51666717817069\*cos(r)+1\*0.51666717817069\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); plot\[domain=2.3506836341863977:3.9339293327304574,variable=\]([1\*0.719075436642927\*cos(r)+0\*0.719075436642927\*sin(r)]{},[0\*0.719075436642927\*cos(r)+1\*0.719075436642927\*sin(r)]{}); plot\[domain=-0.5830776988158339:0.5924723398848708,variable=\]([1\*0.9211182116186638\*cos(r)+0\*0.9211182116186638\*sin(r)]{},[0\*0.9211182116186638\*cos(r)+1\*0.9211182116186638\*sin(r)]{}); (0.09852713674962052,1.250536628718387) circle (0.7574954102991686cm); (2.343671284658925,1.25040048923337) circle (0.7574954102991686cm); (5.7123130455605535,1.2427485893272738) circle (0.7574954102991686cm); (7.957457193469859,1.2426124498422568) circle (0.7574954102991686cm); plot\[domain=1.5737998535724984:4.715392507162291,variable=\]([1\*1.2273007109475733\*cos(r)+0\*1.2273007109475733\*sin(r)]{},[0\*1.2273007109475733\*cos(r)+1\*1.2273007109475733\*sin(r)]{}); (0.09262754998570155,2.454590350211306)– (9.6,2.449787628); (0.1,0)– (9.6,0); plot\[domain=2.651526337357502:3.6330012492875543,variable=\]([1\*2.5981656743405397\*cos(r)+0\*2.5981656743405397\*sin(r)]{},[0\*2.5981656743405397\*cos(r)+1\*2.5981656743405397\*sin(r)]{}); plot\[domain=-0.41005552442894366:0.4005426772305154,variable=\]([1\*3.1087631516065994\*cos(r)+0\*3.1087631516065994\*sin(r)]{},[0\*3.1087631516065994\*cos(r)+1\*3.1087631516065994\*sin(r)]{}); plot\[domain=4.302386010838742:5.078168584050612,variable=\]([1\*1.0481989230313646\*cos(r)+0\*1.0481989230313646\*sin(r)]{},[0\*1.0481989230313646\*cos(r)+1\*1.0481989230313646\*sin(r)]{}); plot\[domain=0.969994260950464:2.146348567799668,variable=\]([1\*0.7109635065368765\*cos(r)+0\*0.7109635065368765\*sin(r)]{},[0\*0.7109635065368765\*cos(r)+1\*0.7109635065368765\*sin(r)]{});
(3.647040179783018,1.251684065629662) circle (1pt); (4.002494967436406,1.248870822572264) circle (1pt); (4.357949755089795,1.246057579514866) circle (1pt);
(0.20600112035952642,-0.3) node [$[e_1]$]{}; (0.19669607174480683,2.7) node [$[a_1]$]{}; (-1.55,1.4) node [$[d_1]$]{}; (0.73, 2.15) node [$[b_1]$]{}; (2.4392127878922274,2.7) node [$[a_2]$]{}; (2.448517836506947,-0.3) node [$[e_2]$]{}; (3.2, 1.9547606120430459) node [$[b_2]$]{}; (5.891385823953194,2.7) node [$[a_{g-1}]$]{}; (5.900690872567914,-0.3) node [$[e_{g-1}]$]{}; (6.5,2.15) node [$[b_{g-1}]$]{}; (8.05015710256814,2.7) node [$[a_g]$]{}; (8.059462151182858,-0.3) node [$[e_g]$]{}; (8.8,1.9454555634283333) node [$[b_g]$]{}; (1.294691808281718,1.75) node [$[d_2]$]{}; (6.924246220187068,1.75) node [$[d_g]$]{};
The Dehn twists $\tau_{e_2}, \tau_{b_i}, \tau_{d_i}$ are called the Humphries generators. Then $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ is generated by the Humphries generators together with four families of relations called disjointness relations, braid relations, 3-chain relation and lantern relation, see [@FM Theorem 5.3]. The presentation of $\mathrm{MCG}(\Sigma_g)$ is obtained as the quotient of $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ by the hyperelliptic relation: $$\label{hyperelliptic}
\left( \tau_{b_g} \tau_{d_g} \ldots \tau_{b_1} \tau_{d_1} \tau_{d_1} \tau_{b_1} \ldots \tau_{d_g} \tau_{b_g} \right) \omega = \omega \left( \tau_{b_g} \tau_{d_g} \ldots \tau_{b_1} \tau_{d_1} \tau_{d_1} \tau_{b_1} \ldots \tau_{d_g} \tau_{b_g} \right)$$ where $\omega$ is any word in the Humphries generators which equals $\tau_{a_g}$.\
The action of the Humphries generators on the fundamental group is easily computed. We just indicate the non-trivial actions: $$\label{actionPi1}
\begin{split}
& \tau_{e_2}(a_1) = e_2^{-1} a_1 e_2, \:\: \tau_{e_2}(b_1) = e_2^{-1} b_1 e_2, \:\: \tau_{e_2}(b_2) = e_2^{-1} b_2, \\
& \tau_{b_i}(a_i) = b^{-1}_i a_i, \\
& \tau_{a_1}(b_1) = b_1 a_1 \:\:\:\: (\text{recall that } a_1 = d_1),\\
& \tau_{d_i}(a_{i-1}) = d_i^{-1} a_{i-1} d_i, \:\: \tau_{d_i}(b_{i-1}) = b_{i-1} d_i, \:\: \tau_{d_i}(b_i) = d_i^{-1} b_i \:\:\: \text{(with } i \geq 2\text{)}.
\end{split}$$ In the sequel, we will be concerned with positively oriented, non-separating simple loops in $\pi_1(\Sigma_g^{\mathrm{o}})$. We say that a simple loop is positively oriented if its orientation is clockwise, as indicated in Figure \[figureCourbeOriente\]. Recall that a loop is non-separating if it does not cut the surface into two connected components and that it is simple if it does not contains self-crossings (up to homotopy). It is clear that these properties are preserved by Dehn twists, hence the set of such loops is stable under the action of $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$. Note that the loops $a_i, b_i, d_i, e_i$ satisfy these properties.
(1.9956673267530805,1.4577673242750369) to \[bend left=30\] (1.7414945936295918,2.234406231041254); (1,1)– (3,1); (1.9956673267530805,1.4577673242750369) to \[bend right=20\] (3,1); (5,1)– (3,1); (4.004332673246919,1.4577673242750366) to \[bend left=20\] (3,1); (4.004332673246919,1.4577673242750366) to \[bend right=30\] (4.258505406370409,2.234406231041254); (1.7414945936295918,2.234406231041254) to \[bend left=50\] (4.258505406370409,2.234406231041254); (3,1) circle (2.5pt); (3,1.8) node[$\circlearrowright$]{};
Dehn twists as automorphisms of $\mathcal{L}_{g,0}(H)$
------------------------------------------------------
In $\pi_1(\Sigma_g^{\mathrm{o}})$ we have the curves $a_i, b_i$ while in $\mathcal{L}_{g,0}(H)$ we have the matrices $\overset{I}{A}(i), \overset{I}{B}(i)$. Using this, we can lift the action of the Humphries generators on $\pi_1(\Sigma_g^{\mathrm{o}})$ to $\mathcal{L}_{g,0}(H)$ by replacing loops by matrices, up to some normalization, as we shall see now.\
First, we lift some of the loops introduced above. We replace the generators of $\pi_1(\Sigma_g^{\mathrm{o}})$ by matrices of generators of $\mathcal{L}_{g,0}(H)$ (see Figure \[figureIntro\]), up to some normalization. More precisely, we define $$\overset{I}{D}_j = \overset{I}{v}{^2}\overset{I}{A}(j-1) \overset{I}{B}(j) \overset{I}{A}(j){^{-1}} \overset{I}{B}(j){^{-1}}, \:\:\:\:\: \overset{I}{E}_2 = \overset{I}{v}{^4}\overset{I}{B}(1) \overset{I}{A}(1){^{-1}} \overset{I}{B}(1){^{-1}} \overset{I}{A}(1) \overset{I}{B}(2) \overset{I}{A}(2){^{-1}} \overset{I}{B}(2){^{-1}}$$ with $2 \leq j \leq g$. Then $\overset{I}{D}_j$ and $\overset{I}{E}_2$ satisfy the fusion relation of $\mathcal{L}_{0,1}(H)$: $$(\overset{I \otimes J}{D_j})_{12} = (\overset{I}{D}_j)_1 \overset{IJ}{(R')}_{12} (\overset{J}{D}_j)_2 (\overset{IJ}{R'})^{-1}_{12}, \:\:\:\:\: (\overset{I \otimes J}{E_2})_{12} = (\overset{I}{E}_2)_1 \overset{IJ}{(R')}_{12} (\overset{J}{E}_2)_2 (\overset{IJ}{R'})^{-1}_{12}.$$ This is easy to check: we observe that $\overset{I}{D}_j = \overset{I}{A}(1) \overset{I}{C}(2) \overset{I}{A}(2){^{-1}}$, $\overset{I}{E}_2 = \overset{I}{C}(1) \overset{I}{C}(2) \overset{I}{A}(2){^{-1}}$ and we use Lemma \[decGauss\] and relations to write the fusion and reorder the matrices. Note that the normalizations by powers of $v$ are necessary to have the $\mathcal{L}_{0,1}(H)$-fusion relation on these elements (see the proof of Proposition \[liftHumphries\] below for an example of computation).\
Now, we lift the action of the Humphries generators on the fundamental group . More precisely, let us define maps $\widetilde{\tau_{e_2}}, \widetilde{\tau_{b_i}}, \widetilde{\tau_{d_j}} : \mathcal{L}_{g,0}(H) \to \mathcal{L}_{g,0}(H)$ by: $$\label{courbesDeviennentMatrices}
\begin{split}
& \widetilde{\tau_{e_2}}(\overset{I}{A}(1)) = \overset{I}{E}{_2^{-1}} \overset{I}{A}(1) \overset{I}{E}_2, \:\: \widetilde{\tau_{e_2}}(\overset{I}{B}(1)) = \overset{I}{E}{_2^{-1}} \overset{I}{B}(1) \overset{I}{E}_2, \:\: \widetilde{\tau_{e_2}}(\overset{I}{B}(2)) = \overset{I}{v}\overset{I}{E}{_2^{-1}} \overset{I}{B}(2), \\
& \widetilde{\tau_{b_i}}(\overset{I}{A}(i)) = \overset{I}{v} \overset{I}{B}(i){^{-1}} \overset{I}{A}(i), \\
& \widetilde{\tau_{a_1}}(\overset{I}{B}(1)) = \overset{I}{v}{^{-1}} \overset{I}{B}(1) \overset{I}{A}(1) \:\:\:\: (\text{recall that } a_1 = d_1),\\
& \widetilde{\tau_{d_j}}(\overset{I}{A}(j-1)) = \overset{I}{D}{_j^{-1}} \overset{I}{A}(j-1) \overset{I}{D}_j, \:\: \widetilde{\tau_{d_j}}(\overset{I}{B}(j-1)) = \overset{I}{v}{^{-1}}\overset{I}{B}(j-1) \overset{I}{D}_j, \:\: \widetilde{\tau_{d_j}}(\overset{I}{B}(j)) = \overset{I}{v}\overset{I}{D}{_j^{-1}}\overset{I}{B}(j),
\end{split}$$ for $j \geq 2$, and the other matrices are fixed.
\[liftHumphries\] 1) The maps $\widetilde{\tau_{e_2}}, \widetilde{\tau_{b_i}}, \widetilde{\tau_{d_i}}$ are automorphisms of $\mathcal{L}_{g,0}(H)$.\
2) The assignment $$\tau_{e_2} \mapsto \widetilde{\tau_{e_2}}, \:\:\: \tau_{b_i} \mapsto \widetilde{\tau_{b_i}}, \:\:\: \tau_{d_i} \mapsto \widetilde{\tau_{d_i}}$$ extends to a morphism of groups $\mathrm{MCG}(\Sigma_g^{\mathrm{o}}) \to \mathrm{Aut}(\mathcal{L}_{g,0}(H))$.
[*Proof:* ]{}1) We have to check that these maps are compatible with the defining relations . This relies on straightforward but tedious computations. For instance, let us show that $\widetilde{\tau_{d_j}}(\overset{I}{B}(j-1))$ satisfies the fusion relation. First, it is easy to establish the following exchange relation: $$(\overset{IJ}{R'})_{12} \overset{J}{B}(j-1)_2 \overset{IJ}{R}_{12} (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} = (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} \overset{J}{B}(j-1)_2.$$ Hence $$\begin{aligned}
\overset{I \otimes J}{B}\!(j-1)_{12} (\overset{I \otimes J}{D_j})_{12} &= \overset{I}{B}(j-1)_1 (\overset{IJ}{R'})_{12} \overset{J}{B}(j-1)_2 (\overset{IJ}{R'})^{-1}_{12} (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} (\overset{J}{D}_j)_2 (\overset{IJ}{R'})^{-1}_{12}\\
&= \overset{I}{B}(j-1)_1 (\overset{IJ}{R'})_{12} \overset{J}{B}(j-1)_2 (\overset{IJ}{R'})^{-1}_{12} \overset{I \otimes J}{v}_{\!\!\! 12} (\overset{IJ}{R'})_{12} \overset{IJ}{R}_{12} \overset{I}{v}{_1^{-1}} \overset{J}{v}{_2^{-1}} (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} (\overset{J}{D}_j)_2 (\overset{IJ}{R'})^{-1}_{12}\\
&= \overset{I \otimes J}{v}_{\!\!\! 12} \overset{I}{v}{_1^{-1}} \overset{J}{v}{_2^{-1}} \overset{I}{B}(j-1)_1 (\overset{IJ}{R'})_{12} \overset{J}{B}(j-1)_2 \overset{IJ}{R}_{12} (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} (\overset{J}{D}_j)_2 (\overset{IJ}{R'})^{-1}_{12}\\
&= \overset{I \otimes J}{v}_{\!\!\! 12} \overset{I}{v}{_1^{-1}} \overset{I}{B}(j-1)_1 (\overset{I}{D}_j)_1 (\overset{IJ}{R'})_{12} \overset{J}{v}{_2^{-1}} \overset{J}{B}(j-1)_2 (\overset{J}{D}_j)_2 (\overset{IJ}{R'})^{-1}_{12}.\end{aligned}$$ For the second equality we applied a trick based on . The aim is to replace $R'^{-1}$ by $R$ in order to apply the previously established exchange relation. It follows that $\overset{I}{v}{^{-1}} \overset{I}{B}(j-1) \overset{I}{D}_j$ satisfies the fusion relation, as desired. This computation shows how the normalizations by powers of $v$ arises in order to satisfy the fusion relation. These normalizations have no importance when one checks the compatibility with the other defining relations of $\mathcal{L}_{g,0}(H)$.\
2) Straightforward verification using Wajnryb’s relations. [[$\Box$]{} ]{}
The lift of an element $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$, denoted by $\widetilde{f}$, is its image by the morphism of Proposition \[liftHumphries\].
Let $u_i$ be one of the generators of $\pi_1(\Sigma_g^{\mathrm{o}})$, let $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ and let $f(u_i) = a_{i_1}^{m_1} b_{j_1}^{n_1} \ldots a_{i_k}^{m_k} b_{j_k}^{n_k}$ with $m_{\ell}, n_{\ell} \in \mathbb{Z}$. Then it follows from the definition of $\widetilde{\tau_{e_2}}, \widetilde{\tau_{b_i}}, \widetilde{\tau_{d_i}}$ that $$\label{expressionLift}
f(\overset{I}{U}(i)) = \overset{I}{v}{^N}\overset{I}{A}(i_1)^{m_1} \overset{I}{B}(j_1)^{n_1} \ldots \overset{I}{A}(i_k)^{m_k} \overset{I}{B}(j_k)^{n_k}$$ where $U(i)=A(i)$ (resp. $U(i)=B(i)$) if $u_i = a_i$ (resp. $u_i = b_i$) and for some $N \in \mathbb{Z}$. In other words, $f$ and $\widetilde{f}$ are formally identical except for the normalizations by some power of $v$.\
Recall that $\mathcal{L}_{g,0}(H) \cong \mathrm{End}_{\mathbb{C}}\!\left((H^*)^{\otimes g}\right)$ is a matrix algebra. By the Skolem-Noether theorem, every automorphism of $\mathcal{L}_{g,0}(H)$ is inner. Hence to each $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ is associated an element $\widehat{f} \in \mathcal{L}_{g,0}(H)$, unique up to scalar, such that $$\label{conjugaison}
\forall\, x \in \mathcal{L}_{g,0}(H), \:\:\: \widetilde{f}(x) = \widehat{f}x\widehat{f}^{-1}.$$ We now determine the elements $\widehat{\tau_{\gamma}}$ associated to Dehn twists.
\[lemmeA1\] We have $\widehat{\tau_{a_1}} = v_{A(1)}^{-1}$. In other words: $$\forall\, x \in \mathcal{L}_{g,0}(H), \:\: \widetilde{\tau_{a_1}}(x) = v_{A(1)}^{-1} \, x \, v_{A(1)}.$$
[*Proof:* ]{}We have $v_{A(1)}^{-1} \overset{I}{A}(1) = \overset{I}{A}(1) v_{A(1)}^{-1} = \widetilde{\tau_{a_1}}(\overset{I}{A}(1))v_{A(1)}^{-1}$. Indeed, since $v^{-1}$ is central in $H$, $v_{A(1)}^{-1}$ is central in the subalgebra generated by the coefficients of the matrices $\overset{I}{A}(1)$. Next, let $j_1 : \mathcal{H}(\mathcal{O}(H)) \to \mathcal{H}(\mathcal{O}(H))^{\otimes g}$ be the canonical embedding on the first copy. Observe that for all $x \in H$, $\Psi_{g,0}(x_{A(1)}) = j_1(x)$. Then: $$\begin{aligned}
\Psi_{g,0}(v_{A(1)}^{-1} \overset{I}{B}(1)) &= j_1(v^{-1} \overset{I}{L}{^{(+)}} \, \overset{I}{T} \, \overset{I}{L}{^{(-)-1}}) = j_1(\overset{I}{L}{^{(+)}} \, \overset{I}{T} \, \overset{I}{(v'^{-1})} \, v''^{-1} \,\overset{I}{L}{^{(-)-1}})\\
& = j_1(\overset{I}{L}{^{(+)}} \, \overset{I}{T} \, \overset{I}{v}{^{-1}}\overset{I}{b_i} \overset{I}{a_j} a_i b_j \, v^{-1} \,\overset{I}{L}{^{(-)-1}}) = j_1(\overset{I}{v}{^{-1}} \overset{I}{L}{^{(+)}} \, \overset{I}{T} \, \overset{I}{L}{^{(-)-1}} \, \overset{I}{L}{^{(+)}} \,\overset{I}{L}{^{(-)-1}} v^{-1})\\
& = \Psi_{g,0}(\overset{I}{v}{^{-1}} \overset{I}{B}(1) \overset{I}{A}(1) v_{A(1)}^{-1}) = \Psi_{g,0}( \widetilde{\tau_{a_1}}(\overset{I}{B}(1)) v_{A(1)}^{-1}).\end{aligned}$$ We used the exchange relation of $\mathcal{H}(\mathcal{O}(H))$ together with and the definition of the matrices $\overset{I}{L}{^{(\pm)}}$. Finally, recall the matrices which occur in the definition of the Alekseev isomorphism. The same argument as in the proof of Lemma \[expressionM\] shows that $$\Psi_{1,0}^{\otimes g}(\overset{I}{\Lambda}_i) = \overset{I}{S^{-1}(b_{\ell})} \: \widetilde{a_{\ell}^{(2i -1)}} a_{\ell}^{(2i)} \otimes \ldots \otimes \widetilde{a_{\ell}^{(1)}}b_{\ell}^{(2)}.$$ From this we see that $j_1(v^{-1})$ commutes with $\Psi_{1,0}^{\otimes g}(\overset{I}{\Lambda}_i)$. Eventually it follows that $\Psi_{g,0}(v^{-1}_{A(1)})$ commutes with $\Psi_{g,0}(\overset{I}{U}(i)) = \Psi_{g,0}(\widetilde{\tau_{a_1}}(\overset{I}{U}(i)))$, where $U$ is $A$ or $B$. [[$\Box$]{} ]{}
If $\gamma_1, \gamma_2$ are non-separating circles on a surface, it is well known that there exists a homeomorphism $f$ such that $f(\gamma_1)$ is freely homotopic to $\gamma_2$ (see e.g. [@FM Sect. 1.3.1]). Here we need to consider fixed-point homotopies.
\[transfoLoop\] Let $\gamma_1, \gamma_2$ be positively oriented, non-separating simple loops in $\pi_1(\Sigma_g^{\mathrm{o}})$, then there exists $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ such that $f(\gamma_1) = \gamma_2$ in $\pi_1(\Sigma_g^{\mathrm{o}})$.
[*Proof:* ]{}We know that there exists $\eta \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ such that $\eta(\gamma_1) = \gamma_2' = \alpha^{\varepsilon} \gamma_2^{\pm 1} \alpha^{-\varepsilon}$ in $\pi_1(\Sigma_g^{\mathrm{o}})$ for some loop $\alpha$ and some $\varepsilon \in \{\pm 1\}$. $\gamma'_2$ is positively oriented, non-separating and simple since $\gamma_1$ is, and thus we can assume that $\alpha$ is simple and does not intersect $\gamma_2$ (except at the basepoint). There are six possible configurations for the loops $\alpha$ and $\gamma_2$ in a neighbourhood of the basepoint:
[l l l]{} 1.
(1,1)– (5,1); (3,1) to \[bend left=10\] (1.9983888861853318,1.332201709197648); (1.9983888861853318,1.332201709197648) to \[bend left=30\] (1.7552819764080068,2.26861350982142); (1.7552819764080068,2.26861350982142) to \[bend left=50\] (4.3484223473661405,2.2596095502000373); (3,1) to \[bend right=10\] (4.096311477967434,1.332201709197648); (4.096311477967434,1.332201709197648) to \[bend right=30\] (4.3484223473661405,2.2596095502000373); (3,1) to \[bend left=30\] (2.4575908268758346,2.103680132957095); (3,1) to \[bend right=30\] (3.5290620218203412,2.103680132957095); (2.4575908268758346,2.103680132957095) to \[bend left=50\] (3.5290620218203412,2.103680132957095); (3,1) circle (2.5pt); (3.5290620218203412+0.3,2.103680132957095) node[$\gamma_2$]{}; (4.3484223473661405+0.3,2.2596095502000373) node[$\alpha$]{};
& 2.
(1,1)– (5,1); (3,1) to \[bend left=10\] (1.9983888861853318,1.332201709197648); (1.9983888861853318,1.332201709197648) to \[bend left=30\] (1.7552819764080068,2.26861350982142); (1.7552819764080068,2.26861350982142) to \[bend left=50\] (4.3484223473661405,2.2596095502000373); (3,1) to \[bend right=10\] (4.096311477967434,1.332201709197648); (4.096311477967434,1.332201709197648) to \[bend right=30\] (4.3484223473661405,2.2596095502000373); (3,1) to \[bend left=30\] (2.4575908268758346,2.103680132957095); (3,1) to \[bend right=30\] (3.5290620218203412,2.103680132957095); (2.4575908268758346,2.103680132957095) to \[bend left=50\] (3.5290620218203412,2.103680132957095); (3,1) circle (2.5pt); (3.5290620218203412+0.3,2.103680132957095) node[$\alpha$]{}; (4.3484223473661405+0.4,2.2596095502000373) node[$\gamma_2$]{};
& 3.
(1,1)– (5,1); (3,1) to \[bend left=30\] (1.26,2.18); (3,1) to \[bend right=20\] (2.38,2.54); (1.26,2.18) to \[bend left=50\] (2.38,2.54); (3,1) to \[bend left=20\] (3.54,2.56); (3.54,2.56) to \[bend left=50\] (4.74,1.94); (3,1) to \[bend right=30\] (4.74,1.94); (3,1) circle (2.5pt); (1,2.18) node[$\alpha$]{}; (5.1,1.94) node[$\gamma_2$]{};
\
4.
(1,1)– (5,1); (3,1) to \[bend left=30\] (1.26,2.18); (3,1) to \[bend right=20\] (2.38,2.54); (1.26,2.18) to \[bend left=50\] (2.38,2.54); (3,1) to \[bend left=20\] (3.54,2.56); (3.54,2.56) to \[bend left=50\] (4.74,1.94); (3,1) to \[bend right=30\] (4.74,1.94); (3,1) circle (2.5pt); (1-0.1,2.18) node[$\gamma_2$]{}; (5,1.94) node[$\alpha$]{};
& 5.
(1,1)– (5,1); (3,1) to \[bend left=50\] (1.5121750666306817,2.3046293483069493+0.2); (1.5121750666306817,2.3046293483069493+0.2) to \[bend left=20\] (2.3315353921764808,2.5837520965698046+0.2); (2.3315353921764808,2.5837520965698046+0.2) to \[bend left=20\] (3.060856121508456,2.23259767133589+0.2); (3.060856121508456,2.23259767133589+0.2) to \[bend left=30\] (3,1); (3,1) to \[bend left=30\] (2.3585472710406283,2.0705263981510065+0.2); (2.3585472710406283,2.0705263981510065+0.2) to \[bend left=20\] (3.141891758100898,2.5207243792201277+0.2); (3.141891758100898,2.5207243792201277+0.2) to \[bend left=30\] (3.943244164403932+0.3,1.8994511653447406+0.2); (3.943244164403932+0.3,1.8994511653447406+0.2) to \[bend left=40\] (3,1); (3,1) circle (2.5pt); (1.5121750666306817-0.4,2.3046293483069493+0.2) node[$\gamma_2$]{}; (3.943244164403932+0.6,1.8994511653447406+0.2) node[$\alpha$]{};
& 6.
(1,1)– (5,1); (3,1) to \[bend left=50\] (1.5121750666306817,2.3046293483069493+0.2); (1.5121750666306817,2.3046293483069493+0.2) to \[bend left=20\] (2.3315353921764808,2.5837520965698046+0.2); (2.3315353921764808,2.5837520965698046+0.2) to \[bend left=20\] (3.060856121508456,2.23259767133589+0.2); (3.060856121508456,2.23259767133589+0.2) to \[bend left=30\] (3,1); (3,1) to \[bend left=30\] (2.3585472710406283,2.0705263981510065+0.2); (2.3585472710406283,2.0705263981510065+0.2) to \[bend left=20\] (3.141891758100898,2.5207243792201277+0.2); (3.141891758100898,2.5207243792201277+0.2) to \[bend left=30\] (3.943244164403932+0.3,1.8994511653447406+0.2); (3.943244164403932+0.3,1.8994511653447406+0.2) to \[bend left=40\] (3,1); (3,1) circle (2.5pt); (1.5121750666306817-0.4,2.3046293483069493+0.2) node[$\alpha$]{}; (3.943244164403932+0.6,1.8994511653447406+0.2) node[$\gamma_2$]{};
In case 1, $\gamma_2' = \alpha \gamma_2 \alpha^{-1}$, and then $\tau_{\alpha}(\gamma'_2) = \alpha^{-1} \gamma'_2 \alpha = \gamma_2$. Case 2 is impossible because none of the four possible loops $\alpha^{\varepsilon} \gamma_2^{\pm 1} \alpha^{-\varepsilon}$ is simple. In case 3, $\gamma'_2 = \alpha \gamma_2 \alpha^{-1}$. For $\beta = \alpha \gamma_2$, we have $\tau_{\beta}(\alpha) = \beta^{-1}\alpha \beta$, $\tau_{\beta}(\gamma_2) = \beta^{-1} \gamma_2 \beta$, and thus $\tau_{\beta}(\gamma'_2) = \gamma_2$. In case 4, $\gamma'_2 = \alpha^{-1} \gamma_2 \alpha$. For $\delta = \gamma_2 \alpha$, we get similarly to case 3 that $\tau_{\delta}^{-1}(\gamma'_2) = \delta \gamma'_2 \delta^{-1} = \gamma_2$. In case 5, $\gamma'_2 = \alpha^{-1} \gamma_2^{-1} \alpha$. Observe that $\tau_{\alpha}(\gamma_2) = \gamma_2 \alpha$, $\tau_{\gamma_2}(\alpha) = \gamma_2^{-1}\alpha$, and then $$\tau_{\alpha}^{-1} \tau_{\gamma_2}^{-2} \tau_{\alpha}^{-1}(\alpha^{-1} \gamma^{-1}_2 \alpha) = \tau_{\alpha}^{-1} \tau_{\gamma_2}^{-2}(\gamma^{-1}_2 \alpha) = \tau_{\alpha}^{-1}(\gamma_2 \alpha) = \gamma_2.$$ In case 6, $\gamma'_2 = \alpha \gamma_2^{-1} \alpha^{-1}$, and we get similarly to case 5 that $\tau_{\alpha} \tau_{\gamma_2}^{2} \tau_{\alpha}(\gamma'_2) = \gamma_2$. [[$\Box$]{} ]{}
We have $$\tau_{b_i} \tau_{a_i}(b_i) = a_i, \:\:\:\:\:
\tau_{d_i}^{-1} \tau_{b_{i-1}}^{-1}(d_i) = b_{i-1}, \:\:\:\:\:
\tau_{y_i}^{-1} \tau_{a_i}^{-1} \tau_{b_{i-1}}^{-1} \tau_{y_i}^{-1}(a_i) = b_{i-1}, \:\:\:\:\:
\tau_{y_2}^{-1}\tau_{b_1}^{-1}\tau_{e_2}\tau_{y_2}(e_2) = b_1$$ where $y_i=a_{i-1}b_i$. This allows to transform any of the loops $a_i, b_i, d_i, e_2$ into $a_1$.
\[liftUnique\] Let $f, g \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ such that $f(a_1) = g(a_1)$. Then $\widetilde{f}(\overset{I}{A}(1)) = \widetilde{g}(\overset{I}{A}(1))$.
[*Proof:* ]{}Let $\eta \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ be such that $\eta(a_1) = a_1$. A priori, $\widetilde{\eta}(\overset{I}{A}(1)) = \overset{I}{v}{^N}\overset{I}{A}(1)$ (see ) and we must show that $N=0$. Let $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})_{[a_1]}$ be the stabilizer of the free homotopy class $[a_1]$. There is a surjection $p : \mathrm{MCG}(\Sigma_g^{\mathrm{o}} \setminus [a_1]) \to \mathrm{MCG}(\Sigma_g^{\mathrm{o}})_{[a_1]}$. $\mathrm{MCG}(\Sigma_g^{\mathrm{o}} \setminus [a_1])$ is generated by $\tau_{d_i}, \tau_{b_i}, \tau_{e_2}, \tau_{a_g}, \tau_{e_g}, \tau_{\beta}$ with $i \geq 2$, where $\tau_{\beta}$ is such that $p(\tau_{\beta}) = \tau_{e_g}$ (see [@FM Figure 4.10]). It follows that $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})_{[a_1]} = \langle \tau_{d_i}, \tau_{b_i}, \tau_{e_2}, \tau_{a_g}, \tau_{e_g} \rangle_{i \geq 2}$. For each of these generators $h$, it is possible to verify directly that $$\begin{aligned}
&h(a_1) = \gamma_h a_1 \gamma_h^{-1} \: \text{ in } \: \pi_1(\Sigma_g^{\mathrm{o}}), \: \text{ with } \: \gamma_h = a_{i_1}^{m_1} b_{j_1}^{n_1} \ldots a_{i_k}^{m_k} b_{j_k}^{n_k} \:\: (m_{\ell}, n_{\ell} \in \mathbb{Z}),\\
&\widetilde{h}(\overset{I}{A}(1)) = \overset{I}{\Gamma}_h \overset{I}{A}(1) \overset{I}{\Gamma}{^{-1}_h} \: \text{ in } \: \mathcal{L}_{g,0}(H), \: \text{ with } \: \Gamma_h = \overset{I}{v}{^r}\overset{I}{A}(i_1)^{m_1} \overset{I}{B}(j_1)^{n_1} \ldots \overset{I}{A}(i_k)^{m_k} \overset{I}{B}(j_k)^{n_k} \:\: (r, m_{\ell}, n_{\ell} \in \mathbb{Z}).\end{aligned}$$ In other words, $h(a_1)$ and $\widetilde{h}(\overset{I}{A}(1))$ are formally identical, without any power of $v$ in the expression of $\widetilde{h}(\overset{I}{A}(1))$ (note that the normalization of $\Gamma_h$ by a power of $v$ vanishes in the conjugation). We deduce that this property is true for any $h \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})_{[a_1]}$. Since $\eta(a_1) = a_1$ in $\pi_1(\Sigma_g^{\mathrm{o}})$, then in particular $\eta \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})_{[a_1]}$, and thus $\widetilde{\eta}(\overset{I}{A}(1)) = \overset{I}{A}(1)$, as desired. [[$\Box$]{} ]{}\
This lemma justifies the following definition.
Let $\gamma \in \pi_1(\Sigma_g^{\mathrm{o}})$ be a positively-oriented, non-separating simple loop, and let $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ be such that $f(a_1) = \gamma$. The lift of $\gamma$ is $\overset{I}{\widetilde{\gamma}} = \widetilde{f}(\overset{I}{A}(1))$.
Some comments are in order. First, if $\gamma = a_{i_1}^{m_1} b_{j_1}^{n_1} \ldots a_{i_k}^{m_k} b_{j_k}^{n_k}$ with $m_{\ell}, n_{\ell} \in \mathbb{Z}$ is a positively oriented, non-separating simple loop, then $\overset{I}{\widetilde{\gamma}} = \overset{I}{v}{^N}\overset{I}{A}(i_1)^{m_1} \overset{I}{B}(j_1)^{n_1} \ldots \overset{I}{A}(i_k)^{m_k} \overset{I}{B}(j_k)^{n_k}$ with $N \in \mathbb{Z}$. In other words, $\gamma$ and $\widetilde{\gamma}$ are formally identical except for the normalization by a power of $v$. Note that by definition every lift satisfies the fusion relation of $\mathcal{L}_{0,1}(H)$. We mention again that the normalization by a power of $v$ is required to satisfy the fusion relation (see e.g. the proof of Proposition \[liftHumphries\]).\
We can now answer the question of what are the elements implementing lifting of Dehn twists by conjugation. We use the notation introduced at the end of subsection 3.2.
\[propDehnTwist\] For any non-separating circle $\gamma$ on $\Sigma_g^{\mathrm{o}}$, we have $\widehat{\tau_{\gamma}} = v_{\widetilde{\gamma}}^{-1}$. In other words: $$\forall\, x \in \mathcal{L}_{g,0}(H), \:\: \widetilde{\tau_{\gamma}}(x) = v_{\widetilde{\gamma}}^{-1} \, x \, v_{\widetilde{\gamma}}.$$ If $\gamma, \delta \in \pi_1(\Sigma_g^{\mathrm{o}})$ are positively oriented non-separating simple loops such that $[\gamma] = [\delta]$, then $v_{\widetilde{\gamma}}$ is proportional to $v_{\widetilde{\delta}}$
[*Proof:* ]{}We represent the circle $[\gamma]$ by a positively-oriented, non-separating simple loop $\gamma \in \pi_1(\Sigma_g^{\mathrm{o}})$. Let $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ be such that $f(a_1) = \gamma$, then $$\widetilde{\tau_{\gamma}} = \widetilde{\tau_{f(a_1)}} = \widetilde{f \tau_{a_1}f^{-1}} = \widetilde{f} \, \widetilde{\tau_{a_1}} \, \widetilde{f}^{-1}.$$ Hence, by Lemma \[lemmeA1\], $$\widetilde{\tau_{\gamma}}\!\left(\widetilde{f}(x)\right) = \widetilde{f}\!\left(\widetilde{\tau_{a_1}}(x)\right) = \widetilde{f}\!\left(v_{A(1)}^{-1} x v_{A(1)}\right) = v_{\widetilde{\gamma}}^{-1} \widetilde{f}(x) v_{\widetilde{\gamma}}.$$ Replacing $x$ by $\widetilde{f}^{-1}(x)$, we get the result. The second claim follows from a similar reasoning together with the fact that $\tau_{\gamma}$ depends only of the free homotopy class of $\gamma$. [[$\Box$]{} ]{}
Note that an analogous result in the semi-simple setting has been stated without proof in [@AS]. The notation $v_{\widetilde{\gamma}}^{-1}$ does not appear in their work; instead, they express this element in a basis of characters, which is possible in the semi-simple case only.
For all $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$, it holds $\widehat{f} \in \mathcal{L}_{g,0}^{\mathrm{inv}}(H)$.
[*Proof:* ]{}Let $\gamma$ be a positively-oriented, non-separating simple loop. Then $\overset{I}{\widetilde{\gamma}}$ satisfies the fusion relation of $\mathcal{L}_{0,1}(H)$, and thus $j_{\widetilde{\gamma}}$ is a morphism of $H$-module-algebras (Lemma \[injectionFusion\]). Hence, since $v^{-1} \in \mathcal{Z}(H) = \mathcal{L}_{0,1}^{\mathrm{inv}}(H)$, we have $v_{\widetilde{\gamma}}^{-1} \in \mathcal{L}_{g,0}^{\mathrm{inv}}(H)$. In particular, the statement is true for the Humphries generators thanks to Proposition \[propDehnTwist\] and thus for any $f$. [[$\Box$]{} ]{}
Representation of the mapping class group {#sectionRepMCG}
-----------------------------------------
The only additional fact needed is the following lemma.
\[lemmevA1Moins1\] It holds: $v_{A(g)}^{-1} = v_{A(g)^{-1}}^{-1}$.
[*Proof:* ]{}Denote as usual $X_i \otimes Y_i = RR'$, $\overline{X}_i \otimes \overline{Y}_i = (RR')^{-1}$ and let $\mu^l$ be the left integral on $H$ (unique up to scalar). Using basic facts about integrals and , we have (see [@Fai18 Prop. 5.3]): $$\mu^l(vX_i)Y_i = \mu^l(v\overline{X}_i)\overline{Y}_i = \mu^l(v)v^{-1}.$$ Let us write $\mu^l(v)^{-1}\mu^l(v?) = \sum_{i,j,I} c_{I,i}^j \overset{I}{T}{^i_j}$ with $c_{I,i}^j \in \mathbb{C}$. Then, using the identification $\overset{I}{M} = (\overset{I}{X_i})Y_i$ between $\mathcal{L}_{0,1}(H)$ and $H$, the fact that $\overset{I}{M}{^{-1}} = (\overset{I}{\overline{X}_i})\overline{Y}_i$ and the equalities above, we get $$v_{A(g)}^{-1} = j_{A(g)}\!\left( \sum_{i,j,I} c_{I,i}^j \overset{I}{M}{^i_j} \right) = j_{A(g)}\!\left( \sum_{i,j,I} c_{I,i}^j (\overset{I}{M}{^{-1}})^i_j \right) = j_{A(g)^{-1}}\!\left( \sum_{i,j,I} c_{I,i}^j \overset{I}{M}{^i_j} \right) = v_{A(g)^{-1}}^{-1}$$ where the morphisms $j_{\bullet}$ are defined at the end of subsection \[defLgn\]. We used that $j_{A(g)}$ is a morphism of algebras (see Lemma \[injectionFusion\]). [[$\Box$]{} ]{}
It is clear that the lemma holds for the lift of any positively oriented, non-separating simple loop, but we do not need this.\
Recall that we have a representation of $\mathcal{L}_{g,0}(H)$ on $(H^*)^{\otimes g}$, let us denote it $\rho$. We also have the induced representation of $\mathcal{L}^{\mathrm{inv}}_{g,0}(H)$ on $\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$, let us denote it $\rho_{\mathrm{inv}}$. Also recall that the elements $\widehat{f}$ are defined in . We can now state the representation of the mapping class groups $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ and $\mathrm{MCG}(\Sigma_g)$. An analogous result in the semi-simple setting has been given without proof in [@AS].
\[thmRepMCG\] 1) The map $${ \begin{array}{rcl} \mathrm{MCG}(\Sigma_g^{\mathrm{o}}) & \rightarrow & \mathrm{GL}\!\left((H^*)^{\otimes g}\right) \\ f &\mapsto & \rho(\widehat{f}) \end{array}}$$ is a projective representation.\
2) The map $${ \begin{array}{rcl} \mathrm{MCG}(\Sigma_g) & \rightarrow & \mathrm{GL}\!\left(\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)\right) \\ f &\mapsto & \rho_{\mathrm{inv}}(\widehat{f}) \end{array}}$$ is a projective representation.
[*Proof:* ]{}1) This is an immediate consequence of Proposition \[liftHumphries\].\
2) We must show that the hyperelliptic relation is projectively satisfied. The word $\omega$ can be constructed as follows: take $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ such that $f(a_1) = a_g$ and express it as a word in the Humphries generators $f = \tau_{\gamma_1} \ldots \tau_{\gamma_n}$. Then $\tau_{a_g} = f \tau_{a_1} f^{-1}$. The automorphism $\widetilde{\tau_{a_g}}$ is implemented by conjugation by $\widehat{f} v_{A(1)}^{-1} \widehat{f}^{-1}$ and also by conjugation by $v_{A(g)}^{-1}$ (Proposition \[propDehnTwist\]). Hence, $\widehat{f} v_{A(1)}^{-1} \widehat{f}^{-1} \sim v_{A(g)}^{-1}$, where $\sim$ means proportional. Now, let $H = \tau_{b_g} \tau_{d_g} \ldots \tau_{b_1} \tau_{d_1} \tau_{d_1} \tau_{b_1} \ldots \tau_{d_g} \tau_{b_g}$. A computation gives $\widetilde{H}(\overset{I}{A}(g)) = \overset{I}{A}(g){^{-1}} \overset{I}{C}_{g,0}$. Thus $$\widehat{H} \widehat{f} v_{A(1)}^{-1} \widehat{f}^{-1} \widehat{H}^{-1} \sim \widehat{H} v_{A(g)}^{-1} \widehat{H}^{-1} = \widetilde{H}(v_{A(g)}^{-1}) = v_{A(g)^{-1}C_{g,0}}^{-1}.$$ By definition of $\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$ and Lemma \[lemmevA1Moins1\], we have $$\rho_{\mathrm{inv}}(v_{A(g)^{-1}C_{g,0}}^{-1}) = \rho_{\mathrm{inv}}(v_{A(g)^{-1}}^{-1}) = \rho_{\mathrm{inv}}(v_{A(g)}^{-1}).$$ It follows that $\rho_{\mathrm{inv}}\!\left(\widehat{H} \left(\widehat{f} v_{A(1)}^{-1} \widehat{f}^{-1}\right) \widehat{H}^{-1}\right) \sim \rho_{\mathrm{inv}}\!\left(\widehat{f} v_{A(1)}^{-1} \widehat{f}^{-1}\right)$. This shows that the map is well-defined since $\mathrm{MCG}(\Sigma_g)$ is the quotient of $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ by the hyperelliptic relation and that it is a projective representation. [[$\Box$]{} ]{}
Discussion for the case $n > 0$ {#CasGeneral}
-------------------------------
Let us consider the general case $n>0$, see Figure \[figureIntro\]. Denote $\Sigma_{g,n}^{\mathrm{o}} = \Sigma_{g,n} \setminus D$, where $D$ is an embedded open disk. Recall that by definition the mapping class group fixes pointwise the boundary.\
In general, $\mathcal{L}_{g,n}(H)$ is not a matrix algebra and we cannot claim directly the existence and unicity up to scalar of the elements $\widehat{f}$. Nevertheless, we now describe an extension of the previous construction which should not be difficult to apply.\
Consider a generating set $\tau_{c_1}, \ldots, \tau_{c_k}$ of $\mathrm{MCG}(\Sigma^{\mathrm{o}}_{g,n})$ (this consists only of Dehn twists when $g > 1$, see [@FM Figure 4.10]) and compute the action on $\pi_1(\Sigma_{g,n}^{\mathrm{o}})$. Here the $c_i$ are loops in $\pi_1(\Sigma_{g,n}^{\mathrm{o}})$ written in terms of the generators depicted in Figure \[figureIntro\].\
Determine the lifts $\widetilde{c_i}$ of the loops $c_i$ (*i.e.* replace generators of $\pi_1(\Sigma_{g,n}^{\mathrm{o}})$ by matrices of generators of $\mathcal{L}_{g,n}(H)$ and then determine the normalisations by powers of $v$ needed to satisfy the fusion relation).\
Determine the lifts $\widetilde{\tau_{c_i}}$ of the generators as automorphisms of $\mathcal{L}_{g,n}(H)$ (*i.e.* replace generators of $\pi_1(\Sigma_{g,n}^{\mathrm{o}})$ by matrices of generators of $\mathcal{L}_{g,n}(H)$ in the action of $\tau_{c_1}, \ldots, \tau_{c_k}$ on $\pi_1(\Sigma_{g,n}^{\mathrm{o}})$, and then determine the normalisations by powers of $v$ needed to satisfy the fusion relation and check that the other relations of are satisfied).\
Show that the assignment $\tau_{c_k} \mapsto \widetilde{\tau_{c_k}}$ extends to a morphism of groups $\mathrm{MCG}(\Sigma_{g,n}^{\mathrm{o}}) \to \mathrm{Aut}(\mathcal{L}_{g,n}(H))$ (this is a just a tedious verification using a presentation of $\mathrm{MCG}(\Sigma_{g,n}^{\mathrm{o}})$). Thus the lift $\widetilde{f}$ of $f \in \mathrm{MCG}(\Sigma_{g,n}^{\mathrm{o}})$ is still defined.\
It is clear that Lemma \[transfoLoop\] still holds, so in particular for each $i$ there exists $f_i \in \mathrm{MCG}(\Sigma^{\mathrm{o}}_{g,n})$ such that $f_i(a_1) = c_i$.\
It is clear that Lemma \[lemmeA1\] still holds, so that $\widetilde{\tau}_{c_i}(x) = v_{\widetilde{c_i}}^{-1} x v_{\widetilde{c_i}}$ (by reproducing the proof of Proposition \[propDehnTwist\] with the $f_i$). Since the $\tau_{c_i}$ are a generating set, it follows that for each $f \in \mathrm{MCG}(\Sigma_{g,n}^{\mathrm{o}})$, there exists an element $\widehat{f} \in \mathcal{L}^{\mathrm{inv}}_{g,n}(H)$, unique up to an invertible central element such that $\widetilde{f}(x) = \widehat{f} x \widehat{f}^{-1}$.\
Since $\mathcal{Z}\!\left(\mathcal{H}({\mathcal{O}}(H))^{\otimes g}\right) \cong \mathbb{C}$, we have for all $c \in \mathcal{Z}\!\left(\mathcal{L}_{g,n}(H)\right)$: $$\Psi_{g,n}(c) = 1 \otimes \ldots \otimes 1 \otimes c_1 \otimes \ldots \otimes c_n$$ with $c_i \in \mathcal{Z}(H)$. Let $V = (H^*)^{\otimes g} \otimes S_1 \otimes \ldots \otimes S_n$, where $S_1, \ldots, S_n$ are simple representations of $H$. Then $\Psi_{g,n}(c)$ acts by scalar on $V$ thanks to Schur lemma. Let $\rho$ (resp. $\rho_{\mathrm{inv}}$) be the representation of $\mathcal{L}_{g,n}(H)$ (resp. $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$) on $V$ (resp. $\mathrm{Inv}(V)$). Then the elements $\rho(\widehat{f})$ and thus $\rho_{\mathrm{inv}}(\widehat{f})$ are unique up to scalar. It should not be difficult to check that the corresponding generalisation of Theorem \[thmRepMCG\] is true.
Explicit formulas for the representation of some Dehn twists {#sectionFormulesExplicites}
------------------------------------------------------------
We will compute explicitly the representation on $(H^*)^{\otimes g}$ of the Dehn twists $\tau_{\gamma}$, where the curves $\gamma$ are represented in Figure \[figureCourbesCanoniques\]. Thanks to Proposition \[propDehnTwist\], this amounts to compute the action of $v_{\widetilde{\gamma}}^{-1}$ on $(H^*)^{\otimes g}$.\
We recall that the action $\triangleright$ of $\mathcal{L}_{g,0}(H)$ on $(H^*)^{\otimes g}$ is defined using $\Psi_{g,0}$ in and that we denote the associated representation by $\rho$. Also recall the definition of the elements $\widetilde{h}$ in and the notation $RR' = X_i \otimes Y_i$. Note that $$\label{coproduitRR}
X_i \otimes Y_i' \otimes Y_i'' = a_j X_i b_k \otimes Y_i \otimes b_j a_k.$$
Recall from [@Fai18] the elements $v_A^{-1}, v_B^{-1} \in \mathcal{L}_{1,0}(H)$ and their action on $H^*$: $$\label{actionvAvB}
\begin{split}
v_A^{-1} \triangleright \varphi &= \varphi^{v^{-1}} = \varphi(v^{-1} ?),\\
v_B^{-1} \triangleright \varphi &= \mu^l(v)^{-1}\left(\mu^l\!\left(g^{-1}v\,?\right) \varphi^v\right)^{v^{-1}} \end{split}$$ where $\varphi^h = \varphi(h?)$ for $h \in H$ and $\mu^l$ is the left integral on $H$.\
We will need the following generalization of [@Fai18 Lemma 5.7] (in which we restricted to $\varphi \in \mathrm{SLF}(H)$).
\[lemmeActionTresseAB\] For all $\varphi \in H^*$: $$\begin{aligned}
\left( v_{A}^{-1} v_{B}^{-1} v_{A}^{-1} \right)^2 \triangleright \varphi &= \frac{\mu^l(v^{-1})}{\mu^l(v)} \varphi\!\left( S^{-1}(a_i) g^{-1}v^{-1} S(?) b_i \right)\\
\left( v_{A}^{-1} v_{B}^{-1} v_{A}^{-1} \right)^{-2} \triangleright \varphi &= \frac{\mu^l(v)}{\mu^l(v^{-1})} \varphi\!\left( b_j S^{-1}(?) a_j g^{-1}v \right)\end{aligned}$$
[*Proof:* ]{}Write $\varphi = \sum_{I,i,j} \Phi_{I,i}^j \overset{I}{T}{_j^i} = \sum_I \mathrm{tr}\!\left( \Phi_I \overset{I}{T} \right)$ with $\Phi_{I,i}^j \in \mathbb{C}$ and let $z(\varphi) = \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{M} \right)$ $\in \mathcal{L}_{0,1}(H)$. Then $z(\varphi)_B \triangleright \varepsilon = \varphi$ (where $z(\varphi)_B = j_B(z(\varphi))$, see notation at the end of section \[defLgn\], and $\varepsilon$ is the counit of $H$). Indeed $$z(\varphi)_B \triangleright \varepsilon
= \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{L}{^{(+)}} \overset{I}{T} \overset{I}{L}{^{(-)-1}} \triangleright \varepsilon \right)
= \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{a_j} \overset{I}{T} \overset{I}{b_j} \right) = \sum_I \mathrm{tr}\!\left(\Phi_I \overset{I}{T} \right) = \varphi.$$ We simply used , , the cyclicity of the trace and the equality $S^{-1}(a_i)a_j \otimes b_j b_i = 1 \otimes 1$. Observe that $$\left(\widetilde{\tau}_a \widetilde{\tau}_b \widetilde{\tau}_a\right)^2 (\overset{I}{B}) = \overset{I}{v}{^2}\overset{I}{A}{^{-1}}\overset{I}{B}{^{-1}}\overset{I}{A} = \overset{I}{B}{^{-1}} \overset{I}{C}$$ where $\overset{I}{C} = \overset{I}{C}_{1,0}$ is defined in . Hence: $$\begin{aligned}
\left( v_{A}^{-1} v_{B}^{-1} v_{A}^{-1} \right)^2 \triangleright \varphi &= \left( v_{A}^{-1} v_{B}^{-1} v_{A}^{-1} \right)^2 z(\varphi)_{B} \triangleright \varepsilon = z(\varphi)_{B^{-1}C} \left( v_{A}^{-1} v_{B}^{-1} v_{A}^{-1} \right)^2 \triangleright \varepsilon\\
&= \frac{\mu^l(v^{-1})}{\mu^l(v)} z(\varphi)_{B^{-1}C} \triangleright \varepsilon = \frac{\mu^l(v^{-1})}{\mu^l(v)} z(\varphi)_{B^{-1}} \triangleright \varepsilon.\end{aligned}$$ We used Proposition \[propDehnTwist\], the formula of [@Fai18 Lemma 5.7] applied to $\varepsilon$, and the fact that $\overset{I}{C} \triangleright \varepsilon = \mathbb{I}_{\dim(I)}\varepsilon$ (which follows from \[actionH\]). Now we compute $$\begin{aligned}
z(\varphi)_{B^{-1}} \triangleright \varepsilon &= \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{L}{^{(-)}} S(\overset{I}{T}) \overset{I}{L}{^{(+)-1}} \triangleright \varepsilon \right)
= \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{S^{-1}(b_j)}a_j \triangleright S(\overset{I}{T}) \right)\\
&= \sum_I \mathrm{tr}\!\left( \overset{I}{b_i} \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{S^{-1}(b_j)} \overset{I}{S(a_j)} S(\overset{I}{T}) \right) = \sum_I \mathrm{tr}\!\left( \Phi_I \overset{I}{S^{-1}(a_i)} \overset{I}{g}{^{-1}} \overset{I}{v}{^{-1}} S(\overset{I}{T}) \overset{I}{b_i} \right)\\
&= \varphi\!\left( S^{-1}(a_i) g^{-1}v^{-1} S(?) b_i \right).\end{aligned}$$ We used and . The second formula is easily checked. [[$\Box$]{} ]{}
\[formulesExplicites\] The following formulas hold: $$\begin{aligned}
v_{A(i)}^{-1} \triangleright \left(\varphi_1 \otimes \ldots \otimes \varphi_g\right) &= \varphi_1 \otimes \ldots \otimes \varphi_{i-1} \otimes (v_A^{-1} \triangleright \varphi_i) \otimes \varphi_{i+1} \otimes \ldots \otimes \varphi_g, \\
v_{B(i)}^{-1} \triangleright \left(\varphi_1 \otimes \ldots \otimes \varphi_g\right) &= \varphi_1 \otimes \ldots \otimes \varphi_{i-1} \otimes (v_B^{-1} \triangleright \varphi_i) \otimes \varphi_{i+1} \otimes \ldots \otimes \varphi_g, \\
v_{D_i}^{-1} \triangleright \left(\varphi_1 \otimes \ldots \otimes \varphi_g\right) &= \varphi_1 \otimes \ldots \otimes \varphi_{i-2} \otimes \varphi_{i-1}\!\left(S^{-1}(a_j)a_k?b_k v''^{-1} b_j\right) \otimes \varphi_i\!\left( S^{-1}(a_l) S^{-1}(v'^{-1}) a_m ? b_m b_l \right)\\
& \:\:\:\:\: \otimes \varphi_{i+1} \otimes \ldots \otimes \varphi_g, \\
v_{E_i}^{-1} \triangleright \left(\varphi_1 \otimes \ldots \otimes \varphi_g\right) &= \varphi_1\!\left(S^{-1}\!\left(v^{(2i-2)-1}\right) ? v^{(2i-1)-1} \right) \otimes \ldots \otimes \varphi_{i-1}\!\left(S^{-1}\!\left( v^{(2)-1} \right) ? v^{(3)-1} \right)\\
&\:\:\:\:\: \otimes \varphi_i\!\left(S^{-1}(a_j) S^{-1}\!\left(v^{(1)-1}\right) a_k ? b_k b_j \right) \otimes \varphi_{i+1} \otimes \ldots \otimes \varphi_g,\end{aligned}$$ with $i \geq 2$ for the two last formulas.
The rest of the section is devoted to the proof of that theorem. First, it is useful to record that $$\label{formuleLambda}
\begin{split}
\Psi_{1,0}^{\otimes g}(\overset{I}{\Lambda}_i) = \Psi_{1,0}^{\otimes g}\!\left( \overset{I}{\underline{C}}{^{(-)}}(1) \ldots \overset{I}{\underline{C}}{^{(-)}}(i-1) \right) &= \overset{I}{S^{-1}(b_j)} \: \widetilde{a_j^{(2i-3)}} a_j^{(2i-2)} \otimes \ldots \otimes \widetilde{a_j^{(1)}}a_j^{(2)} \otimes 1^{\otimes g-i+1} \\
\Psi_{1,0}^{\otimes g}\!\left( \overset{I}{\underline{C}}{^{(+)}}(1) \ldots \overset{I}{\underline{C}}{^{(+)}}(i-1) \right) &= \overset{I}{a_j} \: \widetilde{b_j^{(2i-3)}} b_j^{(2i-2)} \otimes \ldots \otimes \widetilde{b_j^{(1)}}b_j^{(2)} \otimes 1^{\otimes g-i+1}
\end{split}$$ where the matrix $\overset{I}{\Lambda}_k$ is defined in . The proof is a simple computation analogous to that of Lemma \[expressionM\]. Second, recall from the proof of Lemma \[lemmevA1Moins1\] that $$\label{VIntegrale}
\mu^l(v)^{-1}\mu^l(vX_i)Y_i = v^{-1}.$$ We will write $\mu^l(v)^{-1}\mu^l(v?) = \sum_{I} \mathrm{tr}\!\left(c_I \overset{I}{T}\right)$. Then $v^{-1} = \sum_{I} \mathrm{tr}\!\left(c_I \overset{I}{M}\right)$ under the identification $\mathcal{L}_{0,1}(H) = H$.\
[*Proof of the formula for the action of $v_{A(i)}^{-1}$.*]{} By definition and by , we have $$\begin{aligned}
&\Psi_{g,0}(v_{A(i)}^{-1}) = \sum_{I} \mathrm{tr}\!\left(c_I \Psi_{1,0}^{\otimes g}(\overset{I}{\Lambda}_i \, \overset{I}{\underline{A}}(i) \, \overset{I}{\Lambda}_i{^{-1}})\right)\\
& = \sum_I \mathrm{tr}\!\left(c_I \overset{I}{S^{-1}(b_j)} \overset{I}{X_k} \overset{I}{b_l}\right) \widetilde{a_j^{(2i-3)}}\widetilde{a_l^{(2i-3)}} a_j^{(2i-2)} a_l^{(2i-2)} \otimes \ldots \otimes \widetilde{a_j^{(1)}}\widetilde{a_l^{(1)}} a_j^{(2)}a_l^{(2)} \otimes Y_k \otimes 1^{\otimes g-i}\\
&=\mu^l(v)^{-1}\mu^l\!\left( v S^{-1}(b_j) X_k b_l\right) \widetilde{a_j^{(2i-3)}}\widetilde{a_l^{(2i-3)}} a_j^{(2i-2)} a_l^{(2i-2)} \otimes \ldots \otimes \widetilde{a_j^{(1)}}\widetilde{a_l^{(1)}} a_j^{(2)}a_l^{(2)} \otimes Y_k \otimes 1^{\otimes g-i}\\
&= \mu^l(v)^{-1}\mu^l\!\left(v S^{-1}\!\left(b_jS^{-1}(b_l)\right) X_k\right) \widetilde{a_j^{(2i-3)}}\widetilde{a_l^{(2i-3)}} a_j^{(2i-2)} a_l^{(2i-2)} \otimes \ldots \otimes \widetilde{a_j^{(1)}}\widetilde{a_l^{(1)}} a_j^{(2)}a_l^{(2)} \otimes Y_k \otimes 1^{\otimes g-i}\\
&= \mu^l(v)^{-1}\mu^l\!\left( v X_k\right) 1^{\otimes i-1} \otimes Y_k \otimes 1^{\otimes g-i} = 1^{\otimes i-1} \otimes v^{-1} \otimes 1^{\otimes g-i}\end{aligned}$$ and the formula follows. We used , the formula $R^{-1} = a_l \otimes S^{-1}(b_l)$ and .\
[*Proof of the formula for the action of $v_{B(i)}^{-1}$.*]{} This the same proof as for $v_{A(i)}^{-1}$ (the conjugation by $\overset{I}{\Lambda}_i$ vanishes thanks to ).\
[*Proof of the formula for the action of $v_{D_i}^{-1}$, $i \geq 2$.*]{} We first compute the action of $\overset{I}{A}(i-1)\overset{I}{A}(i)$. We have $$\begin{aligned}
\Psi_{g,0}\!\left( \overset{I}{A}(i-1)\overset{I}{A}(i) \right) &= \Psi_{1,0}^{\otimes g}\!\left(\overset{I}{\Lambda}_{i-1} \, \overset{I}{\underline{A}}(i-1) \, \overset{I}{\Lambda}{_{i-1}^{-1}} \, \overset{I}{\Lambda}_{i} \, \overset{I}{\underline{A}}(i) \, \overset{I}{\Lambda}{_{i}^{-1}}\right)\\
&= \Psi_{1,0}^{\otimes g}\!\left(\overset{I}{\Lambda}_{i-1} \, \overset{I}{\underline{A}}(i-1) \, \overset{I}{\underline{C}}{^{(-)}}(i-1) \, \overset{I}{\underline{A}}(i) \, \overset{I}{\underline{C}}{^{(-)}}(i-1)^{-1} \, \overset{I}{\Lambda}{_{i-1}^{-1}}\right).\end{aligned}$$ Hence: $$\begin{aligned}
&\Psi_{g,0}\!\left( v_{A(i-1)A(i)}^{-1} \right) = \sum_{I} \mathrm{tr}\!\left(c_I \Psi_{1,0}^{\otimes g}\!\left(\overset{I}{\Lambda}_{i-1} \, \overset{I}{\underline{A}}(i-1) \, \overset{I}{\underline{C}}{^{(-)}}(i-1) \, \overset{I}{\underline{A}}(i) \, \overset{I}{\underline{C}}{^{(-)}}(i-1)^{-1} \, \overset{I}{\Lambda}{_{i-1}^{-1}}\right)\right)\\
&= \mu^l(v)^{-1}\mu^l\!\left( v S^{-1}(b_j) X_k S^{-1}(b_l) X_m b_n b_o \right) \widetilde{a_j^{(2i-5)}}\widetilde{a_o^{(2i-5)}} a_j^{(2i-4)} a_o^{(2i-4)} \, \otimes \, \ldots \, \otimes \, \widetilde{a_j^{(1)}}\widetilde{a_o^{(1)}} a_j^{(2)}a_o^{(2)}\\
& \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\otimes \widetilde{a_l'}\widetilde{a_n'} Y_k a_l'' a_n'' \otimes Y_m \otimes 1^{\otimes g-i}\\
&= \mu^l(v)^{-1}\mu^l\!\left( v X_k S^{-1}(b_l) X_m b_n \right) 1^{\otimes i-2} \otimes \widetilde{a_l'}\widetilde{a_n'} Y_k a_l'' a_n'' \otimes Y_m \otimes 1^{\otimes g-i}\\
&= \mu^l(v)^{-1}\mu^l\!\left( v a_k S^{-1}(b_l) X_m b_n \right) 1^{\otimes i-2} \otimes \widetilde{a_l}\widetilde{a_n'} b_k a_n'' \otimes Y_m \otimes 1^{\otimes g-i}\\\end{aligned}$$ We used and the fact that $X_k S^{-1}(b_l) \otimes a_l' \otimes Y_k a_l'' = X_k S^{-1}(b_p)S^{-1}(b_l) \otimes a_l \otimes Y_k a_p = a_kS^{-1}(b_l) \otimes a_l \otimes b_k$. We see that we can assume without loss of generality that $g=2$, $i=2$ since the action is “local”. Moreover, this can be simplified. Let $F : H^* \to H^*$ be the map defined by $$F(\varphi) = \varphi\!\left(a_j ? b_j\right), \:\:\:\:\: F^{-1}(\varphi) = \varphi\!\left(S^{-1}(a_j) ? b_j\right).$$ We compute: $$\begin{aligned}
&(F^{-1} \otimes \mathrm{id}) \circ \rho\!\left( v_{A(1)A(2)}^{-1} \right) \circ (F \otimes \mathrm{id})(\varphi \otimes \psi)\\
&= \mu^l(v)^{-1}\mu^l\!\left( v a_k b_l X_m b_n \right) \varphi\!\left( a_j S^{-1}(a_n')a_l S^{-1}(a_o) ? b_o b_k a_n''b_j \right) \otimes \psi(?Y_m)\\
&= \mu^l(v)^{-1}\mu^l\!\left(v a_k b_l X_m b_n \right) \varphi\!\left( S^{-1}(a_n'') a_j a_l S^{-1}(a_o) ? b_o b_k b_j a_n' \right) \otimes \psi(?Y_m) = (\star).\end{aligned}$$ We used the formula $R\Delta = \Delta^{\mathrm{op}}R$. Now, we have a Yang-Baxter identity $$a_k b_l \otimes a_j a_l \otimes b_k b_j = R_{13} R_{23} R_{21} = R_{21} R_{23} R_{13} = b_l a_k \otimes a_l a_j \otimes b_j b_k$$ which allows us to continue the computation: $$\begin{aligned}
(\star) &= \mu^l(v)^{-1}\mu^l\!\left(v b_l a_k X_m b_n \right) \varphi\!\left( S^{-1}(a_n'') a_l a_j S^{-1}(a_o) ? b_o b_j b_k a_n' \right) \otimes \psi(?Y_m)\\
&= \mu^l(v)^{-1}\mu^l\!\left(v S^{-2}(b_p) b_l a_k X_m b_n \right) \varphi\!\left( S^{-1}(a_p) a_l ? b_k a_n \right) \otimes \psi(?Y_m)\\
&= \mu^l(v)^{-1}\mu^l\!\left(v a_k X_m b_n \right) \varphi\!\left( ? b_k a_n \right) \otimes \psi(?Y_m)\\
&= \mu^l(v)^{-1}\mu^l\!\left(v X_m \right) \varphi\!\left( ? Y_m'' \right) \otimes \psi(? Y_m') = \varphi\!\left( ? v''^{-1} \right) \otimes \psi(? v'^{-1}).\end{aligned}$$ We used basic properties of the $R$-matrix and relations , . We have thus shown that $$v_{A(1)A(2)}^{-1} \triangleright \varphi \otimes \psi = \varphi\!\left( S^{-1}(a_j)a_k ? b_k v''^{-1} b_j \right) \otimes \psi(?v'^{-1}).$$ Recall that $\overset{I}{D}_2 = \overset{I}{v}{^2} \overset{I}{A}(1)\overset{I}{B}(2)\overset{I}{A}(2){^{-1}} \overset{I}{B}(2){^{-1}}$. Hence $\left(\widetilde{\tau}_{a_2} \widetilde{\tau}_{b_2} \widetilde{\tau}_{a_2} \right)^{-2}(\overset{I}{A}(1)\overset{I}{A}(2)) = \overset{I}{D}_2$. It follows that $\left(\widetilde{\tau}_{a_2} \widetilde{\tau}_{b_2} \widetilde{\tau}_{a_2} \right)^{-2}(v_{A(1)A(2)}^{-1}) = v_{D_2}^{-1}$, and thus by Proposition \[propDehnTwist\] and Lemma \[lemmeActionTresseAB\]: $$\begin{aligned}
v_{D_2}^{-1} \triangleright \varphi \otimes \psi &= \left( v_{A(2)}^{-1} v_{B(2)}^{-1} v_{A(2)}^{-1} \right)^{-2} v_{A(1)A(2)}^{-1} \left( v_{A(2)}^{-1} v_{B(2)}^{-1} v_{A(2)}^{-1} \right)^2 \triangleright \varphi \otimes \psi\\
&= \left( v_{A(2)}^{-1} v_{B(2)}^{-1} v_{A(2)}^{-1} \right)^{-2} \triangleright \varphi\!\left( S^{-1}(a_j)a_k ? b_k v''^{-1} b_j \right) \otimes \psi\!\left( S^{-1}(a_l) g^{-1} v^{-1} S(v'^{-1}) S(?) b_l \right)\\
&= \varphi\!\left( S^{-1}(a_j)a_k ? b_k v''^{-1} b_j \right) \otimes \psi\!\left( S^{-1}(a_l) g^{-1} v^{-1} S(v'^{-1}) S\!\left(b_m S^{-1}(?) a_m g^{-1}v\right) b_l \right)\\
&= \varphi\!\left( S^{-1}(a_j)a_k ? b_k v''^{-1} b_j \right) \otimes \psi\!\left( S^{-1}(a_l) S^{-1}(v'^{-1}) a_m ? b_mb_l \right)\end{aligned}$$ which is the announced formula.\
[*Proof of the formula for the action of $v_{E_i}^{-1}$, $i \geq 2$.*]{} We first compute the action of $\overset{I}{C}(1) \ldots \overset{I}{C}(i-1) \overset{I}{A}(i)$. We have $$\begin{aligned}
\Psi_{g,0}\!\left(\overset{I}{C}(1) \ldots \overset{I}{C}(i-1) \overset{I}{A}(i) \right) &= \Psi_{1,0}^{\otimes g}\!\left( \overset{I}{\underline{C}}{^{(+)}}(1) \ldots \overset{I}{\underline{C}}{^{(+)}}(i-1) \left(\overset{I}{\underline{C}}{^{(-)}}(1) \ldots \overset{I}{\underline{C}}{^{(-)}}(i-1)\right)^{-1} \overset{I} \Lambda_i \overset{I}{\underline{A}}(i) \Lambda_i^{-1} \right)\\
&= \Psi_{1,0}^{\otimes g}\!\left( \overset{I}{\underline{C}}{^{(+)}}(1) \ldots \overset{I}{\underline{C}}{^{(+)}}(i-1) \overset{I}{\underline{A}}(i) \left(\overset{I}{\underline{C}}{^{(-)}}(1) \ldots \overset{I}{\underline{C}}{^{(-)}}(i-1)\right)^{-1} \right)\\
&= \overset{I}{a_j} \overset{I}{X_k} \overset{I}{b_l} \, \widetilde{b_j^{(2i-3)}} \widetilde{a_l^{(2i-3)}} b_j^{(2i-2)} a_l^{(2i-2)} \otimes \ldots \otimes \widetilde{b_j^{(1)}} \widetilde{a_l^{(1)}} b_j^{(2)} a_l^{(2)} \otimes Y_k \otimes 1^{\otimes g-i}\\
&= \overset{I}{X_k} \, \widetilde{Y_k^{(2i-2)}} Y_k^{(2i-1)} \otimes \ldots \otimes \widetilde{Y_k^{(2)}} Y_k^{(3)} \otimes Y_k^{(1)} \otimes 1^{\otimes g-i}\end{aligned}$$ thanks to and . Hence, by : $$\begin{aligned}
\Psi_{g,0}\!\left(v_{C(1) \ldots C(i-1) A(i)}^{-1} \right) &= \mu^l(v)^{-1}\mu^l(v X_k)\, \widetilde{Y_k^{(2i-2)}} Y_k^{(2i-1)} \otimes \ldots \otimes \widetilde{Y_k^{(2)}} Y_k^{(3)} \otimes Y_k^{(1)} \otimes 1^{\otimes g-i}\\
&= \widetilde{v^{(2i-2)-1}} v^{(2i-1)-1} \otimes \ldots \otimes \widetilde{v^{(2)-1}} v^{(3)-1} \otimes v^{(1)-1} \otimes 1^{\otimes g-i},\end{aligned}$$ which means that $$\begin{aligned}
v_{C(1) \ldots C(i-1) A(i)}^{-1} \triangleright \left(\varphi_1 \otimes \ldots \otimes \varphi_g\right) &= \varphi_1\!\left(S^{-1}\!\left(v^{(2i-2)-1}\right) ? v^{(2i-1)-1} \right) \otimes \ldots \otimes \varphi_{i-1}\!\left(S^{-1}\!\left( v^{(2)-1} \right) ? v^{(3)-1} \right)\\
&\:\:\:\:\: \otimes \varphi_i\!\left( ? v^{(1)-1} \right) \otimes \varphi_{i+1} \ldots \otimes \varphi_g. \end{aligned}$$ Recall that $\overset{I}{E}_i = \overset{I}{v}{^2} \overset{I}{C}(1) \ldots \overset{I}{C}(i-1)\overset{I}{B}(i)\overset{I}{A}(i){^{-1}} \overset{I}{B}(i){^{-1}}$. Hence $\left(\widetilde{\tau}_{a_i} \widetilde{\tau}_{b_i} \widetilde{\tau}_{a_i} \right)^{-2}(\overset{I}{C}(1) \ldots \overset{I}{C}(i-1)\overset{I}{A}(i)) = \overset{I}{E}_i$. As previously, it follows from Proposition \[propDehnTwist\] and Lemma \[lemmeActionTresseAB\] that $$\begin{aligned}
v_{E_i}^{-1} \triangleright \left( \varphi_1 \otimes \ldots \otimes \varphi_g \right) &= \left( v_{A(i)}^{-1} v_{B(i)}^{-1} v_{A(i)}^{-1} \right)^{-2} v_{C(1) \ldots C(i-1)A(i)}^{-1} \left( v_{A(i)}^{-1} v_{B(i)}^{-1} v_{A(i)}^{-1} \right)^2 \triangleright \left( \varphi_1 \otimes \ldots \otimes \varphi_g \right) \\
&= \varphi_1\!\left(S^{-1}\!\left(v^{(2i-2)-1}\right) ? v^{(2i-1)-1} \right) \otimes \ldots \otimes \varphi_{i-1}\!\left(S^{-1}\!\left( v^{(2)-1} \right) ? v^{(3)-1} \right)\\
&\:\:\:\:\: \otimes \varphi_i\!\left(S^{-1}(a_j) S^{-1}\!\left(v^{(1)-1}\right) a_k ? b_k b_j \right) \otimes \varphi_{i+1} \ldots \otimes \varphi_g,\end{aligned}$$ which is the announced formula. [[$\Box$]{} ]{}
Equivalence with the Lyubashenko representation
===============================================
In a series of papers [@lyu95a; @lyu95b; @lyu96], V. Lyubashenko has constructed projective representations of $\mathrm{MCG}(\Sigma_{g,n})$ by categorical techniques based on the coend of a ribbon category. Our assumptions on $H$ allow to apply his construction to $\mathrm{mod}_l(H)$, the ribbon category of finite-dimensional left $H$-modules. Here we will show that these two representations are equivalent. For the case of the torus $\mathrm{MCG}(\Sigma_{1,0}^{\mathrm{o}})$ and $\mathrm{MCG}(\Sigma_{1,0})$, we have already shown in [@Fai18] that the projective representation obtained thanks to $\mathcal{L}_{1,0}(H)$ is equivalent to the Lyubashenko-Majid representation [@LM].
The Lyubashenko representation for $\mathrm{mod}_l(H)$
------------------------------------------------------
Let us first quickly recall the Lyubashenko representation in the general framework of a ribbon category $\mathcal{C}$ satisfying some assumptions (see [@lyu95b]).\
Let $K = \int^X \! X^* \otimes X$ be the coend of the functor $F : \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{C}$, $F(X,Y) = X^* \otimes Y$ and let $i_X : X^* \otimes X \to K$ be the associated dinatural transformation (see [@ML IX.6]). Thanks to the universal property of the coend $K$, Lyubashenko defined several morphisms; we will need some of them which we recall now. The first is an algebra structure $K \otimes K \to K$. Consider the following family of morphisms (for each $X,Y \in \mathcal{C}$) $$\label{dinatProduit}
\begin{split}
d_{X,Y} : X^* \otimes X \otimes Y^* \otimes Y \xrightarrow{\mathrm{id}_{X^*} \otimes c_{X,Y^*} \otimes \mathrm{id}_Y} X^* \otimes Y^* \otimes X \otimes Y &\xrightarrow{\mathrm{id}_{X^*} \otimes \mathrm{id}_{Y^*} \otimes c_{X,Y}} X^* \otimes Y^* \otimes Y \otimes X\\
&\:\:\:\:\:\:\:\:\: \xrightarrow{\sim} (Y \otimes X)^* \otimes Y \otimes X \xrightarrow{i_{Y \otimes X}} K.
\end{split}$$ Since the family $d_{X,Y}$ is dinatural in $X$ and $Y$, it exists a unique $m_K : K \otimes K \to K$ such that $d_{X,Y} = m_K \circ (i_X \otimes i_Y)$, which is in fact an associative product on $K$. Actually, $K$ is endowed with a Hopf algebra structure whose structure morphisms are similarly defined using the universal property, but we do not need this here.\
Next, consider the following families of morphisms $$\label{dinatRep}
\begin{split}
&\alpha_X : X^* \otimes X \xrightarrow{\theta_{X^*} \otimes \mathrm{id}_X} X^* \otimes X \xrightarrow{i_X} K,\\
&\beta_{X,Y} : X^* \otimes X \otimes Y^* \otimes Y \xrightarrow{\mathrm{id}_{X^*} \otimes \left(c_{Y^*, X} \circ c_{X, Y^*}\right) \otimes \mathrm{id}_Y} X^* \otimes X \otimes Y^* \otimes Y \xrightarrow{i_X \otimes i_Y} K,\\
&\gamma_X^Y : X^* \otimes X \otimes Y \xrightarrow{\mathrm{id}_{X^*} \otimes \left(c_{Y, X} \circ c_{X, Y}\right)} X^* \otimes X \otimes Y \xrightarrow{i_X \otimes \mathrm{id}_Y} K \otimes Y.
\end{split}$$ The families $\alpha_X$ and $\gamma_X^Y$ (with $Y$ fixed) are dinatural in $X$, and the family $\beta_{X,Y}$ is dinatural in $X,Y$. Hence by the universal property of $K$, there exists unique morphisms $\mathcal{T} : K \to K, \mathcal{O} : K \otimes K \to K \otimes K, \mathcal{Q}_Y : K \otimes Y \to K \otimes Y$ such that $$\label{morphismesRep}
\alpha_X = \mathcal{T} \circ i_X, \:\:\:\:\: \beta_{X,Y} = \mathcal{O} \circ (i_X \otimes i_Y), \:\:\:\:\: \gamma_X^Y = \mathcal{Q}_Y \circ (i_X \otimes \mathrm{id}_Y).$$
Finally, the morphism $\mathcal{S} : K \to K$ is defined by $\mathcal{S} = (\varepsilon_K \otimes \mathrm{id}_K) \circ \mathcal{O} \circ (\mathrm{id}_K \otimes \Lambda_K)$, where $\Lambda_K$ is the two-sided cointegral on $K$.\
Let $X$ be any object of $\mathcal{C}$ and $V_X = \mathrm{Hom}_{\mathcal{C}}(X, K^{\otimes g})$. The Lyubashenko representation $Z_X : \mathrm{MCG}(\Sigma_g^{\mathrm{o}}) \to \mathrm{PGL}(V_X)$ [@lyu95b Section 4.4] takes the following values: $$\label{repLyubashenkoC}
\begin{split}
Z_X(\tau_{a_i}) &= \mathrm{Hom}_{\mathcal{C}}\!\left(X, \mathrm{id}_K^{\otimes g-i} \otimes \mathcal{T} \otimes \mathrm{id}_K^{\otimes i-1}\right),\\
Z_X(\tau_{b_i}) &= \mathrm{Hom}_{\mathcal{C}}\!\left(X, \mathrm{id}_K^{\otimes g-i} \otimes (\mathcal{S}^{-1} \circ \mathcal{T} \circ \mathcal{S}) \otimes \mathrm{id}_K^{\otimes i-1}\right),\\
Z_X(\tau_{d_i}) &= \mathrm{Hom}_{\mathcal{C}}\!\left(X, \mathrm{id}_K^{\otimes g-i} \otimes (\mathcal{O} \circ (\mathcal{T} \otimes \mathcal{T})) \otimes \mathrm{id}_K^{\otimes i-2}\right) \:\:\: \text{for } i \geq 2, \\
Z_X(\tau_{e_i}) &= \mathrm{Hom}_{\mathcal{C}}\!\left(X, \mathrm{id}_K^{\otimes g-i} \otimes \left((\mathcal{T} \otimes \theta_{K^{\otimes i-1}}) \circ \mathcal{Q}_{K^{\otimes i-1}}\right) \right) \:\:\: \text{for } i \geq 2.
\end{split}$$ Recall that the curves $a_i, b_i, d_i, e_i$ are represented in Figure \[figureCourbesCanoniques\]. Since these Dehn twists are a generating set, we have an operator $Z_X(f)$ for all $f \in \mathrm{MCG}(\Sigma_g^{\mathrm{o}})$. If moreover we take $X = \mathbf{1\!\!\!1}$, the unit object of $\mathcal{C}$, then this defines a representation $Z_{\mathbf{1\!\!\!1}} : \mathrm{MCG}(\Sigma_g) \to \mathrm{PGL}(V_{\mathbf{1\!\!\!1}})$ of the mapping class group of $\Sigma_g$.\
Now, let us explicit the above formulas to the case of $\mathcal{C} = \mathrm{mod}_l(H)$. Recall that the category $\mathrm{mod}_l(H)$ has braiding $c_{I,J} : X \otimes Y \to Y \otimes X$ and twist $\theta_X : X \to X$ given by $$c_{X,Y}(x \otimes y) = b_i \cdot y \otimes a_i \cdot x, \:\:\:\: \theta_X(x) = v^{-1} \cdot x$$ and that the action on the dual module $V^*$ is $h \cdot \varphi = \varphi(S(h) \cdot ?)$ for all $\varphi \in V^*, h \in H$, see [@kassel Chap. XIII–XIV] for more details.\
It is well-known (and not difficult to see) that $K$ is $H^*$ endowed with the coadjoint action: $$\forall \, h \in H, \:\forall \, \varphi \in K, \:\: h \varphi = \varphi(S(h')?h'')$$ and that the dinatural transformation of $K$ is $$i_X(\psi \otimes x) = \psi(? \cdot x) \in K.$$ Note that $\psi(? \cdot x)$ is just a matrix coefficient of the module $X$. The dinatural family $d_{X,Y}$ of is $$d_{X,Y}(\varphi \otimes x \otimes \psi \otimes y) = \psi(S(b_i)?b_j \cdot y) \varphi(?a_ja_i \cdot x)$$ where in the right of the equality it is the usual product in $H^*$: $\langle fg, h\rangle = f(h')g(h'')$. To compute the product $m_K$ in $K$ explicitly, observe that $i_{H_{\mathrm{reg}}}(\varphi \otimes 1) = \varphi$, where $H_{\mathrm{reg}}$ is the regular representation of $H$. Thus $$\begin{aligned}
m_K(\varphi \otimes \psi) = m_K \circ (i_{H_{\mathrm{reg}}} \otimes i_{H_{\mathrm{reg}}})\!\left( \varphi \otimes 1 \otimes \psi \otimes 1 \right) = d_{H_{\mathrm{reg}}, H_{\mathrm{reg}}}(\varphi \otimes 1 \otimes \psi \otimes 1)
&= \psi(S(b_i)?b_j) \varphi(?a_ja_i)\\
&= \varphi(a_j ? a_i) \psi(S(b_i)b_j ?)\end{aligned}$$ where we used $R \Delta = \Delta^{\mathrm{op}} R$ for the last equality. Moreover, the unit element of $K$ is $1_K = \varepsilon$, the counit of $H$. We record the following lemma, already given in [@lyu95b].
\[cointegraleBilatere\] Assume $\mathcal{C} = \mathrm{mod}_l(H)$, and let $\mu^r \in H^*$ be the right integral on $H$ (unique up to scalar). Then $\mu^r$ is the two-sided cointegral in $K$ (unique up to scalar): $$\forall\, \varphi \in K, \:\:\: m_K(\mu^r \otimes \varphi) = m_K(\varphi \otimes \mu^r) = \varepsilon_K(\varphi)\mu^r$$ where $\varepsilon_K(\varphi) = \varphi(1)$.
[*Proof:* ]{}Using , we get $$m_K(\mu^r \otimes \varphi) = \mu^r(a_j ? a_i) \varphi(S(b_i)b_j ?) = \mu^r(S^2(a_i)a_j ?) \varphi(S(b_i)b_j ?) = \mu^r \varphi = \varphi(1)\mu^r.$$ We used and the basic properties of $R$ [@kassel VIII.2]. Similarly: $$m_K(\varphi \otimes \mu^r) = \mu^r(S(b_i)?b_j) \varphi(?a_ja_i) = \mu^r(S^2(b_jS^{-1}(b_i))?) \varphi(?a_ja_i) = \mu^r \varphi = \varphi(1) \mu^r.$$ [[$\Box$]{} ]{}
The dinatural families of are $$\begin{split}
&\alpha_X(\varphi \otimes x) = \varphi(v^{-1}? \cdot x), \\
&\beta_{X,Y}(\varphi \otimes x \otimes \psi \otimes y) = \varphi(?b_j a_i \cdot x) \otimes \psi(S(a_jb_i)? \cdot y) = \varphi(?v'^{-1}v \cdot x) \otimes \psi(S(v''^{-1})v? \cdot y), \\
&\gamma_X^Y(\varphi \otimes x \otimes y) = \varphi(?b_j a_i \cdot x) \otimes a_j b_i \cdot y = \varphi(?v'^{-1}v \cdot x) \otimes v''^{-1}v \cdot y
\end{split}$$ where we used . It follows that the morphisms defined in are $$\begin{split}
&\mathcal{T}(\varphi) = \mathcal{T} \circ i_{H_{\mathrm{reg}}}(\varphi \otimes 1) = \alpha_{H_{\mathrm{reg}}}(\varphi \otimes 1) = \varphi(v^{-1}?), \\
&\mathcal{O}(\varphi \otimes \psi) = \mathcal{O} \circ (i_{H_{\mathrm{reg}}} \otimes i_{H_{\mathrm{reg}}})(\varphi \otimes 1 \otimes \psi \otimes 1) = \beta_{H_{\mathrm{reg}}, H_{\mathrm{reg}}}(\varphi \otimes 1 \otimes \psi \otimes 1) = \varphi(?v'^{-1}v) \otimes \psi(S(v''^{-1})v?), \\
&\mathcal{Q}_Y(\varphi \otimes y) = \mathcal{Q}_Y \circ (i_{H_{\mathrm{reg}}} \otimes \mathrm{id}_Y)(\varphi \otimes 1 \otimes y) = \gamma_{H_{\mathrm{reg}}}^Y(\varphi \otimes 1 \otimes y) = \varphi(?v'^{-1}v) \otimes v''^{-1}v \cdot y.
\end{split}$$ Note that $(\mathcal{T} \otimes \theta_Y) \circ \mathcal{Q}_Y(\varphi \otimes y) = \varphi(? v'^{-1}) \otimes v''^{-1} \cdot y$ (see ). Finally, thanks to Lemma \[cointegraleBilatere\], the morphism $\mathcal{S}$ is $$\mathcal{S}(\varphi) = \varphi\!\left(v'^{-1}v\right) \mu^r\!\left(S(v''^{-1})v?\right) = \varphi\!\left(S^{-1}(v''^{-1})v\right) \mu^r\!\left(v'^{-1}v?\right)$$ where the second equality is due to the equality $v'^{-1} \otimes S(v''^{-1}) = S^{-1}(v''^{-1}) \otimes v'^{-1}$ (which follows from $S(v^{-1}) = v^{-1}$). Moreover, we will need the following lemma to prove the equivalence of the representations.
\[lemmeST\] Let $\rho$ be the representation of $\mathcal{L}_{1,0}(H)$ on $H^*$, then the following formulas hold: $$\begin{split}
&\mathcal{T} = \rho(v_A^{-1}) = (v^{-1})_*, \:\:\:\: \mathcal{S} = \mu^l(v^{-1}) g^{-1}_* \circ \rho(v_A^2v_B) \circ g_*,\\
&\mathcal{S}^{-1} \circ \mathcal{T} \circ \mathcal{S} = (g^{-1} v)_* \circ \rho(v_B^{-1}) \circ (g v^{-1})_*,
\end{split}$$ where $h_*(\varphi) = \varphi(?h)$ for all $h \in H$ and $\varphi \in H^*$.
[*Proof:* ]{}The formula for $\mathcal{T}$ is obvious. Propositions 4.10 and 5.3 of [@Fai18] give $\rho(v_B)$ and then we compute using and : $$\begin{aligned}
\rho(v_B)(\varphi) &= v_B \triangleright \varphi = \mu^l(v^{-1})^{-1}\left( \mu^l(g^{-1}v^{-1}?)\varphi^v \right)^{v^{-1}} = \mu^l(v^{-1})^{-1}\left( \mu^r(gv^{-1}?)\varphi^v \right)^{v^{-1}}\\
&= \mu^l(v^{-1})^{-1} \mu^r\!\left(v'^{-1}?gv^{-1}\right) \varphi\!\left(S^{-1}(v''^{-1})g^{-1}v\right)\\
&= \mu^l(v^{-1})^{-1} \left( gv^{-2} \right)_*\!\left(\mu^r(v v'^{-1}?)\right) \left\langle g^{-1}_*(\varphi), S^{-1}(v''^{-1})v \right\rangle\\
&= \mu^l(v^{-1})^{-1} \left( gv^{-2} \right)_* \circ \mathcal{S} \circ g^{-1}_*(\varphi) = \mu^l(v^{-1})^{-1} \rho(v_A^{-2}) \circ g_* \circ \mathcal{S} \circ g^{-1}_*(\varphi)\end{aligned}$$ where $\varphi^h = \varphi(h?)$ for $h \in H$. The last claimed formula follows from $\mathcal{S} = \mu^l(v^{-1}) (g^{-1}v)_* \circ \rho(v_Av_Bv_A) \circ (gv^{-1})_*$ and the fact that $v_A, v_B \in \mathcal{L}_{1,0}(H)$ satisfy the braid relation $v_A v_B v_A = v_B v_A v_B$ (see [@Fai18 Prop. 5.5]). [[$\Box$]{} ]{}
For the representation space, we take $X = H_{\mathrm{reg}}$, so that $V_X = \mathrm{Hom}_H(H_{\mathrm{reg}}, K^{\otimes g}) \cong K^{\otimes g}$. Then by the previous formulas, we get the Lyubashenko projective representation of $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ applied to $\mathrm{mod}_l(H)$: $$\label{formulesLyubashenko}
\begin{split}
Z_{H_{\mathrm{reg}}}(\tau_{a_i})(\varphi_1 \otimes \ldots \otimes \varphi_g) &= \varphi_1 \otimes \ldots \otimes \varphi_{g-i+1}(v^{-1}?) \otimes \ldots \otimes \varphi_g, \\
Z_{H_{\mathrm{reg}}}(\tau_{b_i})(\varphi_1 \otimes \ldots \otimes \varphi_g) &= \varphi_1 \otimes \ldots \otimes (g^{-1}v)_* \circ \rho(v_B^{-1}) \circ (gv^{-1})_*(\varphi_{g-i+1}) \otimes \ldots \otimes \varphi_g, \\
Z_{H_{\mathrm{reg}}}(\tau_{d_i})(\varphi_1 \otimes \ldots \otimes \varphi_g) &= \varphi_1 \otimes \ldots \otimes \varphi_{g-i+1}\!\left(? v'^{-1}\right) \otimes \varphi_{g-i+2}\!\left(S(v''^{-1})?\right) \otimes \ldots \otimes \varphi_g, \\
Z_{H_{\mathrm{reg}}}(\tau_{e_i})(\varphi_1 \otimes \ldots \otimes \varphi_g) &= \varphi_1 \otimes \ldots \otimes \varphi_{g-i} \otimes \varphi_{g-i+1}\!\left(? v^{(1)-1}\right) \otimes \varphi_{g-i+2}\!\left( S(v^{(2)-1}) ? v^{(3)-1} \right) \otimes \ldots \\
&\:\:\:\:\: \otimes \varphi_g\!\left( S(v^{(2i-2)-1}) ? v^{(2i-1)-1} \right),
\end{split}$$ with $i \geq 2$ for the two last formulas. If we take $X = \mathbb{C}$, we get $$V_{\mathbb{C}} = \mathrm{Hom}_H(\mathbb{C}, K^{\otimes g}) = (K^{\otimes g})^{\mathrm{inv}} = \left\{ f \in K^{\otimes g} \left| \, \forall\, h \in H, \:\: h \cdot f = \varepsilon(h)f\right.\right\}$$ where by definition of the action of $H$ on $K$, the action of $H$ on $K^{\otimes g}$ is $$\label{actionHKg}
h \cdot \varphi_1 \otimes \ldots \otimes \varphi_g = \varphi_1\!\left(S(h^{(1)}) ? h^{(2)}\right) \otimes \ldots \otimes \varphi_g\!\left(S(h^{(2g-1)}) ? h^{(2g)}\right).$$ Then $Z_{\mathbb{C}}$ is a projective representation of $\mathrm{MCG}(\Sigma_g)$ (note that $Z_{\mathbb{C}}$ is just $Z_{H_{\mathrm{reg}}}$ restricted to $(K^{\otimes g})^{\mathrm{inv}}$).
To conclude this section, we explain how to see $\mathcal{L}_{0,1}(H)$ as a coend. Interpreting slightly differently the fusion relation of Definition \[defL01\], we can view $\mathcal{L}_{0,1}(H)$ as $H^*$ endowed with a new product. Indeed, we know that $\mathcal{L}_{0,1}(H)$ is generated by matrix coefficients $\overset{I}{M}{^i_j}$ and due to , $\dim(\mathcal{L}_{0,1}(H)) = \dim(H^*)$; hence $\mathcal{L}_{0,1}(H) \cong H^*$ as vector spaces and we identify them: $\overset{I}{M} = \overset{I}{T}$. To avoid confusion, we exceptionally denote by $\star$ (resp. $\ast$) the product of $\mathcal{O}(H)$ (resp. $\mathcal{L}_{0,1}(H)$); both are products on $H^*$ thanks to the identification. Due to and to obvious commutation relations, the $\mathcal{L}_{0,1}$-fusion relation on $H^*$ is given by $$(\overset{J}{a_i})_2 \overset{I}{T}_1 \ast \overset{J}{T}_2 (\overset{I}{b_i})_1 \overset{IJ}{(R')}{^{-1}_{12}}= \overset{I}{T}_1 \star \overset{J}{T}_2.$$ Using the relation $S^{-1}(a_i)a_j \otimes b_j b_i = 1 \otimes 1$ together with obvious commutation relations, we get $$\overset{I}{T}_1 \ast \overset{J}{T}_2 = \overset{J}{S^{-1}(a_i)}_2 \overset{I}{T}_1 \star \overset{J}{T}_2 (\overset{I}{b_j})_1 (\overset{J}{a_j})_2 (\overset{I}{b_i})_1 = \overset{I}{T}\!\left( ? b_j b_i \right)_1 \star \overset{J}{T}\!\left(S^{-1}(a_i) ? a_j\right)_2.$$ Since every element of $H^*$ is a linear combination of matrix elements $\overset{I}{T}{^i_j}$ for certain $I,i,j$, the product in $\mathcal{L}_{0,1}(H)$ is $$\label{produitL01Explicite}
\varphi \ast \psi = \varphi\!\left( ? b_j b_i \right) \star \psi\!\left(S^{-1}(a_i) ? a_j\right).$$ Moreover, we define a left $H$-module structure on $\mathcal{L}_{0,1}(H)$ by $h \cdot \varphi = \varphi \cdot S^{-1}(h) = \varphi\!\left( S^{-1}(h'') ? h' \right)$ (see ). Since $h \cdot (\varphi \ast \psi) = (h'' \cdot \varphi) \ast (h' \cdot \psi)$, $\mathcal{L}_{0,1}(H)$ is an algebra in $\mathrm{mod}_l(H^{\mathrm{cop}})$, where $H^{\mathrm{cop}}$ is $H$ with opposite coproduct. Moreover, in $H^{\mathrm{cop}}$, we replace $\Delta$ by $\Delta^{\mathrm{op}}$, $R$ by $R'$ and $S$ by $S^{-1}$ so that the formulas for the product and the $H$-action in the coend of $\mathrm{mod}_l(H^{\mathrm{cop}})$ are exactly those of $\mathcal{L}_{0,1}(H)$. We state this as a proposition.
It holds: $$\mathcal{L}_{0,1}(H) = \int^{X \in \mathrm{mod}_l(H^{\mathrm{cop}})} X^* \otimes X.$$
Equivalence of the representations
----------------------------------
Recall that we denote by $\rho$ (resp. $\rho_{\mathrm{inv}}$) the representation of $\mathcal{L}_{g,0}(H)$ on $(H^*)^{\otimes g}$ (resp. $\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$). Also recall the map $F : H^* \to H^*$ $$F(\varphi) = \varphi\!\left(a_i ? b_i\right), \:\:\:\:\: F^{-1}(\varphi) = \varphi\!\left(S^{-1}(a_i) ? b_i\right)$$ (already used in the proof of Theorem \[formulesExplicites\]) and let $\sigma : (H^*)^{\otimes g} \to (H^*)^{\otimes g}$ be the permutation $$\sigma(\varphi_1 \otimes \varphi_2 \otimes \ldots \otimes \varphi_{g-1} \otimes \varphi_g) = \varphi_g \otimes \varphi_{g-1} \otimes \ldots \otimes \varphi_2 \otimes \varphi_1.$$ It satisfies $\sigma^{-1} = \sigma$.
\[thmEquivalenceReps\] The representation of Theorem \[thmRepMCG\] and the Lyubashenko representation of $\mathrm{MCG}(\Sigma_g^{\mathrm{o}})$ and $\mathrm{MCG}(\Sigma_g)$ are equivalent. More precisely:\
1) The isomorphism of vector spaces $${ \begin{array}{crll}(F \circ S)^{\otimes g} \circ \sigma :& K^{\otimes g} & \rightarrow & (H^*)^{\otimes g} \\ &\varphi_1 \otimes \ldots \otimes \varphi_g &\mapsto & \varphi_g\!\left(b_i S(?) a_i\right) \otimes \ldots \varphi_1\!\left(b_i S(?) a_i\right) \end{array}}$$ is an intertwiner between the two representations:$$\left[(F \circ S)^{\otimes g} \circ \sigma \right] \circ Z_{H_{\mathrm{reg}}}(f) = \rho(\widehat{f}) \circ \left[(F \circ S)^{\otimes g} \circ \sigma \right].$$ 2) The isomorphism of vector spaces $$(F \circ S)^{\otimes g} \circ \sigma : (K^{\otimes g})^{\mathrm{inv}} \rightarrow \mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$$ is an intertwiner between the two representations:$$\left[(F \circ S)^{\otimes g} \circ \sigma \right] \circ Z_{\mathbb{C}}(f) = \rho_{\mathrm{inv}}(\widehat{f}) \circ \left[(F \circ S)^{\otimes g} \circ \sigma \right].$$
[*Proof:* ]{}1) We show that this isomorphism intertwines the formulas of Theorem \[formulesExplicites\] and of . Thanks to the properties of $v$ , it is clear that $(F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{a_i}) = \rho(v_{A(i)}^{-1}) \circ (F \circ S)^{\otimes g} \circ \sigma$. Next, thanks to , and , we have $$v_B^{-1} \triangleright \varphi = \mu^l(v)^{-1} \mu^r\!\left(gv^{-1} v'?\right) \varphi\!\left( vS^{-1}(gv'') \right).$$ Hence, for $\varphi \in H^*$, $$\begin{aligned}
\rho(v_B^{-1}) \circ (F \circ S)(\varphi) &= \mu^l(v)^{-1} \mu^r\!\left(gv^{-1}v'?\right) \varphi\!\left( v b_i g v'' a_i \right) = \mu^l(v)^{-1} \mu^r\!\left(g\overline{Y}_j?\right) \varphi\!\left( v^2 b_i g \overline{X}_j a_i \right)\\
&= \mu^l(v)^{-1} \mu^r\!\left(g S(a_j)S^{-1}(b_k)?\right) \varphi\!\left( v^2 g S^{-2}(b_i) b_j a_k a_i \right) = (\star)\end{aligned}$$ with $\overline{X}_i \otimes \overline{Y}_i = (RR')^{-1}$. We have a Yang-Baxter relation $$\begin{aligned}
S(a_j)S^{-1}(b_k) \otimes S^{-2}(b_i)b_j \otimes a_k a_i &= a_jS^{-1}(b_k) \otimes S^{-1}\!\left(b_j S^{-1}(b_i)\right) \otimes a_k a_i = (\mathrm{id} \otimes S^{-1} \otimes \mathrm{id})(R_{12} R_{31}^{-1} R_{32}^{-1})\\
&= (\mathrm{id} \otimes S^{-1} \otimes \mathrm{id})( R_{32}^{-1} R_{31}^{-1} R_{12}) = S^{-1}(b_k)S(a_j) \otimes b_j S^{-2}(b_i) \otimes a_i a_k\end{aligned}$$ which allows us to continue the computation: $$\begin{aligned}
(\star) &= \mu^l(v)^{-1} \mu^r\!\left(g S^{-1}(b_k)S(a_j)?\right) \varphi\!\left( v^2 g b_j S^{-2}(b_i) a_i a_k \right) = \mu^l(v)^{-1} \mu^r\!\left(g S^{-1}(b_k)S(a_j)?\right) \varphi\!\left( v S^2(b_j) a_k \right)\\
&= \mu^l(v)^{-1} \mu^r\!\left(g S^{-1}(a_j b_k)?\right) \varphi\!\left( v b_j a_k \right) = \mu^l(v)^{-1} \mu^r\!\left(g v S^{-1}(v''^{-1})?\right) \varphi\!\left( v^2 v'^{-1} \right).\end{aligned}$$ We used and . On the other hand, we compute $$\begin{aligned}
(F \circ S) \circ Z(\tau_b)(\varphi) &= (F \circ S) \circ (\mathcal{S}^{-1} \circ \mathcal{T} \circ \mathcal{S})(\varphi) = (F \circ S) \circ (g^{-1} v)_* \circ \rho(v_B^{-1}) \circ (g v^{-1})_*(\varphi)\\
&= F \circ S\!\left( \mu^l(v)^{-1} \mu^r\!\left( v'?\right) \varphi\!\left( S^{-1}(v'') \right) \right) = \mu^l(v)^{-1} \mu^r\!\left( v' b_i S(?) a_i\right) \varphi\!\left( S^{-1}(v'') \right)\\
&= \mu^l(v)^{-1} \mu^r\!\left( v S^2(a_i) \overline{Y}_j b_i S(?)\right) \varphi\!\left( v S^{-1}(\overline{X}_i) \right)\\
&= \mu^l(v)^{-1} \mu^r\!\left( v S^2(a_i) S(a_j) S^{-1}(b_k) b_i S(?)\right) \varphi\!\left( v S^{-1}(b_ja_k) \right) = (\star\star).\end{aligned}$$ We used Lemma \[lemmeST\], and . As previously, we have a Yang-Baxter relation $$S^2(a_i) S(a_j) \otimes S^{-1}(b_k) b_i \otimes b_ja_k = S(a_j)S^2(a_i) \otimes b_i S^{-1}(b_k) \otimes a_k b_j$$ which allows us to continue the computation: $$\begin{aligned}
(\star \star) &= \mu^l(v)^{-1} \mu^r\!\left( v S(a_j) S^2(a_i) b_i S^{-1}(b_k) S(?)\right) \varphi\!\left( v S^{-1}(a_k b_j) \right)\\
& = \mu^l(v)^{-1} \mu^r\!\left( S(a_j) g S^{-1}(b_k) S(?)\right) \varphi\!\left( v S^{-1}(a_k b_j) \right) = \mu^l(v)^{-1} \mu^r\!\left( g a_j b_k S(?)\right) \varphi\!\left( v b_j a_k \right)\\
&= \mu^l(v)^{-1} \mu^r\!\left( g v v''^{-1} S(?)\right) \varphi\!\left( v^2 v'^{-1} \right) = \mu^l(v)^{-1} \mu^r \circ S\!\left( ? S^{-1}(v''^{-1})vg^{-1}\right) \varphi\!\left( v^2 v'^{-1} \right)\\
&= \mu^l(v)^{-1} \mu^l\!\left( ? S^{-1}(v''^{-1})vg^{-1}\right) \varphi\!\left( v^2 v'^{-1} \right) = \mu^l(v)^{-1} \mu^r\!\left( gv S^{-1}(v''^{-1})? \right) \varphi\!\left( v^2 v'^{-1} \right).\end{aligned}$$ We used to simplify $S^2(a_i) b_i = S(S^{-1}(b_i)S(a_i)) = gv^{-1}$ and the properties of $\mu^l$ and $\mu^r$ recorded in section \[preliminaries\]. Hence, it holds $\rho(v_B^{-1}) \circ (F \circ S) = (F \circ S) \circ Z(\tau_b)$, which clearly implies that $\rho(v_{B(i)}^{-1}) \circ (F \circ S)^{\otimes g} \circ \sigma = (F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{b_i})$. Let us now proceed with $\tau_{d_i}$ ($i \geq 2$): $$\begin{aligned}
&(F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{d_i}) \circ \sigma \circ (S^{-1} \circ F^{-1})^{\otimes g}\!\left(\varphi_1 \otimes \ldots \varphi_g\right)\\
&=(F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{d_i})\!\left( \varphi_g\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \otimes \ldots \otimes \varphi_1\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \right)\\
&= (F \circ S)^{\otimes g} \circ \sigma \!\left( \varphi_g\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \otimes \ldots \otimes \varphi_i\!\left(S^{-1}(a_j) S^{-1}(v'^{-1})S^{-1}(?) b_j\right) \otimes \varphi_{i-1}\!\left(S^{-1}(a_j) S^{-1}(?) v''^{-1} b_j\right) \right.\\
&\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \left. \otimes \ldots \otimes \varphi_1\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \right)\\
&= \varphi_1 \otimes \ldots \otimes \varphi_{i-1}\!\left(S^{-1}(a_j)a_k ? b_k v''^{-1} b_j \right) \otimes \varphi_i\!\left(S^{-1}(a_j) S^{-1}(v'^{-1})a_k ? b_k b_j \right) \otimes \ldots \otimes \varphi_g\\
&= \rho(v_{D_i}^{-1})\!\left(\varphi_1 \otimes \ldots \varphi_g\right).\end{aligned}$$ Finally, for $\tau_{e_i}$ ($i \geq 2$): $$\begin{aligned}
&(F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{e_i}) \circ \sigma \circ (S^{-1} \circ F^{-1})^{\otimes g}\!\left(\varphi_1 \otimes \ldots \varphi_g\right)\\
&= (F \circ S)^{\otimes g} \circ \sigma \circ Z(\tau_{e_i})\!\left( \varphi_g\!\left(S^{-1}(a_j) S^{-1}(?) b_j \right) \otimes \ldots \otimes \varphi_1\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \right)\\
&= (F \circ S)^{\otimes g} \circ \sigma\!\left( \varphi_g\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \otimes \ldots \otimes \varphi_{i+1}\!\left(S^{-1}(a_j) S^{-1}(?) b_j\right) \otimes \varphi_{i}\!\left(S^{-1}(a_j)S^{-1}(v^{(1)-1}) S^{-1}(?) b_j\right) \right. \\
& \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \left. \otimes \, \varphi_{i-1}\!\left( S^{-1}(a_j) S^{-1}(v^{(3)-1}) S^{-1}(?) v^{(2)-1} b_j \right) \otimes \ldots \otimes \varphi_1\!\left( S^{-1}(a_j) S^{-1}(v^{(2i-1)-1}) S^{-1}(?) v^{(2i-2)-1} b_j \right) \right)\\
&= \varphi_1\!\left( S^{-1}(a_j) S^{-1}(v^{(2i-1)-1}) a_k ? b_k v^{(2i-2)-1} b_j \right) \otimes \ldots \otimes \varphi_{i-1}\!\left( S^{-1}(a_j) S^{-1}(v^{(3)-1}) a_k ? b_k v^{(2)-1} b_j \right) \\
& \:\:\:\: \otimes \varphi_{i}\!\left(S^{-1}(a_j)S^{-1}(v^{(1)-1}) a_k ? b_k b_j\right) \otimes \varphi_{i+1} \otimes \ldots \otimes \varphi_g\\
&= \varphi_1\!\left( S^{-1}(v^{(2i-2)-1}) ? v^{(2i-1)-1} \right) \otimes \ldots \otimes \varphi_{i-1}\!\left( S^{-1}(v^{(2)-1}) ? v^{(3)-1} \right) \otimes \varphi_{i}\!\left(S^{-1}(a_j)S^{-1}(v^{(1)-1}) a_k ? b_k b_j\right) \otimes \varphi_{i+1}\\
&\:\:\:\: \otimes \ldots \otimes \varphi_g\\
&= \rho(v_{E_i}^{-1})\!\left(\varphi_1 \otimes \ldots \varphi_g\right).\end{aligned}$$ We used $\Delta^{\mathrm{op}}R = R\Delta$ for the last equality.\
2) It is not difficult to see that $(F \circ S)^{\otimes g} \circ \sigma : K^{\otimes g} \to (H^*)^{\otimes g}$ is a morphism of $H$-modules, where $K^{\otimes g}$ is endowed with the action and $(H^*)^{\otimes g}$ is endowed with the action (with $n=0$). Hence, the restriction of $(F \circ S)^{\otimes g} \circ \sigma$ to $(K^{\otimes g})^{\mathrm{inv}}$ indeed takes values in $\mathrm{Inv}\!\left((H^*)^{\otimes g}\right)$. Since $$Z_{\mathbb{C}}(f) = \left(Z_{H_{\mathrm{reg}}}(f)\right)_{\vert \, (K^{\otimes g})^{\mathrm{inv}}} \:\:\: \text{ and } \:\:\: \rho_{\mathrm{inv}}(\widehat{f}) = \rho(\widehat{f})_{\vert \, \mathrm{Inv}\!\left((H^*)^{\otimes g}\right)},$$ the result follows from the first part of the theorem. [[$\Box$]{} ]{}
[30]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Strong wind-wind collisions in massive binaries generate a very hot plasma that frequently produces a moderately strong iron line. The morphology of this line depends upon the properties of the wind interaction zone and its orientation with respect to the line of sight. As the binary components revolve around their common centre of mass, the line profiles are thus expected to vary. With the advent of the next generation of X-ray observatories ([*Astro-H*]{}, [*Athena*]{}) that will offer high-resolution spectroscopy above 6keV, it will become possible to exploit these changes as the most sensitive probe of the inner parts of the colliding wind interaction. Using a simple prescription of the wind-wind interaction in an early-type binary, we have generated synthetic line profiles for a number of configurations and orbital phases. These profiles can help constrain the properties of the stellar winds in such binary systems.'
address:
- 'Groupe d’Astrophysique des Hautes Energies, Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 Août, 19c, Bât B5c, 4000 Liège, Belgium'
- 'Observatoire Astronomique de Strasbourg, Université de Strasbourg, CNRS, UMR 7550, 11 rue de l’Université, 67000 Strasbourg, France'
- 'Groupe d’Astrophysique des Hautes Energies, Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 Août, 19c, Bât B5c, 4000 Liège, Belgium'
author:
- Gregor Rauw
- Enmanuelle Mossoux
- Yaël Nazé
title: 'Fe [xxv]{} line profiles in colliding wind binaries'
---
Stars: early-type; binaries: spectroscopic; line: profiles; X-rays: stars
=0.5 cm
Introduction
============
[*Chandra*]{} and [*XMM-Newton*]{} have opened up the era of high-resolution X-ray spectroscopy for moderately bright X-ray sources. In the case of massive stars, HETG and RGS spectra unveiled for the first time some details of the morphology of the spectral lines [see e.g. @GN]. For presumably single massive stars, where the X-ray emission arises from hot plasma embedded in the stellar wind, a number of observational and theoretical studies were undertaken with the goal to connect the line morphologies to the properties of the stellar wind [e.g. @Feldmeier; @OC2; @Oskinova; @Cohen; @Herve; @lamCep].
In massive binary systems, the X-ray emission not only arises from the individual winds of both stars, but also from shock-heated plasma inside the wind interaction zone between the stars [e.g. @SBP]. At the shock front between the winds, the kinetic energy normal to the shock is converted into heat, thereby generating a very hot plasma. Since the plasma temperature exceeds the values typically reached in the shocks that prevail in the winds of most single massive stars, the detection of such a hot plasma provides strong hints of a wind-wind interaction. The most obvious signature of such a high-temperature plasma is the moderately strong iron line near 6.7keV, commonly called the Fe K line, that can be observed in some systems. Whilst numerous studies have been devoted to colliding winds over the last two decades [for a general review on colliding winds see @RN], most of them focused on broadband medium-resolution spectroscopy. Based on density and velocity maps obtained from hydrodynamical simulations, @Henley presented calculations of theoretical X-ray line profiles of colliding wind binaries and of their variations along the orbit for a range of orbital and wind parameters. Their study focused on the Ly$\alpha$ transitions of O [viii]{}, Ne [x]{}, Mg [xii]{}, Si [xiv]{} and S [xvi]{}. These lines are located at energies below 3keV, in the sensitivity band of the current high-resolution spectrographs. However, most of these lines are also emitted by the plasma in the winds of each individual binary component outside the wind-wind interaction zone. Their profiles are thus often contaminated, or even dominated, by contributions from the intrinsic emissions of the stars. Moreover, as shown by @Henley, photoelectric absorption by the cool unshocked winds affects the morphology of the lines at longer wavelengths, especially in the case of the O, Ne, and Mg L$\alpha$ lines. Both effects render the interpretation of the observed line profiles in terms of colliding winds rather difficult.
The Fe K feature offers a promising way out of this dilemma. Indeed, except for some stars featuring a strong magnetic field, this line is not seen in the X-ray spectra of single massive stars. Furthermore, at energies near 6.7 – 7.0keV, it is essentially unaffected by absorption in the unshocked winds. Moreover, it is rather isolated, thereby limiting the problems of blends with other species. Unfortunately though, the iron line falls outside the sensitivity range of the current generation of high-resolution spectrographs. However, this situation is about to change in the near future thanks to the JAXA mission [*Astro-H*]{} [@AstroH] and, in the more distant future, thanks to the ESA observatory [*Athena*]{} [@Nandra]. Each of these missions will carry a high-resolution bolometric spectrograph optimized for a high-sensitivity and high-resolution coverage of the spectral region around 6.0 – 7.0keV. [*Astro-H*]{} will be equiped with the [*Soft X-ray Spectrometer*]{} [SXS, @SXS], whilst [*Athena*]{} will host the [*X-ray Integral Field Unit*]{} [XIFU, @XIFU; @Ravera]. Whilst these instruments are primarily designed to address other scientific questions, they can also be efficiently used for the purpose of the study of colliding wind binaries [@SR].
In this context, we address here the question of the morphology of the Fe line at 6.7keV and its variations with orbital phase.
The iron line in the spectra of colliding wind binaries
=======================================================
Before we turn to the discussion of our models, we need to briefly consider the nature of the Fe K line that is observed in the spectra of colliding wind binaries at energies around 6.7keV. As shown by spectral fits, this line is compatible with emission from highly ionized iron in a thermal plasma of high temperature $T$. The emissivity of the triplet of helium-like Fe[xxv]{} between 6.64 and 6.70keV peaks near $\log{T} \sim 7.8$ (i.e. 63MK or $kT = 5.4$keV). Conversely, the emissivity of the Ly$\alpha$ doublet of hydrogen-like Fe[xxvi]{} peaks at $\log{T} \sim 8.2$ (i.e. 158MK or $kT = 13.7$keV). Considering that typical wind velocities of massive stars range between 1000 and 3000kms$^{-1}$, the Rankine-Hugoniot condition for strong shocks implies typical post-shock temperatures between 1.2 and 10.5keV. This leads us to conclude that the iron line of the majority of the colliding wind binaries should be associated with the Fe[xxv]{} helium-like triplet.
Helium-like triplets consist of a resonance line, an intercombination doublet and a forbidden line. In the case of Fe [xxv]{}, the energies of these lines, taken from the AtomDB database [@AtomDB], are 6.7004keV (resonance line), 6.6823 and 6.6676keV (intercombination doublet), and 6.6366keV (forbidden line).
![Modification of the ${\cal R}$ ratio between the forbidden and intercombination lines in the Fe[xxv]{} triplet formed in the wind of an O-type star with $T_{\rm eff} = 55\,000$K and $\log{g} = 4.0$. \[figure1\]](fir.eps){width="45.00000%"}
As shown e.g. by @Blumenthal and @Porquet, the forbidden line of the helium-like triplets of light elements can be strongly suppressed as a result of either collisional excitation or radiative pumping of the electrons from its upper level (2$^3S_1$) to the upper levels of the intercombination doublet (2$^3P_{1,2}$). The latter effect has been used extensively to study the location of the X-ray emitting plasma in the winds of massive stars [e.g. @Leutenegger06]. Does a similar effect also exist for the Fe[xxv]{} triplet? To address this question, we follow the formalism of @Blumenthal [@Porquet] and @Leutenegger06. These authors have shown that the ratio ${\cal R}$ between the forbidden and intercombination lines can be written $$\label{eq:1}
{\cal R} = \frac{{\cal R}_0}{1 + 2\,w(r)\,\frac{\psi_*}{\psi_c} + \frac{N_e}{N_c}}$$ Here ${\cal R}_0$ is the ratio between the forbidden and intercombination line in the absence of photospheric light and collisional excitation. $N_e$ is the local electron density and $N_c$ is the critical electron density above which collisional excitation from 2$^3S_1$ to 2$^3P_{1,2}$ becomes important. The photoexcitation rate $\psi_*$ due to the photospheric radiation field is multiplied by the dilution factor $$w(r) = \frac{1}{2}\,\left(1 - \sqrt{1 - \frac{R_*^2}{r^2}}\right)$$ at a distance $r$ from the centre of the star of radius $R_*$. This value is compared to a critical rate $\psi_c$. In the case of the Fe[xxv]{} triplet, @Blumenthal quote values of $N_c = 4.7 \times 10^{16}$cm$^{-3}$ and $\psi_c = 1.23 \times 10^8$s$^{-1}$.
For a solar composition, $N_c$ corresponds to $\rho_c = 2.3 \times 10^{-8}$gcm$^{-3}$. This value is several orders above the typical density in adiabatic post-shock regions [e.g. @Pittard09]. The suppression of the forbidden line by collisional excitation can thus be neglected.
Concerning the impact of photoexcitation, the relevant transitions that will pump the electrons are located in the EUV at wavelengths of 271.4 and 396.7Å [@NIST]. We use the fluxes at the stellar surface predicted by the non-LTE plane-parallel TLUSTY stellar atmosphere model and available via the Ostar2002 grid [@Ostar2002] to estimate $\psi_*$. If we consider a very hot main-sequence star with $T_{\rm eff} = 55\,000$K, we find that the impact of photospheric radiation on the Fe[xxv]{} triplet is rather marginal (of order 8% at the stellar surface, see Fig.\[figure1\]) and drops rapidly when we move out into the stellar wind, at radii $\geq 5\,R_*$, where the wind-wind collision occurs. The main reason for this weak effect is of course the high value of $\psi_c$: even such a hot star does not provide enough radiation at the relevant wavelengths to produce a strong pumping. We can thus conclude that we can safely neglect the impact of photospheric radiation on the relative strengths of the components of the Fe[xxv]{} triplet.
A simplified model for the wind-wind interaction
================================================
Depending on the efficiency of radiative cooling in the post-shock plasma, colliding wind binaries are frequently divided into two broad categories [@SBP].
The plasma is said to be in the adiabatic regime when the post-shock densities are sufficiently low for the cooling time ($t_{\rm cool}$) to be much longer than the escape time ($t_{\rm esc}$) from the shock region. The efficiency of radiative cooling is thus expressed via the parameter $\chi = t_{\rm cool}/t_{\rm esc}$. This situation mainly occurs in wide, long-period binaries and/or systems consisting of stars with relatively low mass-loss rates. The corresponding wind interaction zone is quite smooth and can usually be well modelled with hydrodynamical simulations. The X-ray emission should scale as $\dot{M}^2\,v^{-3.2}\,d^{-1}$ where $\dot{M}$, $v$ and $d$ are the mass-loss rate, the pre-shock wind velocity and the separation between the stars, respectively.
Conversely, if the post-shock plasma density is high, radiative cooling happens very quickly and the wind interaction zone is then said to be radiative. This concerns close binary systems hosting stars with large mass-loss rates, but can also occur in wide, eccentric systems around periastron passage. From first principles, the X-ray emission of these systems should scale with the kinetic power of the incoming stellar winds. However, the observed level of X-ray emission is usually much lower than theoretically expected. This could be due to radiative inhibition [@SP], radiative braking [@Gayley] or the strong hydrodynamical instabilities that strongly distort the wind interaction zone [@Kee].
In our present work, we consider mainly systems that can be described as undergoing an adiabatic wind interaction. We further assume that the winds collide at their terminal velocity $v_{\infty}$, thereby neglecting wind acceleration, radiative inhibition and braking. Since the emission of interest here arises in the inner parts of the interaction zone, we neglect the impact of the orbital motion and the shocks are considered to be axisymmetric about the binary axis. We then use the analytical solution of @Canto to describe the properties of the wind interaction zone. In principle, the formalism of @Canto holds for situations where the shocked plasma forms a thin shell around the contact discontinuity, which is the case if both winds are highly radiative. Although we consider here mainly situations where the shocks are adiabatic, the position of the contact discontinuity is unlikely to drastically differ from the predictions of @Canto, as was shown by @PP08 and @Pittard09.
For a given value of the wind momentum ratio $$\eta = \frac{\dot{M_2}\,v_{\infty,2}}{\dot{M_1}\,v_{\infty,1}}$$ solving equation (28) of @Canto by means of a Newton-Raphson scheme allows us to estimate the asymptotic opening angle of the shock $\theta_{\infty}$ as seen from the star with the weaker wind. The interval $[0,\theta_{\infty}]$ is then discretized into 200 steps $\theta_k$. For each angle, equations (23) and (24) of @Canto are solved to compute the shape of the contact discontinuity, i.e. to establish the relation between $r$ and $\theta$ (see Fig.\[figure2\]). Along with the adopted mass-loss rates and wind velocities, equations (29) and (30) of @Canto further allow us to estimate the mass surface density $\sigma$ of the wind interaction zone and the tangential velocity along the contact dicontinuity. The resulting shape and velocity vector are then rotated about the binary axis to generate an axisymmetric shock cone, discretized into $200 \times 360$ 2-D cells, along with the associated velocity field. The thickness of these cells is computed as the ratio of the surface density $\sigma$ divided by the post-shock density $\rho_s$. Since we are dealing with an adiabatic wind interaction zone, we use the Rankine-Hugoniot condition for strong shocks $\rho_s = 4\,\rho_w$ where $\rho_w$ is the pre-shock wind density. The volume associated with a cell is then calculated following $$\label{eq:2}
dV = \frac{r^2\,\sin{\theta}\,d\theta\,d\phi}{\sin{(\alpha - \theta)}}\,\frac{\sigma}{\rho_s}$$ (see Fig.\[figure2\]). Here $d\phi = 1^{\circ}$ is the incremental step of the azimuthal angle used to build the axisymmetric shock cone. $\alpha$ is the angle between the tangential velocity and the direction of the binary axis.
![Geometry of our model of the wind interaction zone. The star with the weaker wind is on the left. The contact discontinuity between the winds is shown by the thick black line. The blue vector indicates the tangential velocity at the point of coordinates $(r, \theta)$. $\beta = \pi/2 - \alpha + \theta$ is the angle between the tangential velocity and the normal to the position vector. \[figure2\]](canto.eps){width="45.00000%"}
The associated contribution to the line emission is given by $$\label{eq:3}
d\epsilon = Q\,n_e\,n_H\,dV$$ where $n_e$ and $n_H$ are the electron and hydrogen number densities in the post-shock region and $Q$ is the line emissivity at the temperature of the post-shock plasma inside the cell assuming ionization equilibrium. The post-shock temperature is computed assuming that the velocity normal to the shock is entirely thermalized [see @SBP] and that the electrons and ions have equalized their temperatures. For wind-wind interactions in wide binary systems, these two points, ionization equilibrium and equal electron and ion temperatures ($T_e = T_{\rm ion}$), are clearly approximations. Indeed, as pointed out by several authors [e.g. @Usov; @ZS; @Zhekov; @Pollock], the shocked plasma is probably out of ionization equilibrium and the shocks are likely collisionless, implying $T_e \leq T_{\rm ion}$. However, these approximations are frequently used [including in the work of @Henley]. Moreover, they mainly affect the strengths of the simulated lines and should be much less important for their profiles.
For a given pair of orbital inclination $i$ and orbital phase $\Phi$, we determine the direction of the line-of-sight and project the tangential velocity of each cell onto this direction to obtain the line-of-sight velocity of the gas in the cell. If aberration of the shock due to orbital motion can be neglected, then the orientation of the line-of-sight can be characterized via a single angle $\Theta$ with $$\cos(\Theta) = \sin(i)\,\cos(M)
\label{eq:Theta}$$ Here $M = v + \omega -\frac{3\,\pi}{2}$ where $v$ is the true anomaly and $\omega$ is the longitude of periastron of the primary star, which we consider to be the star with the stronger wind (see Fig.\[figure3\]).
![Illustration of the angles used to define $\Theta$. The star with the weaker wind is the one at the origin of the axes. The $x$ and $y$ axes are inside the orbital plane and rotate as the stars move around each other on their orbit. \[figure3\]](geometry.eps){width="45.00000%"}
Although the effect of photoelectric absorption by the winds is very low at the energies of interest here, we have nevertheless accounted for the optical depth along the line-of-sight. Using the ionized wind opacity model of @HD108, we estimate the opacity of the wind at 6.7keV to be 0.7cm$^2$g$^{-1}$ for solar abundances. This is about a factor 100 and 40 lower than the equivalent opacity at the energies of the O [viii]{} L$\alpha$ and Ne [x]{} L$\alpha$ lines, respectively. To compute the optical depth for a given sightline, we first check whether it intersects any of the stars and whether it crosses the wind of one or two stars. If the cell is occulted by one of the stars, its contribution to the observed profile is set to zero. Otherwise, the optical depth along the sightline is evaluated accounting for the appropriate opacities of each wind that is crossed.
Thermal broadening is accounted for by distributing the flux of the cell over a Gaussian profile centered on the line-of-sight velocity. Given the high mass of the Fe ions, its effect remains quite moderate though (typically $\sigma_{\rm therm} = 86$kms$^{-1}$ for a 50MK plasma).
Finally the synthetic line profile is obtained as an histogram of the contributions of all the cells according to their line-of-sight velocity. We assume a constant energy resolution of the instrument of 3eV (FWHM) at the relevant energies. The synthetic profiles are computed with line-of-sight velocity steps of 50kms$^{-1}$, and are then binned onto line-of-sight velocity bins of 150kms$^{-1}$ corresponding to a bit more than one resolution element.
Results
=======
In this section, we present the output of our model for several sets of parameters, including systems with circular or elliptical orbits. As pointed out above, the Fe [xxv]{} line consists of four components which are closely spaced in wavelength and will thus be blended over parts of the parameter space. To illustrate some effects, we thus start our discussion with the simulation of a simpler case, where we consider a single line.
Systems with circular orbits \[circular\]
-----------------------------------------
Let us consider early-type binaries with circular orbits. As a first step, we compare the line profiles obtained for three different values of the wind momentum ratio $\eta$ and seen under five different viewing angles. To ease comparison with the work of @Henley, we adopt the same values of $\Theta$ as done by these authors.
------------------------------------------- ------------ ------ ------- -------
Parameter Primary
I II III
$\dot{M}$ ($10^{-6}$M$_{\odot}$yr$^{-1}$) 1.0 0.98 0.3 0.1
$v_{\infty}$ (kms$^{-1}$) 2000 2000 2000 2000
$R_*$ (R$_{\odot}$) 20 20 10 10
$M_*$ (M$_{\odot}$) 30 30 30 30
$\eta$ - 0.98 0.3 0.1
$\chi$ $\geq 5.6$ 5.7 13.1 26.8
$\theta_{\infty}$($^{\circ}$) 90.4 111.5 128.8
------------------------------------------- ------------ ------ ------- -------
: Parameters of the test models for circular binary systems with an orbital separation of 100R$_{\odot}$. All systems feature the same primary star, but differ by the properties of the secondary.
\[table1\]
{width="80.00000%"}
{width="80.00000%"}
Because of numerical singularities our code cannot deal with the case where $\eta$ is strictly equal to one. Case I thus rather considers $\eta = 0.98$. Figure\[figure4\] illustrates the profiles of individual lines as a function of viewing angle for the three cases described in Table\[table1\]. Figure\[figure5\] illustrates the expected morphologies of the full Fe [xxv]{} complex.
We start by considering Case I, which corresponds to a situation where the wind interaction zone is almost planar and located midway between the stars. Therefore, for conjunction phases ($\Theta = 0^{\circ}$ for the conjunction with the primary in front and $\Theta = 180^{\circ}$ for the opposite situation), the velocity vectors of the material flowing out of the wind interaction zone are essentially perpendicular to the line of sight. This results in very narrow profiles, centered on zero velocity. For these configurations, one hence expects to resolve the individual components of the Fe [xxv]{} complex (see Fig.\[figure5\]). We note that our results slightly underestimate the line width at these viewing angles. In a real system, the velocity vectors in the wind interaction zone are not simply tangent to the contact discontinuity, but their direction spans a range of values around the tangent. The hydrodynamic simulations of @Henley indeed yield somewhat wider lines for $\eta = 1$ and $\Theta = 0$ or $180^{\circ}$. For other viewing angles, Case I shows very large variations of the width of the line. The width is maximum for $\Theta = 90^{\circ}$, i.e. when the line of sight crosses the plane of the wind interaction region. For this orientation, the full-width of the line could reach $2 \times v_{\infty}$ if all cells of the wind interaction zone were emitting at the same level. The fact that the line width does not reach this value reflects the variations of the plasma temperature and hence emissivity along the shock: the further we move away from the binary axis, the more oblique the shock with respect to the inflowing wind and thus the lower the post-shock temperature and hence the lower the contribution to the line emission. We further see that the profile remains roughly symmetric and centered on zero velocity. Unlike the lines simulated by [@Henley], the profiles in Fig.\[figure4\] do not display strong asymmetries between the blue and red side. This is because the impact of photoelectric absorption by the cool unshocked winds is very small at the energies of the Fe [xxv]{} lines considered here. Photoelectric absorption mainly affects the lines at longer wavelengths that are modelled by @Henley. Its net effect is to skew these lines towards the blue. In our simulations, the only slight asymmetries between the blue and red wings are seen for $\Theta = 45$ and $135^{\circ}$, where we note a slight depression in the red wing which stems from occultation of receding material by the star that is in front.
Another difference with the results of @Henley concerns the fact that, in their simulations, the unabsorbed profiles of the lighter elements display a double-peaked morphology for viewing angles away from conjunction. The reason for this behaviour is that the emissivities of the Ly$\alpha$ lines of lighter elements peak at some distance away from the binary axis, leading to an emission region that has essentially the shape of a cone with its apex truncated [this is similar to the situation of the optical line profiles modelled by @Luehrs]. Conversely, the emissivity of Fe [xxv]{} reaches its maximum near the apex and progressively decreases along the cone. This situation leads to profiles that exhibit a single maximum peak near the line-of-sight velocity of the apex.
We finally note that the strong line broadening for viewing angles away from conjunction leads to severe blending of the various Fe [xxv]{} components. The resulting profiles are thus rather complex (see Fig.\[figure5\]). In practice, this result highlights the need to implement a least-squares deconvolution technique [LSD, @LSD] to recover the profiles of individual lines from observed data. In the remainder of this paper, we will mainly focus on the individual line profiles, assuming that they have been extracted via LSD.\
A different situation prevails for $\eta = 0.3$ and $\eta = 0.1$ (Cases II and III, respectively). At conjunction phases, we still observe a relatively narrow line, though it is now much wider than in Case I. We also note that the profile, although narrow, is now markedly skewed towards the red for $\Theta = 0^{\circ}$ and towards the blue for $\Theta = 180^{\circ}$. This is of course because the wind interaction zone is now wrapped around the star with the weaker wind (the secondary) and the post-shock material is thus moving away from the observer at $\Theta = 0^{\circ}$ and moving towards the observer at $\Theta = 180^{\circ}$. Again, the maximum width (for the five values of $\Theta$ considered here) is observed at quadrature ($\Theta = 90^{\circ}$), where our calculations predict a symmetric profile centered on the rest wavelength of the lines. At intermediate phases, highly asymmetric line profiles with a marked peak near zero velocity are predicted, especially for Case III. This peak results from the fact that at these values of $\Theta$ and for the specific value of $\theta_{\infty}$, one of the arms of the interaction region is almost perpendicular to the observer’s sightline.\
As a next step, we have simulated phase-resolved synthetic line profiles and spectra for the three cases above, but now assuming binary systems seen under an orbital inclination of $i = 75^{\circ}$. At this stage, we also account for the orbital motion by requesting the wind interaction zone to rotate with the stars on their orbit. The resulting profiles of individual lines at ten distinct orbital phases are shown in the top row of Fig.\[figure6\]. The variations of the line profiles are clearly seen. In the following subsection, we analyse these profile variations by means of the so-called Doppler tomography technique.
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
Doppler tomography \[tomography\]
---------------------------------
Doppler tomography is an elegant and powerful technique to study the dynamics of gas flows in spectroscopic binaries with circular orbits [@Horne; @kaitchuk]. This method is nowadays widely applied to optical spectra, but to the best of our knowledge, it has so-far never been used in the X-ray domain.
Doppler tomography translates phase-locked emission line profile variations into a map of the line formation region in velocity space. To do so, one adopts a reference frame centered on the centre of mass of the binary with the $x$-axis pointing from the secondary to the primary and the positive $y$-axis pointing along the direction of the primary’s orbital motion. The fundamental assumption of the Doppler tomography technique is that the phase dependence of the radial velocity $v(\Phi)$ of any cell of emitting gas that is stationary in the rotating frame of reference of the binary can be described by the simple relation: $$v(\Phi) = -v_x\,\cos{(2\,\pi\,\Phi)} + v_y\,\sin{(2\,\pi\,\Phi)}
\label{eq:4}$$ Here $\Phi$ stands for the orbital phase, with $\Phi = 0$ at conjunction with the primary star in front. The pair $(v_x, v_y)$ yields the velocity coordinates of the gas cell projected along the $x$ and $y$ axes: $v_x = V_x\,\sin{i}$ and $v_y = V_y\,\sin{i}$, where $V_x$ and $V_y$ are the actual velocities in the orbital plane of the binary and $i$ is the orbital inclination.
The Doppler map $DM(v_x,v_y)$ yields a measure of the flux that is carried across the line profile for each particular $(v_x,v_y)$ pair. If we adopt the back-projection method, the Doppler map can be expressed as $$DM(v_x,v_y) = \frac{\int D(v(v_x,v_y,\Phi),\Phi)\,W(\Phi)\,d\Phi}{\int W(\Phi)\,d\Phi}
\label{eq:5}$$ where $D(v,\Phi)$ is the observed flux density at radial velocity $v(v_x,v_y,\Phi)$ (given by equation\[eq:4\]) at phase $\Phi$. $W(\Phi)$ is the weight assigned to the observation at phase $\Phi$ [see e.g. @Horne for a detailed discussion of the method]. The implementation of the Doppler tomography that we use here is adapted from the one used by @WR20a. The method is based on a Fourier filtered back-projection algorithm [@Horne]. The Point Spread Function (PSF) of a pure back-projection technique in the $(v_x,v_y)$ plane has a Gaussian core with extended wings having a $1/\sqrt{v_x^2 + v_y^2}$ profile [@Horne]. To sharpen the PSF we apply a Fourier filter to the spectra prior to the back projection. For this purpose, we first compute the Fourier transform of the trailed spectrogram and we then multiply the result by a filter $\frac{\omega}{\omega_N}\,\exp{\left[-\frac{(\omega/\omega_C)^2}{2}\right]}$ to suppress the $1/\sqrt{v_x^2 + v_y^2}$ tail and to prevent the amplification of high-frequency noise in the spectra, where $\omega_N$ is the Nyquist frequency of the spectra, $\omega_C$ is set by the spectral resolution [for further details see e.g. @Horne]. Finally, an inverse Fourier transformation is performed to recover the filtered spectra that are then back-projected.
Back-projection produces stripes across the Doppler map at angles corresponding to the binary phases sampled by the data. Therefore, the back-projection method can lead to the so-called ‘radial-spoke artefact’ if the data do not provide a uniform coverage of the orbital cycle. To avoid this problem, we have sampled our synthetic spectra at 20 equally-spaced orbital phases.
The results of applying our method to the three sets of simulations are shown in Fig.\[figure6\]. It becomes clear from these examples that the aspect of the Doppler map strongly depends on the shock opening angle and hence the value of the wind momentum ratio $\eta$. Doppler tomography of the Fe [xxv]{} line thus provides a sensitive diagnostics of the relative strengths of the winds in colliding wind binaries with circular orbits.
------------------------------------------- ------------ ------------- ------------ -------------
Parameter
Prim. Seco. Prim. Seco.
$\dot{M}$ ($10^{-6}$M$_{\odot}$yr$^{-1}$) 12.0 5.0 57.0 1.8
$v_{\infty}$ (kms$^{-1}$) 2400 2500 2860 3200
$R_*$ (R$_{\odot}$) 16 19 13 12
$M_*$ (M$_{\odot}$) 44 50 16 41
$i$($^{\circ}$)
$e$
$\omega$($^{\circ}$) 12 192 224.6 44.6
$a$ (R$_{\odot}$)
$\eta$
$\chi$ $\geq 5.8$ $\geq 10.8$ $\geq 2.4$ $\geq 21.9$
$\theta_{\infty}$($^{\circ}$)
------------------------------------------- ------------ ------------- ------------ -------------
: Parameters of the test models for eccentric binary systems.
\[table2\]
Systems with eccentric orbits
-----------------------------
In this section we use our code to simulate line profiles for a sample of eccentric colliding wind binary systems. In addition to the wind parameters, the orbital inclination and major axis, such systems are characterized by two new parameters: the eccentricity $e$ and the longitude of periastron $\omega$.
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
Table\[table2\] lists the parameters of the systems that we consider here. They are directly inspired from CygOB2\#9 [Case IV, see @Naze] and WR140 [Case V, see @Pollock; @Fahed].
For the eccentric systems, we predict strong variations of the integrated line flux with orbital phase. In fact, since we are dealing with simulations that assume adiabatic wind interaction zones, we actually recover the $1/d$ flux variation that is expected for such systems [@SBP].
Figure\[figure7\] illustrates the results of our simulations for the two eccentric systems. For Case IV ($e = 0.71$, $\eta = 0.43$), the shock opening angle is relatively large, leading to rather broad lines at most orbital phases. The individual profiles have a single-peaked morphology. Their intensity, centroid and skewness all change with orbital phase. The full Fe [xxv]{} complex displays a rather complicated morphology indicating that LSD might again be needed to disentangle the variations of the individual profiles from apparent variations due to the blending.
For Case V ($e = 0.896$, $\eta = 0.035$), the primary wind overwhelms that of the secondary and, in our simple model, it is only the wide separation of the stars that allows for the shock region to remain detached from the secondary’s surface. The adopted pre-shock velocities are very high, leading to post-shock temperatures near 139MK. This is above the temperature of maximum emissivity of the Fe [xxv]{} complex. Hence, in our simulation, the maximum of the emission of the line occurs at positions slightly away from the shock apex, thereby leading to a double-peaked line morphology (see Fig.\[figure7\]), as discussed in Sect.\[circular\]. This renders the resulting profiles of the full Fe [xxv]{} blend even more complex.
{width="75.00000%"}
Comparison with observations
----------------------------
Current instruments in X-ray astrophysics lack the spectral resolution in the hard energy band needed to confront the profiles obtained in our simulations with real observations. Yet, we can compare the relative variations of the integrated line fluxes of several eccentric systems with the predictions of models for adiabatic wind interaction zones. For this purpose, we have considered five systems which exhibit a clear Fe [xxv]{} feature in their broadband X-ray spectra. In increasing order of the orbital period, these are CygOB2 \#8a [$P_{\rm orb} = 21.9$days, $e = 0.24$ @DeBecker], WR21a [$P_{\rm orb} = 31.7$days, $e = 0.69$ @Tramper], WR25 [$P_{\rm orb} = 208$days, $e = 0.48$ @Gamen], CygOB2 \#9 [$P_{\rm orb} = 2.36$yrs, $e = 0.71$ @Naze], and WR140 [$P_{\rm orb} = 7.9$yrs, $e = 0.89$ @Fahed]. We have analysed [*XMM-Newton*]{} spectra for these systems. For CygOB2 \#8a, CygOB2 \#9 and WR21a, the description of the data is given by @Cazorla, @Naze, and @Gosset, respectively. For WR140 and WR25, the data were retrieved from the [*XMM-Newton*]{} archive and processed with the Science Analysis Software version 14.0. The EPIC spectra of each object were fitted between 6 and 8keV using a powerlaw with zero slope for the continuum and a single Gaussian for the Fe K line[^1]. The fluxes of the Fe K line are displayed as a function of orbital phase in Fig.\[figure8\]. In this figure, they are compared against the $1/r$ trend expected for an adiabatic wind-wind interaction zone.
As we can see on this figure, there are clear deviations from the $1/r$ trend. For the shortest period systems, CygOB2 \#8a and WR21a, these deviations are likely due to the shocked gas becoming radiative around periastron passage and/or the shock collapsing onto the surface of the star with the weaker wind [@Cazorla; @Gosset]. The agreement is better for the longer period systems, although also here there are significant deviations. For instance, in the case of WR140, the data follow the relation rather well up to $\Phi = 0.986$, but fall short of the expected emission level afterwards. WR140, CygOB2 \# 8a and CygOB2 \#9 are prominent non-thermal radio emitters and, in the particular case of WR140, [@PD] suggested that the deviations from the $1/r$ behaviour for the overall X-ray flux could reflect the impact of particle acceleration on the energy budget and the properties of the shocks [see also the case of 9 Sgr discussed by @9Sgr]. Whatever the exact reason for the behaviour seen in Fig.\[figure8\], it is clear that high-resolution X-ray spectra around the Fe K with next generation observatories will shed new light on the properties of the wind interaction zone.
Conclusions and future prospects
================================
In this paper, we have used a rather simple model, based on the analytical solution of @Canto, to predict the morphology of the Fe [xxv]{} lines in the X-ray spectra of colliding wind massive binaries in the adiabatic regime. Our results are in qualitative agreement with those of @Henley. The latter authors computed synthetic line profiles based on the density and velocity fields from snapshots of 2-D axisymmetric hydrodynamical simulations of adiabatic wind collisions and assuming that the winds collide at their terminal velocity. Compared to @Henley, the advantage of our method is its simplicity and low computational cost, allowing to compute large grids of models for comparison with actual observations.
We have shown that the morphology of the Fe [xxv]{} line and its orbital changes provide direct diagnostics of the colliding wind interaction and thus of the properties of the stellar winds. This line offers the cleanest probe of the conditions near the apex of the shock region. On the one hand, it does not suffer from significant absorption by the cool unshocked winds. On the other hand, it is not affected by contributions from the intrinsic emission of the stars that make up the binary system. In this regard, there is a great potential for observational studies of colliding wind binary systems with the bolometric spectrographs that will fly on the coming X-ray observatories [*Astro-H*]{} and [*Athena*]{}.
In the future, we will try to generalize this work also to systems where radiative cooling in the shocked winds is efficient and where the Coriolis force leads to a significant aberration. An interesting case of such a system is V444 Cyg (WN5 + O6, $P_{\rm orb} = 4.21$days) which should be largely in the radiative regime and exhibits a rather prominent Fe [xxv]{} line in its [*XMM-Newton*]{} spectra [@Lomax]. These properties make V444 Cyg an ideal case to apply the Doppler tomography technique that we have outlined above.
Acknowledgements {#acknowledgements .unnumbered}
================
The Liège team acknowledges support through an ARC grant for Concerted Research Actions, financed by the Federation Wallonia-Brussels, from the Fonds de la Recherche Scientifique (FRS/FNRS), as well as through an XMM PRODEX contract.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In realistic situations, black hole spacetimes do not admit a global timlike Killing vector field. However, it is possible to describe the horizon in a quasi-local setting by introducing the notion of a quasi-local boundary with certain properties which mimic the properties of a black hole inner boundary. Isolated horzons and Killing horizons are examples of such kind. In this paper, we construct such a boundary of spacetime which is null and admits a conformal Killing vector field. Furthermore we construct the space of solutions (in general relativity) which admits such quasi-local conformal Killing boundaries. We also establish a form of first law for these quasi-local horizons.'
author:
- Ayan Chatterjee
- Avirup Ghosh
title: 'Quasi-local conformal Killing horizons: Classical phase space and the first law'
---
Introduction
============
A black hole is described to be a region of spacetime where the gravitational attraction is high enough to prevent even light from escaping to infinity. In asymptotically flat spacetimes, the impossibility of light escaping to future null infinity form the appropriate characterization of a black hole. In other words, this region lies outside the causal past of the future null infinity $\mathscr{I}^{+}$. The boundary of such a region is called the event horizon $\mathscr{H}$ [@Hawking:1973uf; @Wald:1984rg]. To be more precise, consider a strongly asymptotically predictable spacetime ($\mathcal{M}, g_{ab}$). The spacetime is said to contain a black hole if $\cal{M}$ is not contained in $J^{-}(\mathscr{I}^{+})$. The black hole region is denoted by $\mathscr{B}=\mathcal{M}-J^{-}(\mathscr{I}^{+})$ and the event horizon is the boundary of $\mathscr{B}$ (alternatively it may also be defined as the future boundary of past of future null infinity: $\mathscr{H}=\partial[\,J^{-}(\mathscr{I}^{+})\,]$). The definition of event horizon thus requires that we are able to construct the future null infinity $\mathscr{I}^{+}$. This implies that the entire future of the spacetime needs to be known beforehand to ensure the existence of an event horizon. Indeed, the condition of strong asymptotic predictibility of spacetime signifies that we have a complete knowledge of the future evolution. From the above consideration, it is clear that $\mathscr{H}$ is a global concept and it becomes difficult to proceed much further using this definition. However, the notions simplify for stationary spacetimes which are expected states of black holes in equilibrium. In equilibrium, these spacetimes admit Killing symmetries and thus exhibit a variety of interesting features. Indeed, the strong rigidity theorem implies that the event horizon of a stationary black hole is a Killing horizon [@Hawking:1971vc]. However not all Killing horizons are event horizons. Killing horizons only require a timelike Killing vector field in the neighbourhood of the horizon whereas construction of a stationary event horizon requires a global timelike Killing vector field.
The identification of the event horizon of a stationary black hole to a Killing horizon was useful to prove the laws of mechanics for event horizons [@Bardeen:1973gs]. It was shown that in general relativity, the surface gravity $\kappa_{H}$ of a stationary black hole must be a constant over the event horizon. The first law of black hole mechanics refers to stationary space-times admitting an event horizon and small perturbations about them. This law states that the differences in mass $M$, area $A$ and angular momentum $J$ to two nearby stationary black hole solutions are related through $\delta M=\kappa_{H} \delta A/8\pi + \Omega_{H}\delta J.$ One gets additional terms like charge if matter fields are present. Hawking’s proof that due to quantum particle creation, black holes radiate to infinity, particles of all species at a temperature $\kappa_{H}/2\pi$, implied that laws of black hole mechanics are the laws of thermodynamics of black holes [@Hawking:1974sw]. Moreover, the entropy of the black holes must be proportional to it’s area [@Bekenstein:1973ur; @Bekenstein:1974ax].
However, it was realised very soon that this identification of entropy to area leads to new difficulties. Classical general relativity gives rise to infinite number of degrees of freedom but it is not clear if the laws of thermodynamics can arise out of a statistical mechanical treatment of these classical information (see [@Wald:1995yp]). One must find ways to extract quantum degrees of freedom of general relativity. The framework of Killing Horizon was broadened to understand the origin of entropy and black hole thermodynamics [@Wald:1993nt; @Iyer:1994ys; @Jacobson:1993vj; @Youm:1997hw; @Carlip:1999cy; @Dreyer:2013noa; @Ghosh:2014pha]. It turned out that the framework of Isolated Horizons (IH) was more suited to address these questions from the perspective of loop quantum gravity [@Ashtekar:1998sp; @Ashtekar:2000sz; @Ashtekar:2000hw; @Ashtekar:2001is; @Ashtekar:2001jb; @Chatterjee:2006vy; @Chatterjee:2008if]. It is argued that the effective quantum degrees of freedom which capture the thermodynamic information of black holes are localised, more precisely, reside on the horizon. Isolated horizons are suited for this description since they capture only the local information; isolated horizons are local descriptions of horizons and unlike event horizons, do not require the global history of spacetime [@Smolin:1995vq; @Krasnov:1996tb; @Rovelli:1996dv; @Ashtekar:1997yu; @Ashtekar:1999wa; @Ghosh:2006ph; @Ghosh:2008jc; @Ghosh:2011fc; @Ghosh:2013iwa]. It arises that the effective field theory induced on a IH is a Chern- Simons theory whose quantisation and counting of states is consistent with the results of Bekenstein and Hawking. Moreover, since IH replaced the global notion of event horizons with a local description, the requirement of a knowledge of full space-time history as well as the asymptotics is avoided (see [@Corichi:2013zza; @Corichi:2014zoa] for a first order description of theories with topological terms). The underlying spacetime therefore might not admit a global Killing vector at all in the isolated framework. While this has been a significant development in the understanding of black hole mechanics, generalizations to dynamically evolving horizons has also been reported [@Ashtekar:2002ag; @Ashtekar:2003hk; @Ashtekar:2004cn]. These dynamical horizons are closely related to the notion of trapping horizons developed earlier [@Hayward:1993wb; @Hayward:1994yy]. Using the boundary conditions for dynamical horizon it was shown that a flux balance law, relating the change of area of the dynamical horizon to the flux of the matter energy, exists, reproducing an integrated version of a first law [@Ashtekar:2002ag; @Ashtekar:2003hk; @Ashtekar:2004cn]. Moreoever, it has also been shown that if the horizon is slowly evolving, a form of the first law arises [@Booth:2003ji; @Booth:2006bn; @Booth:2007wu]. The construction of a phase space for these horizons has also been carried out in the metric variables.
Another class of horizons that has been of interest are conformal Killing horizons (CKH). Though not a trapping horizon it essentially captures a dynamical situation. The notion of CKH and it’s properties were developed in [@DyerHonig; @Suldyer; @Sultana:2005tp; @Jacobson:1993pf; @Nielsen:2012xu]. These are null hypersurfaces whose null geodesics are orbits of a conformal Killing field. If $\xi^{a}$ is a vector field which satisfies $\lie_{\xi}g_{ab}=2fg_{ab}$, and is null, it generates a CKH for the metric $g_{ab}$. It has been shown that an analogue of the zeroth law holds for a conformal Killing horizon as well. More precisely, since $\xi^{a}$ generates a null surface, it is geodesic and one can define an accelration through $\xi^{b}\nabla_{b}\xi^{a}=\kappa_{\xi} \xi^{a}$. Then, the quantity $(\kappa_{\xi}-2f)$ which essentially is a combination of the acceleration of the conformal Killing vector and the conformal factor, can be shown to be Lie dragged along the horizon and can therefore be interpreted as a temperature. An analogue of the first law is therefore expected to hold in this case as well but has not been established in the literature. In this paper, we address the question if a form of the first law can be established at all for a CKH. As we discuss below, if such a law exists, it may lead to some important clues for a dynamically evolving horizon.
The plan of the paper is as follows. We start by developing the geometry of a quasi-local conformal Killing horizon. We assume that a spacetime time region $\mathcal{M}$ has a null boundary $\Delta$ which however may have non- zero expansion ($\theta =-2\rho\neq 0$). In other words we take the null generators of $\Delta$ to be only shear-free. We observe that these conditions are enough to ensure that the null generators $l^{a}$ are conformal Killing vectors on $\Delta$. Now, since these null surfaces are not expansion free, they may be growing; in fact $\lie_l ~{}^{2}\epsilon=\theta~{}^{2}\epsilon$ and hence, are good candidates for growing horizons. The situation in some sense mimics what one has at null infinity in an asymptotically flat space-time. However, we are more interested in an inner horizon. The physical situation for these horizons can be visualised as follows. Suppose matter falls in through a horizon as a result of which it grows (supposing that matter satisfies standard energy conditions) and hence has a non- zero positive expansion. When this matter flux stops to fall in through the horizon, by the Raychaudhuri equation, an initially positively expanding horizon will slow down it’s expansion and after some time reach the state of equilibrium. This equilibrium state has zero expansion and it’s geometrical set- up has been developed through the Isolated horizon formulation. We are interested to construct the space of solutions of only those dynamically evolving horizons which can be generated by a conformal Killing vector field. By construction, the CKH admit a limit to the IH formulation. We suppose that the matter flux across $\Delta$ be a real scalar field ($\varphi$) satisfying the condition $\lie_l\varphi =-2\rho\varphi$ on the horizon. The geometrical conditions ensures that a form of zeroth law exists. In the next section, we show that the action for general relativity admits a well defined variational principle in presence of the conformal Killing horizon boundary and proceed to construct the symplectic structure. An interesting outcome is the construction of the phase space, identification of a boundary symplectic structure and the existence of a first law. Further, it arises that gravity and matter together gives a well defined phase- space provided a balance condition holds. This balance condition turns out to be nothing but Einstein’s equation contracted with the null generators $l^a$ (say). We thus get a quasi-local analogue of a conformal Killing horizon.
Geometrical setting and boundary conditions
===========================================
In this section, we introduce the minimal set of boundary conditions which are suitable for a quasilocal conformal Killing horizon. We assume that all fields under consideration are smooth. Let $\mathcal{M}$ be a $4$- manifold equipped with a metric $g_{ab}$ of signature $(-,+,+,+)$. Consider a null hypersurface $\Delta$ of $\mathcal M$ with $l^a$ being it’s future directed null normal. Given this null normal $l^{a}$, one can introduce another future directed null vector field $n^{a}$ which is transverse to $\Delta$. Further, one has a set of complex null vector field $(m,\,\bar{m})$, which are tangential to $\Delta$. This null tetrad $(l,\, n\, m\, \bar{m})$ constitutes the Newman- Penrose basis. The vector fields satisfy the condition that $l.n=-1=-m.\bar{m}$, while all other scalar products vanish. Let $q_{ab}$ be the degenerate metric on the hypersurface. The expansion $\theta_{l}$ of the null normal is given by $q^{ab}\nabla_{a} l_b$. In terms of the Newman- Penrose co-effecients, $\theta_l=-2\rho$ (see appendix A and [@Chandrasekhar:1985kt] for details). The accelaration of $l^a$ follows from the expression $l^a\nabla_a~l_b=(\epsilon+\bar\epsilon)l_b$ and is given by $\kappa_l :=\epsilon+\bar\epsilon$. To avoid cumbersome notation, we will do away with the subscripts $(l)$ from now on if no confusion arises. It would be useful to define an equivalance class of null normals $[l^{a}]$ such that two null normals $l$ and $l'$ will be said to belong to the same equivalance class if $l'=cl$ where $c$ is a constant on $\Delta$.\
Quasi-local conformal horizon {#quasi-local-conformal-horizon .unnumbered}
-----------------------------
[*[Definition]{}*]{}: A null hypersurface $\Delta$ of $\mathcal{M}$ will be called quasi-local conformal horizon if the following conditions hold.:
1. $\Delta$ is topologically $S^2\times R$ and null.
2. The shear $\sigma$ of $l$ vanishes on $\Delta$ for any null normal $l$.
3. All equations of motion hold at $\Delta$ and the stress- energy tensor $T_{ab}$ on $\Delta$ is such that $-T^{a}{}_{b}\,l^b$ is future directed and causal.
4. If $\varphi$ is a matter field then it must satisfy $\lie_l\varphi=-2\rho\,\varphi$ on $\Delta$ for all null normals $l$.
5. The quantity $\left[2\rho+\epsilon + \bar{\epsilon}\right]$ is Lie dragged for any null-normal $l$.
Some comments on the boundary conditions are in order. The first condition imposes restrictions on the topology of the hypersurface. It is natural to motivate this condition from Hawking’s theorem on the topology of black holes in asymptotically flat stationary spacetimes or it’s extension [@Hawking:1971vc; @Galloway:2005mf]. But, we are also interested in spacetimes which are aymptotically non- flat or that they are non- stationary for which, these theorems may not hold true. However it is not unnatural to argue that since black hole horizons forming out of gravitational collapse have spherical topologies, such conditions might exist. This condition is also assumed in the Isolated Horizon formalism. For these isolated hypersurfaces, the expansion $\theta$ of the null normal $l^{a}$ vanishes (which is not true in our case). It is possible that cross- sections of such quasilocal horizons may admit other topologies. For the time being, we would not include such generalities and only retain the condition that the cross- sections of the hypersurfaces are spherical.
The second boundary condition on the shear is a simplification. Shear measures the amount of gravitational flux flowing across the surface, and we put the gravity flux to be vanishing. This boundary condition on the shear $\sigma$ of null normal $l^{a}$ has several consequences. First, since $l_a$ is hypersurface orthogonal, the Frobenius theorem implies that $\rho$ is real and $\kappa=0$. Secondly, the Ricci identity can be written as D-=(+|+3-| )-(-|+|+3)+\_0, where $D=l^{a}\nabla_{a}$, $\delta=m^{a}\nabla_{a}$, $\Psi_{0}$ is one of the Weyl scalars and the other quantities are the Newman- Penrose scalars (see [@Chandrasekhar:1985kt] for details). If $\sigma\stackrel{\Delta}{=}0$, it implies $\Psi_0\stackrel{\Delta}{=}0$. Next, since $\l^{a}$ is null normal to $\Delta$, it is twist- free and a geodetic vector field. The implications of $\l^{a}$ being twist- free has already been shown above. The accelaration of $l^a$ follows from the expression $l^a\nabla_a~l^b=(\epsilon+\bar\epsilon)l^b$ and is given by $\kappa_l :=\epsilon+\bar\epsilon$. The acceleration of the null normal varies over the equivalence class $[cl]$ where $c$ is a constant on $
\Delta$ . This is only natural that the acceleration varies in the class since in the absence of the knowledge of asymptotics, the acceleration cannot be fixed.
Further, it can be seen that the null normal $l^{a}$ is such that -2 m\_[(a]{}|[m]{}\_[b)]{} which implies that $l^a$ is a conformal Killing vector on $\Delta$. Moreover, the Raychaudhuri equation implies that $R_{ab}l^{a}l^{b}\neq 0$ and hence $-R^{a}{}_{b}l^{b}$ can have components which are tangential as well as transverse to $\Delta$.
The third boundary condition only implies that the field equations of gravity be satisfied and that the matter fields be such that their energy momentum tensor satisfies some mild energy conditions. The fourth and the fifth boundary conditions are somewhat adhoc but can be motivated. Let us first look at the fourth boundary condition. We have kept open the possibility that matter fields may cross the horizon and the horizon may grow. The matter field is taken to be a massless scalar field $\varphi$ which behaves in a certain way which mimics it’s conformal nature. The fifth condition is motivated by the fact that surface gravity remains invariant under conformal transformations [@Jacobson:1993pf; @Sultana:2005tp]. It can be shown that the quantity that is constant for these horizons is $\left(2\rho+\epsilon+\bar{\epsilon}\right)$. A conformal transformation of the metric amounts to a conformal transformation of the two-metric on $\Delta$. Under a conformal transformation $g_{ab}\rightarrow \Omega^2\,g_{ab}$ and one needs a new covariant derivative operator which annihilates the conformally transformed metric. Under such a conformal transformation $l^a\rightarrow l^a,l_a\rightarrow \Omega^2 l_a,n^a\rightarrow \Omega^{-2} n^a,n_a\rightarrow n_a,m^a\rightarrow \Omega^{-1}m^a,m_a\rightarrow \Omega m_a$. The new derivative operator is such that it transforms as $$\nabla_al_b\rightarrow\Omega^2\nabla_al_b+2\Omega\partial_a\Omega~l_b-\Omega^2\left[l_c\delta^c_a\partial_b\log{\Omega}+l_c
\delta^c_b\partial_a\log{\Omega}-g_{ab}g^{cd}l_c\partial_d\log{\Omega}\right]$$ If one defines a one- form $\omega_a\=-n^b\underleftarrow{\nabla_a}l_b$, it transforms under the conformal transformation as $$\tilde{\omega}_a\=\omega_a+2\partial_a\log{\Omega}-\partial_a\log{\Omega}-n_al^c\partial_c\log{\Omega}$$ It follows that the Newman- Penrose scalars transform in the following way $$\begin{aligned}
\widetilde{\left(\epsilon+\bar\epsilon\right)}&\=&(\epsilon+\bar\epsilon)
+2\lie_l\log{\Omega}\\
\tilde{\rho}&\=&\rho-\lie_l\log{\Omega}\\
\tilde{\sigma}&\=&\sigma\end{aligned}$$ where, $\rho=-m^a\bar{m}^b\nabla_al_b$ and $\sigma=-\bar{m}^a\bar{m}^b\nabla_al_b$. Thus it follows that $2\rho + \epsilon+\bar\epsilon$ remains invariant under a conformal transformation.
At this point, it would be useful to recall the boundary conditions of a weakly isolated horizon and note the important differences. A weakly isolated horizon is a null hypersurface which satisfies the first and the third boundary conditions given here and that the expansion of the null normal $l^{a}$ be zero. On such surface, there exists a one- form $\omega_{a}$ which is also assumed to be Lie dragged by the vector field $l^{a}$. Thus, instead of the condition on shear, for a WEH, the expansion of the null normal $l^{a}$ is taken to be vanishing, $\theta=0=2\rho$. By the Raychaudhuri equation, the boundary conditions imply that the shear is zero and that no matter field crosses the horizon (and hence the name isolated). However, here, we impose only the condition that the shear vanishes and keep the possibility that matter fields may fall through the surface (but no gravitational flux) and that the hypersurface may grow along the affine parameter. As we shall show, removing our last condition does not restrict one to define a well defined phase space, but is essential to get a first law. It is an analogue of the condition $\lie_l(\epsilon+\bar\epsilon)\stackrel{\Delta}{=}0$ assumed in the case of weakly isolated horizon. It may be useful to note that the fifth boundary condition as given above, can be recast is a form which is an analogue of that for a weakly isolated horizon by setting $\lie_l\tilde{\omega}=0$, where $\tilde{\omega}_a\=\omega_a+\partial_a\log{\Omega}-n_al^c\partial_c\log{\Omega}$ and the conformal factor is set such that $\lie_l\log{\Omega}=\rho$.\
Action principle and the classical phase space
==============================================
We are interested in constructing the space of solutions of general relativity, and we use the first order formalism in terms of tetrads and connections. This formalism is naturally adapted to the nature of the problem in the sense that the boundary conditions are easier to implement. Moreover it has the advantage that the construction of the covariant phase- space becomes simpler. For the first order theory, we take the fields on the manifold to be ($e_{a}{}^{I},\, A_{aI}{}^{J},\, \varphi$), where $e_{a}{}^{I}$ is the co- tetrad, $A_{aI}{}^{J}$ is the gravitational connection and $\varphi$ is the scalar field. The Palatini action in first order gravity with a scalar field is given by: $$\label{lagrangian1}
S_{G+M}=-\frac{1}{16\pi G}\int_{\mathcal{M}}\left(\Sigma^{I\!J}\wedge
F_{I\!J}\right)-\frac{1}{2}\int_{\mathcal{M}}d\varphi\wedge {}{\star}
d\varphi\;$$ where $\Sigma^{IJ}=\half\,\epsilon^{IJ}{}_{KL}e^K\wedge e^L$, $A_{IJ}$ is a Lorentz $SO(3,1)$ connection and $F_{IJ}$ is a curvature two-form corresponding to the connection given by $F_{IJ}=dA_{IJ}+A_{IK}\wedge A^{K}~_{J}$. The action might have to be supplemented with boundary terms to make the variation well defined.
Variation of the action {#variation-of-the-action .unnumbered}
-----------------------
For the variational principle, we consider the spacetime to be bounded by a null surface $\Delta$, two Cauchy surfaces $M_{+}$ and $M_{-}$ which extend to the asymptotic infinity. The boundary conditions on the fields are the following. At the asymptotic infinity, the fields satisfy appropriate boundary conditions. The fields on the hypersurfaces $M_{+}$ and $M_{-}$ are fixed so that their variations vanish. On the surface $\Delta$, we fix a set of internal null- tetrad $(l^{I}, n^{I}, m^{I}, \bar{m}^{I})$ such that the flat connection annihilates them. The fields on the manifold ($e_{a}{}^{I},\, A_{aI}{}^{J},\, \varphi$), must satisfy the following conditions. First, on $\Delta$, the configurations of the tetrads be such that $l^{a}=e_{I}^{a}l^{I}$ are the null vectors which satisfy the boundary conditions for quasi- local conformal horizon. Second, the possible connnections also satisfy the boundary conditions and be such that $\left(2\rho+\epsilon + \bar{\epsilon}\right)$ is constant. Thirdly, we consider all those configurations of scalar field which, on $\Delta$, satisfy $\lie_l\,\varphi=-2\rho\,\varphi$.
We now check that the variational principle is well- defined if the boundary conditions on the fields, as given above, hold. However, we need some expressions for tetrads and connections on $\Delta$, details of which are given in the appendix A. On the conformal horizon, the $\Sigma^{IJ}$ is given by $$\underleftarrow{\Sigma}^{IJ}\=2l^{[I}n^{J]}~^{2}\epsilon+2n\wedge(im~l^{[I}\bar{
m}^{J]}-i\bar{m}~l^{[I}m^{J]}),$$ and the connection is given by $$\begin{aligned}
\label{connection_delta}
\underleftarrow{A_{a}{}_{IJ}}&\stackrel{\Delta}{=}&
2\left[(\epsilon+\bar{\epsilon})n_a
-(\bar{\alpha}+\beta)\bar{m}_a-(\alpha+\bar{\beta})m_a\right]\,
l_{[I}n_{J]}+2(-\bar{\kappa}n_a
+\bar{\rho}\bar{m}_a)\, m_{[I}n_{J]}+2(-{\kappa}n_a
+{\rho}{m}_a)\,\bar{m}_{[I}n_{J]}\nn
&+& 2(\pi n_a+-\mu\bar{m}_a-\lambda m_a)\, m_{[I}l_{J]}+2(\bar{\pi} n_a
-\bar{\mu}{m}_a-\bar{\lambda}\bar{m}_a)\,\bar{m}_{[I}l_{J]}\nn
&+& 2\left[-(\epsilon-\bar{\epsilon})n_a +(\alpha-\bar{\beta}) m_a+(\beta-\bar{\alpha})\bar{m}_a
\right]\, m_{[I}\bar{m}_{J]}..\end{aligned}$$ The Lagrangian $4$- form for the fields ($e_{a}{}^{I},\, A_{aI}{}^{J},\,
\varphi$) is given in the following way. $$L_{G+M}=-\frac{1}{16\pi G}\left(\Sigma^{I\!J}\wedge
F_{I\!J}\right)-\frac{1}{2}d\varphi\wedge \star d\varphi .$$ The first variation of the action leads to equations of motion and boundary terms. The equations of motion consist of the following equations. First, variation of the action with respect to the connection implies that the curvature $F^{IJ}$ is related to the Riemann tensor $R^{cd}$, through the relation $F_{ab}{}^{IJ}=R_{ab}{}^{cd}\,e^{I}_{c}e^{J}_{d}$. Second, variation with respect to the tetrads lead to the Einstein equations and third, the first variation of the matter field gives the equation of motion of the matter field. On- shell, the first variation is given by the following boundary terms $$\delta L_{G+M} := d\Theta(\delta)=-\frac{1}{16\pi
G}d\left(\Sigma^{IJ}\wedge\delta A_{IJ}\right)-d(\delta\varphi\star d\varphi),$$ which are to be evaluated on the boundaries $M_{-}$, $M_{+}$, asymptotic infinity and $\Delta$. However, since fields are set fixed on the initial and the final hypersurfaces they vanish. The boundary conditions at infinity are assumed to be appropriately chosen and they can be suitably taken care of. The only terms which are of relevance for this case are the terms on the internal boundary. On the internal boundary $\Delta$, the boundary terms give (see appendix \[appb\] for details) 16G L\_[G+M]{}=-( n)\^2-(2 n\^2) + 8G( n) \^2Since Einstein’s equations give $R_{11}=8\pi G \, T_{11}$, the first and the third term cancel and only $\left(2\rho\, n\wedge\,^2\epsilon\right)$ remains. Thus, if one adds the term $16\pi G\,S^{'}= \int_{\Delta}\left(2\rho\, n\wedge\,^2\epsilon\right)$ to the action, it is well defined for the set of boundary conditions on $\Delta$. As we shall see below, since this is a boundary term, it does not contribute to the symplectic structure.
Covariant phase- space and the symplectic Structure {#covariant-phase--space-and-the-symplectic-structure .unnumbered}
---------------------------------------------------
For a general Lagrangian, the on-shell variation gives $\delta
L=d\Theta(\delta)$ where $\Theta$ is called the symplectic potential. It is a $3$-form in space-time and a $0$-form in phase space. Given the symplectic potential, one can construct the symplectic structure $\Omega (\delta_{1}, \,\delta_{2})$ on the space of solutions. One first constructs the symplectic current $J(\delta_1,\delta_2)=
\delta_1\Theta(\delta_2)-\delta_2\Theta(\delta_1)$, which, by definition, is closed on-shell. The symplectic structure is then defined to be: $$\Omega(\delta_1,\delta_2)=\int_{M}J(\delta_1,\delta_2)
$$ where $M$ is a space-like hypersurface. It follows that $dJ=0$ provided the equations of motion and linearized equations of motion hold. This implies that when integrated over a closed region of spacetime bounded by $M_+\cup M_-\cup
\Delta$ (where $\Delta$ is the inner boundary considered), $$\int_{M_+}J-\int_{M_-}J~+~\int_{\Delta}J=0,$$ where $M_+,M_-$ are the initial and the final space-like slices, respectively. If the third term vanishes then the bulk symplectic structure is independent of choice of hypersurface. However, if it does not vanish but turns out to be exact, $\int_{\Delta}J=\int_{\Delta}dj $ then the hypersurface independent symplectic structure is given by: $$\Omega(\delta_{1}, \,\delta_{2})= \int_MJ-\int_{S_\Delta}j$$ where $S_\Delta$ is the 2-surface at the intersection of the hypersurface $M$ with the boundary $\Delta$. The quantity $j(\delta_1,\delta_2)$ is called the boundary symplectic current and symplectic structure is also independent of the choice of hypersurface.
Our strategy shall be to construct the symplectic structure for the action given in eqn. . Let us first look at the Lagrangian for gravity. The symplectic potential in this case is given by, $16\pi G\Theta(\delta)=-\Sigma^{I\!J}\wedge \delta A_{I\!J}$. The symplectic current is therefore given by, $$\label{symplectic_current1}
J_G(\delta_1,\delta_2)=-\frac{1}{8\pi
G}\,\delta_{[1}\Sigma^{IJ}\wedge~\delta_{2]}A_{IJ}$$ The above expression eqn. , when pulled back and rescticted to the surface $\Delta$ gives \[symplec\_pulled\_back\] &&-2\_[\[1]{} \^2[****]{} \_[2\]]{}{(+|)n-(+| )m-(|+)|[m]{}}&&+2\_[\[1]{}(nim)\_[2\]]{}(||[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m) It can be shown that the symplectic current pulled back on to $\Delta$ for the gravity sector is given by (see the appendix for details)[^1]\
(\_1,\_2)&&- The first term in the above expression is exact but not others. Therefore the phase is well defined for our boundary conditions $\sigma\stackrel{\Delta}{=}0$ provided, if either $\Phi_{00}=0$, there is no matter flux across the horizon or if $\Phi_{00}/\rho$ gets cancelled with a contribution from the matter degrees of freedom through Einstein’s equation. We deal with a more general case. We show that the contribution of the scalar field is such that the symplectic current on $\Delta$ is again exact.
The symplectic current for the real scalar field is given by, $J_M(\delta_1,\delta_2)=2\,\delta_{[1}\varphi~\delta_{2]}\,{}\star d\varphi$. The symplectic current on the hypersurface $\Delta$ can be obtained as $$\underleftarrow{J_M}(\delta_1,\delta_2)=2\delta_{[1}\varphi~\delta_{2]}
(D\varphi~ n\wedge im\wedge\bar{m}),$$ where $D=l^{a}\nabla_{a}$. The boundary condition on the scalar field implies $D\varphi=-2\rho \,\varphi$ and hence, we get that $$\begin{aligned}
\underleftarrow{J_M}(\delta_1,\delta_2)&=&4\delta_{[1}\varphi~\delta_{2]}
(-\varphi\,\rho~n\wedge im\wedge\bar{m})\\
&=&-d\left\{\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\right\}+\delta_{[1}\frac
{D\varphi D\varphi}{\rho}n\wedge~\delta_{2]}
~^2\epsilon\nn
&=&-d\left\{\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\right\}+\delta_{
[1}~^2\epsilon\wedge~\delta_{2]}
\left(\frac{{\bf T}_{11}}{\rho}n\right)\end{aligned}$$ The combined expression is then given by: $$\underleftarrow{J_{M+G}}(\delta_1,\delta_2) \stackrel{\Delta}{=}-\frac{1}{4\pi
G}\left\{d\left(\delta_{[1}~^2{\bf\epsilon}~
\delta_{2]}\log{\rho}\right)\right\}-d\,\left\{\delta_{[1}\varphi^2~\delta_{2]}
~^2\epsilon\right\}$$ It follows that the hypersurface independent symplectic structure is given by: $$\begin{aligned}
\Omega(\delta_{1},
\delta_{2})=\int_{\mathcal{M}}J_{M+G}(\delta_1,\delta_2)-\int_{S_\Delta}
j&=&-\frac{1}{8\pi G}
\int_{\mathcal{M}}\delta_{[1}\Sigma^{IJ}\wedge~\delta_{2]}A_{IJ}+2\int_{\mathcal
{M}}\delta_{[1}\varphi~\delta_{2]}
(\star d\varphi)\nn
&+&\frac{1}{4\pi
G}\int_{S_{\Delta}}\left\{\delta_{[1}~^2{\bf\epsilon}~\delta_{2]}\log{\rho}
\right\}
+\int_{S_{\Delta}}\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\end{aligned}$$ In the next section, we shall use this expression to derive the first law of mechanics for the conformal Killing horizon.
Hamiltonian evolution and the first law {#hamiltonian-evolution-and-the-first-law .unnumbered}
---------------------------------------
Given the symplectic structure, we can proceed to study the evolution of the system. We assume that there exists a vector which gives the time evolution on the spacetime. Given this vector field, one can define a corresponding vector field on the phase- space which can be interpreted as the infinitesimal generator of time evolution in the covariant phase- space. The Hamiltonian $H_l$ generating the time evolution is obtained as $\delta\, \tilde{H}_{l}= \Omega(\delta, \delta_{l})$, for all vector fields $\delta$ on the phase- space. Using the Einstein equations, we get that $$\begin{aligned}
\label{firstlaw2}
\Omega(\delta,\delta_l)&=&-\frac{1}{16\pi
G}\int_{S_{\Delta}}\left[(l.A_{IJ})\delta\Sigma^{IJ}-(l.\Sigma^{IJ})\wedge
\delta A_{IJ}\right]
+\int_{S_{\Delta}}\delta\varphi~(l\cdotp{}{\star} d\varphi)\nn
\nn
&&\hspace{1cm}+\frac{1}{8\pi
G}\int_{S_{\Delta}}\left(\delta~^2{\bf\epsilon}~\delta_{l}\log{\rho}-\delta_l~^2
{\bf\epsilon}
~\delta\log{\rho}\right)+\int_{S_{\Delta}}\frac{1}{2}(\delta\varphi^2~\delta_{l}
~^2\epsilon-\delta_l\varphi^2\delta~^2\epsilon)\nn\end{aligned}$$ We now need to impose a few conditions on the fields to make a well defined Hamiltonian. These conditions are to be imposed since the action of $\delta_{l}$ on some phase- space fields is not like $\lie_{l}$. This is because of $\rho, \epsilon+\bar\epsilon$ and $\varphi$ all cannot be free data on $\Delta$. First, we note the following equalities $$\begin{aligned}
\lie_{l}\left(\frac{1}{4\pi G}\log{\rho}-\frac{1}{8\pi
G}\log{\varphi}-\varphi^2\right)&=& \frac{1}{4\pi
G}(2\rho+\epsilon+\bar{\epsilon})\\
\lie_{l}\left(\frac{~^2\epsilon}{\varphi}\right)&=&0\end{aligned}$$ We assume that $\delta_l$ acts on $(2\rho+\epsilon+\bar{\epsilon})$ and $\left(\frac{~^2\epsilon}{\varphi}\right)$ like $\lie_l$. This can also be argued in the following fashion. Since $\delta_{l}\lie_{l}(2\rho+\epsilon+\bar{\epsilon})=0$ it immediately implies that $\lie_{l}\delta_{l}(2\rho+\epsilon+\bar{\epsilon})=0$. Hence, choosing $\delta_{l}(2\rho+\epsilon+\bar{\epsilon})=0$ at the initial cross-section implies that it remains zero throughout $\Delta$. Furthermore if we set $\delta_l\left(\frac{1}{4\pi G}\log{\rho}-\frac{1}{8\pi
G}\log{\varphi}-\varphi^2\right)$=0 at the initial cross-section, it remains zero everywhere on $\Delta$ and so, $$\begin{aligned}
\label{eqn_no1}
\frac{\delta_l\rho}{\rho}-8\pi
G\varphi\delta_l\varphi-\frac{\delta_l\varphi}{2\varphi}=0\end{aligned}$$ Another condition can be derived from the equation above $$\begin{aligned}
\delta_l\left(\frac{~^2\epsilon}{\varphi}\right)=\frac{1}{\varphi}
\delta_l~^2\epsilon-~^2\epsilon\frac{1}{\varphi^2}\delta_l\varphi=0\end{aligned}$$ The variations $\delta_l$ satisfy the following differential equations, which can be checked to be consistent with each other: $$\begin{aligned}
\label{eqn_no2}
\lie_l\delta_l\varphi&=&-2\delta_l\rho\varphi-2\rho\delta_l\varphi\\
\lie_l\delta_l~^2\epsilon&=&-2\delta_l\rho~^2\epsilon-2\rho\delta_l~^2\epsilon\end{aligned}$$ Putting condition $\eqref{eqn_no1}$ in $\eqref{eqn_no2}$, we get $$\delta_l\varphi=C(\theta,\phi)\exp\left[-{\int\left(16\pi
G\varphi^2+3\right)\rho dv}\right],$$ where $C(\theta,\phi)$, is a constant of integration. If we choose this constant $C(\theta,\phi)=0$, it immediately implies that $\delta_{l}\varphi=0=\delta_{l}{}^{2}\epsilon.$ With the choice of $\delta_l$ only the bulk symplectic structure survives and one gets from eq. $\eqref{firstlaw2}$ [^2] $$\begin{aligned}
\delta H_{l}&=&-\frac{1}{8\pi
G}\int_{S_\Delta}(2\rho+\epsilon+\bar{\epsilon})\delta~^2\epsilon+
\frac{1}{8\pi G}\int_{S_\Delta}~^2\epsilon~(-\delta\rho-8\pi G\,\delta\varphi
D\varphi)+\delta E^\infty\end{aligned}$$ where we have redefined our Hamiltonian $H_l=\tilde{H}_l+\int_{S_\Delta}\rho~^2\epsilon$. This redefination is possible since the definition of the Hamiltonian is ambiguous upto a total variation. Further, as expected $\Omega(\delta_{l},\delta_l)=0$. Next we define, $E^l_{\Delta}=E^\infty-H_l,$ as the horizon energy. It is clear from above that for $\rho\rightarrow 0$ (i.e in the isolated horizon limit) it matches with the definition in [@Ashtekar:2002ag; @Ashtekar:2003hk] if asymptotics is flat and $E^\infty=E_{ADM}$. It therefore follows that: -E\^l\_=-\_[S\_]{}(2++ |) \^2-\_[S\_]{}.
To recover the the more familiar form of first law known for a dynamical situation, we assume there is a vector field $\tilde{\delta}$ on phase space which acts only on the fields on $\Delta$ (and not in the bulk) such that it’s action on the boundary variables is to evolve the boundary fields along the affine parameter $v$ (it may be interpreted to be a time evolution, like $\lie$). Now demanding that $\tilde{\delta}$ to be Hamiltonian would give an integrability condition which also ensures that $\delta_l$ is Hamiltonian. So one can calculate $\Omega(\tilde{\delta},\delta_l):=\tilde{\delta}H_{l}$ which can be written in the following form[^3] $$\label{firstlaw}
\dot{E^l_\Delta}=\,\frac{1}{8\pi G}\left(2\rho+\epsilon+\bar{\epsilon}\right)\dot{A}+\frac{1}{8\pi
G}\int_{S_\Delta}\left[~^2\epsilon~(\dot\rho+8\pi
G\,\dot\varphi D\varphi)\right]$$ where dots imply changes in the variables produced by the action of $\tilde{\delta}$. Note that if $\tilde{\delta} =\lie_{l}$, then, $\tilde{\delta}\varphi D\varphi$ gives the expression $T_{ab}l^{a}l^{b}$. Equation is the form of evolution for the conformal Killing horizons. The first term in the above expression is the usual $TdS$ term while the second term is a flux term which takes into account the non-zero matter flux across $\Delta$.
Discussions
===========
In this paper, we have developed the geometrical set-up for a quasi-local description of a conformal Killing horizon. Further, we have also shown that one can understand these horizons to have a zeroth law (as was also discussed in [@Suldyer]) and a first law. This development of a notion of quasi-local conformal horizon should be taken in the same spirit as the development of the notion of isolated horizon from Killing horizons. A conformal Killing horizon is one which has a conformal Killing vector in the neighbourhood of the horizon. In contrast, a quasi-local conformal horizon only requires the existence of a null hypersurface generating vector which is shear free on the null hypersurface. The number of solutions of Einsteins’s equation for gravity and matter that admits a conformal Killing horizon may be small (examples of such kind has been constructed by [@Sultana:2005tp]). However the solutions admitting a quasi-local conformal horizon may be large. We do not comment on the nature of solutions that admits a quasi-local conformal horizon, we think that significant amount of insights may be obtained by numerical simulations and therefore falls in the regime of numerical relativity. The most useful application of these geometrical structures are in the dynamical evolution of black holes. Indeed, as matter falls in through the horizon and the black hole horizon grows, the expansion is non- zero. In such cases, it is important to understand if in this dynamical situation one can prove the existence of laws for black hole mechanics in some form.
We have taken a real scalar field as the matter field in question. The flux balance law is seen to be successfully implemented if it satisfies a condition $\lie_l\varphi\stackrel{\Delta}{=}-2\rho\,\varphi$. This assumption is motivated through the fact that $l^a$ is a conformal Killing vector on $\Delta$. Taking other matter fields will therefore be an immediate extension of our work. Further, from the onset we have ignored any space-like axial conformal Killing vector on $S_{\Delta}$. So a generalization to the rotational case seems to be another plausible extension. Since the case of an isolated horizon appears as a special case $\rho\rightarrow0$, the consistency of our analysis can actually be checked by taking the isolated horizon limit. In fact we perform this consistency check and find that the final expressions and the first law does give back the results obtained for an isolated horizon.
We should mention at this point that our construction does not capture the most general dynamical situation, as constructed in [@Hayward:1993wb; @Ashtekar:2002ag]. The horizons discussed in these references are spacelike boundaries foliated by partially trapped two surfaces which may not be shear-free. Further, an integrated version of the first law has been demonstrated to exist, which captures the dynamics of growing black hole horizons in full generality. However in these constructions, which use metric variables, the existence of a well defined phase- space has not been established and consequently the first law does not follow directly from the symplectic structure. In our case we have assumed that there is no gravitational flux (shear is zero) but only matter flows across the null boundary $\Delta$. In this simplified geometry, we have demonstrated that a space of solutions of Einstein’s equations exists which admit the boundary conditions of CKH and that a differential version of the first law of black hole mechanics can be obtained. Also, we have used the first order formalism for the construction of this symplectic structure. We do not know if one may get a well defined symplectic structure for boundary conditions discussed in [@Hayward:1993wb; @Ashtekar:2002ag]. Even if one is able to construct a phase- space, it is not possible to obtain a differential version the first law since there is no analogue of the zeroth law for such boundaries, but an integrated version of the first law is expected to hold.
Given a form of the first law, it is obvious to compare with the first law of thermodynamics. However, since the horizon is growing, it describes a non- equilibrium situation and hence may differ considerably from equilibrium thermodynamics where one studies the transition from one equilibrium state to a nearby equilibrium state. One should keep in view that thermodynamics arises out of microscopic dynamics of the underlying degrees of freedom and have universal validity (that are independent of the underlying dynamics of a particular system). For a general dynamical spacetime (when the gravitational degrees of freedom are excited), there is no time translation symmetry and hence no definition of entropy may be possible. Also in non- equilibrium cases, a system may not get enough time to relax back to the equilibrium state and hence no canonical definition of temperature exists. But, in the present scenario, though the horizon makes transition between two states which are far from equilibrium, because there exits a conformal Killing vector, this leads to a definite identification of temperature and a first law and possibly entropy. One may then enquire if dynamically growing horizons is attributed some entropy that can arise from some counting of microstates. The boundary symplectic structure has a natural interpretation of being the symplectic structure of a field theory residing on the boundary. In the case of an isolated horizon it turns out to be an $SU(2)$ or an $U(1)$ Chern-Simons theory. A quantization of the boundary theory therefore provides a microscopic description of the entropy of the isolated horizon. Since we explicitly construct the boundary symplectic structure it will be interesting to see if it does coincide with any known topological field theory. A complete answer to such questions shall have important implications for thermodynamics as well as black hole physics.
The Connection in terms of Newman-Penrose co-effecients
=======================================================
Fix a set a internal null vectors $(l_I,n_I,m_I,\bar m_I) $ on $\Delta$ such that $\partial_a (l_I,n_I,m_I,\bar m_I)\stackrel{\Delta}{=}0$. Given any tetrad $e^I_a$, the null tetrad $(l_a,n_a,m_a,\bar m_a)$ can be expanded as $l_a=e^I_a~l_I$. The expression for $\Sigma^{IJ}$ can now be readily calculated and is given as. \^[IJ]{}&=&2l\^[\[I]{}n\^[J\]]{} \^[2]{}+2n(im l\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} l\^[\[I]{}m\^[J\]]{})&-&2i l n m\^[\[I]{}|[m]{}\^[J\]]{}-2l(im n\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} n\^[\[I]{}m\^[J\]]{})
This is the full expression for $\Sigma^{IJ}$ where nothing has been been assumed regarding the nature of the boundary $\Delta$. If $\Delta$ is a null surface and $l_{a}$ is the null normal, we get that \^[IJ]{}&&2l\^[\[I]{}n\^[J\]]{} \^[2]{}+2n(im l\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} l\^[\[I]{}m\^[J\]]{})
The covariant derivative is defined to be compatible with the tetrad *i.e.* $\nabla_b~e^I_a=0$. The covariant derivatives on the null tetrads can be written in terms of the Newman-Penrose coeffecients and are given by the following,
\_al\_b&=&-(+|)n\_al\_b+|n\_am\_b+n\_a|[m]{}\_b-(+|)l\_al\_b+|l\_am\_b+l\_a|[m]{}\_b&&+\[(|+)|[m]{}\_al\_b -||[m]{}\_am\_b-|[m]{}\_a|[m]{}\_b+(+| )m\_al\_b-m\_a|[m]{}\_b-|m\_a m\_b\]\
\_a n\_b&=&(+|)n\_an\_b- n\_am\_b-|n\_a|[m]{}\_b+(+|)l\_an\_b-l\_am\_b-| l\_a|[m]{}\_b&&-\[(|+)|[m]{}\_an\_b-|[m]{}\_am\_b-||[m]{}\_a|[ m]{}\_b+(+|)m\_an\_b-| m\_a|[m]{}\_b-m\_a m\_b\]\
\_am\_b&=&-|n\_al\_b+n\_an\_b-(-|)n\_am\_b-|l\_al\_b+ l\_an\_b-(-|)l\_am\_b&&+\[||[m]{}\_al\_b-|[m]{}\_an\_b+(-|)|[m]{} \_am\_b+| m\_al\_b-m\_an\_b+(-|)m\_a[m]{}\_b\]
Next, once we have fixed a set of null internal vectors on $\Delta$, the connection can be expanded in terms of these Newman- Penrose coefficients. Note that $\nabla_a~l_I=\partial_a~l_I+A_{aI}^J~l_J$. Therefore on $\Delta$, we have $e^b_I\nabla_al_b\stackrel{\Delta}{=}A_{aI}~^Jl_J$ and hence
A\^[(l)]{}\_[aI]{} \^Jl\_J&&-(+|)n\_al\_I+|n\_am\_I+n\_a|[m]{}\_I-(+|)l\_al\_I+|l\_am\_I+l\_a|[m]{}\_I&&+\[(|+)|[m]{}\_al\_I-||[m]{}\_am\_I-|[m]{}\_a|[ m]{}\_I+(+|)m\_al\_I-m\_a|[m]{}\_I-|m\_a m\_I\]\
A\^[(l)]{}\_[aIJ]{}&& 2l\_[\[I]{}n\_[J\]]{}&&+ 2m\_[\[I]{}n\_[J\]]{}+ 2|[m]{}\_[\[I]{}n\_[J\]]{}where the subscript $l$ in $A^{(l)}$ indicates that the only the vector field $l^{a}$ has been used to evaluate the connection. Similarly, we can proceed for other vector fields $n^{a}, m^{a}$ and $\bar{m}^{a}$. The resulting connections are given as follows A\^[(n)]{}\_[aIJ]{}&& 2n\_[\[I]{}l\_[J\]]{}&&+(n\_a+l\_a-|[m]{}\_a-m\_a) 2m\_[\[I]{}l\_[J\]]{}+(| n\_a+|l\_a-|[m]{}\_a-||[m]{}\_a) 2|[m]{}\_[\[I]{}l\_[J\]]{}\
A\^[(m)]{}\_[aIJ]{}&&(-|n\_a-|l\_a+||[m]{}\_a+| m\_a) 2l\_[\[I]{}|[m]{}\_[J\]]{}+(n\_a+l\_a-|[m]{}\_a-m\_a) 2n\_[\[I]{}|[m]{}\_[J\]]{}&&+ 2m\_[\[I]{}|[m]{}\_[J\]]{}
The full connection is then given by: A\_[aIJ]{}&&2 l\_[\[I]{}n\_[J\]]{}&&+2 m\_[\[I]{}n\_[J\]]{}+2 |[m]{}\_[\[I]{}n\_[J\]]{}&&+2 m\_[\[I]{}l\_[J\]]{}+2 |[m]{}\_[\[I]{}l\_[J\]]{}&&+2 m\_[\[I]{}|[m]{}\_[J\]]{}
Note that as in in the case of $\Sigma_{IJ}$ no boundary condition has been assumed in the above expression. In the main part of the paper this expression for the connection eqn shall be used but with the boundary conditions.
Further, we would be requiring the exterior derivatives on the null tetrads. We therefore give the expressions here. dn=\_[a]{}n\_[b]{} dx\^adx\^b&=&-n m-|n|[m]{}+(+|)ln-l m-| l|[m]{}&&-\[(|+)|[m]{}n-|[m]{} m+(+|)mn-| m|[m]{}\]\
dl=\_[a]{}l\_[b]{} dx\^adx\^b&=&-(+|)n l+|nm+n|[m]{}+|lm+ l|[m]{}&&+\[(|+)|[m]{}l-||[m]{} m+(+|)ml-m|[m]{}\]\
dm=\_[a]{}m\_[b]{} dx\^adx\^b&=&-|n l-(-|)nm+l n-(-|)lm&&+\[||[m]{}l-|[m]{} n+(-|)|[m]{}m+| ml-mn\]From the above expressions, it follows that for the area two- form which is given by ${}^{2}\epsilon=im\wedge\bar{m}$, we get that $d~^2\epsilon=2\rho \,n\wedge ~^2\epsilon \,\mbox{and} \, \lie_l
~^2\epsilon=-2\rho~~^2\epsilon$.
Variation of the action {#appb}
=======================
Since the boundary symplectic structure turned out to be exact, it is at once evident that the variation of the action should be well-defined with the the boundary conditions considered. However one may need to add an additional boundary term in order to it. As has been pointed out that such terms won’t affect the symplectic structure though. Therefore for completeness we consider the variation of the action and find out the necessary boundary term needed to make the variation well defined. We consider the action for gravity and a scalar field without any boundary terms a priori. The expression for $\Theta$ on $\Delta$ is calculated imposing the boundary conditions and the required boundary term can be obtained. We have L\_[M+G]{}=-(\^[IJ]{} F\_[IJ]{})-dd; It follows that d()=-d(\^[IJ]{} A\_[IJ]{})-d(d) Consider the gravity terms first[^4] &&-2 \^2+2(nim)(|[m]{})-2(n i|[m]{})()&=&-2 \^2+2(nim)(|[m]{})-2(n i|[m]{})()&=&d-4n\^2+2 \^2+4n\^2 +2 n \^2&=&d+2 \^2+(2 n \^2) The matter term gives
\(d) &=&-d(\^2 \^2)+( d) \^2&=&-d(\^2 \^2)-(n) \^2Adding everything up, one finds that, d()=-d(\^[IJ]{} A\_[IJ]{})-d(d)=-d( n \^2) So one needs to add $\frac{1}{8\pi G}\int_{\Delta}\left(\rho n\wedge~^2\epsilon\right)$ to the action to make the variation well-defined.
Boundary Symplectic Structure for Gravity
=========================================
The symplectic current in first order gravity is therefore given by, J\_G(\_1,\_2)&=&-\_[\[1]{}\^[IJ]{} \_[2\]]{}A\_[IJ]{}
We need to pull back the above expression on to the boundary and check if it is exact.
&&-2\_[\[1]{} \^2[ ****]{} \_[2\]]{}((+|)n-(+|)m-(|[ ]{}+)|[m]{})&&+2\_[\[1]{}(nim)\_[2\]]{}(||[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)
We consider the first term in the above expression. By using the Ricci identity in terms of Newman-Penrose co-effecients D=\^2+(+|)+\_[00]{} we find that the first term can be written in the following form, -2\_[\[1]{} \^2[****]{}\_[2\]]{}((+|)n)&=&-2\_[\[1]{} \^2[****]{}\_[2\]]{}(( -- )n)&&=d(2\_[\[1]{} \^2[****]{} \_[2\]]{}log )-(2\_[\[1]{} d\^2[****]{}\_[2\]]{}log)&&+2\_[\[1]{} \^2[****]{}\_[2\]]{}((+)n)
Since the first term in the above expression is already exact, we leave it for the the moment and check if there is any simplication of the other terms when combined with the rest of the third and forth term in the symplectic current.
-2\_[\[1]{} d\^2[****]{}\_[2\]]{}log &&=-4\_[\[1]{} inm|[m]{} \_[2\]]{}log&&=-2\_[\[1]{} (nim)|[m]{} \_[2\]]{}-2(n im)\_[\[1]{}|[m]{} \_[2\]]{}&&+2\_[\[1]{} (ni|[m]{}) m \_[2\]]{}+2(ni|[m]{})\_[\[1]{} m \_[2\]]{}
The third and the fourth term in the symplectic current gives: &&2\_[\[1]{}(nim)\_[2\]]{}(|[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)&&=2\_[\[1]{}(n im)|[m]{} \_[2\]]{}()+2\_[\[1]{}(n im)\_[2\]]{}|[m]{}&&-2\_[\[1]{}(ni|[m]{}) m \_[2\]]{}()-2\_[\[1]{}(ni|[m]{})\_[2\]]{}m
Adding the above two equations and then simplifying gives:
&&-2\_[\[1]{} d\^2[****]{}\_[2\]]{}log+2\_[\[1]{}(n im)\_[2\]]{}(|[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)&&=-2n\_[\[1]{} \^2\_[2\]]{}+2\_[\[1]{} (n)\_[2\]]{} \^2&&=-2\_[\[1]{} \^2\_[2\]]{}(n)
So the boundary term becomes d(2\_[\[1]{} \^2[****]{} \_[2\]]{}log)+2\_[\[1]{} \^2[ ****]{}\_[2\]]{}(n)
Bulk Symplectic structure
=========================
For any vector field $\xi$ generating diffeomorphisms, the corresponding phase space variation $\delta_\xi$ acts in the bulk like $\lie_\xi$. It can then be shown that J\_G(,\_)&=&-+()e\^I Similarly for the matter fields, we get that J\_M(,\_) &=&d--.d (d)\
The second and the third term in the last expression enters Einstein’s equation. Therefore the full bulk symplectic structure is,
\_[M]{}J(,\_)&=&-\_[ M]{}+\_[M]{} (.d)
Acknowedgements {#acknowedgements .unnumbered}
---------------
The authors acknowledge the discussions with Amit Ghosh. The authors also thank the anonymous referee for suggestions that made the presentation better. AC is partially supported through the UGC- BSR start-up grant vide their letter no. F.20-1(30)/2013(BSR)/3082. AG is supported by Department of Atomic-Energy, Govt. Of India.
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[^1]: The entire construction and whatever follows goes through for negative $\rho$ with the replacement $\lvert\rho\rvert$ in place of $\rho$ in the argument of $\log$
[^2]: We assume that the contribution from the boundary at asymptotic infinity is a total variation $\delta E^\infty$.
[^3]: If the stress tensor satisfies the dominant enegy condition then $(2\rho+\epsilon+\bar\epsilon)$ is a constant on $\Delta$ [@Sultana:2005tp].
[^4]: In our case it might not be possible to define a unique covariant derivative on $\Delta$. However, since in the the calculations $l^a\nabla_a$ acts only on functions, the amibiguity do not play a role.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We present the full public release of all data from the TNG100 and TNG300 simulations of the IllustrisTNG project. IllustrisTNG is a suite of large volume, cosmological, gravo-magnetohydrodynamical simulations run with the moving-mesh code [Arepo]{}. TNG includes a comprehensive model for galaxy formation physics, and each TNG simulation self-consistently solves for the coupled evolution of dark matter, cosmic gas, luminous stars, and supermassive blackholes from early time to the present day, $z=0$. Each of the flagship runs – TNG50, TNG100, and TNG300 – are accompanied by halo/subhalo catalogs, merger trees, lower-resolution and dark-matter only counterparts, all available with 100 snapshots. We discuss scientific and numerical cautions and caveats relevant when using TNG.
The data volume now directly accessible online is $\sim$750 TB, including 1200 full volume snapshots and $\sim$80,000 high time-resolution subbox snapshots. This will increase to $\sim$1.1 PB with the future release of TNG50. Data access and analysis examples are available in IDL, Python, and Matlab. We describe improvements and new functionality in the web-based API, including on-demand visualization and analysis of galaxies and halos, exploratory plotting of scaling relations and other relationships between galactic and halo properties, and a new JupyterLab interface. This provides an online, browser-based, near-native data analysis platform enabling user computation with local access to TNG data, alleviating the need to download large datasets.
address:
- 'Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany'
- 'Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany'
- 'Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA'
- 'Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA'
- 'Kavli Institute for Astrophysics and Space Research, Department of Physics, MIT, Cambridge, MA, 02139, USA'
- 'Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany'
- 'Zentrum für Astronomie der Universität Heidelberg, ARI, Mönchhofstr. 12-14, 69120 Heidelberg, Germany'
- 'Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Apdo. Postal 72-3, 58089 Morelia, Mexico'
- 'University of Florida, Department of Physics, 2001 Museum Rd., Gainesville, FL 32611, USA'
- 'Department of Physics and Astronomy, University of Bologna, Piero Gobetti 93/2, I-40129 Bologna, Italy'
- 'Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, IL 60208, USA'
- 'Center for Astrophysics and Cosmology, Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland'
- 'Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK'
author:
- Dylan Nelson
- Volker Springel
- Annalisa Pillepich
- 'Vicente Rodriguez-Gomez'
- Paul Torrey
- Shy Genel
- Mark Vogelsberger
- Ruediger Pakmor
- Federico Marinacci
- Rainer Weinberger
- Luke Kelley
- Mark Lovell
- Benedikt Diemer
- Lars Hernquist
bibliography:
- 'refs.bib'
nocite: '[@label]'
title: |
The IllustrisTNG Simulations:\
Public Data Release’
---
$\dagger$ Permanently available at [www.tng-project.org/data](http://www.tng-project.org/data/)
methods: data analysis ,methods: numerical ,galaxies: formation ,galaxies: evolution ,data management systems ,data access methods, distributed architectures
Main Text {#main-text .unnumbered}
=========
Introduction
============
Some of our most powerful tools for understanding the origin and evolution of large-scale cosmic structure and the galaxies which form therein are cosmological simulations. From pioneering beginnings [@ps74; @davis85], dark matter, gravity-only simulations have evolved into cosmological hydrodynamical simulations [@katz92]. These aim to self-consistently model the coupled evolution of dark matter, gas, stars, and blackholes at a minimum, and are now being extended to also include magnetic fields, radiation, cosmic rays, and other fundamental physical components. Such simulations provide foundational support in our understanding of the $\Lambda$CDM cosmological model, including the nature of both dark matter and dark energy.
Modern large-volume simulations now capture cosmological scales of tens to hundreds of comoving megaparsecs, while simultaneously resolving the internal structure of individual galaxies at $\lesssim\,$1 kpc scales. Recent examples reaching $z=0$ include Illustris [@vog14a; @genel14], EAGLE [@schaye15; @crain15], Horizon-AGN [@dubois14], Romulus [@tremmel17], Simba [@dave19], Magneticum [@dolag16], among others. These simulations produce observationally verifiable outcomes across a diverse range of astrophysical regimes, from the stellar and gaseous properties of galaxies, galaxy populations, and the supermassive blackholes they host, to the expected distribution of molecular, neutral, and ionized gas tracers across interstellar, circumgalactic, and intergalactic scales, in addition to the expected distribution of the dark matter component itself.
Complementary efforts, although not the focus of this data release, include high redshift reionization-era simulations such as BlueTides [@feng16], Sphinx [@rosdahl18], and CoDa II [@ocvirk19], among others. In addition, ‘zoom’ simulation campaigns include NIHAO [@wang15], FIRE-2 [@hopkins18], and Auriga [@grand17], in addition to many others. These have provided numerous additional insights into many questions in galaxy evolution [recent progress reviewed in @fg18]. For instance, reionization simulations may be able to include explicit radiative transfer, and zoom simulations may be able to reach higher resolutions and/or more rapidly explore model variations, in comparison to large cosmological volume simulations.
Observational efforts studying the properties of galaxies across cosmic time provide ever richer datasets. Surveys such as SDSS [@york00], CANDELS [@grogin11], 3D-HST [@brammer12], LEGA-C [@vanderwel16], SINS/zC-SINF and KMOS3D [@genzel14; @wisnioski15], KBSS [@steidel14], and MOSDEF [@kriek15] provide local and high redshift measurements of the statistical properties of galaxy populations. Complementary, spatially-resolved data has recently become available from large, $z=0$ IFU surveys such as MANGA [@bundy15], CALIFA [@sanchez12] and SAMI [@bryant15].
In order to inform theoretical models using observational constraints, as well as to interpret observational results using realistic cosmological models, public data dissemination from both observational and simulation campaigns is required. Observational data release has a successful history dating back at least to the SDSS SkyServer [@szalay00; @szalay02a], which provides tools for remote users to query and acquire large datasets [@gray02; @szalay02b]. The still-in-use approach is based on user written SQL queries, which provide search results as well as data acquisition. From the theoretical community, the public data release of the Millennium simulation [@spr05c] was the first attempt of similar scale. Modeled on the SDSS approach, data was stored in a relational database, with an immediately recognizable SQL-query interface [@lemson06]. It has since been extended to include additional simulations, including Millennium-II [@boylan09; @guo11], and a first attempt at the idea of a “virtual observatory” (VO) was realized [@overzier13]. The Theoretical Astrophysical Observatory [TAO; @bernyk14] was similarly focused around mock observations of simulated galaxy and galaxy survey data. Explorations continue on how to best deliver theoretical results within the existing VO framework [@lemson09; @lemson14].
Other dark-matter only simulations have released data with similar approaches, including Bolshoi and MultiDark [CosmoSim; @klypin11; @riebe13], DEUS [@rasera10], and MICE [Cosmohub; @crocce10]. In contrast, some recent simulation projects have made group catalogs and/or snapshots available for direct download, including MassiveBlack-II [@khandai14], the Dark Sky simulation [@skillman14], $\nu^2$GC [@makiya16], and Abacus [@lehman18]. Skies and Universes [@klypin18] organizes a number of such data releases. With respect to Illustris, the most comparable in simulation type, data complexity, and scientific scope is the recent public data release of the Eagle simulation, described in [@mcalpine15] [see also @camps18]. The initial group catalog release was modeled on the Millennium database architecture, and the raw snapshot data was also subsequently made available [@eagle17]. More recently, significant infrastructure research and development has focused on providing remote computational resources, including the NOAO Data Lab [@fitzpatrick14] and the SciServer project [@medvedev16; @raddick17]. Web-based orchestration projects also include [@ragagnin17], Tangos [@pontzen18], and Jovial [@araya18].
{width="7.0in"}
The public release of IllustrisTNG (hereafter, TNG) follows upon and further develops tools and ideas pioneered in the original Illustris data release. We offer direct online access to all snapshot, group catalog, merger tree, and supplementary data catalog files. In addition, we develop a web-based API which allows users to perform many common tasks without the need to download any full data files. These include searching over the group catalogs, extracting particle data from the snapshots, accessing individual merger trees, and requesting visualization and further data analysis functions. Extensive documentation and programmatic examples (in the IDL, Python, and Matlab languages) are provided.
This paper is intended primarily as an overview guide for TNG data users, describing updates and new features, while exhaustive documentation will be maintained online. In Section \[sSims\] we give an overview of the simulations. Section \[sDataProducts\] describes the data products, and Section \[sDataAccess\] discusses methods for data access. In Section \[sRemarks\] we present some scientific remarks and cautions, while in Section \[sCommunity\] we discuss community considerations including citation requests. Section \[sImplementation\] describes technical details related to the data release architecture, while Section \[sConclusions\] summarizes. Appendix A provides a few additional data details, while Appendix B shows several examples of how to use the API.
Description of the Simulations {#sSims}
==============================
IllustrisTNG is a suite of large volume, cosmological, gravo-magnetohydrodynamical simulations run with the moving-mesh code [<span style="font-variant:small-caps;">A</span>REPO]{} [@spr10]. The TNG project is made up of three simulation volumes: TNG50, TNG100, and TNG300. The first two simulations, TNG100 and TNG300, were recently introduced in a series of five presentation papers [@springel18; @pillepich18b; @naiman18; @nelson18a; @marinacci18], and these are here publicly released in full. The third and final simulation of the project is TNG50 [@pillepich19; @nelson19b] which will also be publicly released in the future. TNG includes a comprehensive model for galaxy formation physics, which is able to realistically follow the formation and evolution of galaxies across cosmic time [@weinberger17; @pillepich18a]. Each TNG simulation solves for the coupled evolution of dark matter, cosmic gas, luminous stars, and supermassive blackholes from a starting redshift of $z=127$ to the present day, $z=0$.
The IllustrisTNG project[^1] is the successor of the original Illustris simulation[^2] [@vog14a; @vog14b; @genel14; @sijacki15] and its associated galaxy formation model [@vog13; @torrey14]. Illustris was publicly released in its entirety roughly three and a half years ago [@nelson15b]. TNG incorporates an updated ‘next generation’ galaxy formation model which includes new physics and numerical improvements, as well as refinements to the original model. TNG newly includes a treatment of cosmic magnetism, following the amplification and dynamical impact of magnetic fields, as described below.
The three flagship runs of IllustrisTNG are each accompanied by lower-resolution and dark-matter only counterparts. Three physical simulation box sizes are employed: cubic volumes of roughly 50, 100, and 300 Mpc side length, which we refer to as TNG50, TNG100, and TNG300, respectively. The three boxes complement each other by enabling investigations of various aspects of galaxy formation. The large physical volume associated with the largest simulation box (TNG300) enables, for instance, the study of galaxy clustering, the analysis of rare and massive objects such as galaxy clusters, and provides the largest statistical galaxy sample. In contrast, the smaller physical volume simulation of TNG50 enables a mass resolution which is more than a hundred times better than in the TNG300 simulation, providing a more detailed look at, for example, the structural properties of galaxies, and small-scale gas phenomena in and around galaxies. Situated in the middle, the TNG100 simulation uses the same initial conditions (identical phases, adjusted for the updated cosmology) as the original Illustris simulation. This facilitates robust comparisons between the original Illustris results and the updated TNG model. For many galaxy evolution analyses, TNG100 provides an ideal balance of volume and resolution, particularly for intermediate mass halos. Despite these strengths, each volume still has intrinsic physical and numerical limitations – for instance, TNG300 is still small compared to the scale of the BAO for precision cosmology, and lacks statistics for the most massive halos at $\sim 10^{15}$ M$_\odot$, while TNG50 is still too low-resolution to resolve ultra-faint dwarf galaxies with $M_\star \lesssim 10^5$ M$_\odot$, globular clusters, or small-scale galactic features such as nuclear star clusters. We provide an overview and comparison between the specifications of all the TNG runs in .
![ Spatial resolution of the three high-resolution TNG simulations at $z\sim0$. The dark regions of the distributions highlight star-forming gas inside galaxies, the corresponding median values marked by dark vertical dotted lines. \[fig\_res\]](figures/TNGSeries_GasCellSizes_099_v3c.pdf){width="3.35in"}
This data release includes the TNG100 and TNG300 simulations in full. It will, in the future, also include the final TNG50 simulation. For each, snapshots at all 100 available redshifts, halo and subhalo catalogs at each snapshot, and two distinct merger trees are released. This includes three resolution levels of the 100 and 300 Mpc volumes, and four resolution levels of the 50 Mpc volume, decreasing in steps of eight in mass resolution (two in spatial resolution) across levels. The highest resolution realizations, TNG50-1, TNG100-1 and TNG300-1, include $2 \times 2160^3$, $2 \times 1820^3$ and $2 \times 2500^3$ resolution elements, respectively (see Table \[table\_sims\]). As the actual spatial resolution of cosmological hydrodynamical simulations is highly adaptive, it is poorly captured by a single number. Figure \[fig\_res\] therefore shows the distribution of Voronoi gas cell sizes in these three simulations, highlighting the high spatial resolution in star-forming gas – i.e., within galaxies themselves. In contrast, the largest gas cells occur in the low-density intergalactic medium.
All ten of the baryonic runs invoke, without modification and invariant across box and resolution, the fiducial “full” galaxy formation physics model of TNG. All ten runs are accompanied by matched, dark matter only (i.e. gravity-only) analogs. In addition, there are multiple, high time-resolution “subboxes”, with up to 8000 snapshots each and time spacing down to one million years.
This paper serves as the data release for IllustrisTNG as a whole, including the future public release of TNG50.
--------------- --------------------------------- --------------------------- -------------------------- -------------------- ---------------------- ------------------------- ---------------------- ---------------------
Run Volume $L_{\rm box}$ $N_{\rm GAS,DM}$ $N_{\rm TRACER}$ $m_{\rm baryon}$ $m_{\rm DM}$ $m_{\rm baryon}$ $m_{\rm DM}$
\[cMpc$^3$\] \[cMpc/$h$\] - - \[M$_\odot / h$\] \[M$_\odot / h$\] \[10$^6$M$_\odot$\] \[10$^6$M$_\odot$\]
TNG50-1 $51.7^3$ 35 $2160^3$ $1\times2160^3$ $5.7 \times 10^4$ $3.1 \times 10^5$ 0.08 0.45
TNG50-2 $51.7^3$ 35 $1080^3$ $1\times1080^3$ $4.6 \times 10^5$ $2.5 \times 10^6$ 0.68 3.63
TNG50-3 $51.7^3$ 35 $540^3$ $1\times540^3$ $3.7 \times 10^6$ $2.0 \times 10^7$ 5.4 29.0
TNG50-4 $51.7^3$ 35 $270^3$ $1\times270^3$ $2.9 \times 10^7$ $1.6 \times 10^8$ 43.4 232
TNG100-1 $106.5^3$ 75 $1820^3$ $2\times1820^3$ $9.4 \times 10^5$ $5.1 \times 10^6$ 1.4 7.5
TNG100-2 $106.5^3$ 75 $910^3$ $2\times910^3$ $7.6 \times 10^6$ $4.0 \times 10^7$ 11.2 59.7
TNG100-3 $106.5^3$ 75 $455^3$ $2\times455^3$ $6.0 \times 10^7$ $3.2 \times 10^8$ 89.2 478
TNG300-1 $302.6^3$ 205 $2500^3$ $1\times2500^3$ $7.6 \times 10^6$ $4.0 \times 10^7$ 11 59
TNG300-2 $302.6^3$ 205 $1250^3$ $1\times1250^3$ $5.9 \times 10^7$ $3.2 \times 10^8$ 88 470
TNG300-3 $302.6^3$ 205 $625^3$ $1\times625^3$ $4.8 \times 10^8$ $2.5 \times 10^9$ 703 3760
TNG50-1-Dark $51.7^3$ 35 $2160^3$ - - $3.7 \times 10^5$ - 0.55
TNG50-2-Dark $51.7^3$ 35 $1080^3$ - - $2.9 \times 10^6$ - 4.31
TNG50-3-Dark $51.7^3$ 35 $540^3$ - - $2.3 \times 10^7$ - 34.5
TNG50-4-Dark $51.7^3$ 35 $270^3$ - - $1.9 \times 10^8$ - 275
TNG100-1-Dark $106.5^3$ 75 $1820^3$ - - $6.0 \times 10^6$ - 8.9
TNG100-2-Dark $106.5^3$ 75 $910^3$ - - $4.8 \times 10^7$ - 70.1
TNG100-3-Dark $106.5^3$ 75 $455^3$ - - $3.8 \times 10^8$ - 567
TNG300-1-Dark $302.6^3$ 205 $2500^3$ - - $7.0 \times 10^7$ - 47
TNG300-2-Dark $302.6^3$ 205 $1250^3$ - - $3.8 \times 10^8$ - 588
TNG300-3-Dark $302.6^3$ 205 $625^3$ - - $3.0 \times 10^9$ - 4470
Run $\epsilon_{\rm DM,\star}^{z=0}$ $\epsilon_{\rm DM,\star}$ $\epsilon_{\rm gas,min}$ $r_{\rm cell,min}$ $\bar{r}_{\rm cell}$ $\bar{r}_{\rm cell,SF}$ $\bar{n}_{\rm H,SF}$ $n_{\rm H,max}$
\[kpc\] \[ckpc/$h$\] \[ckpc/$h$\] \[pc\] \[kpc\] \[pc\] \[cm$^{-3}$\] \[cm$^{-3}$\]
TNG50-1 0.29 0.39 $\rightarrow$ 0.195 0.05 8 5.8 138 0.8 650
TNG50-2 0.58 0.78 $\rightarrow$ 0.39 0.1 19 12.9 282 0.7 620
TNG50-3 1.15 1.56 $\rightarrow$ 0.78 0.2 65 25.0 562 0.6 80
TNG50-4 2.30 3.12 $\rightarrow$ 1.56 0.4 170 50.1 1080 0.5 35
TNG100-1 0.74 1.0 $\rightarrow$ 0.5 0.125 14 15.8 355 1.0 3040
TNG100-2 1.48 2.0 $\rightarrow$ 1.0 0.25 74 31.2 720 0.6 185
TNG100-3 2.95 4.0 $\rightarrow$ 2.0 0.5 260 63.8 1410 0.5 30
TNG300-1 1.48 2.0 $\rightarrow$ 1.0 0.25 47 31.2 715 0.6 490
TNG300-2 2.95 4.0 $\rightarrow$ 2.0 0.5 120 63.8 1420 0.5 235
TNG300-3 5.90 8.0 $\rightarrow$ 4.0 1.0 519 153 3070 0.4 30
--------------- --------------------------------- --------------------------- -------------------------- -------------------- ---------------------- ------------------------- ---------------------- ---------------------
Physical Models and Numerical Methods
-------------------------------------
All of the TNG runs start from cosmologically motivated initial conditions, assuming a cosmology consistent with the [@planck2015_xiii] results ($\Omega_{\Lambda,0}=0.6911$, $\Omega_{m,0}=0.3089$, $\Omega_{b,0}=0.0486$, $\sigma_8=0.8159$, $n_s=0.9667$ and $h=0.6774$), with Newtonian self-gravity solved in an expanding Universe. All of the baryonic TNG runs include the following additional physical components: (1) Primordial and metal-line radiative cooling in the presence of an ionizing background radiation field which is redshift-dependent and spatially uniform, with additional self-shielding corrections. (2) Stochastic star formation in dense ISM gas above a threshold density criterion. (3) Pressurization of the ISM due to unresolved supernovae using an effective equation of state model for the two-phase medium. (4) Evolution of stellar populations, with associated chemical enrichment and mass loss (gas recycling), accounting for SN Ia/II, AGB stars, and NS-NS mergers. (5) Stellar feedback: galactic-scale outflows with an energy-driven, kinetic wind scheme. (6) Seeding and growth of supermassive blackholes. (7) Supermassive blackhole feedback: accreting BHs release energy in two modes, at high-accretion rates (‘quasar’ mode) and low-accretion rates (‘kinetic wind’ mode). Radiative proximity effects from AGN affect nearby gas cooling. (8) Magnetic fields: amplification of a small, primordial seed field and dynamical impact under the assumption of ideal MHD.
Simulation Aspect Illustris TNG (50/100/300)
------------------------- -------------------------------------------------------------- ----------------------------------------
Magnetic Fields no ideal MHD [@pakmor11]
BH Low-State Feedback ‘Radio’ Bubbles BH-driven wind (kinetic kick)
BH Accretion Boosted Bondi-Hoyle ($\alpha=100$) Un-boosted Bondi-Hoyle
BH Seed mass $10^5$ M$_\odot$/h $8 \times 10^5$ M$_\odot$/h
Winds (Directionality) bi-polar ($\vec{v}_{\rm gas} \times \nabla \phi_{\rm grav}$) isotropic
Winds (Thermal Content) cold warm (10%)
Winds (Velocity) $\propto \sigma_{\rm DM}$ + scaling with H(z), and $v_{\rm min}$
Winds (Energy) constant per unit SFR + metallicity dependence in $\eta$
Stellar Evolution Illustris Yields TNG Yields
Metals Tagging - SNIa, SNII, AGB, NSNS, FeSNIa, FeSNII
Shock Finder no yes [@schaal15]
For complete details on the behavior, implementation, parameter selection, and validation of these physical models, see the two TNG methods papers: [@weinberger17] and [@pillepich18a]. Table \[table\_model\] provides an abridged list of the key differences between Illustris and TNG. We note that the TNG model has been designed (i.e. ‘calibrated’, or ‘tuned’) to broadly reproduce several basic, observed galaxy properties and statistics. These are: the galaxy stellar mass function and the stellar-to-halo mass relation, the total gas mass content within the virial radius ($r_{500}$) of massive groups, the stellar mass – stellar size and the BH–galaxy mass relations all at $z=0$, in addition to the overall shape of the cosmic star formation rate density at $z\lesssim10$ [see @pillepich18a for a discussion].
The TNG simulations use the moving-mesh [Arepo]{} code [@spr10] which solves the equations of continuum magnetohydrodynamics [MHD; @pakmor11; @pakmor13] coupled with self-gravity. The latter is computed with the Tree-PM approach, while the fluid dynamics employs a Godunov (finite-volume) type method, with a spatial discretization based on an unstructured, moving, Voronoi tessellation of the domain. The Voronoi mesh is generated from a set of control points which move with the local fluid velocity modulo mesh regularization corrections. Assuming ideal MHD, an 8-wave Powell cleaning scheme maintains the zero divergence constraint. The previous MUSCL-Hancock scheme has been replaced with a time integration approach following Heun’s method, and the original Green-Gauss method for gradient estimation of primitive fluid quantities has been replaced with a least-squares method, obtaining second order convergence in the hydrodynamics [@pakmor16]. The long-range FFT calculation employs a new column-based MPI-parallel decomposition, while the gravity solver has been rewritten based on a recursive splitting of the N-body Hamiltonian into short- and long- timescale systems (as in [Gadget-4]{}, ). The code is second order in space, and with hierarchical adaptive time-stepping, also second order in time. Of order 10 million individual timesteps are required to evolve the high-resolution runs to redshift zero.
During the simulation we employ a Monte Carlo tracer particle scheme [@genel13] to follow the Lagrangian evolution of baryons. An on-the-fly cosmic shock finder is coupled to the code [@schaal15; @schaal16]. Group catalogs are computed during the simulations using the [FoF]{} and [Subfind]{} [@spr01] substructure identification algorithms.
Model Validation and Early Findings {#sec_earlyresults}
-----------------------------------
TNG has been shown to produce observationally consistent results in several regimes beyond those adopted to calibrate the model. Some examples regarding galaxy populations, galactic structural and stellar population properties include: the shapes and widths of the red sequence and blue cloud of SDSS galaxies [@nelson18a]; the shapes and normalizations of the galaxy stellar mass functions up to $z\sim4$ [@pillepich18b]; the spatial clustering of red vs. blue galaxies from tens of kpc to tens of Mpc separations [@springel18]; the spread in Europium abundance of metal-poor stars in Milky Way like halos [@naiman18]; the emergence of a population of quenched galaxies both at low [@weinberger18] and high redshift [@habouzit18]; stellar sizes up to $z\sim2$, including separate star-forming and quiescent populations [@genel18]; the $z=0$ and evolution of the gas-phase mass-metallicity relation [@torrey18]; the dark matter fractions within the extended bodies of massive galaxies at $z=0$ in comparison to e.g. SLUGGS results [@lovell18]; and the optical morphologies of galaxies in comparison to Pan-STARRS observations [@rodriguezgomez18].
The IllustrisTNG model also reproduces a broad range of unusual galaxies, tracing tails of the galaxy population, including low surface brightness galaxies [@zhu18] and jellyfish, ram-pressure stripped galaxies [@yun18]. The large-volume of TNG300 helps demonstrate reasonable agreement in several galaxy cluster, intra-cluster and circumgalactic medium properties – for example, the scaling relations between total radio power and X-ray luminosity, total mass, and Sunyaev-Zel’dovich parameter of massive haloes [@marinacci18]; the distribution of metals in the intra-cluster plasma [@vog18a]; the observed fraction of cool core clusters [@barnes18]; and the OVI content of the circumgalactic media around galaxies from surveys at low redshift including COS-Halos and eCGM [@nelson18b].
IllustrisTNG is also producing novel insights on the formation and evolution of galaxies. For instance, halo mass alone is a good predictor for the entire stellar mass profile of massive galaxies [@pillepich18b]; the metal enrichment of cluster galaxies is higher than field counterparts at fixed mass and this enhancement is present pre-infall [@gupta18]; star-forming and quenched galaxies take distinct evolutionary pathways across the galaxy size-mass plane [@genel18] and exhibit systematically different column densities of OVI ions [@nelson18b] and different magnetic-field strengths [@nelson18a] at fixed galaxy stellar mass, as well as different magnetic-field topologies [@marinacci18]. Galaxies oscillate around the star formation main sequence and the mass-metallicity relations over similar timescales and often in an anti-correlated fashion [@torrey18b]; the presence of jellyfish galaxies is signaled by large-scale bow shocks in their surrounding intra-cluster medium [@yun18]; baryonic processes affect the matter power spectrum across a range of scales [@springel18] and steepen the inner power-law total density profiles of early-type galaxies [@wang18]; a significant number of OVII, OVIII [@nelson18b] and NeIX [@martizzi18] absorption systems are expected to be detectable by future X-ray telescopes like ATHENA.
IllustrisTNG has also been used to generate mock 21-cm maps [@villaescusa18] and estimates of the molecular hydrogen budget [@diemer18] in central and satellite galaxies in the local [@stevens18] as well as in the high-redshift Universe as probed by ALMA [@popping19]. Finally, TNG provides a test bed to explore future observational applications of machine learning techniques: for example, the use of Deep Neural Networks to estimate galaxy cluster masses from Chandra X-ray mock images [@ntampaka18] or optical morphologies versus SDSS [@huertascompany19].
See the up to date list of results[^3] for additional references. Please note that on this page we provide, and will continue to release, data files accompanying published papers as appropriate. For instance, electronic versions of tables, and data points from key lines and figures, to enable comparisons with other results. These are available with small links next to each paper.
Breadth of Simulated Data
-------------------------
{width="6.5in"}
All of the observational validations and early results from TNG100 and TNG300 demonstrate the broad applications of the IllustrisTNG simulations. To give a sense of the expansive scope, the richness of the resulting data products, and the potential for wide applications across many areas of galaxy formation and cosmology, Figure \[fig\_fields\] visualizes the TNG100 simulation at redshift zero. Each slice reveals a view into the synthetic IllustrisTNG universe. Together, they range from purely theoretical quantities to directly observable signatures, spanning across the baryonic and non-baryonic matter components of the simulation: dark matter, gas, stars, and blackholes.
The wealth of available information in the simulation outputs translates directly into the wide range of astrophysical phenomena which can be explored with the TNG simulations.
Data Products {#sDataProducts}
=============
We release all 100 snapshots of the IllustrisTNG cosmological volumes. These include up to five types of resolution elements (dark matter particles, gas cells, gas tracers, stellar and stellar wind particles, and supermassive blackholes). The same volumes are available at multiple resolutions: high (-1 suffix, e.g. TNG100-1), intermediate (-2 suffix), and low (-3 suffix), always separated by a factor of two (eight) in spatial (mass) resolution. At each resolution, these ‘baryonic’ runs include the fiducial TNG model for galaxy formation physics. Each baryonic run is matched to its dark matter only analog (-Dark suffix).
For all runs, at every snapshot, two types of group catalogs are provided: friends-of-friends (FoF) halo catalogs, and [Subfind]{} subhalo catalogs. In postprocessing, these catalogs are used to generate two distinct merger trees, which are both released: [SubLink]{}, and [LHaloTree]{}. Finally, supplementary data catalogs containing additional computations and modeling, and focusing on a variety of topics, are being continually created and released. All these data types are described below.
Snapshots
---------
### Snapshot Organization
There are 100 snapshots stored for every run. These include all particles/cells in the whole volume. The complete snapshot listings, spacings and redshifts can be found online. Note that, unlike in Illustris, TNG contains two different types of snapshots: ‘full’ and ‘mini’. While both encompass the entire volume, ‘mini’ snapshots only have a subset of particle fields available (detailed online). In TNG, twenty snapshots are full, while the remaining 80 are mini. The 20 full snapshots are given in Table \[table\_snaps\]. Every snapshot is stored on-disk in a series of ‘chunks’, which are more manageable, smaller HDF5 files – additional details are provided in Table \[table\_chunks\] of the appendix.
Snap $a$ $z$ Snap $a$ $z$
------ -------- ----- ------ -------- -----
2 0.0769 12 33 0.3333 2
3 0.0833 11 40 0.4 1.5
4 0.0909 10 50 0.5 1
6 0.1 9 59 0.5882 0.7
8 0.1111 8 67 0.6667 0.5
11 0.125 7 72 0.7143 0.4
13 0.1429 6 78 0.7692 0.3
17 0.1667 5 84 0.8333 0.2
21 0.2 4 91 0.9091 0.1
25 0.25 3 99 1 0
: Abridged snapshot list for TNG runs: snapshot number together with the corresponding scalefactor and redshift. The twenty snapshots shown here are the ‘full’ snapshots, while the remaining eighty are ‘mini’ snapshots with a subset of fields.[]{data-label="table_snaps"}
Note that, just as in Illustris, the snapshot data is not organized according to spatial position. Rather, particles within a snapshot are sorted based on their group/subgroup memberships, according to the [FoF]{} or [Subfind]{} algorithms. Within each particle type, the sort order is: group number, subgroup number, and then binding energy. Particles/cells belonging to the group but not to any of its subhalos (“inner fuzz”) are included after the last subhalo of each group. In Figure \[fig\_snap\_schematic\] we show a schematic of the particle organization [as in @nelson15b], for *one particle type*. Note that halos may happen to be stored across multiple, subsequent file chunks, and different particle types of a halo are in general stored in different sets of file chunks.
![ Illustration of the organization of particle/cell data within a snapshot for one particle type (e.g dark matter). Therein, particle order is set by a global sort of the following fields in this order: FoF group number, [Subfind]{} subhalo number, binding energy. As a result, FOF halos are contiguous, although they can span file chunks. [Subfind]{} subhalos are only contiguous within a single group, being separated between groups by an “inner fuzz” of all FOF particles not bound to any subhalo. “Outer fuzz” particles outside all halos are placed at the end of each snapshot. \[fig\_snap\_schematic\]](figures/schematic_snapshots_new2.pdf){width="3.3in"}
### Snapshot Contents
Each HDF5 snapshot contains several groups: ‘Header’, ‘Parameters’, ‘Configuration’, and five additional ‘PartTypeX’ groups, for the following particle types (DM only runs have a single PartType1 group):
- PartType0 - GAS
- PartType1 - DM
- PartType2 - (unused)
- PartType3 - TRACERS
- PartType4 - STARS & WIND PARTICLES
- PartType5 - BLACK HOLES
The ‘Header’ group contains a number of attributes giving metadata about the simulation and snapshot. The ‘Parameters’ and ‘Configuration’ groups provide the complete set of run-time parameter and compile-time configuration options used to run TNG. That is, they encode the fiducial “TNG Galaxy Formation Model”. Many will clearly map to Table 1 of [@pillepich18a], while others deal with more numerical/technical options. In the future, together with the release of the TNG initial conditions and the TNG code base, this will enable any of the TNG simulations to be reproduced.
The complete snapshot field listings of the ‘PartTypeX’ groups, including dimensions, units and descriptions, are given online. The general system of units is ${\rm kpc}/h$ for lengths, $10^{10} {\rm M}_\odot/h$ for masses, and ${\rm km/s}$ for velocities. Comoving quantities can be converted to the corresponding physical ones by multiplying by the appropriate power of the scale factor $a$. New fields in TNG, not previously available in the original Illustris, are specially highlighted.
With respect to Illustris, the following new fields are generally available in the snapshots: (i) EnergyDissipation and Machnumber, giving the output of the on-the-fly shock finder, (ii) GFM\_Metals, giving the individual element abundances of the nine tracked species (H, He, C, N, O, Ne, Mg, Si, Fe), (iii) GFM\_MetalsTagged, metal tracking as described below, (iv) MagneticField and MagneticFieldDivergence, providing the primary result of the MHD solver.
### Tagged Metals
Subbox Environment Center Position \[Code Units\] Box Size $f_{\rm vol}$ \[%\]
----------------- -------------------------------------------------------------------- -------------------------------- -------------------- --------------------- --
TNG100 Subbox-0 Crowded, including a $5 \times 10^{13} {\rm M}_\odot$ halo (9000, 17000, 63000) 7.5 ${\rm cMpc}/h$ 0.1
TNG100 Subbox-1 Less crowded, including several $> 10^{12} {\rm M}_\odot$ halos (37000, 43500, 67500) 7.5 ${\rm cMpc}/h$ 0.1
TNG300 Subbox-0 Massive cluster ($\sim2 \times 10^{15} M_{\odot}$) merging at z=0 (44, 49, 148) \* 1000 15 ${\rm cMpc}/h$ 0.04
TNG300 Subbox-1 Crowded, above average \# of halos above $10^{13} M_{\odot}$ (20, 175, 15) \* 1000 15 ${\rm cMpc}/h$ 0.04
TNG300 Subbox-2 Semi-underdense, one local group analog at z=0 (169, 97.9, 138) \* 1000 10 ${\rm cMpc}/h$ 0.01
TNG50 Subbox-0 Somewhat-crowded ($\sim$6 MWs) (26000, 10000, 26500) 4.0 ${\rm cMpc}/h$ 0.15
TNG50 Subbox-1 Low-density, many dwarfs, no halos $>5 \times 10^{10}$ M$_{\odot}$ (12500, 10000, 22500) 4.0 ${\rm cMpc}/h$ 0.15
TNG50 Subbox-2 Most massive cluster ($2 \times 10^{14}$ M$_{\odot}$ at z=0) (7300, 24500, 21500) 5.0 ${\rm cMpc}/h$ 0.3
Run $N_{\rm snap}$ $\Delta t_{(z=6)}$ $\Delta t_{(z=2)}$ $\Delta t_{(z=0)}$
---------- ---------------- -------------------- -------------------- -------------------- --
TNG100-3 2431 $\sim$4 Myr $\sim$7 Myr $\sim$19 Myr
TNG100-2 4380 $\sim$2 Myr $\sim$4 Myr $\sim$10 Myr
TNG100-1 7908 $\sim$1 Myr $\sim$1.5 Myr $\sim$2.5 Myr
TNG300-3 2050 $\sim$6 Myr $\sim$11 Myr $\sim$8 Myr
TNG300-2 3045 $\sim$3 Myr $\sim$6 Myr $\sim$4 Myr
TNG300-1 2449 $\sim$1.5 Myr $\sim$4 Myr $\sim$6 Myr
TNG50-4 2333 $\sim$7 Myr $\sim$6 Myr $\sim$8 Myr
TNG50-3 4006 $\sim$2 Myr $\sim$3 Myr $\sim$4 Myr
TNG50-2 1895 $\sim$3 Myr $\sim$6 Myr $\sim$8 Myr
TNG50-1 $\sim$3600 $\sim$3 Myr $\sim$2 Myr $\sim$2 Myr
: Details of the subbox snapshots: the number and approximate time resolution $\Delta t$ at three redshifts: $z=6$, $z=2$, and $z=0$. Every subbox for a given volume and resolution combination has the same output times.[]{data-label="table_subbox2"}
The units of all the entries of GFM\_MetalsTagged field, except for NSNS, are the same as GFM\_Metals: dimensionless mass ratios. Summing all elements of GFM\_Metals heavier than Helium recovers the sum of the three tags SNIa+SNII+AGB. Likewise, the Fe entry of GFM\_Metals roughly equals the sum of FeSNIa+FeSNII, modulo the small amount of iron consumed (i.e. negative contribution) by AGB winds. The fields are (in order):
- SNIa (0): The total metals ejected by Type Ia SN.
- SNII (1): The total metals ejected by Type II SN.
- AGB (2): the total metals ejected by stellar winds, which is dominated by AGB stars.
- NSNS (3): the total mass ejected from NS-NS merger events, which are modeled stochastically (i.e. no fractional events) with a DTD scheme similar to that used for SNIa, except with a different $\tau$ value. Note that the units of NSNS are arbitrary. To obtain physical values in units of solar masses, this field must be multiplied by $\alpha / \alpha_0$ where $\alpha$ is the desired mass ejected per NS-NS merger event, and $\alpha_0$ is the base value (arbitrary) used in the simulation, e.g. [@shen15] take $\alpha = 0.05 \rm{M}_{\odot}$. The value of $\alpha_0$ varies by run, and it is 0.05 for all TNG100 runs, and 5000.0 for all TNG300 and TNG50 runs. See [@naiman18] for more details.
- FeSNIa (4): The total iron ejected by Type Ia SN.
- FeSNII (5): The total iron ejected by Type II SN.
Note a somewhat subtle but fundamental detail: these tags do not isolate where a given heavy element was created, but rather identify the last star it was ejected from. This can be problematic since, for example, AGB winds create little iron, but eject a significant amount of iron which was previously created by SnIa and SNII at earlier epochs. The FeSNIa field is, for example, more accurately described as ‘the total iron ejected by type Ia supernovae not yet consumed and re-ejected from another star’.
### Subboxes
Separate ‘subbox’ cutouts exist for each baryonic run. These are spatial cutouts of fixed comoving size and fixed comoving coordinates, and the primary benefit is that their time resolution is significantly better than that of the main snapshots – details are provided in Tables \[table\_subbox1\] and \[table\_subbox2\]. These snapshots are useful for some types of analysis and science questions requiring high time-resolution data, and for creating time-evolving visualizations. There are two subboxes for TNG100 (corresponding to the original Illustris subboxes \#0 and \#2, the latter increased in size), and three subboxes for TNG50 and TNG300. Note that subboxes, unlike full boxes, are not periodic.
The subboxes sample different areas of the large boxes, roughly described by the environment column in Table \[table\_subbox1\]. The particle fields are all identical to the main snapshots, except that the particles/cells are not sorted by their group membership, since no group catalogs exist for subbox snapshots.
Group Catalogs
--------------
Group catalogs give the results of substructure identification, and broadly contain two types of objects: dark matter halos (either [FoF]{} halos or central subhalos) and galaxies themselves (the inner stellar component of subhalos, either centrals or satellites). There is one group catalog produced for each snapshot, which includes both [FoF]{} and [Subfind]{} objects. The group files are split into a small number of sub-files, just as with the raw snapshots. In TNG, these files are called [fof\_subhalo\_tab\_]{}, whereas in original Illustris they were called [groups\_]{} (they are otherwise essentially identical). Every HDF5 group catalog contains the following groups: Header, Group, and Subhalo. The IDs of the member particles of each group/subgroup are not stored in the group catalog files. Instead, particles/cells in the snapshot files are ordered according to group membership.
In order to reduce confusion, we adopt the following terminology when referring to different types of objects. “Group”, “FoF Group”, and “FoF Halo” all refer to halos. “Subgroup”, “Subhalo”, and “Subfind Group” all refer to subhalos. The first (most massive) subgroup of each halo is the “Primary Subgroup” or “Central Subgroup”. All other following subgroups within the same halo are “Secondary Subgroups”, or “Satellite Subgroups”.
**FoF Groups.** The Group fields are derived with a standard friends-of-friends (FoF) algorithm with linking length $b=0.2$. The FoF algorithm is run on the dark matter particles, and the other types (gas, stars, BHs) are attached to the same groups as their nearest DM particle. **Subfind Groups.** The Subhalo fields are derived with the [Subfind]{} algorithm. In identifying gravitationally bound substructures the method considers all particle types and assigns them to subhalos as appropriate.
Complete documentation for the TNG group catalogs, comprising FoF halos as well as Subfind subhalos, is available online. Differences and additions with respect to original Illustris are highlighted.
Merger Trees
------------
Merger trees have been created for the TNG simulations using [SubLink]{} [@rodriguezgomez15] and [LHaloTree]{} [@spr05]. In the population average sense the different merger trees give similar results. In more detail, the exact merger history or mass assembly history for any given halo may differ. For a particular science goal, one type of tree may be more or less useful, and users are free to use whichever they prefer. We generally recommend use of the [SubLink]{} trees as a first option, as they are more efficiently stored and accessible.
Trees can be ‘walked’, i.e. the descendants or progenitors of a given subhalo can be determined, thus linking objects across snapshots saved at different points in time. Main branches, such as the main progenitor branch (MPB), as well as full trees can be extracted. Examples of walking the tree are provided in the example scripts. For the technical details, algorithmic descriptions, and storage structures of the trees, please refer to [@nelson15b] and the online documentation – we omit these details here.
### SubLink
The [SubLink]{} merger tree is one large data structure split across several sequential HDF5 files named , where goes from e.g. 0 to 19 for the TNG100-1 run, and 0 to 125 for the TNG300-1 run.
### LHaloTree
The [LHaloTree]{} merger tree is one large data structure split across several HDF5 files named , where TNG100-1 has for instance 80 chunks enumerated by , while TNG300-1 has 320. Within each file there are a number of HDF5 groups named “TreeX”, each of which represents one disjoint merger tree.
### Offsets Files
As described above, snapshot particle data is ordered by the subhalo each particle belongs to. To facilitate rapid loading of snapshot data, particle ‘offset’ numbers provide the location where particles belonging to each subhalo begin. Most simply, offsets describe where in the group catalog files to find a specific halo/subhalo, and where in the snapshot files to find the start of the particles of a given halo/subhalo.
To use the helper scripts (provided online) for working with the actual data files (snapshots or group catalogs) on a local machine, then it **is required** to download the offset file(s) for the snapshot(s) of interest. The offsets are **not required** when using the web-based API or analyzing the particle cutouts it provides, for instance.
Note that in Illustris, offsets were embedded inside the group catalog files for convenience. In TNG however, we have kept offsets as separate files called [offsets\_.hdf5]{} (one per snapshot), which must be downloaded as well.
### The ‘simulation.hdf5’ file
Each run has a single file called ‘simulation.hdf5’ which is purely optional, for convenience, and not required by any of the public scripts. Its purpose is to encapsulate all data of an entire simulation into a single file.
To accomplish this, we make advantage of a new feature of the HDF5 library called “virtual datasets”. A virtual dataset is a collection of symbolic links to one or more datasets in other HDF5 file(s), where these symlinks can refer to subsets of a dataset, in either the source or target of the link. The simulation.hdf5 is thus a large collection of links, which refer to other files which actually contain data. In order to use it, the corresponding files must also be downloaded (e.g. of snapshots, group catalogs, or supplementary data catalogs).
Using this resource, the division of snapshots and group catalogs over multiple file chunks is no longer relevant. Loading particle data from snapshots or subhalo or halo fields from group catalogs become one line operations. It also makes loading the particles of a given halo or subhalo using the offset information trivial. Finally, supplementary data catalogs (either those we provide, or similar user-run computations) can be ‘virtually’ inserted as datasets in snapshots or group catalogs. This provides a clean way to organize post-processing computations which produce additional values for halos, subhalos, or individual particles/cells. Such data can then be loaded with the same scripts (and same syntax) as ‘original’ snapshot/group catalog fields.
We refer to the online documentation for examples of each use case as well as technical requirements, namely a relatively new version (1.10+) of the HDF5 library.
Initial Conditions
------------------
We provide as part of this release the initial conditions for all TNG volumes as well as the original Illustris volumes. These were created with the Zeldovich approximation and the [N-GenIC]{} code [@springel15]. Each particular realization was chosen from among tends of random realizations of the same volume as the most average, based on sinspection of the $z=0$ power spectrum and/or dark matter halo mass function – see [@vog14a] and for details. Each IC is a single HDF5 file with self-evident structure: the coordinates, velocities, and IDs of the set of total matter particles at $z=127$, the starting redshift for all runs. These ICs were used as is for dark-matter only simulations, while for baryonic runs total matter particles were split upon initialization in the usual way, into dark matter and gas, according to the cosmic baryon fraction and offsetting in space by half the mean interparticle spacing. These ICs can be run by e.g. [Gadget]{} or [Arepo]{} as is, or easily converted into other data formats.
Supplementary Data Catalogs
---------------------------
Many additional data products have been computed in post-processing, based on the raw simulation outputs. These are typically in support of specific projects and analysis in a published paper, after which the author makes the underlying data catalog public. Many such catalogs have been made available for the original Illustris simulation, and the majority of these will also be recalculated for TNG. We provide a list of TNG supplementary data catalogs which are now available or which we anticipate to release in the near future:
1. Tracer Tracks – time-evolution of Monte Carlo tracer properties for TNG100 ().
2. Stellar Mass, Star Formation Rates – multi-aperture and resolution corrected masses, time-averaged SFRs [@pillepich18b].
3. Stellar Circularities, Angular Momenta, and Axis Ratios – for the stellar components of galaxies, as for Illustris [@genel15].
4. Subhalo Matching Between Runs – cross-matching subhalos between baryonic and dark-matter only runs, between runs at different resolutions, and between TNG100 and Illustris [@lovell18; @nelson15b; @rodriguezgomez15; @rodriguezgomez17].
5. Stellar Projected Sizes – half-light radii of TNG100 galaxies [@genel18].
6. Blackhole Mergers and Details – records of BH-BH mergers and high time-resolution BH details, as for Illustris [@kelley17; @blecha16], and with an updated approach ().
7. Stellar Assembly – in-situ versus ex-situ stellar growth, as for Illustris [@rodriguezgomez16; @rodriguezgomez17].
8. Subbox Subhalo List – record of which subhalos exist in what subboxes across particular redshift ranges, and interpolated properties [@nelson19b]
9. Molecular and Atomic Hydrogen (HI+H2) – decomposition of the neutral hydrogen in gas cells and galaxies into HI/H2 masses [@diemer18; @stevens18].
10. Halo/galaxy angular momentum and baryon content – measurements of spherical overdensity values, as for Illustris [@zjupa17].
11. SDSS Photometry and Mock Fiber Spectra – broadband colors and spectral mocks including dust attenuation effects [@nelson18a].
12. SKIRT Synthetic Images and Optical Morphologies – dust radiative-transfer calculations using SKIRT to obtain broadband images, automated morphological measurements [@rodriguezgomez18].
13. DisPerSE Cosmic Web – topological classification of the volume into sheets, filaments, nodes, and voids ().
14. Particle-level lightcones – in a variety of configurations, from small field of view ‘deep fields’ to all-sky projections, across the different matter components, to facilitate lensing, x-ray, Sunyaev-Zeldovich, and related explorations ().
Several of these were previously available for the original Illustris simulation and will be re-computed for TNG. We would plan to provide a number of ‘pre-defined’ galaxy samples, particularly with respect to common observational selection techniques, current and/or upcoming surveys, and other distinct classes of interest. This can include, for example, red versus blue galaxies, luminous red galaxies (LRGs) and emission-line galaxies (ELGs) of SDSS, damped Lyman-alpha (DLA) host halos, and ultra-diffuse or low surface brightness (LSB) galaxies. Such samples would facilitate rapid comparisons to certain types of observational samples, and can be included as supplementary data catalogs as they become available.
Data Access {#sDataAccess}
===========
There are three complementary ways to access and analyze TNG data products.
1. **(Local data, local analysis).** Raw files can be directly downloaded, and example scripts are provided as a starting point for local analysis.
2. **(Remote data, local analysis).** The web-based API can be used, either through a web browser or programmatically in a script, to perform search, data extraction, and visualization tasks, followed by local analysis of these derivative products.
3. **(Remote data, remote analysis).** A web-based JupyterLab (or Jupyter notebook) session can be instantiated to explore the data, develop analysis scripts with persistent storage, run data-intensive and compute-intensive tasks, and make final plots for publication.
These different approaches can be combined. For example, by downloading the full redshift zero group catalog to perform a complex search which cannot be easily done with the API. After determining a sample of interesting galaxies (i.e. a set of subhalo IDs), one can then extract their individual merger trees (and/or raw particle data) without needing to download the full simulation merger tree (or a full snapshot).
These approaches are described below, while “getting started” tutorials for several languages (currently: Python, IDL, and Matlab) can be found online.
Direct File Download and Example Scripts
----------------------------------------
**Local data, local analysis.** All of the primary outputs of the TNG simulations are released in HDF5 format, which we use universally for all data products. This is a portable, self-describing, binary specification (similar to FITS), suitable for large numerical datasets. File access libraries, routines, and examples are available in all common computing languages. We typically use only basic features of the format: attributes for metadata, groups for organization, and large datasets containing one and two dimensional numeric arrays. To maintain reasonable filesizes for transfer, most outputs are split across multiple files called “chunks”. For example, each snapshot of TNG100-1 is split into 448 sequentially numbered chunks. Links to the individual file chunks for a given simulation snapshot or group catalog are available under their respective pages on the main data release page.
The provided example scripts (in IDL, Python, and Matlab) give basic I/O functionality, and we expect they will serve as a useful starting point for writing any analysis task, and intend them as a ‘minimal working examples’ which are short and simple enough that they can be quickly understood and extended. For a getting-started guide and reference, see the online documentation.
Web-based API
-------------
**Remote data, local analysis.** For TNG we enhance the web-based interface (API) introduced with the original Illustris simulation, augmented by a number of new features and more sophisticated functionality. At its core, the API can respond to a variety of user requests and queries. It provides a well-defined interface between the user and simulation data, and the tools it provides are independent, as much as possible, from any underlying details of data structure, heterogeneity, storage format, and so on. The API can be used as an alternative to downloading large data files for local analysis. Fundamentally, the API allows a user to **search**, **extract**, **visualize**, or **analyze** a simulation, a snapshot, a group catalog, or a particular galaxy/halo. By way of example, the following requests can be handled by the current API:
- Search across subhalos with numeric range(s) over any field(s) present in the Subfind catalogs.
- Retrieve a snapshot cutout for all the particles/cells within a given subhalo/halo, optionally restricted to a subset of specified particle/cell type(s) and fields(s).
- Retrieve the complete merger history or main branches for a given subhalo.
- Download subsets of snapshot files, containing only specified particle/cell type(s), and/or specific field(s) for each type.
- Traverse links between halos and subhalos, for instance from a satellite galaxy, to its parent FoF halo, to the primary (central) subhalo of that group, as well as merger tree progenitor/descendant connections.
- Render visualizations of any field(s) of different components (e.g. dark matter, gas, stars) of a particular halo/subhalo.[^4]
- Download actual data from such a halo/subhalo visualization, e.g. maps of projected gas density, O VI column density, or stellar luminosity in a given band.^^
- Render a static visualization of the complete merger tree (assembly history) of any subhalo.^^
- Plot the relationship between quantities in the group catalogs, e.g. fundamental scaling relations such as the star-forming main sequence of TNG.^^
- Plot tertiary relationships between group catalog quantities, e.g. the dependence of gas fraction on offset from the main sequence.^^
The IllustrisTNG data access API is available at the following permanent URL:
- <http://www.tng-project.org/api/>
For a getting-started guide for the API, as well as a cookbook of common examples and the complete reference, see the online documentation.
Remote Data Analysis
--------------------
**Remote data, remote analysis.** Coincident with the TNG public data release we introduce a new, third option for working with and analyzing large simulation datasets. Namely, an online, browser-based scientific computing environment which enables researchers’ computations to “be brought to” the data. It is similar in spirit to the NOAO Data Lab [@fitzpatrick14] and SciServer services [@raddick17], i.e. simultaneously hosting petabyte-scale datasets as well as a full-featured analysis platform and toolset. This alleviates the need to download any data, or run any calculations locally, thereby facilitating broad, universal, open access to large cosmological simulation datasets such as TNG.
To enable this functionality we make use of extensive development on Jupyter and JupyterLab over the last few years. JupyterLab is the evolution of the Jupyter Notebook [@kluyver16], previously called IPython [@perez07]. It is a next-generation, web-based user interface suitable for scientific data analysis. In addition to the previous ‘notebook’ format, JupyterLab also enables a traditional workflow based around a collection of scripts on a filesystem, text editors, a console, and command-line execution. It provides an experience nearly indistinguishable from working directly on a remote computing cluster via SSH.
Computation is language agnostic, as ‘kernels’ are supported in all common languages, including Python 2.x, Python 3.x, IDL, Matlab, R, and Julia. Development, visualization, and analysis in any language or environment practically available within a Linux environment is possible, although we focus at present on Python 3.x support.
Practically, this service enables direct access to one of the complete mirrors of the Illustris\[TNG\] data, which is hosted at the Max Planck Computing and Data Facility (MPCDF) in Germany. Users can request a new, on-demand JupyterLab instance, which is launched on a system at MPCDF and connected to the user web browser. All Illustris\[TNG\] data is then directly available for analysis. A small amount of persistent user storage is provided, so that under-development scripts, intermediate analysis outputs, and in-progress figures for publication all persist across sessions. Users can log out and pick up later where they left off. A base computing environment is provided, which can be customized as needed (e.g. by installing new python packages with either `pip` or `conda`). Users can synchronize their pre-existing tools, such as analysis scripts, with standard approaches (`git`, `hg`, `rsync`) or via the JupyterLab interface. Results, such as figures or data files, can be viewed in the browser or downloaded back to the client machine with the same tools.
For security and resource allocation, users must specifically request access to the JupyterLab TNG service. At present we anticipate providing this service on an experimental (beta) basis, and only to active academic users.
Further Online Tools
--------------------
### Subhalo Search Form
We provide the same, simple search form to query the subhalo database as was available in the Illustris data release. It exposes the search capabilities of the API in a user-friendly interface, enabling quick exploration without the need to write a script or URL by hand. As examples, objects can be selected based on total mass, stellar mass, star formation rate, or gas metallicity. The tabular output lists the subhalos which match the search, along with their properties. In addition, each result contains links to a common set of API endpoints and web-based tools for inspection and visualization.
### Explore: 2D and 3D
The 2D Explorer and 3D Explorer interfaces are experiments in the *interactive* visualization and exploration of large data sets such as those generated by the IllustrisTNG simulations. They both leverage the approach of thin-client interaction with derived data products. The 2D Explorer exposes a Google Maps$-$like tile viewer of pre-computed imagery from a slice of the TNG300-1 simulation at redshift zero, similar to the original Illustris explorer. Multiple views of different particle types (gas, stars, dark matter, and blackholes) can be toggled and overlaid, which is particularly useful in exploring the spatial relationships between different phenomena of these four matter components.
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The 3D Explorer introduces a new interface, showing a highly derivative (although spatially complete) view of an entire snapshot. That is, instead of particle-level information, we facilitate interactive exploration of the group catalog output in three-dimensional space. This allows users to rotate, zoom, and move around the cubic box representing the simulation domain, where the largest dark matter halos are represented by wireframe spheres of size equal to their virial radii, while the remaining smaller halos are represented by points. User selection of a particular halo, via on-click ray cast and sphere intersection testing, launches an API query and returns the relevant halo information and further introspection links. At present, both Explorers remain largely proof of concept interfaces for how tighter integration of numeric, tabular, and visual data analysis components may be combined in the future for the effective exploration and analysis of large cosmological datasets [see also @dykes18 and the Dark Sky simulation].
### Merger Tree Visualization
In the Illustris data release we demonstrated a rich-client application built on top of the API, in the form of an interactive visualization of merger trees. The tree is vector based, and client side, so each node can be interacted with individually. The informational popup provides a link, back into the API, where the details of the selected progenitor subhalo can be interrogated. This functionality is likewise available for all new simulations. Furthermore, we have added a new, static visualization of the complete merger tree of a subhalo. This allows a quick overview of the assembly history of a given object, particularly its past merger events and its path towards quiescence. In the fiducial configuration, node size in the tree is scaled with the logarithm of total halo mass, while color is mapped to instantaneous sSFR.
### Plot Group Catalog
The first significant new feature of the API for the TNG public data release is a plotting routine to examine the group catalogs. Since the objects in the catalogs are either galaxies or dark matter halos, plotting the relationships among their various quantities is one of most fundamental explorations of cosmological simulations. Classically observed scaling relations, such as Tully-Fisher (rotation velocity vs. stellar mass), Faber-Jackson (stellar velocity dispersion vs. luminosity), the stellar size-mass relation, the star-formation main sequence, or the Magorrian relation (blackhole mass versus bulge mass) are all available herein. Such relations can be used to assess the outcome of the simulations by comparison to observational data. More complex relations, those involving currently unobserved properties of galaxies/halos, and/or those only currently observed with very limited statistics or over limited dynamic range, represent a powerful discovery space and predictive regime for simulations such as TNG. At the level of the galaxy (or halo) population, i.e. with tens to hundreds of thousands of simulated objects, many such relationships reveal details of the process of galaxy formation and evolution, as well as the working mechanisms of the physical/numerical models.
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The ‘group catalog plotter’ is an API endpoint which returns publication quality figures (e.g. PNG or PDF outputs). In Figure \[fig\_plotex\] we show several examples of its output, taken from TNG300-1 and TNG100-1 at $z=0$. Many options exist to control the behavior and structure of the plots, all of which are detailed in the online documentation. As for the subhalo search form, we also provide a new web-based interface to assist in interactively constructing plots from this service. Fundamentally, the quantities to be plotted against each other on the x- and y-axes can be selected. In this case, a two-dimensional histogram showing the density of subhalos in this space is overlaid with the median relation and bounding percentiles. Optionally, a third quantity can be added, which is then used to color every pixel in the histogram according to a user-defined statistic (e.g. median) of all the objects in that bin. For example, plotting the stellar-mass halo-mass relation, colored by stellar metallicity, reveals one reason for the scatter in this relation. This third quantity can optionally be normalized relative to the median value at each x-axis value (e.g. as a function of stellar mass), highlighting the ‘relative’ deviation of that property compared to its evolving median value. The types of subhalos included can be chosen, for example selecting only centrals or only satellite galaxies, and the subhalos to be included can be filtered based on numeric range selections on a fourth quantity. We expect that this tool will enable rapid, initial exploration of interesting relationships among galaxy (or halo) integral properties, and serve as a starting point for more in-depth analysis [see also @desouza15]. Complete usage documentation is available online.
### Visualize Galaxies and Halos
The second significant new feature of the API for the TNG public data release is an on-demand visualization service. Primarily, this API endpoint renders projections of particle-level quantities (of gas cells, dark matter particles, or stellar particles) for a given subhalo or halo. For example, it can produce gas column density projections, gas temperature projections, stellar line-of-sight velocity maps, or dark matter velocity dispersion maps. Its main rendering algorithm is based on the standard SPH kernel projection technique, with adaptive kernel sizes for all particle types, although alternatives are available. In Figure \[fig\_visex\] we show several examples of output, at both the halo-scale (circle indicating virial radius), and the galaxy scale (outer circle showing twice the stellar half mass radius).
The visualization service returns publication quality figures (e.g. PNG or PDF outputs). It can also return the raw data used to construct any image, in scientifically accurate units (HDF5 output). For instance, a user can request not only an image of the gas density projection of an ongoing galaxy merger, but also the actual grid of density values in units of e.g. $\rm{M}_{\odot}\, \rm{kpc}^{-2}$. Numerous options exist to control the behavior of the rendered projections, as well as the output style, all of which are detailed in the online documentation. All parameters of the rendering can be specified – as an example, the view direction can be a rotation into face-on or edge-on orientations. Most properties available in the snapshots can be visualized, for any galaxy/halo, at any snapshot, for any run.
Beyond snapshot level information, the visualization service currently has two more advanced features. First, it is coupled to the [CLOUDY]{} photoionization code [@ferland17], following [@nelson18b]. This enables ionic abundance calculations for gas cells on the fly. For example, a user can request a column density map of the O VI or C IV ions. All relevant atoms are supported, assuming solar abundances for non-tracked elements, typically up to the tenth ionization state (Al, Ar, Be, B, Ca, C, Cl, Cr, Co, Cu, F, He, H, Fe, Li, Mg, Mn, Ne, Ni, N, O, P, K, Sc, Si, Na, S, Ti, V, Zn). Emission line luminosities are also available – a surface brightness map of metal-line emission from O VIII at 22.1012Å, for example. Secondly, this service is also coupled to the [FSPS]{} stellar population synthesis code [@conroy09; @conroy10] through [python-fsps]{} [@foremanmackey14], following [@nelson18a]. This enables emergent stellar light calculations for stellar population particles on the fly, with optional treatments of dust attenuation effects. For example, a user can request a map of stellar surface brightness, or luminosity, either rest frame or observed frame, for any of the $\sim$140 available bands, including common filters on surveys/instruments such as SDSS, DES, HST, and JWST.
We expect that this tool will enable rapid, initial exploration of many interesting facets of galaxies and halos across the simulations, and serve as a starting point for more in-depth analysis. We caution that, used improperly, this tool can easily return nonsensical results (e.g. requesting OI emission properties from ISM gas), and users should understand the relevant scientific limitations. In this particular case, we refer to the effective two-phase ISM model used in TNG [@spr03] which intentionally avoids resolving the cold, dense phases of the ISM. Complete usage documentation is available online.
Scientific Remarks and Cautions {#sRemarks}
===============================
In the original Illustris simulation we identified a number of non-trivial issues in the simulated galaxy and halo populations in comparison to observational constraints [see @nelson15b for a summary]. These disagreements motivated a series of important caveats against drawing certain strong scientific conclusions in a number of regimes.
In contrast, our initial explorations of TNG (specifically, of the TNG100-1 and TNG300-1 simulations) have revealed no comparably significant tensions with respect to observable comparisons. With this data release we invite further detailed observational comparisons and scrutiny. The TNG simulations have been shown to realistically resolve numerous aspects of galactic structure and evolution, including many internal properties of galaxies (though, clearly, not all) as well as their co-evolution within the cosmic web of large-scale structure (see Section \[sec\_earlyresults\]). TNG reproduces various observational details and scaling relations of the demographics and properties of the galaxy population, not only at the present epoch ($z=0$), but also at earlier times (see likewise Section \[sec\_earlyresults\]). This has been achieved with a physically plausible although necessarily simplified galaxy formation model. The TNG model is intended to account for most, if not all, of the primary processes that are believed to be important for the formation and evolution of galaxies.
IllustrisTNG: Possible Observational Tensions
---------------------------------------------
We therefore do not specifically caution against the use of TNG in any of the regimes where the original Illustris simulation was found to be less robust. However, the enormous spatial and temporal dynamic range of these simulations, as well as the multi-scale, multi-physics nature of the complex physical phenomena involved, implies modeling approximations and uncertainties. Early comparisons of TNG against observations have identified a number of interesting regimes in which possible tensions exist.
Our ability to make any stronger statement is frequently limited by the complexity of the observational comparison, i.e. the need to accurately reproduce (or ‘mock’) the observational measurement closely and with care. In the qualitative sense, however, these regimes may plausibly indicate areas where the TNG model has shortcomings or is less physically realistic. It will be helpful for any user of the public data to be aware of these points, which should be carefully considered when advancing strong scientific conclusions or making claims based on observational comparisons. Possible tensions of interest include the following:
1. The simulated stars in Milky Way-like galaxies are too alpha-enhanced in comparison to observations of the Milky Way [@naiman18].
2. The Eddington ratio distributions of massive blackholes ($> 10^9\, {\rm M}_{\odot}$) at $z=0$ are dominated in TNG by low accretion rates, generically far below the Eddington limit; recent observations favor at least some fraction of higher accretion rate massive blackholes. This is reflected in a steeper hard X-ray AGN luminosity function at $1 \lesssim z \lesssim 4$ [@habouzit18].
3. TNG galaxies may have a weaker connection between galaxy morphology and color than observed at $z\sim0$, reflected in a possible excess of red disk-like galaxies in the simulations [@rodriguezgomez18], although see .
4. TNG galaxies exhibit a somewhat sharper trend than observations in quenched fraction vs. galaxy stellar mass for $M_\star \in 10^{10-11}{\rm M}_{\odot}$ [@donnari19], and similarly in the relation between sSFR and $M_{\rm BH}$ at low redshift ().
5. The locus of the galaxy star-forming main sequence is below the face-value observed SFMS at $1 \lesssim z \lesssim 2$, modulo known inconsistencies with e.g. the observed stellar mass function [@donnari19].
6. Similarly, the H$_2$ mass content of massive TNG galaxies at $z=1-3$ may be lower than implied by ALMA observations [@popping19] and sub-mm galaxy demographics ().
7. The DM fractions within massive elliptical galaxies at $z=0$ are consistent with observations at large galactocentric distances, but may be too high within their effective radii [@lovell18] and likewise are tentatively higher than values inferred from observations of massive $z=2$ star forming galaxies [@lovell18 and ].
With respect to points (III)$-$(IV) there is, in general, an interesting transitional mass regime (galaxy stellar mass $\sim 10^{10.5} {\rm M}_{\odot}$) where central blue vs. red galaxies or star-forming vs. quiescent galaxies co-exist: this reflects the effective quenching mechanism of the TNG model based on SMBH feedback [@nelson18a; @weinberger18] but how precisely such transitional galaxies differ also in other structural and kinematical properties still requires careful examination and consideration.
Note that for the items in this list we have not included more specific quantification of observed tension (i.e. $\chi^2$ or fractional deviation values) – the referenced papers contain more discussion. On the one side, not all observational results are in agreement among each other, making quantitative statements necessarily partial; nor observational statements of different galaxy properties are necessarily consistent within one another, especially across cosmic times. On the other side, excruciating care is often necessary to properly map simulated variables into observationally-derived quantities.
Numerical Considerations and Issues
-----------------------------------
To better inform which features of the simulations are robust when making science conclusions, we note the following points related to numerical considerations:
**1. SubhaloFlag.** Not all objects in the Subfind group catalogs should be considered ‘galaxies’. In particular, not all satellite subhalos have a cosmological origin, in the sense that they may not have formed due to the process of structure formation and collapse. Rather, some satellite subhalos will represent fragments or clumps produced through baryonic processes (e.g. disk instabilities) in already formed galaxies, and the Subfind algorithm cannot *a priori* differentiate between these two cases. Such non-cosmological objects are typically low mass, small in size, and baryon dominated (i.e. with little or no dark matter), residing at small galactocentric distances from their host halos, preferentially at late times ($z<1$). These objects may appear as outliers in scatter plots of typical galaxy scaling relations, and should be considered with care.
We have added a [SubhaloFlag]{} field to the group catalogs to assist in their identification, which was constructed as follows. First, a variant of the SubLink merger tree was used which tracks baryonic, rather than dark matter, particles – namely, star-forming gas cells and stars [@rodriguezgomez15]. The algorithm is otherwise the same, with the same weighting scheme for determining descendants/progenitors, except that this “SubLinkGal” tree allows us to track subhalos which contain little or no dark matter.
Then, we flag a subhalo as non-cosmological if all the following criteria are true: (i) the subhalo is a satellite at its time of formation, (ii) it forms within 1.0 virial radii of its parent halo, and (iii) its dark matter fraction, defined as the ratio of dark matter mass to total subhalo mass, at the time of formation of the subhalo, is less than 0.8.
These are relatively conservative choices, implying a low false positive rate. On the other hand, some spurious subhalos may not be flagged under this definition. A much more aggressive criterion would be to flag a subhalo if its instantaneous dark matter fraction is low, e.g. less than 10% [as used in e.g. @genel18; @pillepich18a]. Such a selection will result in a purer sample, with less contaminating subhalos, but will also exclude more genuine galaxies, such as those which have undergone extensive (i.e. physical) stripping of their dark matter component during infall. We encourage users to enforce the provided SubhaloFlag values as a default, but to carefully consider the implications and details, particularly for analyses focused on satellite galaxy populations or dark-matter deficient systems.
**2. Gas InternalEnergy Corrections.** In all TNG simulations, the time-variable UV-background radiation field [@fg09 FG11 version] is enabled only for $z < 6$. Therefore, the ionization state of the IGM above redshift six should be studied with caution, as the usual density-temperature relation will not be present. Two further technical issues exist for the original [InternalEnergy]{} field (i.e. gas temperature) of all TNG simulations. These have been corrected in post-processing, as described below, and the fiducial [InternalEnergy]{} field of all snapshots in all TNG simulations has been rewritten with updated values. The original dataset has been renamed to [InternalEnergyOld]{} for reference, although we do not recommend its use for any purpose.
The first issue is seen in the low-density, low-temperature regime of the intergalactic medium (IGM). Here, due to a numerical issue in the TNG codebase related to the Hubble flow across gas cells, spurious energy injection could occur in underdense gas. In practice, this only affects extremely low density IGM gas in equilibrium between adiabatic cooling and photoheating from the background. The result is a slight upwards curvature in the usual $(\rho,T)$ phase diagram. To correct this issue, we have used one of the TNG model variant boxes (with side length $25 \rm{Mpc}/h$ and $512^3$ resolution) which includes the fix for this issue. The adiabat was then identified in all TNG runs as well as in the corrected simulation by binning the density-temperature phase diagram and locating the temperature of peak gas mass occupancy as a function of density. A multiplicative correction $f_{\rm corr}$, taken as the ratio between the corrected and uncorrected linear gas temperatures, is then defined and applied as a function of density, for gas with physical hydrogen number density $< 10^{-6} a^{-3} \,\rm{cm}^{-3}$. We further restrict the correction to the low-temperature IGM by smoothly damping $f_{\rm corr}$ to unity by $10^{5.0}$ K as $\log T_{\rm corr} = \log T_{\rm orig} + \log(f_{\rm corr}) w(T_{\rm orig})$ with the window function $w(T_{\rm orig}) = 1 - [\rm{tanh}(5(T_{\rm orig}-5.0))+1]/2$. This issue manifests only towards low redshift, and for simplicity and clarity we apply this correction only for $z \leq 5$ (snapshots 17 and later).
The second issue arises for a very small fraction of low-temperature gas cells with $T < 10^4\,\rm{K}$, the putative cooling floor of the model. Here, due to a numerical issue in the TNG codebase related to the cosmological scaling of the energy source term in the Powell divergence cleaning of the MHD scheme [right-most term in Eqn. 21 of @pakmor13], spurious cooling could occur in gas with high bulk velocity and large, local divergence error ($|\nabla \vec{B}| > 0$). In practice, this affects a negligible number of cells which appear in the usual $(\rho,T)$ phase diagram with temperatures less than 10,000 K and densities between the star-formation threshold and four orders of magnitude lower. To correct this issue we simply update the gas temperature values, for all cells in this density range with $\log(T \rm{[K]}) < 3.9$, to the cooling floor value of $10^{4}\,\rm{K}$, near the background equilibrium value. As this issue also manifests only towards low redshift (being most problematic at intermediate redshifts $1 \lesssim z \lesssim 4$), we likewise apply this correction only for $z \leq 5$ (snapshots 17 and later).
Note that for both issues, we have verified in reruns of smaller volume simulations, by applying the fix in correspondingly corrected TNG model variant simulations, that no properties of galaxies or of the galaxy population are noticeably affected by these fixes.
**3. Unresolved ISM.** The multi-phase model of the interstellar medium in TNG (which is the same as in Illustris) is a necessarily coarse approximation of a complex physical regime. In particular, the cold neutral and molecular phases of the ISM are not resolved in the current generation of cosmological simulations like TNG; giant molecular clouds (GMCs) and the individual birth sites of massive star formation and, for example, the resultant nebular excitation is likewise not explicitly resolved. Modeling observables which arise in dense ISM phases (e.g. CO masses) should be undertaken with care.
The modeling of the star formation process is explicitly designed to reproduce the empirical Kennicutt-Schmidt relation, so the correlation between star formation rate and gas density, at the scale where this scaling is invoked, is not a predictive result. Star formation, as in all computational models of galaxy formation, proceeds at a numerical threshold density which is many orders of magnitude lower than the true density at which stars form. This threshold is $n_{\rm H} \simeq 0.1 \rm{cm}^{-3}$ in TNG, which may have consequences for the spatial clustering of young stars, as one example [@buck18].
**4. Resolution Convergence.** Numerical convergence is a complex issue, and working with simulations at multiple resolutions can be challenging. Analysis which includes more than one TNG box at once (e.g. TNG100 and TNG300 together), or explicitly uses multiple realizations at different resolutions should carefully consider the issue of convergence. The degree to which various properties of galaxies or the simulation as a whole is converged depends on the specific property, as well as the mass regime, redshift, and so on. For example, see [@pillepich18a] for convergence of the stellar mass functions of TNG100 and TNG300, and details on a simple ‘resolution correction’ procedure which may be desirable to apply, particularly when combining the results of multiple flagship boxes together into a single analysis.
Community Considerations {#sCommunity}
========================
Citation Request
----------------
To support proper attribution, recognize the effort of individuals involved, and monitor ongoing usage and impact, the following is requested. Any publication making use of data from the TNG100/TNG300 simulations should cite this release paper [@nelson19a] as well as the five works from the “introductory paper series” of TNG100/300, the order being random:
- [@pillepich18b] (stellar contents),
- [@springel18] (clustering),
- [@nelson18a] (colors),
- [@naiman18] (chemical enrichment),
- [@marinacci18] (magnetic fields).
Any publication making use of the data from the TNG50 simulation should cite this release paper, as well as the two introductory papers of TNG50, the order being random:
- [@nelson19b] (outflows),
- [@pillepich19] (structure & kinematics).
Finally, use of any of the supplementary data products should include the relevant citation. A full and up to date list is maintained on the TNG website.
Collaboration and Contributions
-------------------------------
The full snapshots of TNG100-1, and especially those of TNG300-1, are sufficiently large that it may be prohibitive for most users to acquire or store a large number. We note that transferring $\sim\,$1.5 TB (the size of one full TNG100-1 snapshot) at a reasonably achievable 10 MB/s will take roughly 48 hours, increasing to roughly five days for a $\sim\,$4.1 TB full snapshot of TNG300-1. As a result, projects requiring access to full simulation datasets, or extensive post-processing computations beyond what are being made publicly available, may benefit from closer interaction with members of the TNG collaboration.
We also welcome suggestions, comments, or contributions to the data release effort itself. These could take the form of analysis code, derived data catalogs, etc. For instance, interesting data products can be released as a “supplementary data catalog”. Fast analysis routines which operate on individual halos/subhalos can be integrated into the API, such that the result can be requested on demand for any object.
Future Data Releases
--------------------
We anticipate to release additional data in the future, for which further documentation will be provided online.
### Rockstar and Velociraptor
We plan to derive and release different group catalogs, based on the [Rockstar]{} [@behroozi13] and [Velociraptor]{} [@elahi11] algorithms, and will provide further documentation at that time. Such group catalogs will identify different subhalo populations than found by the Subfind algorithm, particularly during mergers. The algorithm used to construct the ‘Consistent Trees’ assembly histories also has fundamental differences to both [LHaloTree]{} and [SubLink]{}. This can provide a powerful comparison and consistency check for any scientific analysis. We also anticipate that some users will simply be more familiar with these outputs, or need them as inputs to other tools.
### Additional Simulations
The flagship volumes of the IllustrisTNG – TNG50, TNG100, and TNG300 – are accompanied by an additional resource: a large number of ‘TNG Model Variation’ simulations. Each modifies exactly one choice or parameter value of the base, fiducial TNG model. The variations cover every physical aspect of the model, including the stellar and blackhole feedback mechanisms, aspects of the star formation, as well as numerical parameters. They are invaluable in assessing the robustness of a physical conclusion to model changes, as well as in diagnosing the underlying cause or mechanism responsible for a given feature in the primary simulations. They were first presented in the [@pillepich18a] TNG methods paper, and used for example in [@nelson18b] to understand the improvement in OVI column densities, in [@lovell18] to study the impact of baryons on dark matter fractions, and in to probe the origin of quenched galaxies in the TNG model.
Each of the $\sim\,$100 TNG model variants simulates the exact same $25 {\rm Mpc}/h$ volume at a resolution approximately equivalent to the flagship TNG100-1. Individual halos can also therefore be cross-matched between the simulations, although the statistics is necessarily limited by the relatively small volume. We plan to publicly release these variations in the near future, and encourage those interested to get in touch in the meantime.
Finally, we anticipate that ongoing and future simulation projects will also be released through this platform in the future. Most notably, this includes the high-resolution TNG50 simulation [@pillepich19; @nelson19b], the third and final volume of the IllustrisTNG project, and potentially other simulations as well.
### API Functionality Expansion
There is significant room for the development of additional features in the web-based API. In particular, for (i) on-demand visualization tasks, (ii) on-demand analysis tasks, and (iii) client-side, browser based tools for data exploration and visualization. We have two specific services which are anticipated to be developed in the near-term future and made available.
First, the on-demand generation of ‘zoom’ initial conditions (ICs), for individual galaxies/halos, based on any object of interest selected from any simulation box. This will allow a user to select a sample of galaxies, perhaps in analogy to an observed sample, or with a peculiar type of assembly history, and obtain ICs for resimulation. Such resimulations could use other codes or galaxy formation models, or explore model parameter variations, to assess how such changes affected a particular galaxy/halo, or class of galaxies/halos. As IC generation will take several minutes at least, it does not fit within our current framework of ‘responses within a few seconds’, and therefore requires a task-based work queue with delayed execution and subsequent notification (e.g. via email) of completion and the availability of new data products for download.
Second, the on-demand execution of longer running analysis tasks, with similar notification upon completion. Specifically, the ability to request SKIRT radiative-transfer calculations for specific galaxies/halos of interest, leveraging the development efforts of [@rodriguezgomez18]. Other expensive mocks, such as spectral HI [with <span style="font-variant:small-caps;">MARTINI</span>; @oman19] or x-ray datacubes, or intergalactic quasar absorption sightlines, can similarly be generated.
We welcome community input and/or contributions in any of these directions, or comments related to any aspect of the public data release of TNG.
Architectural and Design Details {#sImplementation}
================================
In the development of the original Illustris public data release, many design decisions were made, including technical details related to the release effort, based on expected use cases and methods of data analysis. [@nelson15b] discusses the architectural goals and considerations that we followed and continue to follow with the IllustrisTNG data release, and contrasts against other methodologies, as implemented in other astrophysics simulation data releases. We refer the reader to that paper and present only a few updates here.
Usage of the Illustris Public Data Release
------------------------------------------
Since its release, the original Illustris public data release has seen widespread adoption and use. To date, in the three and a half years since launch, 2122 new users have registered and made a total of 269 million API requests, including 2.7 million ‘mock FITS’ file downloads. For the flagship Illustris-1 run, a total of 1390 full snapshots, 6650 group catalogs, and 180 merger trees have been downloaded. 26 million subhalo ‘cutouts’ of particle-level data, and 3.1 million [SubLink]{} merger tree extractions have been requested. The total data transfer for this simulation to date is $\simeq\,$2.15 PB. Roughly 3100 subbox snapshots of Illustris-1 have been downloaded. The next most accessed simulation is Illustris-3, likely because it is included in the getting started tutorials as an easy, lightweight alternative to Illustris-1. Since launch, there has been a nearly constant number of $\sim 100 - 120$ active users, based on activity within the last 30 days.
To date, 163 publications have directly resulted from, or included analysis results from, the Illustris simulation. While early papers were written largely by the collaboration itself, recent papers typically do not involve members of the Illustris team, representing widespread public use of the data release. Of the 10 most recent papers published on Illustris, only one was from the team. Given the significantly expanded scope of TNG with respect to Illustris, as well as the relatively more robust and reliable physical model and outcomes, we expect that uptake and usage will be similarly broad.
New JupyterLab Interface
------------------------
In the original Illustris data release, we promoted two ways to work with the data: either downloading large simulation data files directly (referred to above as ‘local data, local compute’), or by searching and downloading data subsets using functionality in the web-based API (‘remote data, local compute’). Previously, the backend was focused solely on storage and data delivery, and did not have any system in place to allow guest access to compute resources which were local to the datasets themselves. For the TNG data release we have developed this functionality.
We label this newly introduced, third method of working with the data ‘remote data, remote computation’. Technically, we make use of JupyterHub to manage the instantiation of per-user JupyterLab instances. These are spawned inside containerized Docker instances [@merkel14] to isolate the user from the host systems – Singularity [@kurtzer17] could be used in the future. Read-only mounts to the parallel filesystems hosting simulation data are provided, while the user home directory within the container is made persistent by volume mapping it onto the host. Resource limits on CPU, memory, and storage are controllable and will be adjusted during the initial phase of this service as needed.
The JupyterLab instances themselves provide a familiar environment for the development and execution of user analysis programs. Over the past few years there has been significant recent development on remote, multi-user, rich interfaces to computational kernels, and JupyterLab [the successor of Jupyter, previously called IPython; @perez07] is a mature, full-featured solution we deploy. These instances are launched, on demand, inside the sand-boxed containers, through a web-based portal with authentication integrated into the existing user registration system of the data release. We anticipate that this will be a particularly interesting development for researchers who would otherwise not have the computational resources to use the simulation data for their science.
Retiring the Relational Database
--------------------------------
In the original Illustris data release we noted that the read-only, highly structured nature of simulation output motivates different and more efficient approaches for data search, aggregation, processing, and retrieval tasks. The web-based API uses a representational state transfer architecture [REST, @fielding00], and in TNG we continue to employ a relational database on the backend, although we made a design decision never to expose such a database to direct user query.
Looking forward, instead of bringing the object or group catalog data into a traditional database, one could employ a scheme such as bitmap indexing over HDF5, e.g. [FastQuery](http://www-vis.lbl.gov/Events/SC05/HDF5FastQuery/) [@chou11; @byna12], possibly combined with a SQL-compatible query layer [@wang13]. In this case, the database would be used only to handle light meta-data – fast index-accelerated search queries would be made directly against structured binary data on disk. This improvement would be largely transparent from the user perspective. Most obviously, it would remove a layer of complexity and the need to ingest of order billions of rows of group catalog data into a database. It would also enable a tighter coupling of search capabilities and on-disk data contents.
More efficient API standards such as [GraphQL](https://graphql.org/) represent modern alternatives to REST, whereby users make specific, detailed requests to a single endpoint based on a well-defined query language and typed schema, rather than a number of generic requests to a diversity of endpoints. Resolving these declarative queries efficiently and directly on the simulation output data would unify many of these goals – a clear target for future development.
Summary and Conclusions {#sConclusions}
=======================
We have made publicly available data from the IllustrisTNG simulation project at the permanent URL:
- <http://www.tng-project.org/data/>
IllustrisTNG is a series of large-scale, cosmological simulations ideal for studying the formation and evolution of galaxies. The simulation suite consists of three volumes: TNG50, TNG100, and TNG300. Each flagship run is accompanied by lower-resolution realizations, and a dark-matter only analog of every simulation is also available. The current data release includes TNG100 and TNG300 in their entirety, and TNG50 will be publicly released in the future. Full snapshots, group catalogs (both friends of friends halos and [SubFind]{} subhalos), merger trees, high time-resolution subboxes, and many supplementary data catalogs are made available. The highest resolution TNG300-1 includes more than ten million gravitationally bound structures, and the TNG100-1 volume contains $\sim$20,000 well-resolved galaxies at $z=0$ with stellar mass exceeding $10^{9} {\rm M}_\odot$. The galaxies sampled in these volumes encompass a broad range of mass, type, environment and assembly history, and realize fully representative synthetic universes within the context of $\Lambda$CDM.
The total data volume produced by the Illustris\[TNG\] project is sizeable, $\sim$1.1 PB in total, all directly accessible online. We have developed several tools to make these data accessible to the broader community, without requiring extensive local computational resources. In addition to direct data download, example scripts, web-based API access methods, and extensive documentation previously developed for the original Illustris simulation, we extend the data access functionality in several ways. Namely, with new on-demand visualization and analysis functionality, and with the remote JupyterLab-based analysis interface. By making the TNG data publicly available, we aim to maximize the scientific return from the considerable computational resources invested in the TNG simulation suite.
Abbreviations {#abbreviations .unnumbered}
=============
Not applicable.
Declarations {#declarations .unnumbered}
============
Availability of Data and Material {#availability-of-data-and-material .unnumbered}
---------------------------------
This paper describes an explicit public data release of the relevant material.
Competing Interests {#competing-interests .unnumbered}
-------------------
The authors declare that they have no competing interests.
Funding {#funding .unnumbered}
-------
Not applicable.
Author’s Contributions {#authors-contributions .unnumbered}
----------------------
DN has designed and carried out the data release, authored the presentation and data distribution websites, developed and maintains the technical infrastructure, and composed this document. VS, AP, RP and DN have carried out the TNG simulations. VS, LH, RW and AP have developed the TNG model with further input from SG, PT, MV, FM, RP and DN. VRG, LK, ML, and BD have contributed additional data products.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Prof. Volker Springel, together with the Max Planck Computing and Data Facility (MPCDF) and the Max Planck Institute for Astrophysics (MPA) in Garching, Germany, for significant computational resources and ongoing support, without which the TNG public data release would not be possible. We also thank Prof. Lars Hernquist, together with the Research Computing group of Harvard University, for significant computational resources and continuous support, without which the original Illustris public data release and ongoing activities would not be possible.
In addition, we thank the Center for Computational Astrophysics and the Simons Foundation for a future commitment to host a full mirror of the Illustris\[TNG\] data, particularly Dylan Simon and the Scientific Computing Core at the Flatiron Institute for orchestration efforts. In addition, we thank Greg Snyder and the Mikulski Archive for Space Telescopes (MAST) at STScI for future plans to host relevant datasets based on TNG.
The authors would like to thank many additional people for contributing to analysis and understanding of the TNG simulations and their results: Maria Celeste Artale, Kelly Blumenthal, David Barnes, Martina Donnari, Elena D’Onghia, Min Du, Catalina Gomez, Hong Guo, Anshu Gupta, Melanie Habouzit, Chris Hayward, Ghandali Joshi, Guinevere Kauffmann, Davide Martizzi, Michelle Ntampaka, Ana-Roxana Pop, Gerg[ö]{} Popping, Malin Renneby, Greg Snyder, Adam Stevens, Sandro Tacchella, Bryan Terrazas, Nhut Truong, Hannah [Ü]{}bler, Francisco Villaescusa-Navarro, Sarah Wellons, Po-Feng Wu, Dandan Xu, Kiyun Yun, Qirong Zhu, Elad Zinger, and Jolanta Zjupa.
In the execution of the primary simulations presented herein, the authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for providing computing time for the GCS Large-Scale Projects GCS-ILLU (2014) and GCS-DWAR (2016) on the GCS share of the supercomputer Hazel Hen at the High Performance Computing Center Stuttgart (HLRS). GCS is the alliance of the three national supercomputing centres HLRS (Universit[ä]{}t Stuttgart), JSC (Forschungszentrum J[ü]{}lich), and LRZ (Bayerische Akademie der Wissenschaften), funded by the German Federal Ministry of Education and Research (BMBF) and the German State Ministries for Research of Baden-W[ü]{}rttemberg (MWK), Bayern (StMWFK) and Nordrhein-Westfalen (MIWF). Additional simulations were carried out on the Hydra and Draco supercomputers at the Max Planck Computing and Data Facility (MPCDF, formerly known as RZG), as well as on the Stampede supercomputer at the Texas Advanced Computing Center through the XSEDE project AST140063. Some additional computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
Appendix A: Simulation Data Details {#appendix-a-simulation-data-details .unnumbered}
===================================
Run Alternate Name Total $N_{\rm DM}$ N$_{\rm chunks}$ Full Snapshot Size Avg Groupcat Size Total Data Volume
------------------ ---------------- -------------------- ------------------ -------------------- ------------------- -------------------
L35n270TNG TNG50-4 19,683,000 11 5.2 GB 20 MB 0.6 TB
L35n270TNG\_DM TNG50-4-Dark 19,683,000 4 1.2 GB 10 MB 0.1 TB
L35n540TNG TNG50-3 157,464,000 11 44 GB 130 MB 7.5 TB
L35n540TNG\_DM TNG50-3-Dark 157,464,000 4 9.4 GB 50 MB 0.6 TB
L35n1080TNG TNG50-2 1,259,712,000 128 350 GB 860 MB 18 TB
L35n1080TNG\_DM TNG50-2-Dark 1,259,712,000 85 76 GB 350 MB 4.5 TB
L35n2160TNG TNG50-1 10,077,696,000 680 2.7TB 7.2 GB $\sim$320 TB
L35n2160TNG\_DM TNG50-1-Dark 10,077,696,000 128 600 GB 2.3 GB 36 TB
L75n455TNG TNG100-3 94,196,375 8 27 GB 110 MB 1.5 TB
L75n455TNG\_DM TNG100-3-Dark 94,196,375 4 5.7 GB 40 MB 0.4 TB
L75n910TNG TNG100-2 753,571,000 56 215 GB 650 MB 14 TB
L75n910TNG\_DM TNG100-2-Dark 753,571,000 8 45 GB 260 MB 2.8 TB
L75n1820TNG TNG100-1 6,028,568,000 448 1.7 TB 4.3 GB 128 TB
L75n1820TNG\_DM TNG100-1-Dark 6,028,568,000 64 360 GB 1.7 GB 22 TB
L205n625TNG TNG300-3 244,140,625 16 63 GB 340 MB 4 TB
L205n625TNG\_DM TNG300-3-Dark 244,140,625 4 15 GB 130 MB 1 TB
L205n1250TNG TNG300-2 1,953,125,000 100 512 GB 2.2 GB 31 TB
L205n1250TNG\_DM TNG300-2-Dark 1,953,125,000 25 117 GB 810 MB 7.2 TB
L205n2500TNG TNG300-1 15,625,000,000 600 4.1 TB 14 GB 235 TB
L205n2500TNG\_DM TNG300-1-Dark 15,625,000,000 75 930 GB 5.2 GB 57 TB
Appendix B: Web-Based API Examples {#appendix-b-web-based-api-examples .unnumbered}
==================================
By way of explicit example, the following are absolute URLs for the web-based API which encompass some of its functionality. The type of the request, as well as the data expected in return, should be relatively clear:
- [www.tng-project.org/api/Illustris-2/](www.tng-project.org/api/Illustris-2/)
- [www.tng-project.org/api/Illustris-2/snapshots/68/](www.tng-project.org/api/Illustris-2/snapshots/68/)
- [www.tng-project.org/api/Illustris-1/snapshots/135/subhalos/73664/](www.tng-project.org/api/Illustris-1/snapshots/135/subhalos/73664/)
- [www.tng-project.org/api/Illustris-1/snapshots/80/halos/523312/cutout.hdf5?dm=Coordinates&gas=all](www.tng-project.org/api/Illustris-1/snapshots/80/halos/523312/cutout.hdf5?dm=Coordinates&gas=all)
- [www.tng-project.org/api/Illustris-3/snapshots/135/subhalos?mass\_\_gt=10.0&mass\_\_lt=20.0](www.tng-project.org/api/Illustris-3/snapshots/135/subhalos?mass__gt=10.0&mass__lt=20.0)
- [www.tng-project.org/api/Illustris-2/snapshots/68/subhalos/50000/sublink/full.hdf5](www.tng-project.org/api/Illustris-2/snapshots/68/subhalos/50000/sublink/full.hdf5)
- [www.tng-project.org/api/Illustris-2/snapshots/68/subhalos/50000/sublink/mpb.json](www.tng-project.org/api/Illustris-2/snapshots/68/subhalos/50000/sublink/mpb.json)
- [www.tng-project.org/api/TNG100-2/snapshots/99/subhalos/50000/sublink/mpb.json](www.tng-project.org/api/TNG100-2/snapshots/99/subhalos/50000/sublink/mpb.json)
- [www.tng-project.org/api/TNG300-1/snapshots/99/subhalos?mass\_\_gt=10.0&mass\_\_lt=11.0](www.tng-project.org/api/TNG300-1/snapshots/99/subhalos?mass__gt=10.0&mass__lt=11.0)
- [www.tng-project.org/api/Illustris-2/files/ics.hdf5](www.tng-project.org/api/Illustris-2/files/ics.hdf5)
- [www.tng-project.org/api/Illustris-1/files/groupcat-135.5.hdf5](www.tng-project.org/api/Illustris-1/files/groupcat-135.5.hdf5)
- [www.tng-project.org/api/Illustris-2/files/snapshot-135.10.hdf5?dm=all](www.tng-project.org/api/Illustris-2/files/snapshot-135.10.hdf5?dm=all)
- [www.tng-project.org/api/TNG100-1/snapshots/50/subhalos/plot.png?xQuant=mstar2\_log&yQuant=ssfr](www.tng-project.org/api/TNG100-1/snapshots/50/subhalos/plot.png?xQuant=mstar2_log&yQuant=ssfr)
- [www.tng-project.org/api/TNG300-1/snapshots/7/subhalos/plot.png?xQuant=mstar2&yQuant=delta\_sfms&cQuant=Z\_gas](www.tng-project.org/api/TNG300-1/snapshots/7/subhalos/plot.png?xQuant=mstar2&yQuant=delta_sfms&cQuant=Z_gas)
- [www.tng-project.org/api/TNG100-1/snapshots/99/halos/320/vis.png](www.tng-project.org/api/TNG100-1/snapshots/99/halos/320/vis.png)
- [www.tng-project.org/api/TNG100-2/snapshots/67/halos/0/vis.png?partType=gas&partField=temp](www.tng-project.org/api/TNG100-2/snapshots/67/halos/0/vis.png?partType=gas&partField=temp)
A ‘getting started’ guide for the web-based API is available in the online documention, and this includes a cookbook of common analysis tasks (available in Python, IDL, and Matlab). To give a sense of this method of analyzing TNG data, we include here four short examples, in Python. Each uses of the [get()]{} helper function which performs the actual HTTP request, automatically parsing JSON-type returns.
**Task 1:** For TNG300-1 at $z=0$, get all the information available for the ID = 0 subhalo, print both its total mass and stellar half mass radius.
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{url} \PY{o}{=} \PY{l+s}{\PYZdq{}}\PY{l+s}{http://www.tng\PYZhy{}project.org/api/TNG300\PYZhy{}1/snapshots/99/subhalos/0/}\PY{l+s}{\PYZdq{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{r} \PY{o}{=} \PY{n}{get}\PY{p}{(}\PY{n}{url}\PY{p}{)}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{r}\PY{p}{[}\PY{l+s}{\PYZsq{}}\PY{l+s}{mass}\PY{l+s}{\PYZsq{}}\PY{p}{]}
\PY{l+m+mf}{128335.0}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{r}\PY{p}{[}\PY{l+s}{\PYZsq{}}\PY{l+s}{halfmassrad\PYZus{}stars}\PY{l+s}{\PYZsq{}}\PY{p}{]}
\PY{l+m+mf}{130.065}
**Task 2:** For TNG100-1 at $z=2$, search for all subhalos with total mass $10^{12.1} {\rm M}_\odot < M < 10^{12.2} {\rm M}_\odot$ and print the [Subfind]{} IDs of the first five results.
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{c}{\PYZsh{} convert from log solar masses to group catalog units}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{mass\PYZus{}min} \PY{o}{=} \PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{l+m+mf}{12,1} \PY{o}{/} \PY{l+m+mf}{1e10} \PY{o}{*} \PY{l+m+mf}{0.6774}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{mass\PYZus{}max} \PY{o}{=} \PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{l+m+mf}{12.2} \PY{o}{/} \PY{l+m+mf}{1e10} \PY{o}{*} \PY{l+m+mf}{0.6774}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{params} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s}{\PYZsq{}}\PY{l+s}{mass\PYZus{}\PYZus{}gt}\PY{l+s}{\PYZsq{}}\PY{p}{:}\PY{n}{mass\PYZus{}min}\PY{p}{,} \PY{l+s}{\PYZsq{}}\PY{l+s}{mass\PYZus{}\PYZus{}lt}\PY{l+s}{\PYZsq{}}\PY{p}{:}\PY{n}{mass\PYZus{}max}\PY{p}{\PYZcb{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{c}{\PYZsh{} make the request}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{url} \PY{o}{=} \PY{l+s}{\PYZdq{}}\PY{l+s}{http://www.tng\PYZhy{}project.org/api/TNG100\PYZhy{}1/snapshots/z=2/subhalos/}\PY{l+s}{\PYZdq{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{subhalos} \PY{o}{=} \PY{n}{get}\PY{p}{(}\PY{n}{url}\PY{p}{,} \PY{n}{params}\PY{p}{)}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{ids} \PY{o}{=} \PY{p}{[} \PY{n}{subhalos}\PY{p}{[}\PY{l+s}{\PYZsq{}}\PY{l+s}{results}\PY{l+s}{\PYZsq{}}\PY{p}{]}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{l+s}{\PYZsq{}}\PY{l+s}{id}\PY{l+s}{\PYZsq{}}\PY{p}{]} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{5}\PY{p}{)} \PY{p}{]}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{ids}
\PY{p}{[}\PY{l+m+mi}{13845}\PY{p}{,} \PY{l+m+mi}{16799}\PY{p}{,} \PY{l+m+mi}{23224}\PY{p}{,} \PY{l+m+mi}{24400}\PY{p}{,} \PY{l+m+mi}{12718}\PY{p}{]}
**Task 11:** Download the entire TNG300-1 $z=0$ snapshot including *only the positions, masses, and metallicities of stars* (in the form of 600 HDF5 files). In this example, since we only need these three fields for stars only, we can reduce the download and storage size from $\sim$4.1TB to $\sim$20GB.
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{base\PYZus{}url} \PY{o}{=} \PY{l+s}{\PYZdq{}}\PY{l+s}{http://www.tng\PYZhy{}project.org/api/TNG300\PYZhy{}1/}\PY{l+s}{\PYZdq{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{sim\PYZus{}metadata} \PY{o}{=} \PY{n}{get}\PY{p}{(}\PY{n}{base\PYZus{}url}\PY{p}{)}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{params} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s}{\PYZsq{}}\PY{l+s}{stars}\PY{l+s}{\PYZsq{}}\PY{p}{:}\PY{l+s}{\PYZsq{}}\PY{l+s}{Coordinates,Masses,GFM\PYZus{}Metallicity}\PY{l+s}{\PYZsq{}}\PY{p}{\PYZcb{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{sim\PYZus{}metadata}\PY{p}{[}\PY{l+s}{\PYZsq{}}\PY{l+s}{num\PYZus{}files\PYZus{}snapshot}\PY{l+s}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{:}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{file\PYZus{}url} \PY{o}{=} \PY{n}{base\PYZus{}url} \PY{o}{+} \PY{l+s}{\PYZdq{}}\PY{l+s}{files/snapshot\PYZhy{}99.}\PY{l+s}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n}{i}\PY{p}{)} \PY{o}{+} \PY{l+s}{\PYZdq{}}\PY{l+s}{.hdf5}\PY{l+s}{\PYZdq{}}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{n}{saved\PYZus{}filename} \PY{o}{=} \PY{n}{get}\PY{p}{(}\PY{n}{file\PYZus{}url}\PY{p}{,} \PY{n}{params}\PY{p}{)}
\PY{o}{\PYZgt{}\PYZgt{}}\PY{o}{\PYZgt{}} \PY{k}{print}\PY{n}{(saved\PYZus{}filename)}
[^1]: [www.tng-project.org](www.tng-project.org)
[^2]: [www.illustris-project.org](www.illustris-project.org)
[^3]: [www.tng-project.org/results](www.tng-project.org/results)
[^4]: \[newfeat\]New feature in the TNG data release.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Jing Huang and A. Lee Swindlehurst [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'mybibfile.bib'
title: Cooperative Jamming for Secure Communications in MIMO Relay Networks
---
Wiretap channel, secrecy, relay networks, physical layer security, jamming, interference.
[^1]: This work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant W911NF-07-1-0318. Part of the results in this work have been presented in IEEE GLOBECOM 2010 and ICASSP 2011.
[^2]: The authors are with the Dept. of Electrical Engineering & Computer Science, University of California, Irvine, CA 92697-2625, USA. (Email: {jing.huang; swindle}@uci.edu).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Bas Edixhoven
title: 'On the $p$-adic geometry of traces of singular moduli'
---
The aim of this article is to show that $p$-adic geometry of modular curves is useful in the study of $p$-adic properties of *traces* of singular moduli. In order to do so, we partly answer a question by Ono ([@Ono1 Problem 7.30]). As our goal is just to illustrate how $p$-adic geometry can be used in this context, we focus on a relatively simple case, in the hope that others will try to obtain the strongest and most general results. For example, for $p=2$, a result stronger than Thm. \[thm1\] is proved in [@Boylan1], and a result on some modular curves of genus zero can be found in [@Osburn1] . It should be easy to apply our method, because of its local nature, to modular curves of arbitrary level, as well as to Shimura curves.
\[defi1\] For $d$ a positive integer that is congruent to $0$ or $3$ mod $4$, let $O_d$ be the quadratic order of discriminant $-d$. For $f$ in ${{\mathbb Z}}[j]$ and $E$ an elliptic curve over some ring $R$, let $f(E)$ be the element of $R$ obtained by evaluating the $f$ on the $j$-invariant of $E$. For such $d$ and $f$, let: $$t_f(d) := \sum_{{{\rm End}}(E)\supset O_d} 2f(E)/\#{{\rm Aut}}(E),$$ where the sum ranges over the set of isomorphism classes of complex elliptic curves whose ring of endomorphisms contains $O_d$. We also define an integer $\alpha(d)$ to be $2$ if ${{\mathbb Q}}(\sqrt{-d})={{\mathbb Q}}(\sqrt{-1})$, $3$ if ${{\mathbb Q}}(\sqrt{-d})={{\mathbb Q}}(\sqrt{-3})$, and $1$ otherwise.
For $d$ as above and $m$ a positive integer, the number $t_m(d)$ defined in [@Ono1] is obtained by taking $f:=(j-744)|T_0(m)$.
\[thm1\] Let $d>0$ be an integer that is congruent to $0$ or $3$ modulo $4$, and let $f$ be in ${{\mathbb Z}}[j]$. Let $p$ be a prime not dividing $d$ that splits in ${{\mathbb Q}}(\sqrt{-d})$, and let $n\geq1$. Then $\alpha(d)t_f(p^{2n}d)$ is an integer, and $\alpha(d)t_f(p^{2n}d)\equiv 0$ mod $p^n$.
All that we need from the local moduli theory of ordinary elliptic curves in positive characteristic is summarized in the following proposition. Definitions for the terms occurring in it can be found in [@Silverman1] and [@Katz1].
\[prop2\] Let $p$ be a prime number, $k$ a finite field of characteristic $p$. Let $E_0$ be an ordinary elliptic curve over $k$, and let $A$ be its endomorphism ring. Then $A$ is an order in a quadratic extension of ${{\mathbb Q}}$, and $A$ is split at $p$: ${{\mathbb Z}}_p\otimes
A$ is isomorphic to ${{\mathbb Z}}_p\times {{\mathbb Z}}_p$; in particular, $A$ is maximal at $p$.
Let $k\to {\overline}{k}$ be an algebraic closure, and let $W$ be the ring of Witt vectors of $k$. Let $E/R$ be the universal deformation of $E_0/{\overline}{k}$ over $W$-algebras. Let $\alpha\colon {{\mathbb Q}}_p/{{\mathbb Z}}_p\to
E_0({\overline}{k})[p^\infty]$ be a trivialisation of the group of torsion points of $p$-power order of $E_0({\overline}{k})$. Then $\alpha$ induces a so-called Serre-Tate parameter $q\in R^*$, and $R=W[[q-1]]$. For $n\geq 0$, let $A_n$ be the subring ${{\mathbb Z}}+p^nA$ of $A$, i.e., the order of index $p^n$ in $A$. Then the closed subscheme of $\operatorname{Spec}R$ over which all elements of $A_n$ lift as endomorphisms of $E$ is the closed subscheme defined by the equation $q^{p^n}=1$.
The endomorphism ring $A$ is free of finite rank as a ${{\mathbb Z}}$-module, and is an integral domain because each non-zero element in it is surjective as a morphism from $E_0$ to itself. The $p$-divisible group $E_0[p^\infty]$ is the direct sum of its local and etale parts $E_0[p^\infty]^0$ and $E_0[p^\infty]^{\mathrm{et}}$, hence its endomorphism ring is the ${{\mathbb Z}}_p$-algebra ${{\mathbb Z}}_p\times{{\mathbb Z}}_p$. Then $A$ is commutative because it embeds into ${{\mathbb Z}}_p\times{{\mathbb Z}}_p$. The image of the Frobenius endomorphism of $E_0$ is of the form $(p^mu,v)$, with $u$ and $v$ in ${{\mathbb Z}}_p^*$, and $|k|=p^m$. This proves that $A$ is quadratic over ${{\mathbb Z}}$, and split at $p$.
The construction of $q$ and the statement that $R=W[[q-1]]$ are in [@Katz1 §2]. There it is also shown every $f$ in $A$ determines a closed subscheme $V_f$ of $\operatorname{Spec}R$ given by the condition that $f$ can be lifted as an endomorphism of $E_{V_f}$, and universal for that property. This subscheme $V_f$ coincides with the closed subscheme over which $f$ can be lifted as an endomorphism of $E[p^\infty]$. Let $n\geq0$, and let $V$ be the intersection of the $V_f$ for all $f$ in $A_n$. Then $V$ is defined by the condition that the endomorphism $(p^n,0)$ of $E_0[p^\infty]$ lifts as an endomorphism of the $p$-divisible group. The proof of Part 4 of [@Katz1 Thm 2.1] shows that $V$ is defined by the equation $q^{p^n}=1$.
We can now prove Theorem \[thm1\]. Let $d$, $f$, $p$ and $n$ be as in the statement. For each $E$ with ${{\rm End}}(E)\supset O_{p^{2n}d}$ we have that $\#{{\rm Aut}}(E)/2$ divides $\alpha(d)$, hence $\alpha(d)t_f(p^{2n}d)$ is an integer. Let ${{\overline{{{\mathbb Q}}}}}$ be the algebraic closure of ${{\mathbb Q}}$ in ${{\mathbb C}}$, and choose an embedding of ${{\overline{{{\mathbb Q}}}}}$ into an algebraic closure ${{\overline{{{\mathbb Q}}}}}_p$ of ${{\mathbb Q}}_p$. For $E$ an elliptic curve over ${{\mathbb C}}$ with complex multiplications, let ${\overline}{E}$ denote its reduction over ${{\overline{{{\mathbb F}}}}}_p$. For each $E$ in the sum in Definition \[defi1\] there is a unique $E_0$ such that ${{\rm End}}(E_0)=O_d$ and ${\overline}{E}\cong{\overline}{E_0}$. For each such $E_0$, let ${{\cal E}}(E_0)$ denote the set of $E$ with ${{\rm End}}(E)\supset O_d$ and ${\overline}{E}\cong{\overline}{E_0}$. Then we have: $$\alpha(d)t_f(p^{2n}d) =
\sum_{{{\rm End}}(E_0)=O_d}\frac{2\alpha(d)}{\#{{\rm Aut}}(E_0)}\#{{\rm Aut}}(E_0)
\sum_{E\in{{\cal E}}(E_0)} f(E)/\#{{\rm Aut}}(E).$$ By construction, the $2\alpha(d)/\#{{\rm Aut}}(E_0)$ are integers. Fix an $E_0$ as in the sum, and let $q$ be a $q$-parameter of the deformation space of ${\overline}{E_0}$. Then the relation between deformation spaces and coarse moduli spaces (see [@DeRa I §8.2.1]) and Proposition \[prop2\] imply that: $$\#{{\rm Aut}}(E_0)\sum_{E\in{{\cal E}}(E_0)} f(E)/\#{{\rm Aut}}(E) =
\sum_{x^{p^n}=1}f(x-1),$$ where we can now view $f$ as an element $\sum_{k\geq0}f_kt^k$ of $W[[t]]$, with $t=q-1$ and $W$ the ring of Witt vectors of ${{\overline{{{\mathbb F}}}}}_p$. The observation that $\sum_{x^{p^n}=1}(x-1)^k$ is in $p^nW$ for all $k\geq0$ finishes the proof.
[99]{}
S. Ahlgren and K. Ono. [*Arithmetic of singular moduli and class equations*]{}. To appear in Compositio Mathematica.
M. Boylan. [*$2$-adic properties of Hecke traces of singular moduli.*]{} Preprint, 2004.
P. Deligne, M. Rapoport. [*Les schémas de modules des courbes elliptiques.*]{} In Modular Functions of One Variable II. Springer Lecture Notes in Mathematics 349 (1973).
N. Katz. [*Serre-Tate local moduli.*]{} In “Algebraic surfaces”, Lecture Notes in Mathematics, Vol. 868, 1981.
K. Ono. [*The web of modularity: Arithmetic of the coefficients of modular forms and $q$-series.*]{} Amer. Mat. Soc., CBMS Regional Conf. in Math., vol. 102, 2004.
R. Osburn. [*Congruences for traces of singular moduli.*]{} Preprint, 2004, arxiv math.NT/0412240.
J.H. Silverman. [*The arithmetic of elliptic curves.*]{} Graduate Texts in Mathematics [**106**]{}, Springer-Verlag, 1986.
Bas Edixhoven\
Universiteit Leiden\
Mathematisch Instituut\
Postbus 9512\
2300 RA Leiden\
The Netherlands
[email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'MilkyWay@home is a volunteer computing project that allows people from every country in the world to volunteer their otherwise idle processors to Milky Way research. Currently, more than 25,000 people (150,000 since November 9, 2007) contribute about half a PetaFLOPS of computing power to our project. We currently run two types of applications: one application fits the spatial density profile of tidal streams using statistical photometric parallax, and the other application finds the $N$-body simulation parameters that produce tidal streams that best match the measured density profile of known tidal streams. The stream fitting application is well developed and is producing published results. The Sagittarius dwarf leading tidal tail has been fit, and the algorithm is currently running on the trailing tidal tail and bifurcated pieces. We will soon have a self-consistent model for the density of the smooth component of the stellar halo and the largest tidal streams. The $N$-body application has been implemented for fitting dwarf galaxy progenitor properties only, and is in the testing stages. We use an Earth-Mover Distance method to measure goodness-of-fit for density of stars along the tidal stream. We will add additional spatial dimensions as well as kinematic measures in a piecemeal fashion, with the eventual goal of fitting the orbit and parameters of the Milky Way potential (and thus the density distribution of dark matter) using multiple tidal streams.'
---
Fitting the spatial density of tidal streams with statistical photometric parallax
==================================================================================
It is only in this century that we have known that the Milky Way’s stellar halo has spatial substructure, due to the tidal disruption of dwarf galaxies and globular clusters [@nyetal02; @belokurov (Newberg et al. 2002; Belokurov et al. 2006)]. This discovery was made possible due to the availability of data from high quality, large area surveys of the sky - primarily the Sloan Digital Sky Survey (SDSS; York et al. 2000), but also the Two Micron All Sky Survey (2MASS; Majewski et al. 2003). The spatial substructure was originally found “by eye" in maps of star density, which are constructed from the angular sky positions of the stars in the tracer population, and an estimate of their distance. Streams continue to be found “by eye" even today, though more sophisticated techniques have been developed for pulling out fainter structures by eye [@grillmair4streams (Grillmair 2009)], and more often streams are first detected from velocity substructure [@PSS (Martin et al. 2013)].
Although we have not yet demonstrated successful methods for finding tidal streams in an automated fashion, we have made progress in describing the density structure of individual streams, and in particular the tidal streams from the Sagittarius dwarf galaxy. This has been a difficult task for two reasons: (1) the tidal streams do not have simple geometric shapes, and (2) the highest density tracer populations we have observed are F turnoff stars, which have a large range of absolute magnitudes and are therefore not good indicators of distance.
The second of these problems has been addressed by the technique of statistical photometric parallax [@statphot (Newberg 2013)], in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions that these stars trace. Although the absolute magnitude of the turnoff in general depends on the age and metallicity a stellar population, it turns out that the age-metallicity relationship in the Milky Way leads to a nearly constant absolute magnitude distribution for spheroid turnoff stars [@newby11 (Newby et al. 2011)]; although older populations should have fainter turnoffs, they are also generally more metal-poor, which pushes the turnoff brighter by a nearly equal amount. In addition, there is a fairly uniform distribution of absolute magnitudes in a color-selected sample (assuming negligible color errors) of turnoff stars from the full range of stellar populations in halo globular clusters.
To determine the likelihood of a particular model stellar density, then, one follows the following procedure: (1) For each stellar component, one assumes a parameterized model. (2) The spatial density expected from the model is transformed to $(l,b,g)$ coordinates, where g is the apparent magnitude, assuming all of the stars have the same absolute magnitude. (3) The density is then convolved with the absolute magnitude distribution of the tracer population, which smears the model out along the line-of-sight. (4) The expected distribution is multiplied by the fraction of stars that are expected to be observed, which is usually a function of apparent magnitude (a lower fraction of stars are detected near the survey limit). (5) The resulting distribution is normalized so that the integrated probability of finding a star in the entire observed volume is one. (6) The final probability distribution function (PDF) is the sum of the fraction of stars in each component (these are also model parameters) times the normalized distribution, summed over the number of components in the model. (7) The likelihood of this particular parameterized model is the product of the probability density function evaluations at the location of each observed tracer star in the survey. One must use an optimization technique to find the parameters of the model that best fit the data.
One year after the lumpy nature of the stellar halo was discovered (see [@ntetal02 Newberg et al. 2002], Figure 1), Newberg, Magdon-Ismail and a graduate student attempted to use statistical photometric parallax to determine the density structure of the debris in that dataset. We devised a density model for tidal streams that fit the position, width and density of the stream in $2.5^\circ$-wide segments, matching the width of the SDSS data stripes. This allowed the properties of the stream to vary along its length, and the position and distance of the center of the tidal debris stream to vary arbitrarily across the sky. We eventually determined that using conjugate gradient descent to optimize the likelihood on a single processor would take more than a year. The difficulty was the length of time required to compute the integral of the PDF over the volume observed.
The solution was the parallelization of the algorithm. Then graduate student Travis Desell converted the code to run in parallel on a computer cluster, and on Rensselaer’s IBM BlueGene supercomputer, and on the Berkely Open Infrastructure for Network Computing [@boinc (BOINC, Anderson 2004)]. The algorithm and the proof of concept was published in [@cole08 Cole et al. (2008)]. The density of the Sagittarius (Sgr) dwarf leading tidal tail was published in [@newby13 Newby et al. (2013)]. We did not succeed in automated detection of new tidal streams. However, the maximum likelihood algorithm has given us information about the width of the stream as a function of position along the stream, and a more robust way of calculating the density of stars along the stream, especially in cases where some of the stream stars are missed due to the limiting magnitude of the survey. More importantly, we have a method for separating stars with the density structure of the tidal stream from the smooth component of the stellar halo.
We are currently fitting the Sgr dwarf trailing tidal tail, the bifircated pieces of both the leading and trailing tidal tails, and the Virgo overdensity. We will then fit a smooth model to the remaining spheroid stars that are not in major tidal streams, to genenerate a self-consistent denisty model for the halo.
MilkyWay@home Infrastructure
============================
Our most exciting parallel platform is MilkyWay@home. We began test operations on November 9, 2007. Although most of our volunteers only supply computing power, many are part of a community that follows the scientific progress of a project, participate in a variety of on-line forums, and even help with solving technical and coding issues. A few have donated small amounts of money and hardware. It was a volunteer who first ported our code to GPUs, and showed us the enormous power of these devices. He also caused an outcry on our public forums, since he was able to amass BOINC “credits" at a much faster rate than anyone else. Apparently, the amassing of BOINC credits (which are only posted on the web and cannot be used in any way), is a serious issue. We also have a volunteer moderator who polices the forums to ensure no one is posting inappropriate material, and informs us when the server is malfunctioning. Since inception, 162,382 people from 206 countries (193 members of the United Nations) world-wide have contributed their otherwise unused computer cycles to our project. At any given time, $\sim 25,000$ people from $150$ countries are crunching our work units (Fig. \[fig1\]).
![Worldwide locations of processors donated to MilkyWay@home. This image was generated on June 19th, 2013, using a sample of 10,212 active volunteered computing hosts from MilkyWay@Home, each host having returned valid results within approximately one week. The locations of these hosts’ external IP addresses was determined using a GeoIP database (http://www.hostip.info/), and then plotted on the globe using Google maps (http://maps.google.com/).[]{data-label="fig1"}](Newberg_Heidi_fig1.ps){width="5.25in"}
MilkyWay@home is a tremendous computing resource. We operate a server that generates parameter sets we wish to evaluate, sends one parameter set as a “work unit" to each available volunteer, then receives the result (the likelihood) back from each volunteer when the computation is complete. The result is stored in a database, and another work unit (with a new set of parameters to try) is sent out. It currently delivers about half a PetaFLOPS of computing power, down from a peak of 2 PetaFLOPS shortly after GPU code was released. To put this in perspective, MilkyWay@home had the processing power equivalent to the 45$^{\rm th}$ fastest supercomputer in the world in November 2012.
This computing power comes at a hefty price, however. Our server generates work units (in this case likelihood calculations for one set of model parameters) that are sent out to a very heterogeneous set of processors. We compile our code for sixteen platforms, and respond to questions and problems from our volunteers when there are bugs. It takes us months to release and re-release new algorithms before the bugs are solved on all platforms. In addition to algorithm enhancements, our code must be updated in conjunction with any updates in the BOINC infrastructure. We require fault-tolerant optimization algorithms that work in a highly asynchronous, heterogeneous computing environment. That is, sometimes (either through hardware/software malfunction or malicious intent) the results sent back from a volunteer are incorrect. Also, the time to finish a work unit is highly variable since it depends on the hardware platform that is doing the computation and on the availability of the hardware (which could be turned off for the weekend or working on other tasks). We need to check the validity of a fraction of the work units (by sending them to multiple volunteers) and track the veracity of the results from each volunteer, so that we reject results from consistently incorrect platforms.
The advantages of MilkyWay@home are that we have our very own supercomputer and worldwide outreach program. The only hardware we need to upgrade on a regular basis is a server and a development platform; the volunteers upgrade their hardware at their own expense. We have shown [@newby13 (Newby et al. 2013)] that the results we get from MilkyWay@home are as good, and usually better, than the results we got from one rack of a supercomputer using conjugate gradient descent. The time to converge to the correct solution is about the same for these two platforms, but we can run many more simultaneous jobs on MilkyWay@home (and we don’t have to wait for time in the queue).
Comparing $N$-body simulations to the spatial density of tidal streams
======================================================================
The availability of our own supercomputer has made it possible for us to make bigger plans. We would like to be able to use the tidal debris streams to optimize parameters in the mass distribution of the Milky Way, and learn about the orbital parameters of the dwarf galaxies and globular clusters that are the progenitors of the tidal debris that we see today. Most of the previous work fitting parameters of the Milky Way potential focusses on the Sagittarius dwarf tidal stream [@lm2010 (for example see Law & Majewski 2010)]. [@koposov Koposov, Rix & Hogg (2010)] fit orbits to the stars in the GD-1 stream . [@willettthesis Willett (2010)] attempted to fit halo parameters with simultaneous orbit fits to three tidal streams. Recent work by [@binney; @sanders Binney (2008) and Sanders & Binney (2013)] points out that tidal streams of dwarf galaxies do not follow the orbits of the dwarf galaxy, and they work towards new methods for measuring halo parameters from tidal streams. We believe that with MilkyWay@home we will eventually be able to fit multiple tidal streams to $N$-body simulations to simultaneously constrain the properites of the dwarf galaxies and their orbits, and the distribution of Milky Way dark matter.
We have a test $N$-body simulation running on MilkyWay@home, that uses a version of the [@barnshut Barnes and Hut (1986)] tree code. We currently fit only the properties of the progenitor, which is modeled as a [@plummer Plummer (1911)] sphere with 100,000 bodies. We fit the parameters of both dark matter and stars in the progenitor, and we also fit the simulation time assuming a fixed Milky Way potential, and with a fixed progenitor orbit. The bodies corresponding to stars are sub-selected from the bodies in the original Plummer sphere. We currently fit only the density of stars along the stream (comparing only the stars in the simulation with the observed stars in the stream), using an Earth-Mover Distance method; the similarity of two normalized histograms is measured by the number of items that must be moved and the distances they must move. We also include a cost function for having different numbers of stars in the observed and simulated histograms. In the future we plan to fit the density of the stream in at least two dimensions, and also the radial velocities and velocity dispersions of the stream stars. Recent studies of $N$-body simulations fit by hand to the Cetus Polar Stream [@yamCPS (Yam et al. 2013)] suggest that the dwarf galaxy properties might be better fit if the width or velocity dispersion of the stream is fit in conjunction with the stellar density along the stream.
Future Plans
============
We are developing MilkyWay@home to constrain the potential of the Milky Way galaxy using tidal streams. The application that fits the density distribution of tidal debris is well developed, but we are in the process of implementing an improved algorithm. While the absolute magnitude distributions of F turnoff stars are not expected to change with distance in the Milky Way halo, the stellar population that is sampled in a color-selected sample changes dramatically near the survey limit as the color errors become large [@newby11 (Newby et al. 2011)]. We have recently released an algorithm that includes this effect in the likelihood calculation. This will allow us to create a more accurate description of the spatial density of each stellar substructure in the halo, so that the densities sum to the actual observed spatial density of halo stars.
We can then use the measured spatial densities of the stars in the tidal streams, along with data on the kinematics of the stream stars, to constrain the gravitational potential (and thus the spatial density of dark matter) of the Milky Way and the properties of the progenitor dwarf galaxies (including their dark matter content). We do this by varying the parameters in the $N$-body simulations of the tidal disruption of the dwarf galaxy progenitor until we generate a stream with the correct spatial and kinematic properties. Initially, the only kinematic information we will have is sparsely sampled line-of-sight velocities. Once data from Gaia is available, we will be able to fit proper motions as well.
Fitting models with a large number of parameters to data, such as is described in this proceedings, will be of growing importance as we increase the amount of data that is available, for example from surveys such as LAMOST and Gaia. With small amounts of data, the spatial density of the halo seems to be well fit by a three parameter power law. With millions of stars, many complex substructures are observed; and the stellar halo is shown to be a very poor fit to a simple power law.
Anderson, D. P. 2004, in: (Rajkumar Buyya, ed.), *Fifth IEEE/ACM International Workshop on Grid Computing*, Proc. IEEE Computing Society, p. 4
Barnes, J., & Hut, P. 1986, *Nature*, 324, 446
Belokurov, V., Zucker, D. B., Evans, N. W., et al. 2006, *ApJ* (Letters), 642, L137
Cole, N., Newberg, H. J., Magdon-Ismail, M., et al. 2008, *ApJ*, 683, 750
Grillmair, C. J. 2009, *ApJ*, 693, 1118
Koposov, S. E., Rix, H.-W., & Hogg, D. W. 2010, *ApJ*, 712, 260
Law, D. R., & Majewski, S. R. 2010, *ApJ*, 714, 229
Majewski, S. R., Skrutskie, M. F., Weinberg, M. D., & Ostheimer, J. C. 2003, *ApJ*, 599, 1082
Martin, C., Carlin, J. L., Newberg, H. J., & Grillmair, C. 2013, *ApJ* (Letters), 765, L39
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Newberg, H. J. 2013, in: (Richard de Gris ed.), *Advancing the Physics of Cosmic Distances*, Proc. IAU Symposium No. 289 (Cambridge University Press), p. 74
Newby, M., Newberg, H. J., Simones, J., Cole, N., & Monaco, M. 2011, *ApJ*, 743, 187
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Plummer, H. C. 1911, *MNRAS*, 71, 460
Willett, B. A. 2010, Ph.D. Thesis
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| {
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---
author:
- 'Qixiang Yang and Tao Qian [^1]'
date:
title: '**The duality about function set and Fefferman-Stein Decomposition** '
---
Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D) \bigcap L^{\infty}(\mathbb{R}^D)$.
To get such decomposition, [**(i),**]{} The authors introduce some auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ defined via wavelet expansions. The authors proved $\tls\subsetneqq L^{1}(\rr^D) \bigcup \tls \subset {\rm WE}^{1,\,q}(\rr^D)\subset L^{1}(\rr^D) + \tls$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. [**(ii),**]{} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by function set $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. [**(iii),**]{} We also consider the dual of $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by Banach space $L^{1}(\rr^D) + \tls$.
Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. \[Michigan Math. J. 62 (2013), 691-703\], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $D\ge2$, which needs some new methodology.
Introduction and main results {#s1}
=============================
The Riesz transforms on $\rr^D$ ($D\ge2$), which are natural generalizations of the Hilbert transform on $\rr$, may be the most typical examples of Calderón-Zygmund operators (see, for example, [@g08; @s70; @s93] and references therein). It is well known that the Riesz transforms have many interesting properties, for example, they are the simplest, nontrivial, ¡°invariant¡± operators under the action of the group of rotations in the Euclidean space $\rr^D$, and they also constitute typical and important examples of Fourier multipliers. Moreover, they can be used to mediate between various combinations of partial derivatives of functions. All these properties make the Riesz transforms ubiquitous in mathematics and useful in various fields of analysis such as partial differential equations and harmonic analysis (see [@s70; @s93] for more details on their applications).
The Riesz transform characterization of Hardy spaces plays important roles in the real variable theory of Hardy spaces (see, for example, [@fs; @s70]). Via this Riesz transform characterization of the Hardy space $\hon$ and the duality between $\hon$ and the space of functions with bounded mean oscillation, $\bmo$, Fefferman and Stein [@fs] further obtained the nowadays so-called Fefferman-Stein decomposition of $\bmo$. Later, Uchiyama [@u82] gave a constructive proof of the Fefferman-Stein decomposition of $\bmo$. Since then, many articles focus on the classical Riesz transform characterization and the Fefferman-Stein decomposition of different variants of Hardy spaces and BMO spaces; see, for example, [@ccyy; @cg; @g79; @ll; @yzn] and references therein. Recently, Lin et al. [@lly] established the Hilbert transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr)}$ and the Fefferman-Stein decomposition of Triebel-Lizorkin spaces ${\dot{F}^0_{\fz,\,q'}(\rr)}$ for each $q\in[2,\fz)$. Yang et al. [@yql] obtained the Fefferman-Stein decomposition for $Q$-spaces $Q_\az(\rr^D)$ and the Riesz transform characterization of $P^\az(\rr^D)$, the predual of $Q_\az(\rr^D)$, for any $\az\in[0,\fz)$.
As was pointed out by Lin et al. in [@lly Remark 1.4], the approach used in [@lly] for the Hilbert transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr)}$ can not be applied to $\tls$ when $D\ge2$, which needs to develop some new skills. In this article, motivated by some ideas from [@lly; @yql], we establish the Riesz transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr^D)}$ and the Fefferman-Stein decomposition of Triebel-Lizorkin spaces ${\dot{F}^0_{\fz,\,q'}(\rr^D)}$ for all $D\in\nn:=\{1,\,2,\,\ldots\}$ and $q\in[2,\fz)$.
In order to state the main results of this article, we now recall the definition of the Triebel-Lizorkin space $\tls$ from [@tr1]; see also [@tr2; @tr3; @tr4; @fjw]. Let $\sch$ and $\schd$ be the *Schwartz space* and its *dual* respectively, and $\pd$ the *class of all polynomials on $\rr^D$*. Following [@tr1], we also let $$\schi:=\lf\{\vz\in\sch:\ \int_{\rr^D}\vz(x)x^{\az}\,dx=0
\ {\rm for\ all}\ \az\in\zz^D_+\r\}$$ and $\schid$ be its dual. Here and hereafter, $\zz_+:=\nn\cup\{0\}$, $\zz^D_+:=(\zz_+)^D$ and, for any $\az:=(\az_1,\ldots,\az_D)\in\zz^D_+$ and $x:=(x_1,\ldots,x_D)\in\rr^D$, $x^\az:=x_1^{\az_1}\cdots x_D^{\az_D}$.
\[da.a\] Let $\vz\in\sch$ satisfy $\supp(\widehat{\vz})\st\{\xi\in\rr^D:\
\frac12 \le|\xi|\le 2\}$, $|\widehat{\vz}(\xi)|\ge c>0$ if $\frac35 \le|\xi|\le \frac53$, and $\sum_{j\in\zz}|\widehat{\vz}(2^j\xi)|=1$ if $\xi\neq0$, where $c$ is a positive constant. Write $\vz_j(\cdot):=2^{Dj}\vz(2^{j}\cdot)$ for any $j\in\zz$. Let $q\in(1,\fz)$. Then the *homogeneous Triebel-Lizorkin space* $\tls$ is defined to be the set of all $f\in\schid$ such that $$\|f\|_{\tls}:=\lf\|\lf\{\sum_{j\in\zz}
\lf|\vz_j\ast f\r|^q\r\}^{1/q}\r\|_{\lon}<\fz.$$
\[ra.n\] (i) It is well known that $\schid=\schd/\pd$ with equivalent topologies; see, for example, [@ywy Proposition 8.1] and [@s16 Theorem 6.28] for an exact proof.
\(ii) From [@fjw p.42], it follows that $\dot{F}^0_{1,\,2}(\rr^D)=\hon$ with equivalent norms. Obviously, for any $q\in[2,\fz)$, $\hon\st\tls$.
Now we recall the definition of the dual space of $\tls$, $\dtl$, from [@fj p.70], where $1/q+1/q'=1$.
\[da.b\] Let $q\in(1,\fz)$. Then the *homogeneous Triebel-Lizorkin space* $\dot{F}^{0,\,q}_\fz(\rr^D)$ is defined to be the set of all $f\in\schid$ such that $$\|f\|_{\dot{F}^0_{\fz,\,q}(\rr^D)}:=\sup_{\{Q:\ \rm dyadic\ cube\}}
\lf\{\frac1{|Q|}\int_{Q}\sum_{j=-\log_2\ell(Q)}^\fz\lf|\vz_j\ast f(x)\r|^q
\,dx\r\}^{1/q}<\fz,$$ where the supremum is taken over all dyadic cubes $Q$ in $\rr^D$ and $\ell(Q)$ denotes the *side length* of $Q$.
\[ra.c\] (i) From [@fjw p.42], it follows that $\dot{F}^0_{\fz,\,2}(\rr^D)=\bmo$ with equivalent norms.
\(ii) It was shown in [@fj (5.2)] that, for each $q\in(1,\fz)$, $\dtl$ is the dual space of $\tls$. In particular, $\bmo$ is the dual space of $\hon$, which was proved before in [@fs].
Next we recall the definition of the $1$-dimensional Meyer wavelets from [@w97]; see also [@lly; @m92; @QY] for a different version. Let $\Phi\in C^\fz(\rr)$, the *space of all infinitely differentiable functions on $\rr$*, satisfy $$\label{ph1}
0\le\Phi(\xi)\le\frac1{\sqrt{2\pi}}\quad {\rm for\ any\ }\xi\in\rr,$$ $$\label{ph2}
\Phi(\xi)=\Phi(-\xi)\quad {\rm for\ any\ }\xi\in\rr,$$ $$\label{ph3}
\Phi(\xi)=\frac1{\sqrt{2\pi}}\quad {\rm for\ any\ }\xi\in[-2\pi/3,2\pi/3],$$ $$\label{ph4}
\Phi(\xi)=0\quad {\rm for\ any\ }\xi\in(-\fz,4\pi/3]\cup[4\pi/3,\fz),$$ and $$\label{ph5}
\lf[\Phi(\xi)\r]^2+\lf[\Phi(\xi-2\pi)\r]^2=\frac1{2\pi}
\quad {\rm for\ any\ }\xi\in[0,2\pi].$$ In what follows, the *Fourier transform* and the *reverse Fourier transform* of a suitable function $f$ on $\rr^D$ are defined by $$\widehat{f}(\xi):=(2\pi)^{-D/2}\int_{\rr^D}e^{-i\xi x}f(x)\,dx \quad
{\rm for\ any\ }\xi\in\rr^D,$$ respectively, $$\check{f}(x):=(2\pi)^{-D/2}\int_{\rr^D}e^{ix\xi}f(\xi)\,d\xi \quad
{\rm for\ any\ }x\in\rr^D.$$
From [@w97 Proposition 3.2], it follows that $\phi:=\check{\Phi}$ (the *“farther" wavelet*) is a scaling function of a *multiresolution analysis* defined as in [@w97 Definition 2.2]. The *corresponding function* $m_{\phi}$ of $\phi$, satisfying $\widehat{\phi}(2\cdot)=m_{\phi}(\cdot)\widehat{\phi}(\cdot)$, is a $2\pi$-periodic function which equals $\sqrt{2\pi}\Phi(2\cdot)$ on the interval $[-\pi,\pi)$.
Furthermore, by [@w97 Theorem 2.20], we construct a $1$-dimensional wavelet $\psi$ (the *“mother" wavelet*) by setting $\widehat{\psi}(\xi):=e^{i\xi/2}m_{\phi}(\xi/2+\pi)\Phi(\xi/2)$ for any $\xi\in\rr$. It was shown in [@w97 Proposition 3.3] that $\psi$ is a real-valued $C^{\fz}(\rr)$ function, $\psi(-1/2-x)=\psi(-1/2+x)$ for all $x\in\rr$, and $$\label{ps1}
\supp\lf(\widehat{\psi}\r)\st[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3].$$ Such a wavelet $\psi$ is called a *$1$-dimensional Meyer wavelet*.
Let $D\in\nn\cap[2,\fz)$ and $\vec{0}:=(\overbrace{0,\ldots,0}^{D\ {\rm times}})$. The $D$-dimensional Meyer wavelets are constructed by tensor products as follows. Let $x:=(x_1,\ldots,x_D)\in\rr^D$, $E_D:=\{0,1\}^D\bh\{\vec{0}\}$ and, for any $\lz:=(\lz_1,\ldots,\lz_D)\in E_D$, define $$\psi^{\lz}(x):=\phi^{\lz_1}(x_1)\cdots\phi^{\lz_D}(x_D),$$ with $\phi^{\lz_j}(x_j):=\phi(x_j)$ if $\lz_j=0$ and $\phi^{\lz_j}(x_j):=\psi(x_j)$ if $\lz_j=1$. As in [@w97], for any $(\lz,j,k)\in\blz_D:=\{(\lz,j,k):\ \lz\in E_D,\ j\in\zz,\ k\in\zz^D\}$ and $x\in\rr^D$, we let $\psi^\lz_{j,\,k}(x):=2^{Dj}\psi^{\lz}(2^jx-k)$ and, for $\lz=\vec{0}$ and any $k:=(k_1,\ldots,k_D)$, let $\psi^{\vec{0}}_{j,\,k}(x)
:=2^{Dj}\phi(2^jx_1-k_1)\cdots\phi(2^jx_D-k_D)$ and $\psi^{\vec{0}}(x):=\phi(x_1)\cdots\phi(x_D)$.
By [@w97 Proposition 3.1] and arguments of tensor products, we know that, for any $(\lz,j,k)\in\blz_D$, $\psi^{\lz}_{j,\,k}\in\schi$. Thus, for any $(\lz,j,k)\in\blz_D$ and any $f\in\schid$, let $a^{\lz}_{j,\,k}(f):=\langle f,\psi^{\lz}_{j,\,k}\rangle$, where $\langle \cdot,\cdot\rangle$ represents the duality between $\schid$ and $\schi$. From the proof of [@fjw Theorem (7.20)], it follows that, for any $f\in\schid$, $$\label{x.x}
f=\sum_{\lz\in E_D}\sum_{j\in\zz}\sum_{k\in\zz^D}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}\quad {\rm in}\quad
\schid.$$ Moreover, by [@w97 Proposition 5.2], we know that $\{\psi^{\lz}_{j,\,k}\}_{(\lz,j,k)\in\blz_D}$ is an orthonormal basis of $\ltw$.
For any $\ell\in\{1,\ldots,D\}$ and any $f\in\sch$, denote by $R_{\ell}(f)$ the *Riesz transform* of $f$, which is defined by setting $$\widehat{R_\ell(f)}(\xi):=-i\frac{\xi_\ell}{|\xi|}\widehat{f}(\xi)
\quad {\rm for\ any\ }\xi\in\rr^D.$$ Since and hold true, by [@yql (5.2)], we know that, for any $\ell\in\{1,\ldots,D\}$, $(\lz,j,k),\,(\wz{\lz},\wz{j},\wz{k})\in\blz_D$ and $|j-\wz{j}|\ge2$, we have $$\label{b.d}
\lf( R_{\ell}\lf(\psi^{\lz}_{j,\,k}\r),\psi^{\wz{\lz}}_{\wz{j},\,\wz{k}}
\r)=0,$$ where $(\cdot,\cdot)$ denotes the inner product in $\ltw$.
Now we recall the wavelet characterization of $\tls$ and $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ (see, for example, [@fjw Theorem (7.20)]). For $j\in \mathbb{Z}$ and $k=(k_1,\cdots, k_{D})\in \mathbb{Z}^{D}$, denote $Q_{j,k}=\prod\limits^{D}_{l=1} [2^{-j}k_l, 2^{-j}(1+k_{l})[$.
\[ta.d\] Let $q\in(1,\fz)$. Then
\(i) $f\in\tls$ if and only if $f\in\schid$ and $$\cj_f:=\lf\|\lf\{\sum_{(\lz,\,j,\,k)\in\blz_D}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}<\fz,$$ where $\chi$ denotes the characteristic function of the cube $[0,1)^D$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\cj_f\le C\|f\|_{\tls}.$$
\(ii) $f\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ if and only if $f\in\schid$ and there exists $C>0$ such that for all dyadic cube $Q$, $$\sum_{(\lz,\,j,\,k)\in\blz_D, Q_{j,k} \subset Q} 2^{(q-1) j D} |a^{\lambda}_{j,k}|^{q} \leq C|Q|.$$
\[ra.o\] By Remark \[ra.n\](ii) and Theorem \[ta.d\], we also obtain the wavelet characterization of $\hon$ as in [@m92 p.143].
To consider Fefferman-Stein type decomposition for $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, we need to study some properties relative to frequency. Hence we use Meyer wavelets to introduce the auxiliary function spaces $\loq$. We consider the linear functional on these function sets and consider some exchangeability of Riesz transform and some sums of orthogonal projector operator defined by Meyer wavelets.
Let $q\in(1,\fz)$ and $f\in\schid$. For any $s\in\zz$, $N\in\nn$ and $t\in\{0,\ldots,N+1\}$, let $$\label{b.a}
P_{s,\,N}f:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N\le j\le s\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}
\quad {\rm in}\quad \schid.$$ For each $t\in\{0,\ldots,N+1\}$, let, in $\schid$, $$\label{b.x}
T^{(1)}_{s,\,t,\,N}(f):=\begin{cases}
0, \ \ \ \ &t=0, \\
\displaystyle\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t+1\le j\le s\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}, \ \ \ \
&t\in\{1,\ldots,N+1\}
\end{cases}$$ and $$\label{b.y}
T^{(2)}_{s,\,t,\,N}(f):=\begin{cases}
\displaystyle\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N\le j\le s-t\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}, \ \ \ \ &t\in\{0,\ldots,N\}, \\
0, \ \ \ \ &t=N+1.
\end{cases}$$
\[da.e\] Then the *space $\liq$* is defined to be the space of all $f\in\schid$ such that $$\|f\|_{\liq}:=\sup_{\{s\in\nn,\,N\in\nn\}}
\sup_{t\in\{0,\ldots,N+1\}}
\lf[\lf\|T^{(1)}_{s,\,t,\,N}(f)\r\|_{{\dot{F}^0_{\fz,\,q}(\rr^D)}}
+\lf\|T^{(2)}_{s,\,t,\,N}(f)\r\|_{\li}\r]<\fz.$$
It is easy to see that
For $1<q\leq \infty$, $\liq= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ are Banach spaces.
\[da.e111\] The relative *space $\loq$* is defined to be the space of all $f\in\schid$ such that $$\|f\|_{\loq}:=\sup_{\{s\in\nn,\,N\in\nn\}}
\min_{t\in\{0,\ldots,N+1\}}\lf[\lf\|T^{(1)}_{s,\,t,\,N}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}(f)\r\|_{\lon}\r]<\fz.$$ Further, for $f\in L^{1}(\mathbb{R}^{D}) \bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$, we define $$\|f\|_{\{1, q\}}:= \min (\|f\|_{L^{1}}, \|f\|_{\dot{F}^{0}_{1,q}}).$$ For $f\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{ 1,q}(\mathbb{R}^{D})$, we define $$\|f\|_{1,q}:= \inf\limits_{f+g\in L^{1} + \dot{F}^{0}_{1,q}} \{\|f\|_{L^{1}} + \|g\|_{\dot{F}^{0}_{1,q}}\}.$$
\[ra.k\] The spaces $\loq$ and $\liq$, with $q\in(1,\fz)$, when $D=1$ were introduced by Lin et al. [@lly p.693], respectively, [@lly p.694], which were denoted by $L^{1,\,q}(\rr)$, respectively, $L^{\fz,\,q}(\rr)$. To distinguish these spaces with the well-known Lorentz spaces, we use the notation $\loq$ and $\liq$ which indicate that these spaces are defined via wavelet expansions. Recall also that the space $\loq$ was also called the relative $L^1$ space in [@lly p.693].
We know, $\forall N\geq 1$, the function $P_{N}f(x)= \sum\limits_{(\epsilon,j,k)\in \Lambda_{D}, |j|+|k|\leq 2^{N}} a^{\lambda}_{j,k} \psi^{\lambda}_{j,k}(x)\in \schi$. Set $ A= \loq$ or $ L^{1}(\mathbb{R}^{D}) \bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or $L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. If $f\in A$, then $P_{N}f\in A$. It is easy to see that
\[pr.1.9\] For $1\leq q<\infty$,
\(i) $\loq$ is complete with the above induced norm.
\(ii) The functions in $\schi$ are dense in $A$.
\[re:Banach,function\] Let $q\in[2,\fz)$. It was shown in [@tr3 p.239] that the dual space of $\tls$ is $\dtl$. Further $\dot{F}^{0}_{1,2}(\mathbb{R}^{D})= H^{1}(\mathbb{R}^{D}) \subset L^{1}(\mathbb{R}^{D}).$ Hence $$\label{eq:Hardy}
L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,2}(\mathbb{R}^{D})={\rm WE}^{1,2} (\mathbb{R}^{D})= L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,2}(\mathbb{R}^{D})=L^{1}(\mathbb{R}^{D}).$$ Let $2<q<\infty$. $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ are Banach spaces. $WE^{1,q} (\mathbb{R}^{D})$ are function sets, not Banach spaces. Moreover, the following equalities are [**not**]{} true $$L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})= {\rm WE}^{1,q} (\mathbb{R}^{D})= L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}).$$ In fact, the above two equal signs both have to be changed to the inclusion sign “$\subset$".
For $A$, we can use distributions to define their dual elements.
For $1\leq q<\infty$ and the function set $A\subset L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$, we call $l$ to be a dual element of $A$, if $l\in \schid$ and $$\sup\limits_{f\in \schi, \|f\|_{A}\leq 1} |\langle l, f\rangle|<\infty.$$ We write $l\in A'$.
$A'$ is a linear space. In fact, for $\alpha, \beta\in \mathcal{C}$ and $l_1, l_2\in A'$, we know that $\alpha l_1 +\beta l_2\in A'$. Further, $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ is the linearization function space of the set $L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or the set $ {\rm WE}^{1,q} (\mathbb{R}^{D}).$ The dual elements on the set $L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or on the set ${\rm WE}^{1,q} (\mathbb{R}^{D})$ are the same as which on the linear space $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. Now we are ready to state the first main auxiliary result of this paper.
\[th:111\] For $q\in [2,\infty)$, we have $$\big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= \big({\rm WE}^{1,q} (\mathbb{R}^{D})\big)'=\big( L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
For $q=2$, due to the equation (\[eq:Hardy\]), the above Theorem \[th:111\] is evident. For general $q$, the proof of this theorem will be given in the final section.
Next we state the second main auxiliary result which will be needed in the proof of our Fefferman-Stein type decomposition. We will use certain exchangeability of Meyer wavelets and Riesz transform to prove Theorem \[ta.h\] in section 2.
\[ta.h\] Let $D\in\nn$ and $q\in[2,\fz)$. Then $f\in\schid$ belongs to $\tls$ if and only if $f\in\loq$ and $\{R_{\ell}(f)\}_{\ell=1}^D\st\loq$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{\loq}
\le C\|f\|_{\tls},$$ where $R_0:={\rm Id}$ denotes the *identity operator*.
\[ra.l\] If $D=1$, Theorem \[ta.h\] is just [@lly Theorem 1.3].
Fefferman-Stein decomposition says, for some function space $A$, there exists some space $B$ satisfying $B\nsubseteq A$ such that, for $f\in A$, there exist $f_{l}\in B$ such that $$f=\sum\limits^{D}_{l=0}R_{l} f_{l}.$$ The functions in B have better properties than those in A. But a function in $A$ has been written as a linear combination of a function in $B$ and the $n$ images of functions in $B$ under correspondingly the $n$ Riesz transformations. Such a result brings certain conveniences in PDE and in harmonic analysis. The following theorems \[ta.x\] and \[ta.i\] tell us that we have also Fefferman-Stein decomposition for ${\dot{F}^0_{\fz,\,q}(\rr^D)}$.
By Remark \[ra.k\](iv), we know that, for any $q\in[2,\fz)$, $\tls\st\loq$ and ${\rm WE}^{\fz,\,q'}(\rr^D)\st\dtl$. The following conclusions indicate that the above inclusions of sets are proper, which are extensions of [@lly Remark 1.8]. The proof of theorem \[ta.x\] will be given at section 3.
\[ta.x\] Let $D\in\nn$ and $q\in[2,\fz)$. Then
\(i) $\tls\subsetneqq L^{1}(\rr^D) \bigcup \tls \subset {\rm WE}^{1,\,q}(\rr^D)\subset L^{1}(\rr^D) + \tls$;
\(ii) $ {\rm WE}^{\fz,\,q'}(\rr^D)\subsetneqq\dtl.$
Combining Theorem \[ta.h\], Remark \[ra.k\](iv) and some arguments analogous to those used in the proof of [@lly Theorem 1.7], we obtain the following Fefferman-Stein decomposition of ${\dot{F}^0_{\fz,\,q}(\rr^D)}$, the proof will be given in the final section.
\[ta.i\] Let $D\in\nn$ and $q\in(1,2]$. Then $f\in{\dot{F}^0_{\fz,\,q}(\rr^D)}$ if and only if there exist $\{f_\ell\}_{\ell=0}^D\in \liq
%\st {\dot{F}^0_{\fz,\,q}(\rr^D)}
$ such that $f=f_0+\sum_{\ell=1}^D R_{\ell}\lf(f_\ell\r).$
By Theorems \[th:111\] and \[ta.i\], we have
\[ta.cor\] Let $D\in\nn$ and $q\in[2,\fz)$. Then $f\in\schid$ belongs to $\tls$ if and only if $f\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ and $\{R_{\ell}(f)\}_{\ell=1}^D\st L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})}
\le C\|f\|_{\tls},$$ where $R_0:={\rm Id}$ denotes the *identity operator*.
\[ra.m\] Theorem \[ta.i\] when $D=1$ is just [@lly Theorem 1.7].
The organization of this article is as follows.
In Section \[s2\], via the definition of the space ${\rm WE}^{1,\,q}(\rr^D)$, the boundedness of Riesz transforms on $\tls$, the Riesz transform characterization of $\hon$ and some ideas from [@lly; @yql], we prove Theorem \[ta.h\], namely, establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$. Comparing with the corresponding proof of [@yql Subsection 6.2], the main innovation of this proof is that we regard the corresponding parts of the norms of Riesz transforms $\{R_\ell(f_{s_1,\, N_1})\}_{\ell=1}^D$ in ${\rm WE}^{1,\,q}(\rr^D)$ as a whole to choose $t^1_{s,\, N}\in\{0,\ldots, N+1\}$ such that below holds true, while not to choose $t^\ell_{s,\, N}\in\{0,\ldots, N+1\}$ such that below holds true for each $\ell\in\{1,\ldots,D\}$ separately as in [@yql (6.6)]. Using this technique, we successfully overcome those difficulties described in [@lly Remark 1.4].
In Section \[s3\], we prove Theorem \[ta.x\]. To this end, we first give a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$ (see Example \[ec.a\] below), which is taken from [@w97 Exercise 3.2]. By using such a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$, we then finish the proof of Theorem \[ta.x\] via tensor products and some arguments from the proof of [@lly Remark 1.8]. Comparing with that proof of [@lly Remark 1.8], we make an additional assumption that $\psi(0)\neq0$ here, which is needed in the estimate below.
In Section 4, we give the proof of Theorems \[th:111\], \[ta.i\] and \[ta.cor\].
Finally, we make some conventions on notation. Throughout the whole paper, $C$ stands for a [*positive constant*]{} which is independent of the main parameters, but it may vary from line to line. If, for two real functions $f$ and $g$, $f\le Cg$, we then write $f\ls g$; if $f\ls g\ls f$, we then write $f\sim g$. For $q\in(1,\fz)$, let $q'$ be the *conjugate number* of $q$ defined by $1/q+1/q'=1$. Let $\mathcal{C}$ be the set of complex numbers and $\nn:=\{1,2,\ldots\}$. Furthermore, $\langle\cdot,\cdot\rangle$ and $(\cdot,\cdot)$ represent the duality relation, respectively, the $\ltw$ inner product.
Proof of Theorem \[ta.h\] {#s2}
=========================
In this section, we prove Theorem \[ta.h\]. To this end, we need to recall some well known results.
The following conclusion is taken from [@fjw Corollary (8.21)].
\[ta.f\] Let $D\in\nn$ and $q\in(1,\fz)$. Then the Riesz transform $R_{\ell}$ for each $\ell\in\{1,\ldots,D\}$ is bounded on $\tls$.
\[ra.p\] From Remark \[ra.n\](ii) and Theorem \[ta.f\], it follows that the Riesz transform $R_{\ell}$ for each $\ell\in\{1,\ldots,D\}$ is bounded on $\hon$.
The Riesz transform characterization of $\hon$ can be found in [@s70 p.221].
\[ta.g\] Let $D\in\nn$. The space $\hon$ is isomorphic to the space of all functions $f\in\lon$ such that $\{R_{\ell}(f)\}_{\ell=1}^D\st\lon$. Moreover, there exists a positive constant $C$ such that, for all $f\in\hon$, $$\frac1C\|f\|_{\hon}\le\|f\|_{\lon}+\sum_{\ell=1}^D\|R_{\ell}(f)\|_{\lon}
\le C\|f\|_{\hon}.$$
The following lemma is completely analogous to [@lly Lemma 2.2], the details being omitted.
\[lb.j\] Let $D\in\nn$ and $q\in[2,\fz)$. If $f\in\loq$, then, for any $j\in\zz$, $Q_{j}(f)\in\hon$, where $Q_{j}(f):=\sum_{(\lz,\,k)\in E_D\times\zz^D}a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}$. Moreover, there exists a positive constant $C$ such that, for all $j\in\zz$ and $f\in\loq$, $$\lf\|Q_j(f)\r\|_{\hon}\le C\|f\|_{\loq}.$$
We first show the necessity of Theorem \[ta.h\]. By Remark \[ra.k\](iv) and Theorem \[ta.f\], we have $$\sum_{\ell=0}^D\|R_\ell(f)\|_{\loq}\ls\sum_{\ell=0}^D\|R_\ell(f)\|_{\tls}
\ls\|f\|_{\tls},$$ which completes the proof of the necessity of Theorem \[ta.h\].
Now we show the sufficiency of Theorem \[ta.h\]. To this end, for any $f\in\loq$ such that $\{R_{\ell}(f)\}_{\ell=1}^D\st\loq$, it suffices to show that, for any $s_1\in\zz$, $N_1\in\nn$ and $f_{s_1,\,N_1}:=P_{s_1,\,N_1}f$ defined as in , we have $$\label{b.b}
\lf\|f_{s_1,\,N_1}\r\|_{\tls}\ls\sum_{\ell=0}^D
\lf\|R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\loq},$$ where the implicit constant is independent of $s_1$, $N_1$ and $f$.
Indeed, assume that holds true for the time being. Owing to , for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $$R_{\ell}\lf(f_{s_1,\,N_1}\r)
:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s_1-N_1-1\le j\le s_1+1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\quad{\rm in}\quad \schid.$$ By this and the orthogonality of $\{\psi^{\lz}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}$, we know that, for each $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
\label{b.x1}
&\lf\|R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\loq}\\
&\noz\hs=\sup_{\{\wz{s}\in\zz,\,\wz{N}\in\nn\}}\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}(f)\r\|_{\lon}\r]\\
&\noz\hs\le\lf\|R_{\ell}(f)\r\|_{\loq}<\fz\end{aligned}$$ and, similarly, $$\begin{aligned}
\label{b.x2}
&\lf\|f_{s_1,\,N_1}\r\|_{\loq}\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1,\,
\wz{s}-\wz{N}\ge s_1-N_1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}(f_{s_1,\,N_1})\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}(f_{s_1,\,N_1})\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1,\,
\wz{s}-\wz{N}\ge s_1-N_1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}(f)\r\|_{\lon}\r]\\
&\noz\hs\le\|f\|_{\loq}<\fz.\end{aligned}$$ From , and , we deduce that $$\lf\|f_{s_1,\,N_1}\r\|_{\tls}\ls\sum_{\ell=0}^D
\lf\|R_{\ell}(f)\r\|_{\loq}.$$ This, together with Theorem \[ta.d\] and the Levi lemma, implies that $f\in\tls$ and $$\begin{aligned}
\|f\|_{\tls}
&\ls\lf\|\lf\{\sum_{(\lz,\,j,\,k)\in\blz_D}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}\\
&\sim\lim_{N_1,s_1\to\fz}\lf\|\lf\{\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
s_1-N_1\le j\le s_1\}}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}\\
&\sim\lim_{N_1,\,s_1\to\fz}\lf\|f_{s_1,\,N_1}\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}(f)\r\|_{\loq},\end{aligned}$$ which are the desired conclusions.
Thus, to finish the proof of the sufficiency of Theorem \[ta.h\], we still need to prove . To this end, fix $s_1\in\zz$ and $N_1\in\nn$. In order to obtain the $\loq$-norms of $\{R_{\ell}(f_{s_1,\,N_1})\}_{\ell=0}^D$, by and , it suffices to consider $s:=s_1+1$ and $N:=N_1+2$ in and . For such $s$ and $N$, there exist $t^{(0)}_{s,\,N},\,t^{(1)}_{s,\,N}
\in\{0,\ldots,N+1\}$ such that $$\begin{aligned}
\label{b.g}
&\lf\|T^{(1)}_{s,\,t^{(0)}_{s,\,N},\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t^{(0)}_{s,\,N},\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\\
&\noz\hs=\min_{t\in\{0,\ldots,N+1\}}
\lf[\lf\|T^{(1)}_{s,\,t,\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\end{aligned}$$ and $$\begin{aligned}
\label{b.f}
&\sum_{\ell=1}^D\lf[\lf\|T^{(1)}_{s,\,t^{(1)}_{s,\,N},\,N}R_{\ell}
\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t^{(1)}_{s,\,N},\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\min_{t\in\{0,\ldots,N+1\}}\sum_{\ell=1}^D
\lf[\lf\|T^{(1)}_{s,\,t,\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r].\end{aligned}$$
In the remainder of this proof, to simplify the notation, *we let $g_1:=f_{s_1,\,N_1}$ for any fixed $s_1$ and $N_1$, $t_j:=t^{(j)}_{s,\,N}$ and $T_{i,\,j}:=T^{(i)}_{s,\,t^{(j)}_{s,\,N},\,N}$ for any $i\in\{1,2\}$ and $j\in\{0,1\}$*.
We then consider the following three cases.
**Case I**. $t_0=t_1$. In this case, we write $g_1=a_1+a_2$, where $$a_1:=\sum_{j=s-t_0+1}^s Q_j\lf(g_1\r)
\quad {\rm and}\quad a_2:=\sum_{j=s-N}^{s-t_0}
Q_j\lf(g_1\r).$$ By , we have $a_2=T_{2,\,0}(g_1)\in\lon$ and $$\label{x.o}
\lf\|a_2\r\|_{\lon}
=\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\le\|g_1\|_{\loq},$$ which, together with Lemma \[lb.j\] and $\hon\st\lon$, further implies that $$Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\in\hon$$ and $$\begin{aligned}
\label{b.h}
&\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\hs\le \lf\|Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
+\lf\|Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, by this, $\hon\st\lon$ and , we obtain $$a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\in\lon$$ and $$\begin{aligned}
\label{b.i}
&\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\noz\hs\le\lf\|a_2\r\|_{\lon}
+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Moreover, for each $\ell\in\{1,\ldots,D\}$, we have $$\begin{aligned}
\label{x.y}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(a_2
+Q_{s-t_0+1}\lf(g_1\r)\r)\\
&\noz=T_{2,\,1}R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r)\\
&\noz=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r).\end{aligned}$$ Hence, by , , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that, for any $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
{\rm II}^{(\ell)}:&=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)
+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r)\in\lon\end{aligned}$$ and $$\begin{aligned}
\label{b.j}
\lf\|{\rm II}^{(\ell)}\r\|_{\lon}
&\le\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}\\
&\noz\hs+\lf\|T_{2,\,1}R_{\ell}
\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}\\
&\noz\hs+\lf\|R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}\\
&\noz\hs+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$
From , it follows that, for each $\ell\in\{1,\ldots,D\}$, there exist $\{\tau^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}\st\cc$ such that $$\begin{aligned}
{\rm I}^{(\ell)}:&=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N-1\le j\le s-t_0-1\}}
\tau^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\end{aligned}$$ and $$R_{\ell}Q_{s-t_0+1}\lf(g_1\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t_0\le j\le s-t_0+2\}}
\tau^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}.$$
For any $h\in\li$ and $j_0\in\zz$, let $$P_{j_0}(h):=\sum_{k\in\zz^D}\lf\langle h,\psi^{\vec{0}}_{j_0,\,k}
\r\rangle\psi^{\vec{0}}_{j_0,\,k},$$ where $\langle\cdot,\cdot\rangle$ represents the duality between $\li$ and $\lon$. We claim that $P_{j_0}(h)\in\li$. Indeed, by $
|\langle h,\psi^{\vec{0}}_{j_0,\,k}\rangle|\ls2^{-Dj_0/2}
$ and $\psi^{\vec{0}}\in\sch$, we know that, for all $x\in\rr^D$, $$\lf|P_{j_0}(h)(x)\r|\ls\sum_{k\in\zz^D}2^{-Dj_0/2}
\lf|\psi^{\vec{0}}_{j_0,\,k}(x)\r|
\ls\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}\lf(2^{j_0}x-k\r)\r|\ls1.$$ Let $$h_0:=P_{s-t_0}(h)=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
j\le s-t_0-1\}}a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.$$ Thus, $h_0\in\li$ and $\|h_0\|_{\li}\ls1$ by the above claim.
Moreover, from , we observe that, for any $\ell\in\{1,\ldots,D\}$, $$\lf|\lf\langle {\rm I}^{(\ell)},h\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)},h_0\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)},h_0\r\rangle\r|
\le\lf\|{\rm II}^{(\ell)}\r\|_{\lon}\lf\|h_0\r\|_{\li},$$ which, combined with $\|h_0\|_{\li}\ls1$ and , further implies that $$\lf\|{\rm I}^{(\ell)}\r\|_{\lon}\ls\lf\|{\rm II}^{(\ell)}\r\|_{\lon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ From this, Theorem \[ta.g\] and , it follows that $$a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\in\hon$$ and $$\begin{aligned}
&\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\hs\sim\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\hs\hs+\sum_{\ell=0}^D\lf\|R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which, together with Remark \[ra.n\](ii), and Lemma \[lb.j\], further implies that $$\begin{aligned}
\label{b.l}
\lf\|a_2\r\|_{\tls}&\ls\lf\|a_2\r\|_{\hon}\\
&\noz\ls\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\noz\hs+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Furthermore, by , we find that $$\lf\|a_1\r\|_{\tls}=\lf\|T_{1,\,0}
\lf(g_1\r)\r\|_{\tls}\le\lf\|g_1\r\|_{\tls},$$ which, combined with , implies that $g_1=a_1+a_2\in\tls$ and $$\lf\|g_1\r\|_{\tls}\le\lf\|a_1\r\|_{\tls}
+\lf\|a_2\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This finishes the proof of **Case I**.
**Case II**. $t_0>t_1$. In this case, we write $g_1=b_1+b_2+b_3$, where $$b_1:=\sum_{j=s-t_1+1}^s Q_j\lf(g_1\r),
\quad b_2:=\sum_{j=s-t_0+1}^{s-t_1}
Q_j\lf(g_1\r) \quad {\rm and}\quad b_3:=\sum_{j=s-N}^{s-t_0}
Q_j\lf(g_1\r).$$ Similar to , for any $\ell\in\{1,\ldots,D\}$, we know that $$\begin{aligned}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(b_3+b_2
+Q_{s-t_1+1}\lf(g_1\r)\r)\\
&=R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_1}\lf(g_1\r)+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\\
&\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)
+T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)\\
&=:{\rm I}^{(\ell)}_1+ {\rm I}^{(\ell)}_2+{\rm I}^{(\ell)}_3.\end{aligned}$$
For any $h\in\li$, let $
h_1:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\le s-t_1-1\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.
$ Similar to the proof of $h_0\in\li$, we have $h_1\in\li$ and $$\label{x.z}
\lf\|h_1\r\|_{\li}\ls1.$$ By , we know that, for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $${\rm I}^{(\ell)}_1=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N-1\le j\le s-t_1-1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\quad \mathrm{and}\quad
{\rm I}^{(\ell)}_3=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j=s-t_1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},$$ which imply that $$\lf|\lf\langle {\rm I}^{(\ell)}_1,h\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)}_1,h_1\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)}_1+{\rm I}^{(\ell)}_3,h_1\r\rangle\r|
=\lf|\lf\langle T_{2,\,1}R_{\ell}\lf(g_1\r)
-{\rm I}^{(\ell)}_2,h_1\r\rangle\r|.$$ Hence, by this, , , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that $$\begin{aligned}
\label{b.u}
\lf\|{\rm I}^{(\ell)}_1\r\|_{\lon}
&\ls\lf[\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}
+\lf\|{\rm I}^{(\ell)}_2\r\|_{\lon}\r]\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|{\rm I}^{(\ell)}_2\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, ${\rm I}^{(\ell)}_1\in\lon$. Moreover, by Remark \[ra.p\] and Lemma \[lb.j\], we have $$\begin{aligned}
\label{x.u}
&\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\hs\ls\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ From this, $\hon\st\lon$ and , we deduce that $$\begin{aligned}
\label{x.v}
\lf\|R_{\ell}\lf(b_2+b_3\r)\r\|_{\lon}
&\le\lf\|{\rm I}^{(\ell)}_1\r\|_{\lon}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
On the other hand, for any $h\in\li$, let $
\wz{h}_0:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\ge s-t_0+2\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.
$ Similar to the proof of $h_0\in\li$, we have $h-\wz{h}_0\in\li$ and $\|h-\wz{h}_0\|_{\li}\ls1$, which further implies that $$\label{x.w}
\lf\|\wz{h}_0\r\|_{\li}\le\lf\|h\r\|_{\li}
+\lf\|h-\wz{h}_0\r\|_{\li}\ls1.$$ By , we know that $$\begin{aligned}
&R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t_0+2\le j\le s-t_1+1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},\end{aligned}$$ which implies that $$\begin{aligned}
\label{x.n}
&\lf\langle R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),h\r\rangle\\
&\noz\hs=\lf\langle R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),\wz{h}_0\r\rangle\\
&\noz\hs=\lf\langle R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),\wz{h}_0\r\rangle.\end{aligned}$$ From an argument similar to that used in , it follows that $$\begin{aligned}
\label{x.t}
&\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\hs\ls\lf\|Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, by , , , $\hon\st\lon$ and , we conclude that $$\begin{aligned}
&\lf\|R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|R_{\ell}\lf(b_3+b_2\r)\r\|_{\lon}
+\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\hs\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ Therefore, by this, $\hon\st\lon$ and , we obtain $$\begin{aligned}
\lf\|R_{\ell}\lf(b_2\r)\r\|_{\lon}
&\le\lf\|R_{\ell}\lf(b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs+\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which, together with , implies that $$\begin{aligned}
\label{x.s}
\lf\|R_{\ell}\lf(b_3\r)\r\|_{\lon}
&\le\lf\|R_{\ell}\lf(b_3+b_2\r)\r\|_{\lon}
+\lf\|R_{\ell}\lf(b_2\r)\r\|_{\lon}\\
&\noz\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Furthermore, by , we know that $$\lf\|b_3\r\|_{\lon}=
\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq},$$ which, combined with and Theorem \[ta.g\], implies that $b_3\in\hon$ and $$\begin{aligned}
\label{b.o}
\lf\|b_3\r\|_{\hon}
&\sim\lf\|b_3\r\|_{\lon}
+\sum_{\ell=1}^D\lf\|R_{\ell}\lf(b_3\r)\r\|_{\lon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ By and Remark \[ra.n\](ii), we know that $b_3\in\tls$ and $$\label{b.q}
\lf\|b_3\r\|_{\tls}\ls\lf\|b_3\r\|_{\hon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$
Moreover, by , we obtain $$\lf\|b_1+b_2\r\|_{\tls}=
\lf\|T_{1,\,0}\lf(g_1\r)\r\|_{\tls}
\ls\lf\|g_1\r\|_{\loq},$$ which, together with and Remark \[ra.n\](ii), further implies that $$\lf\|g_1\r\|_{\tls}\le\lf\|b_1+b_2\r\|_{\tls}
+\lf\|b_3\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This finishes the proof of **Case II**.
**Case III**. $t_0<t_1$. In this case, we write $g_1=e_1+e_2+e_3$, where $$e_1:=\sum_{j=s-t_0+1}^s Q_j\lf(g_1\r),
\quad e_2:=\sum_{j=s-t_1+1}^{s-t_0}
Q_j\lf(g_1\r) \quad{\rm and}\quad e_3:=\sum_{j=s-N}^{s-t_1}
Q_j\lf(g_1\r).$$ Similar to , for any $\ell\in\{1,\ldots,D\}$, we have $$\begin{aligned}
\label{x.m}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(e_3
+Q_{s-t_1+1}\lf(g_1\r)\r)\\
&\noz=R_{\ell}\lf(e_3
-\lf[Q_{s-t_1}\lf(g_1\r)+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)\\
&\noz=:{\rm II}^{(\ell)}_1+ {\rm II}^{(\ell)}_2+{\rm II}^{(\ell)}_3.\end{aligned}$$ For any $h\in\li$, let $$h_2:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\le s-t_1-1\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.$$ By an argument similar to that used in the proof of $h_0\in\li$, we conclude that $h_2\in\li$ and $\|h_2\|_{\li}\ls1$. By , we know that, for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $${\rm II}^{(\ell)}_1=\sum_{\{(\lz,\,j,\,k)
\in\blz_D:\ s-N-1\le j\le s-t_1-1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}$$ and $$T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j=s-t_1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},$$ which, together with , imply that $$\lf|\lf\langle {\rm II}^{(\ell)}_1,h\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)}_1,h_2\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)}_1+{\rm II}^{(\ell)}_3,h_2\r\rangle\r|
=\lf|\lf\langle T_{2,\,1}R_{\ell}\lf(g_1\r)
-{\rm II}^{(\ell)}_2,h_2\r\rangle\r|.$$ Hence, by this, $\|h_2\|_{\li}\ls1$, , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that $$\begin{aligned}
\label{b.m}
\lf\|{\rm II}^{(\ell)}_1\r\|_{\lon}
&\ls\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}
+\lf\|{\rm II}^{(\ell)}_2\r\|_{\lon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|{\rm II}^{(\ell)}_2\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, ${\rm II}^{(\ell)}_1\in\lon$.
By , we have $e_2+e_3\in\lon$ and $$\label{b.s}
\lf\|e_2+e_3\r\|_{\lon}
=\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq}.$$ For any $h\in\li$, let $$\wz{h}_1:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
j\le s-t_1-2\}}h^{\lz}_{j,\,k}\psi^{\lz}_{j,\,k}.$$ Similar to the proof of $h_0\in\li$, we have $\|\wz{h}_1\|_{\li}\ls1$. We notice that $$\begin{aligned}
&\lf\langle e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r],h\r\rangle\\
&\hs=\lf\langle e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r],\wz{h}_1\r\rangle
=\lf\langle e_3,\wz{h}_1\r\rangle
=\lf\langle e_3+e_2,\wz{h}_1\r\rangle.\end{aligned}$$ Therefore, by this, and $\|\wz{h}_1\|_{\li}\ls1$, we obtain $$\begin{aligned}
\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}&\ls\lf\|e_3+e_2\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Hence $e_3-[Q_{s-t_1}(g_1)
+Q_{s-t_1-1}(g_1)]\in\lon$. From this, and Theorem \[ta.g\], we deduce that $e_3-[Q_{s-t_1}(g_1)
+Q_{s-t_1-1}(g_1)]\in\hon$ and $$\begin{aligned}
&\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\hs\sim\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\hs\hs+\sum_{\ell=1}^D\lf\|R_{\ell}
\lf(e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ Then, by this, $\hon\st\lon$ and Lemma \[lb.j\], we know that $e_3\in\lon$ and $$\begin{aligned}
\label{b.z}
\lf\|e_3\r\|_{\lon}
&\le\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\noz\hs+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\noz\end{aligned}$$ For each $\ell\in\{1,\ldots, D\}$, we observe that $$\begin{aligned}
T_{1,\,1}R_{\ell}\lf(g_1\r)
=T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r).\end{aligned}$$ By this and , we know that, for any $\ell\in\{1,\ldots, D\}$, $$T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r)\in\tls$$ and $$\lf\|T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r)\r\|_{\tls}\ls
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This, together with $$\label{b.tx}
\lf\|e_1\r\|_{\tls}
=\lf\|T_{1,\,0}\lf(g_1\r)\r\|_{\tls}
\le\lf\|g_1\r\|_{\loq}\quad({\rm see}\ (\ref{b.g})),$$ Theorems \[ta.d\] and \[ta.f\], and , further implies that, for each $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
\label{b.v}
&\lf\|T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\le\lf\|T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}
+\lf\|T_{1,\,1}R_{\ell}\lf(e_1\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(e_1\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|e_1\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|g_1\r\|_{\loq}.\end{aligned}$$
Furthermore, for any $\ell\in\{1,\ldots,D\}$, we notice that $$\begin{aligned}
\label{x.l}
T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_1}\lf(g_1\r)\r)&=R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r).\end{aligned}$$ By Theorems \[ta.d\] and \[ta.f\], Remark \[ra.n\](ii) and Lemma \[lb.j\], we conclude that $$\begin{aligned}
&\lf\|T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\hs\ls\lf\|R_{\ell}\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\hs\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}\\
&\hs\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\tls},\end{aligned}$$ which, together with and , implies that $$\begin{aligned}
\label{y.u}
&\lf\|R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\\
&\noz\hs\le\lf\|T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_1}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\hs+\lf\|T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Now we need a useful identity from [@sw p.224, (2.9)] that, for all $f\in\ltw$, $$\label{y.x}
\sum_{\ell=1}^D R^2_\ell(f)=-f.$$
From $e_2-[Q_{s-t_1+1}(g_1)
+Q_{s-t_0}(g_1)]\in\ltw$ and , we deduce that $$\begin{aligned}
e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]=\sum_{\ell=1}^D R^2_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r]\r)
\in\tls,\end{aligned}$$ which, combined with Theorem \[ta.f\] and , implies that $$\begin{aligned}
\label{y.y}
&\lf\|e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r\|_{\tls}\\
&\noz\hs\le\sum_{\ell=1}^D \lf\|R^2_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D \lf\|R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Again, by Remark \[ra.n\](ii) and Lemma \[lb.j\], we obtain $$\begin{aligned}
\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}\!\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq},\end{aligned}$$ which, together with , implies that $$\begin{aligned}
\label{b.w}
\lf\|e_2\r\|_{\tls}
&\le\lf\|e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r\|_{\tls}\\
&\noz\hs+\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Combining with , and , we obtain $$\begin{aligned}
\lf\|g_1\r\|_{\tls}&\le\sum_{j=1}^3\lf\|e_j\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which completes the proof of **Case III** and hence Theorem \[ta.h\].
Proof of Theorem \[ta.x\] {#s3}
=========================
In this section, we prove Theorem \[ta.x\]. To this end, we first provide a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$, which is taken from [@w97 Exercise 3.2].
\[ec.a\] Let $$f(x):=\begin{cases}
e^{-1/x^2}, \ \ \ \ &x\in(0,\fz),\\
0, \ \ \ \ &x\in(-\fz,0],
\end{cases}$$ $f_1(x):=f(x)f(1-x)$ and $g(x):=\lf[\int_{-\fz}^\fz f_1(t)\,dt\r]^{-1}
\int_{-\fz}^x f_1(t)\,dt$ for all $x\in\rr$. Let $\xi\in\rr$ and $\Phi(\xi):=\frac1{\sqrt{2\pi}}\cos
(\frac{\pi}2 g(\frac{3}{2\pi}|\xi|-1))$. Then, by [@w97 Exercise 3.2], we know that $\Phi\in C^\fz(\rr)$ satisfies through . Following the construction of the $1$-dimensional Meyer wavelet, we obtain the “father" wavelet $\phi$, the corresponding function $m_{\phi}$ of $\phi$, and the “mother" wavelet $\psi$.
By the proof of [@w97 Proposition 3.3(ii)], we know that, for any $x\in\rr$, $$\psi(x)=\frac1{\sqrt{2\pi}}\int_{-\fz}^\fz
\cos\lf([x+1/2]\xi\r)\az(\xi)\,d\xi,$$ where, for each $\xi\in\rr$, $\az(\xi):=m_{\phi}(\xi/2+\pi)\Phi(\xi/2)$ is an even function supported in $[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3]$.
Now we show that $\psi(0)\neq0$. Indeed, by the facts that $\az$ is even, $\supp(\az)\st[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3]$ and $m_{\phi}$ is $2\pi$-periodic, we have $$\begin{aligned}
\psi(0)&=\frac2{\sqrt{2\pi}}\int_0^\fz \cos(\xi/2)\az(\xi)\,d\xi
=\frac2{\sqrt{2\pi}}\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)\az(\xi)\,d\xi\\
&=\frac2{\sqrt{2\pi}}\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)
m_{\phi}(\xi/2+\pi-2\pi)\Phi(\xi/2)\,d\xi\\
&=2\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi\\
&=2\int_{2\pi/3}^{\pi}\cos(\xi/2)\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi
+2\int_{\pi}^{4\pi/3}\cdots+2\int_{4\pi/3}^{8\pi/3}\cdots
=:{\rm J}_1+{\rm J}_2+{\rm J}_3.\end{aligned}$$
We first estimate ${\rm J}_1$. Observe that, for any $\xi\in[2\pi/3,\pi]$, by , we know that $\Phi(\xi/2)=1/\sqrt{2\pi}$. Moreover, from $\frac12\le \frac{3}{2\pi}|\xi-2\pi|-1\le1$ and the construction of $g$, we deduce that $g(\frac12)\le g(\frac{3}{2\pi}|\xi-2\pi|-1)\le g(1)=1$. Hence, by the construction of $\Phi$, we obtain $$0\le\Phi(\xi-2\pi)=\frac1{2\pi}\cos\lf(\frac{\pi}2 g\lf(
\frac{3}{2\pi}|\xi-2\pi|-1\r)\r)\le\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which further implies that $$\begin{aligned}
\lf|{\rm J}_1\r|&\le2\int_{2\pi/3}^{\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int_{2\pi/3}^{\pi}\cos(\xi/2)\,d\xi
=\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Now we deal with ${\rm J}_2$. Observe that, for any $\xi\in[\pi,4\pi/3]$, by , we know that $\Phi(\xi/2)=1/\sqrt{2\pi}$. Moreover, from $0\le \frac{3}{2\pi}|\xi-2\pi|-1\le\frac12$ and the construction of $g$, it follows that $0=g(0)\le g(\frac{3}{2\pi}|\xi-2\pi|-1)\le g(\frac12)$. Hence, by the construction of $\Phi$ again, we have $$1\ge\Phi(\xi-2\pi)\ge\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which further implies that $$\begin{aligned}
\lf|{\rm J}_2\r|&=-2\int^{4\pi/3}_{\pi}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi\\
&\ge-2\int^{4\pi/3}_{\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=-\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int^{4\pi/3}_{\pi}\cos(\xi/2)\,d\xi
=\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Finally, we estimate ${\rm J}_3$. We first write $$\label{x.r}
\lf|{\rm J}_3\r|=-2\int^{8\pi/3}_{4\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi
\ge-2\int^{2\pi}_{4\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi.$$ Observe that, for any $\xi\in[4\pi/3,2\pi]$, by , we know that $\Phi(\xi-2\pi)=1/\sqrt{2\pi}$. Moreover, by $0\le \frac{\xi}2\frac{3}{2\pi}-1\le\frac12$ and the construction of $g$, we conclude that $0=g(0)\le g(\frac{\xi}2\frac{3}{2\pi}-1)\le g(\frac12)$. Hence, from the construction of $\Phi$, we deduce that $$1\ge\Phi(\xi/2)\ge\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which, together with , further implies that $$\begin{aligned}
\lf|{\rm J}_3\r|
&\ge-2\int_{4\pi/3}^{2\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=-\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int_{4\pi/3}^{2\pi}\cos(\xi/2)\,d\xi
=\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Combining the estimates of ${\rm J}_1$, ${\rm J}_2$ and ${\rm J}_3$ and the construction of $g$, we obtain $$\begin{aligned}
|\psi(0)|&\ge\lf|{\rm J}_3+{\rm J}_2\r|-\lf|{\rm J}_1\r|
=\lf|{\rm J}_3\r|+\lf|{\rm J}_2\r|-\lf|{\rm J}_1\r|\\
&\ge\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
+\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
-\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\\
&=\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)>0,\end{aligned}$$ which completes the proof of Example \[ec.a\].
Now we are ready to prove Theorem \[ta.x\].
\(I) Suppose that $q\in[2,\fz)$, $\phi$ and $\Phi$ are defined as in the construction of the $1$-dimensional Meyer wavelets. Moreover, we assume that the $1$-dimensional Meyer wavelet $\psi$ satisfies $\psi(0)\neq0$.
For any $x:=(x_1,\ldots,x_D)\in\rr^D$, let $\psi^{\vec{0}}(x):=\phi(x_1)\cdots\phi(x_D)$. From [@lly (5.1)], we deduce that, for all $x\in\rr^D$, $$\label{c.a}
\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(x-k)\r|
=\prod_{\ell=1}^D\sum_{k_\ell\in\zz}\lf|\phi\lf(x_\ell-k_\ell\r)\r|\ls1.$$
For any $j\in\zz$, $k\in\zz^D$ and $x\in\rr^D$, we write $\psi^{\vec{0}}_{j,\,k}(x):=2^{Dj/2}\psi^{\vec{0}}\lf(2^jx-k\r)$. Let $f\in\lon$. For any $j\in\zz$, define $P_j(f):=\sum_{k\in\zz^D}\langle f,\psi^{\vec{0}}_{j,\,k}\rangle
\psi^{\vec{0}}_{j,\,k}$. Then, for any $j\in\zz$, by , $$\begin{aligned}
\label{c.b}
\lf\|P_j(f)\r\|_{\lon}&\le\int_{\rr^D}\int_{\rr^D}
|f(y)|\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(2^jy-k)\r|
\lf|2^{Dj}\psi^{\vec{0}}(2^jx-k)\r|\,dx\,dy\\
&\noz\ls\int_{\rr^D}\int_{\rr^D}
|f(y)|\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(x-k)\r|\,dy\ls\|f\|_{\lon}.\end{aligned}$$
The inclusion relation in (i) of Theorem \[ta.x\] is easy to see. In order to prove (i) of Theorem \[ta.x\], it suffices to show that $\tls\subsetneqq L^{1}( \rr^D )\bigcup \tls$.We first observe that $\psi^{\vec{0}}\in\lon$. Indeed, $$\lf\|\psi^{\vec{0}}\r\|_{\lon}=\prod_{\ell=1}^D\lf\|\phi\r\|_{\lon}
=\lf\|\phi\r\|_{\lon}^D<\fz.$$ To show $\psi^{\vec{0}}\not\in\tls$, let $a_{j,\,k}(\psi^{\vec{0}}):=(\psi^{\vec{0}},\psi_{j,\,k})$ for any $j\in\zz$ and $k\in\zz^D$, where $(\cdot,\cdot)$ represents the $\ltw$ inner product. Let $\xi:=(\xi_1,\ldots,\xi_D),\,\eta:=(\eta_1,\ldots,\eta_D)\in\rr^D$. Then, by the multiplication formula (see [@sw p.8, Theorem 1.15]), , , and the assumption $\psi(0)\neq0$, we obtain $$\begin{aligned}
\label{3.4x}
\lf|a_{j,\,0}\lf(\psi^{\vec{0}}\r)\r|&=\prod_{\ell=1}^D\lf|\int_{\rr}
\widehat{\phi}\lf(-\xi_\ell\r)
2^{-j/2}\widehat{\psi}\lf(2^{-j}\xi_\ell\r)\,d\xi_\ell\r|\\
&=\prod_{\ell=1}^D\lf|\int_{-4\pi/3}^{4\pi/3}
\Phi\lf(\xi_\ell\r)
2^{-j/2}\widehat{\psi}\lf(2^{-j}\xi_\ell\r)\,d\xi_\ell\r|\noz\\
&=\prod_{\ell=1}^D\lf|\int_{-2^{2-j}\pi/3}^{2^{2-j}\pi/3}
\Phi\lf(2^j\eta_\ell\r)2^{j/2}\widehat{\psi}\lf(\eta_\ell\r)\,d\eta_\ell\r|\noz\\
&=\prod_{\ell=1}^D\lf|\int_{-8\pi/3}^{8\pi/3}
\Phi\lf(2^j\eta_\ell\r)2^{j/2}\widehat{\psi}\lf(\eta_\ell\r)\,d\eta_\ell\r|\noz\\
&\sim2^{Dj/2}\prod_{\ell=1}^D\lf|\int_{-8\pi/3}^{8\pi/3}
\widehat{\psi}\lf(\eta_\ell\r)
\,d\eta_\ell\r|\sim2^{Dj/2}|\psi(0)|^D\gtrsim2^{Dj/2}\noz,\end{aligned}$$ provided that $j<-M$ for some positive integer $M$ large enough. Therefore, we have $$\begin{aligned}
&\int_{\rr^D}\int_{\rr^D}
\lf\{\sum_{j\in\zz,\,k\in\zz^D}\lf[2^{Dj/2}\lf|a_{j,\,k}\lf(\psi^{\vec{0}}\r)\r|
\chi\lf(2^jx-k\r)\r]^q\r\}^{1/q}\,dx\\
&\hs\ge\int_{\rr^D}
\lf\{\sum_{j=-\fz}^{-M-1}2^{Djq/2}\lf|a_{j,\,0}\lf(\psi^{\vec{0}}\r)\r|^q
\chi\lf(2^jx-k\r)\r\}^{1/q}\,dx\\
&\hs\gtrsim\int_{\rr^D}
\lf\{\sum_ {j=-\fz}^{-M-1}2^{Djq}\chi\lf(2^jx-k\r)\r\}^{1/q}\,dx\\
&\hs\gtrsim\sum_{m=M}^\fz\int_{B(0,\,2^{m+1})\bh B(0,\,2^m)}
\lf\{\sum_{j=-\fz}^{-m-1}2^{Djq}\r\}^{1/q}\,dx=\fz,\end{aligned}$$ which, combined with Theorem \[ta.d\], implies that $\psi^{\vec{0}}\not\in\tls$. This finishes the proof of (i) of Theorem \[ta.x\].
\(II) Then we use Daubechies wavelets to proof (ii) of Theorem \[ta.x\]. We know that there exist some integer $M$ and a Daubechies scale function $\Phi^{0}(x)\in C^{D+2}_{0} ([-2^{M}, 2^{M}]^{D})$ satisfying $$\label{5.1}
C_{D}=\int \frac{-y_{1}}{|y|^{n+1}}
\Phi^{0}(y-2^{M+1}e) dy<0,$$ where $e= (1,1,\cdots, 1)$. Let $\Phi(x)= \Phi^{0}(x-2^{M+1}e)$ and let $f$ be defined as $$\label{5.2}
f(x)=\sum\limits_{j\in 2\mathbb{N}} \Phi(2^{j}x).$$
For $j,j'\in 2\mathbb{N}, j\neq j'$, the supports of $\Phi(2^{j}x)$ and $\Phi(2^{j'}x)$ are disjoint. Hence the above $f(x)$ in (\[5.2\]) belongs to $L^{\infty}(\mathbb{R}^{D})$. The same reasoning gives, for any $j'\in \mathbb{N}$, $$\sum\limits_{j\in \mathbb{N}, 2j> j'} \Phi(2^{2j}x)\in
L^{\infty}(\mathbb{R}^{D}).$$ Now we compute the wavelet coefficients of $f(x)$ in (\[5.2\]). For $(\lambda',j',k')\in \Lambda_{D}$, let $f^{\lambda'}_{j',k'}=
\langle f,\ \Phi^{\lambda'}_{j',k'}\rangle$. We divide two cases: $j'<0$ and $j'\geq 0$.
For $j'<0$, since the support of $f$ is contained in $[-3 \cdot
2^{M}, 3 \cdot 2^{M}]^{D}$, we know that if $|k'|> 2^{2M+5}$, then $f^{\lambda'}_{j',k'}=0$. If $|k'|\leq 2^{2M+5}$, we have $$|f^{\lambda'}_{j',k'}| \leq C 2^{Dj'} \int |f(x)| dx
\leq C 2^{Dj'}.$$ For $j'\geq 0$, by orthogonality of the wavelets, we have $$f^{\lambda'}_{j',k'}= \Big\langle f,\ \Phi^{\lambda'}_{j',k'}\Big\rangle
= \Big\langle \sum\limits_{j\in \mathbb{N}, 2j> j'} \Phi(2^{2j}\cdot),\
\Phi^{\lambda'}_{j',k'}\Big\rangle.$$ By the same reasoning, for the case $j'\geq0$, we know that if $|k'|> 2^{2M+5}$, then $f^{\lambda'}_{j',k'}=0$. Since $\sum\limits_{j\in \mathbb{N}, 2j>
j'} \Phi(2^{2j}x)\in L^{\infty}$, if $|k'|\leq 2^{2M+5}$, we have $$|f^{\lambda'}_{j',k'}| \leq C\int
|\Phi^{\lambda'}_{j',k'}(x)| dx \leq C .$$ By the above estimation of wavelet coefficients of $f(x)$ and by the wavelet characterization of $\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$ in (ii) of Theorem \[ta.d\], we conclude that $f\in \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$. That is to say, $$\label{eq:1111}f\in L^{\infty}(\mathbb{R}^{D})\bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
Since $\Phi^{0}\in C^{D+2}_{0}([-2^{M}, 2^{M}]^{D})$, we know that $$\Phi(x)= \Phi^{0}(x-2^{M+1}e)\in C^{D+2}_{0}([2^{M}, 3 \cdot 2^{M}]^{D}).$$ Further, if $|x|\leq 2^{M-1}$ and $y\in [2^{M},
3 \cdot 2^{M}]^{D} $, then $|x-y|> 2^{M-1}$. Hence $R_{1}\Phi(x)$ is smooth in the ball $\{x:\ |x|\leq 2^{M-1}\}$.
Applying (\[5.1\]), there exists a positive $\delta>0$ such that for $|x|<\delta$, there holds $R_{1}\Phi(x)<\frac{C_{D}}{2}<0.$ That is to say, if $2^{2j}|x|<\delta$, then $R_{1}\Phi(2^{2j}x)<\frac{C_{D}}{2}<0$. Hence $$\label{eq:1112} R_{1}f(x)\notin L^{\infty}(\mathbb{R}^{D}).$$
The equations (\[eq:1111\]), (\[eq:1112\]) and the continuity of Riesz operators on $\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$ implies the conclusion (ii).
The proof of Theorems \[th:111\], \[ta.i\] and \[ta.cor\]
=========================================================
We prove first Theorem \[th:111\]
If $l \in (L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D}))'$, then $$\sup\limits_{f\in \schi, \|f\|_{L^{1} \bigcup \dot{F}^{0}_{1,q} }\leq 1} |\langle l, f\rangle|<\infty.$$ That is to say, $$\label{e1e}\sup\limits_{f\in \schi, \|f\|_{L^{1} }\leq 1} |\langle l, f\rangle|<\infty \mbox { and }$$ $$\label{e2e}\sup\limits_{f\in \schi, \|f\|_{ \dot{F}^{0}_{1,q} }\leq 1} |\langle l, f\rangle|<\infty.$$ The condition (\[e1e\]) means $l\in L^{\infty}(\mathbb{R}^{D})$, the condition (\[e2e\]) means $l\in \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$. Hence we have the following inclusion relation: $$\label{eq:inc.1}
\big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)' \subset L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
Further, it is known that $$L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D}) \subset {\rm WE}^{1,q} (\mathbb{R}^{D}) \subset L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}).$$ Hence we have $$\label{eq:inc.2}
\big(L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'\subset \big( {\rm WE}^{1,q} (\mathbb{R}^{D})\big)' \subset \big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'.$$
Moreover, we know that $$\label{eq:inc.3}
\big(L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
The equations (\[eq:inc.1\]) , (\[eq:inc.2\]) and (\[eq:inc.3\]) implies the Theorem \[th:111\].
Then we prove Theorem \[ta.i\].
By the continuity of the Riesz operators on the $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, we know that if $f_{l}\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})\bigcap
L^{\infty}(\mathbb{R}^{D})$, then $$\sum\limits_{0\leq l\leq D}R_{l}f_{l}(x) \in
\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D}).$$
Now we prove the converse result. Let $$\begin{array}{c} B=\Big\{(g_{0},g_{1},\cdots,g_{D}): g_{l}\in {\rm WE}^{1,q'}(\mathbb{R}^{D}),
l=0,\cdots,D\Big\},\\
\tilde{B}=\Big\{(g_{0},g_{1},\cdots,g_{D}): g_{l}\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q'}(\mathbb{R}^{D}),
l=0,\cdots,D\Big\},\end{array}$$ where $B\subset \tilde{B}$. The norm of $B$ and $\tilde{B}$ is defined as follows respectively $$\begin{array}{c}\|(g_{0},g_{1},\cdots,g_{D})\|_{B}=
\sum\limits^{D}_{l=0}\|g_{l}\|_{{\rm WE}^{1,q'}},\\
\|(g_{0},g_{1},\cdots,g_{D})\|_{\tilde{B}}=
\sum\limits^{D}_{l=0}\|g_{l}\|_{L^{1} + \dot{F}^{0}_{1,q'}}.\end{array}$$ We define $$\begin{array}{c}S=\Big\{(g_{0},g_{1},\cdots,g_{D})\in B: g_{l}=R_{l}g_{0}, l=0, 1,\cdots, D\Big\},\\
\tilde{S}=\Big\{(g_{0},g_{1},\cdots,g_{D})\in \tilde{B}: g_{l}=R_{l}g_{0}, l=0, 1,\cdots, D\Big\},\end{array}$$ where $S\subset \tilde{S}.$
By Theorem \[ta.h\], $g_{0}\rightarrow
(g_{0},R_{1}g_{0},\cdots,R_{D}g_{0})$ define a norm preserving map from $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ to $S$. Hence the set of continuous linear functionals $f$ on $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ is equivalent to the set of bounded linear map on the set $S$. According to Theorem \[th:111\], which is also the set of continuous linear functionals on Banach space $\tilde{S}$.
According to Theorem \[th:111\], the continuous linear functionals on $B$ belong to $${\rm WE}^{\infty,q}(\mathbb{R}^{D}) + \cdots + {\rm WE}^{\infty,q}(\mathbb{R}^{D}).$$ $\forall f\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, $f$ defines a continuous linear functional $l$ on $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ and also on $\tilde{S}$. Hence there exist $\tilde{f}_{l}\in {\rm WE}^{\infty,q}(\mathbb{R}^{D}), l=0,1,\cdots,D,$ such that for any $g_{0}\in \dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$, $$\begin{aligned}
& &\int_{\mathbb{R}^{D}}f(x) g_{0}(x) dx \\
&=&\int_{\mathbb{R}^{D}}\tilde{f}_{0}(x) g_{0}(x) dx
+\sum\limits^{D}_{l=1} \int_{\mathbb{R}^{D}}
\tilde{f}_{l}(x) R_{l}g_{0}(x) dx\\
&=&\int_{\mathbb{R}^{D}}\tilde{f}_{0}(x) g_{0}(x) dx
-\sum\limits^{D}_{l=1} \int_{\mathbb{R}^{D}} R_{l}(\tilde{f}_{l})(x)
g_{0}(x) dx.\end{aligned}$$
Hence $f(x)= \tilde{f}_{0}(x) -\sum\limits^{D}_{l=1}
R_{l}(\tilde{f}_{l})(x)$.
Finally, we give the proof of Theorem \[ta.cor\].
By the continuity of Riesz operators on $ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}$, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})}
\le C\|f\|_{\tls}.$$
To prove $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})},$$ it is sufficient to prove $$|\langle f,g\rangle|\leq C\big\{\sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})}\big\}\|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})},
\forall g\in \schi \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$ But $\forall g\in \schi \bigcap\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$, by Theorem \[ta.i\], there exists $g_{l}$ such that $\|g_l\|_{{\rm WE}^{\infty,q'}(\mathbb{R}^{D})}\leq C\|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})}$ and $g=\sum\limits^{D}_{l=0} R_l g_l.$ Hence, we have $$\begin{array}{rcl}
|\langle f, g\rangle | &= |\langle f, \sum\limits^{D}_{l=0} R_l g_l\rangle |\,\,\, \leq \sum\limits^{D}_{l=0} |\langle f, R_l g_l\rangle |
&= \sum\limits^{D}_{l=0} |\langle R_lf, g_l\rangle | \\ &\leq C \sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})} \|g_{l}\|_{{\rm WE}^{\infty,q'}(\mathbb{R}^{D})}
& \leq C \sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})} \|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})}.
\end{array}$$
[**Acknowledgement:**]{} The authors would like to thank Dachun Yang and Xing Fu, who contributed beneficial discussions and useful suggestions to this study.
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Qixiang Yang
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China.
[*E-mail:*]{} `[email protected]`
Tao Qian (Corresponding author)
Department of Mathematics, University of Macau, Macao, China
[*E-mails*]{}: ` [email protected]`
[^1]: Corresponding author
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The transport of the energy contained in electrons, both thermal and suprathermal, in solar flares plays a key role in our understanding of many aspects of the flare phenomenon, from the spatial distribution of hard X-ray emission to global energetics. Motivated by recent [*RHESSI*]{} observations that point to the existence of a mechanism that confines electrons to the coronal parts of flare loops more effectively than Coulomb collisions, we here consider the impact of pitch-angle scattering off turbulent magnetic fluctuations on the parallel transport of electrons in flaring coronal loops. It is shown that the presence of such a scattering mechanism in addition to Coulomb collisional scattering can significantly reduce the parallel thermal and electrical conductivities relative to their collisional values. We provide illustrative expressions for the resulting thermoelectric coefficients that relate the thermal flux and electrical current density to the temperature gradient and the applied electric field. We then evaluate the effect of these modified transport coefficients on the flare coronal temperature that can be attained, on the post-impulsive-phase cooling of heated coronal plasma, and on the importance of the beam-neutralizing return current on both ambient heating and the energy loss rate of accelerated electrons. We also discuss the possible ways in which anomalous transport processes have an impact on the required overall energy associated with accelerated electrons in solar flares.'
author:
- 'Nicolas H. Bian, Eduard P. Kontar, and A. Gordon Emslie'
bibliography:
- 'bian\_et\_al\_turbulent\_transport.bib'
title: 'SUPPRESSION OF PARALLEL TRANSPORT IN TURBULENT MAGNETIZED PLASMAS AND ITS IMPACT ON NON-THERMAL AND THERMAL ASPECTS OF SOLAR FLARES'
---
Introduction
============
A solar flare involves a complex set of energy release and transport mechanisms, with both non-thermal and thermal elements [see, e.g., @1988psf..book.....T; @2011SSRv..159..107H for reviews]. In particular, it is generally accepted [e.g., @2011SSRv..159..301K] that a significant fraction of the energy released is in the form of deka-keV electrons. These electrons *gain* energy through as-yet-not-fully-understood process(es) associated with the reconnection of stressed, current-carrying magnetic fields [see, e.g., @2011SSRv..159..357Z for a review]. They *lose* energy principally through Coulomb collisions [e.g., @1972SoPh...26..441B; @1978ApJ...224..241E], although ohmic losses associated with driving the beam-neutralizing return current through the finite-resistivity ambient medium , may also be significant.
Early impulsive phase models [e.g., @1971SoPh...18..489B] assumed, largely for simplicity, that the electrons are accelerated out of a “point source” at or near the apex of a coronal loop. However, estimates [e.g., @2003ApJ...595L..97H; @2011SSRv..159..107H] of the number of accelerated electrons required to produce a strong hard X-ray burst, combined with even generous estimates of the number density of electrons in the acceleration region, show that an acceleration region extending over a substantial portion of the flare volume is required. Additional evidence for an extended acceleration region includes the following:
- While many observations [e.g. @2002ApJ...569..459P; @2003ApJ...595L.107E] point to an acceleration site in the corona and the production of hard X-ray footpoints by electrons precipitating into the chromosphere, the appearance of coronal hard X-ray sources , in particular extended nonthermal coronal sources shows that the accelerated electrons are, at least in some events, fully confined to the extended coronal region where the acceleration occurs;
- [*RHESSI*]{} [@2002SoPh..210....3L] observations reveal that the accelerated electron distribution is nearly isotropic [e.g., @2006ApJ...653L.149K]. This favors stochastic acceleration mechanisms operating throughout an extended region [e.g., @1992ApJ...398..350H; @1994ApJS...90..623M; @2008ApJ...687L.111B; @2009ApJ...692L..45B; @2012SSRv..173..535P; @2012ApJ...754..103B];
have shown that the number of electrons trapped in extended coronal sources significantly exceeds the number consistent with purely collisional transport of these electrons to the chromospheric footpoints. They thus argue that some form of non-collisional scattering mechanism confines electrons to the coronal part of the loop (and hence to the acceleration region), even in coronal-source-plus-footpoint events. @2014ApJ...787...86J have noted that observed variation of hard X-ray source length with photon energy is not consistent with a purely collisional transport model but rather with one in which parallel transport proceeds through a process involving both collisions and noncollisional scattering. @2014ApJ...780..176K further showed, through analysis of similar events, that the mean free path associated with the noncollisional process is of order $10^8$ cm, an order of magnitude or so less than the collisional mean free path.
Local fluctuations in the magnetic field are already well known to be responsible for angular scattering and isotropization of the particle distribution, leading to spatial diffusion along the guiding magnetic field [@1966ApJ...146..480J; @1989ApJ...336..243S; @2009ASSL..362.....S]. [*Cross-field*]{} diffusion of particles in turbulent magnetic fields has also long been considered as a mechanism for cosmic-ray transport , for the transport of solar energetic particles [@2003ApJ...590L..53M; @2013ApJ...773L..29L], and for transport of thermal electrons in coronal loops . In this context, @2011ApJ...730L..22K and have shown that the width (perpendicular to the guiding magnetic field) of extended coronal hard X-ray sources increases slowly with energy, consistent with transport of energetic electrons across the guiding field lines through collisionless scattering off magnetic inhomogeneities.
In summary, hard X-ray observations of solar flares strongly suggest that electrons are accelerated in an extended region, within which a combination of Coulomb collisions and collisionless pitch-angle scattering operate. In this paper we explore the implications that the addition of such a non-collisional scattering process has for the transport of electrons of [*all*]{} energies, both thermal and non-thermal, in the flaring corona.
Soft X-ray observations [e.g., @2011SSRv..159..107H as a recent review] show that the overall spectrum (and hence the distribution of emitting electrons in the flaring corona) is often near-Maxwellian, with a temperature of a few $\times 10^7$ K. Such fits to flare soft X-ray spectra also provide estimates of the emission measures $EM \equiv \int n^2 \, dV$ of the soft-X-ray-emitting volume; in large flares this can be $\simeq 10^{49}$ cm$^{-3}$ or higher. Since the emitting volume is of order $10^{27}$ cm$^3$, this requires a density of order $10^{11}$ cm$^{-3}$. At such densities and temperatures, the collisional mean free path is of order $10^8$ cm, significantly less than the characteristic source length $L \sim V^{1/3} \simeq 10^9$ cm. Thus, unlike in other situations (e.g., in the solar wind), collisional processes are important in determining the ambient conditions in the plasma, which accounts for the very good fit of Maxwellian forms to observed soft X-ray spectra.
In a collisional environment, transport of quantities such as heat and electric charge are principally determined [e.g. @1962pfig.book.....S] by local gradients in temperature and density. The presence of additional non-collisional scattering processes does not change this essentially local [*nature*]{} of plasma transport phenomena. However, collisionless scattering processes affect the transport coefficients and hence the values of heat flux and current that arise from prescribed values of the local temperature gradient and large-scale electric field. In this paper, we evaluate this effect quantitatively, and we also discuss how substantial deviations from the classical [@1962pfig.book.....S] values of the thermoelectric transport coefficients can have very significant implications for several areas of importance to the solar flare problem. Such implications include:
- [ Suppression of thermal conduction will result in a higher coronal temperature for a prescribed heating rate (e.g., by nonthermal electrons) and so possibly account for very hot sources observed to be confined in the corona ;]{}
- [ A reduction in the thermal conductivity coefficient $\kappa_\parallel$ will lengthen the conductive cooling time $\tau_{\rm cool} \simeq 3nkT/(\kappa_\parallel T^{7/2}/L^2$) from its classical value [@1980sfsl.work..341M] and so offers a possible resolution to the long-standing conundrum of the apparent need for continued energy input to coronal plasma after the impulsive phase.]{}
- [ Nonthermal electrons lose energy in driving a beam-neutralizing return current through the finite resistivity of the ambient atmosphere. For a given beam current density $j_\parallel$, a reduction in the electrical conductivity $\sigma_\parallel$ increases the role of return current losses relative to Coulomb collisions, and this can have significant implications for the spatial distribution of hard X-ray emission and for the energy deposition profile throughout the ambient atmosphere, and hence the hydrodynamic response of the solar atmosphere [e.g., @1984ApJ...279..896N; @1989ApJ...341.1067M; @2005ApJ...630..573A]. Confinement of nonthermal electrons through enhanced resistivity may also offer an alternative explanation for loop-top coronal sources . And, because the hard X-ray production of a population of accelerated electrons depends inversely on the energy loss rates of the accelerated electrons, the results will change the required total energy content in these accelerated electrons, a quantity of considerable importance in the overall energetics of solar eruptive events [@2012ApJ...759...71E].]{}
In Section \[pitch-angle\] we consider the combined effects of collisional and turbulent scattering on the effective mean free path used to compute quantities such as the thermal conductivity $\kappa_\parallel$ and the electrical conductivity $\sigma_\parallel$. In Section \[modified-spitzer\] we study this problem more formally, using a Chapman-Enskog expansion of the kinetic equation for the electron phase-space distribution function $f(z,v,\mu)$, and we derive expressions for the various thermoelectric coefficients that link transport quantities (thermal flux, electrical current density) to the local environment (temperature gradient, electric field). We do this first for a model of isotropic scattering where the explicit dependence of the diffusion coefficient $D_{\mu\mu}$ on pitch angle $\mu$ is, for simplicity, discarded. Scattering by magnetostatic fluctuations, which is generally not isotropic, is addressed in Section \[dependent\]. In Section \[application\] we discuss the application of these results to several aspects of solar flares, including the heating of coronal plasma by non-thermal electrons and the subsequent cooling of this hot plasma by thermal conduction to the chromosphere (Section \[coronal-heating-cooling\]), and the role of return currents in ohmic heating of the corona and in the dynamics of the nonthermal electron population (Section \[return-current\]). In Section \[summary\] we summarize the results and present our conclusions.
Effects of Pitch-Angle Scattering on Particle Transport {#pitch-angle}
=======================================================
The basis of turbulent scattering theory in plasmas was developed some time ago [see @1969npt..book.....S for a review]. Here we adopt the philosophy that turbulence can be thought of as playing a role similar to collisions, resulting in a “rescaling” of the transport coefficients with respect to their collisional values. We also focus on elastic angular scattering, which is the predominant effect for scattering of particles by low-frequency turbulence [@1966JETP...23..145R].
Modeling the scattering frequency and mean free path {#scattering-parameters}
----------------------------------------------------
The pitch-angle diffusion coefficient $D_{\mu\mu}$ of charged particles in a magnetized plasma is related to the angular scattering frequency $\nu $ (s$^{-1}$) by
$$\label{dmumudef}
D_{\mu\mu}=\nu \, \frac{(1-\mu^2)}{2} \,\,\,,$$
where $\mu$ is the cosine of the pitch-angle relative to the guiding magnetic field. In the case where angular scattering is a superposition of two additive processes, collisional and turbulent, the scattering frequency can be written as
$$\label{dmumu-total}
\nu (v) = \nu_{C}(v) + \nu_{T}(v) \,\,\, .$$
Here the collisional contribution is given by
$$\label{dmumu-c}
\nu_{C}(v) = \frac{4\pi n_e \, e^4 \, \ln \Lambda}{m_e^2} \, \frac{1}{v^3} \equiv \frac{v}{\lambda_{\rm C}(v)} \,\,\, ,$$
where we have introduced the collisional mean free path
$$\label{lambdac-def}
\lambda_{\rm C} (v) = \frac{m_e^2}{4\pi n_e \, e^4 \, \ln \Lambda} \, {v^4}\equiv \lambda_{\rm ei} \, \left( \frac{v}{v_{\rm te}} \right)^{4} \,\,\, .$$
Here $e$ and $m_e$ are the electronic charge (esu) and mass (g), respectively, $n_e$ (cm$^{-3}$) is the ambient electron density, $\ln \Lambda$ is the Coulomb logarithm, and the thermal mean free path
$$\label{lambda-ei}
\lambda_{\rm ei} = \frac{(2 k_B)^2}{2 \pi e^4 \, \ln \Lambda} \, \frac{T_e^2}{n_e}\simeq \frac{10^{4} \, T_{e}^{2}}{n_{e}} \,\,\, ,$$
which, by definition, is the collisional mean free path of electrons with thermal speed $v = v_{\rm te} \equiv \sqrt{2 k_B T_e/m_e}$, where $k_B$ is Boltzmann’s constant.
Similarly, the turbulent contribution to the angular scattering frequency may be written
$$\label{dmumu-t}
\nu_{T}(v) = \frac{v}{\lambda_{\rm T}(v)} \,\,\, ,$$
which involves a (generally velocity-dependent) quantity $\lambda_{\rm T}$, identifiable as the “turbulent mean free path.” The physical origin of the turbulence is, for the moment, left unspecified. However, in order to exploit the analogy with collisional scattering we assume a velocity dependence for the turbulent mean free path of the form
$$\label{alpha-def}
\lambda_{T}(v)=\lambda_{0} \left( \frac{v}{v_{\rm te}} \right )^{\alpha},$$
corresponding to a turbulent scattering frequency
$$\label{nut-v}
\nu_{T}(v)=\frac{v} {\lambda_{0}} \, \left ( \frac{v}{v_{\rm te}} \right )^{-\alpha} \,\,\, .$$
Using this approach, we can express the total scattering frequency resulting from a combination of collisions and turbulent scattering in the form
$$\label{lambda-eff}
\nu (v) = \frac{v}{\lambda(v)} \,\,\, ,$$
where the “effective,” or simply *the*, mean free path $\lambda (v)$ is given by
$$\label{lambda-eff-c-t}
\frac{1}{\lambda(v)} = \frac{1}{\lambda_{\rm C}(v)} + \frac{1}{\lambda_{\rm T}(v)} = \frac{1}{\lambda_{\rm ei}} \left ( \frac{v_{\rm te}}{v} \right )^4 + \frac{1}{\lambda_{\rm T}(v)}
= \frac{1}{\lambda_{\rm ei}} \left ( \frac{v_{\rm te}}{v} \right )^4 + \frac{1}{\lambda_0} \left ( \frac{v_{\rm te}}{v} \right) ^{\alpha} \,\,\, .$$
We will find it convenient to introduce the dimensionless ratio
$$\label{r}
R = \frac{\lambda_{\rm ei}}{\lambda_{0}} \,\,\, .$$
By equations (\[dmumu-c\]), (\[lambdac-def\]), (\[nut-v\]), and (\[r\]), the total scattering frequency can be written in the form
$$\label{nu-v}
\nu(v) = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{1+R (v/v_{\rm te})^{4-\alpha}}{(v/v_{\rm te})^3} \,\,\, ,$$
with the corresponding expression
$$\label{lambda-v}
\lambda(v) = \frac{\lambda_{\rm ei} \, (v/v_{\rm te})^4}{1+R (v/v_{\rm te})^{4-\alpha}}$$
for the mean free path.
The parameter $R$ is the ratio of two length scales, collisional and turbulent, and here plays the role of a transport reduction factor. In the presence of turbulence, the scattering frequency for thermal electrons with $v\sim v_{\rm te}$ is $\nu=\nu_{\rm ei}(1+R)$, an increase by a factor $(1+R)$ relative to the collisional value $\nu_{\rm ei}=v_{\rm te}/\lambda_{\rm ei}$. When $R$ is large, this increased scattering frequency corresponds to a decrease of the the mean free path by a factor $\simeq$$R$.
Pitch-angle scattering due to turbulence is in general *not* isotropic; the scattering frequency may also depend on pitch angle, say as
$$\label{nu_t_anisotropic}
\nu_{T}(v,\mu) = \frac{v} {\lambda_{0}} \, \left ( \frac{v}{v_{\rm te}} \right )^{-\alpha}|\mu|^{\beta} \,\,\, .$$
In such cases, the mean free path is related to the scattering frequency by the relation
$$\label{oo}
\lambda_{T} (v,\mu) =
\frac{3v}{4}\int _{-1}^{+1} \, \frac{(1-\mu^{2})}{\nu(v,\mu)} \, d\mu \,\,\, .$$
We caution that in the limit of very large $R$, one cannot compute the mean free path simply by first taking this limit in Equation (\[nu-v\]) and then substituting this into Equation (\[oo\]). For example, it is easily checked that, for pure turbulent scattering ($\nu=\nu_{T}$) and certain values of $\beta$ (e.g., $\beta=1$), the expression (\[oo\]) for the purely turbulent mean free path diverges. Physically, this is because for such values of $\beta$ turbulent scattering alone is incapable of scattering particles through the $90^\circ$ ($\mu=0$) “barrier.” It is therefore essential, in general, to include collisional effects even in the limit of large $R=\lambda_{\rm ei}/\lambda_{0}$, so that the mean free path given by Equation (\[oo\]) remains finite. The corresponding scattering frequency is given (cf. Equation (\[nu-v\])) by
$$\label{nu-v-3}
\nu(v,\mu) = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{1+R \, |\mu|^{\beta} \, (v/v_{\rm te})^{4-\alpha}}{(v/v_{\rm te})^3} \,\,\, .$$
The existence of a turbulent spectrum of magnetic fluctuations transverse to the guiding magnetic field is well-documented [see, e.g., @2009ASSL..362.....S for a review]. Such fluctuations can be considered as quasi-static provided the characteristic particle velocity (a few $v_{\rm te}$ for the electrons that carry the bulk of the thermal flux) is larger than $\lambda _{B}/\tau_{B}$, where $\lambda_{B}$ and $\tau_{B}$ are respectively the correlation length and time associated with the fluctuations. Typically, the turbulent mean free path parameter $\lambda_{0}$ is proportional to the inverse square of the fractional level of the magnetic fluctuations, i.e.,
$$\label{lambda0-lambdaB}
\lambda_{0} \simeq \lambda_{B} \, \left ( \frac{\delta B_{\perp}}{B_{0}} \right )^{-2},$$
so that the value of the ratio $R$ is
$$R = \frac{\lambda_{\rm ei}}{\lambda_{0}} \simeq \left ( \frac{\lambda_{\rm ei}}{\lambda_B} \right ) \,
\left ( \frac{\delta B_\perp}{B_0} \right )^2 = \frac{ 10^4 \, T_e^2 \, (\delta B_\perp/B_0)^2}{n_e \, \lambda_B} \,\,\, .$$
Effect on transport coefficients
--------------------------------
We now consider in a semi-quantitative way the impact of adding turbulent scattering on various transport coefficients. The “collisionality” of a plasma is inversely proportional to the ratio of the collisional mean free path $\lambda_C$ to the temperature gradient length scale $L_T$; low values the collisional Knudsen number ${\rm Kn}_S \equiv \lambda_C/L_T$ imply a high degree of collisionality and vice versa. (Here the subscript $S$ stands for @1962pfig.book.....S.) The collisionality of the solar electron population ranges from quite high (${\rm Kn}_{S}\sim 10^{-2}$) close to the Sun to very low (${\rm Kn}_{S}\sim 1$) in the solar wind at 1 AU. For coronal loops, $L_{\rm T}$ is of the order of the loop length $L$, therefore the collisional Knudsen number ${\rm Kn}_{S} \simeq 10^{4}T_{e}^{2}/n_{e}L$. Taking $n_{e} = 10^{10}$ cm$^{-3}$ and $L=10^{9}$ cm, the collisional Kundsen number ${\rm Kn}_{S}=10^{-15}T_{e}^{2}$, meaning that the range of temperatures $T_{e}=10^{6}-10^{7}$ K corresponds to collisional Knudsen numbers ${\rm Kn}_{S}=10^{-3}-10^{-1}$.
Now let us consider the heat flux density carried by electrons along the field line, which for free-streaming electrons may be straightforwardly written as
$$\label{qparallel}
q_{\parallel} = n_{e} m_e v_{\rm te}^3 = n_e m_e \left ( \frac{2 k_B T_e}{m_e} \right )^{3/2} \,\,\, .$$
However, we know from Fick’s law that when the collisional Knudsen number $\lambda/L_T$ is small, the thermal flux is driven by the local temperature gradient. Hence we write $T^{3/2}_{e} = - \lambda_C \, T_{e}^{1/2}dT_e/dz$, resulting in an approximate expression for the parallel heat flux in a collisional environment:
$$\label{kappa-def-coll}
q_{\parallel} = -\frac{2 n_e \, k_B \, (2 k_B T_e)^{1/2}}{m_e^{1/2}} \, \lambda_C \, \frac{dT_e}{dz} \equiv - \, \kappa_{\parallel,S} \, \frac{dT_e}{dz} \,\,\, .$$
Adding collisionless scattering effects gives
$$\label{kappa-def}
q_{\parallel} = -\frac{2 n_e \, k_B \, (2 k_B T_e)^{1/2}}{m_e^{1/2}} \, \lambda \, \frac{dT_e}{dz} \equiv - \, \kappa_\parallel \, \frac{dT_e}{dz} \,\,\, ,$$
where
$$\kappa_{\parallel}=\frac{2 n_e \, k_B \, (2 k_B T_e)^{1/2}}{m_e^{1/2}} \, \lambda$$
is the thermal conductivity, which, through the form of $\lambda$ (Equation (\[lambda-v\])), includes the effects of both collisional and non-collisional scattering.
When the turbulent transport reduction factor $R = \lambda_{\rm ei}/\lambda_{0}$ is small, the mean free path takes on its collisional value $\lambda \simeq \lambda_{\rm ei}$ and the parallel heat conductivity correspondingly assumes the standard [e.g., @1962pfig.book.....S] collisional form $\kappa_{\parallel S} = 2 n_e k_B \, (2 k_B T_e)^{1/2} \, \lambda_{\rm ei}/m_e^{1/2} = k_B (2 k_B T_e)^{5/2}/(\pi m_e^{1/2} e^4 \ln \Lambda)$. (Note that this is density independent because $\lambda_{ei} \propto T_e^2/n_e$.) On the other hand, when the turbulent reduction factor $R$ is large, the heat conductivity becomes significantly suppressed relative to the collisional value and obeys the (generally density dependent) scaling $\kappa_\parallel = 2 n_e k_B (2 k_B T_e)^{1/2} \lambda_{0}/m_e^{1/2} = R^{-1} \kappa_{\parallel S}$.
Next we consider the [*electrical*]{} conductivity in a turbulent plasma. As usual, this can be obtained by writing the balance between the electric force and the friction force acting on an electron:
$$\label{eqn-motion}
-eE_\parallel - \nu \, m_e v_\parallel = 0 \,\,\, ,$$
where $E_\parallel$ (statvolt cm$^{-1}$) is the component of the electric field in the direction of the background guiding magnetic field. From this we find the parallel current density (defined by $j_\parallel = - e n_e v_\parallel$) to be $j_\parallel = (n_e e^2/\nu \, m_e) \, E_\parallel = (n_e e^2 \lambda/m_e v_{\rm te}) \, E_\parallel$, leading to the Ohm’s law
$$\label{sigma-def}
j_\parallel = \frac{n_{e} e^2 \lambda}{m^{1/2}_e (2 k_B T_e)^{1/2}} \, E_\parallel \equiv \sigma_\parallel \, E_\parallel \,\,\, ,$$
where the electrical conductivity
$$\sigma_\parallel=\frac{n_{e} e^2 \lambda}{m^{1/2}_e (2 k_B T_e)^{1/2}} \,\,\, .$$
Again, when $R\ll 1$, the mean free path takes on its collisional value and the parallel electric conductivity obeys the collisional scaling $\sigma_{\parallel S}=n_e e^2 \lambda_{\rm ei}/m_e v_{\rm Te} = (2 k_B T_e)^{3/2}/(2 \pi m_e^{1/2} e^2 \ln \Lambda)$. On the other hand, when the turbulent reduction factor $R\gg 1$, the electric conductivity becomes significantly suppressed relative to the collisional value and obeys the scaling $\sigma_\parallel = n_e e^2 \lambda_{0}/m_e v_{\rm te} = R^{-1} \, \sigma_{\parallel S}$.
We may summarize the above discussion into one simple and rather obvious expression, valid for $v \simeq v_{\rm te}$,
$$\label{dparallel-ratio}
\frac{\nu_{\rm ei}}{\nu}\sim \frac{\lambda}{\lambda_{\rm ei}}\sim
\frac{{\rm Kn}}{{\rm Kn}_{S}}\sim \frac{D_{\parallel}}{D_{\parallel S}}\sim\frac{\kappa_{\parallel}}{\kappa_{\parallel S}}\sim\frac{\sigma_{\parallel}}{\sigma_{\parallel S}}\sim \frac{1}{1+R} \,\,\, .$$
An enhanced rate of angular scattering yields reduced conductivities compared with the collisional (Spitzer) values.
The modified Spitzer problem {#modified-spitzer}
============================
In this section we will improve upon the dimensional estimates above by using a a more rigorous Chapman-Enskog expansion of the electron kinetic equation, i.e., by solving a standard [@1962pfig.book.....S] collisional transport problem that includes an additional non-collisional source of pitch-angle scattering. The model will involve two important non-dimensional parameters: the Knudsen number ${\rm Kn}$, which measures the degree of collisionality vs. free-streaming and is assumed to be smaller than unity, and the turbulent reduction factor $R$ (as defined above), which measures the additional role of collisionless scattering. Note that these non-dimensional numbers are independent; thus the large-$R$ asymptotic solutions should not be confused with the collisionless limit ${\rm Kn} \rightarrow \infty$: collisions are always essential (even when $R\gg 1$) to drive the electrons toward the Maxwellian distribution and to keep the overall mean free path finite.
Isotropic scattering
--------------------
We first consider the impact of an additional source of *isotropic* scattering on the transport coefficients. (This will set up the framework for later addressing the problem of angular scattering by a spectrum of transverse magnetostatic fluctuations, which is generally not isotropic.) Let us consider, then, the one-dimensional kinetic equation for a gyrotropic ($\partial/\partial \phi = 0$) electron distribution function $f(z,\theta, v, t)$ under the action of an electric field $E_\parallel$ parallel to the ambient magnetic field ${\bf B}$,
$$\label{kinetic}
\frac{\partial f}{\partial t} + v_\parallel \, \mathbf{b}.\nabla f - \frac{eE_\parallel}{m_e} \, \mathbf{b}.\nabla_{\mathbf{v}}f = St^{v}(f) + St^{\theta}(f) \,\,\, ,$$
where $z$ (cm) is the position of the particle gyrocenter along the magnetic field with direction ${\mathbf b} = {\mathbf B_0}/B_0$, $\theta$ is the pitch angle ($\cos \theta = \mathbf {v}.\mathbf{B}_0/vB_0 = v_\parallel/v$) and $v = \sqrt{v_\parallel^2 + v_\perp^2}$ is the electron speed. Transforming to the variables $(z,\mu,v,t)$, with $\mu= \cos \theta$ being the pitch-angle cosine, this Fokker-Planck equation may be rewritten as
$$\label{fokker}
\frac{\partial f}{\partial t} + \mu \, v \, \frac{\partial f}{\partial z} - \frac{e E_\parallel}{m_e} \, \mu\, \frac{\partial f}{\partial v} - \frac{e E_\parallel}{m_e} \, \frac{(1-\mu^2)}{v}
\, \frac{\partial f}{\partial \mu} = St^{v}(f) + St^{\mu}(f) \,\,\, .$$
The velocity scattering operator
$$\label{stv}
St^{v}(f) = \frac{1}{v^{2}} \, \frac{\partial}{\partial v} \left [ v^{2} D(v) \left ( \frac{\partial f}{\partial v} + \frac{m_e}{k_B T_e} \, fv \right ) \right ]$$
describes collisional diffusion in velocity space and collisional drag. The diffusion coefficient in velocity space is given by
$$\label{diff-v}
D(v) = \frac{4 \pi e^{4} \ln \Lambda \, n_e k_B T_e}{m_e^3} \, \frac{1}{v^3} \,\,\, ,$$
whereas the pitch-angle scattering operator takes the form
$$\label{stmu}
St^{\mu}(f) = \frac{\partial}{\partial \mu} \left [ D_{\mu\mu}\, \frac{\partial f}{\partial \mu} \right ] =\frac{\partial}{\partial \mu} \left [ \frac{\nu (v)}{2} \, (1-\mu^{2}) \, \frac{\partial f}{\partial \mu} \right ] \,\,\, .$$
While the velocity-space scattering operator (\[stv\]) is responsible for kinetic energy change, the pitch-angle scattering operator (\[stmu\]) is responsible for momentum change at constant kinetic energy. We recall that the scattering frequency $\nu(v)$, and hence (Equation (\[dmumudef\])) the pitch-angle diffusion coefficient $D_{\mu\mu}$, are sums of two components – collisional and turbulent pitch-angle diffusion; thus $D_{\mu\mu} = D^C_{\mu\mu} + D^T_{\mu\mu}$.
The right hand side of Equation (\[fokker\]) vanishes identically for a Maxwellian distribution, i.e.,
$$\label{st-maxwellian}
St^{v}(f_{0}) + St^{\mu}(f_{0}) =0 \,\,\, ,$$
where
$$\label{maxwellian}
f_{0}(v) = n_e \left ( \frac{m_e}{2\pi k_B T_e} \right )^{3/2} \, e^{-m_e v^2/2k_B T_e} \,\,\, .$$
However, if $T$ and/or $n$ are not uniform – $T_e=T_e(z)$ and/or $n_e=n_e(z)$ – then the (local) Maxwellian does [*not*]{} cancel the spatial transport term on the left hand side of the Fokker-Planck equation (\[fokker\]). This deviation from the homogeneous equilibrium state is responsible for the spatial flux of particles in the plasma and of what they carry (e.g., kinetic energy, electric charge). In this situation, we use a standard Chapman-Enskog expansion of the electron kinetic equation and write the distribution function as the sum of a zeroth order isotropic Maxwellian distribution plus a small flux-carrying anisotropic correction:
$$\label{perturb}
f = f_0(z,v) + \epsilon f_1(z,\mu,v) \,\,\, ,$$
where the expansion parameter $\epsilon$ is of the order of the (small) Knudsen number in the plasma, i.e., $\epsilon \sim$ Kn. All operators being linear, we obtain at order $\epsilon$ an equation for $f_{1}$:
$$\label{first-order}
St^{v}(f_{1}) + \frac{\partial}{\partial \mu} \left [ D_{\mu\mu}(v) \, \, \frac{\partial f_1}{\partial \mu} \right ] =
\mu \, v \, \frac{\partial f_0(z, v)}{\partial z} - \mu \, \left ( \frac{e E_\parallel}{m_e} \right ) \,
\frac{\partial f_0(z, v)}{\partial v} \,\,\, ,$$
where $f_0$ is the Maxwellian distribution (\[maxwellian\]). By solving this equation for the lead anisotropic component $f_1$, we can obtain expressions for both the heat flux and the electric current. This constitutes a standard @1962pfig.book.....S problem which becomes easily tractable in the Lorentz plasma approximation[^1], i.e., if one assumes that the scattering is angular only, i.e., $St^v(f_1)=0$. For such a case, Equation (\[first-order\]) can be immediately integrated once over $\mu$ to give
$$\label{df1dmu-c}
\frac{\partial f_1}{\partial \mu} = \frac{\mu^2}{\nu \, (1-\mu ^2)} \, \left [ v \, \frac{\partial f_0(z, v)}{\partial z}-
\frac{e E_\parallel}{m_e} \, \frac{\partial f_0(z, v)}{\partial v} \right ] + \frac{C}{\nu \, (1-\mu ^2)} \,\,\, .$$
The choice of the integration constant
$$C = - \left [ v \, \frac{\partial f_{0}(z, v)}{\partial z} - \frac{e E_\parallel}{m_e} \, \frac{\partial f_0(z, v)}{\partial v} \right ]$$ yields
$$\label{df1dmu}
\frac{\partial f_1}{\partial \mu} = -\frac{1}{\nu } \left [ v \, \frac{\partial f_0(z, v)}{\partial z}- \frac{e E_\parallel}{m_e} \, \frac{\partial f_0(z, v)}{\partial v} \right ] \,\,\, ,$$
which ensures the regularity of $f_1$ at $\mu=\pm 1$.
Since we are, for now, adopting an isotropic scattering model in which the scattering frequency $\nu$ is independent of the pitch angle cosine $\mu$, one can integrate Equation (\[df1dmu\]) once more to obtain
$$\label{f1-general}
f_1(z, v, \mu) = - \frac{\mu}{\nu} \left [ v \, \frac{\partial f_0(z, v)}{\partial z}- \frac{e E_\parallel}{m_e} \, \frac{\partial f_0(z, v)}{\partial v} \right ] \,\,\, .$$
Evaluating the pertinent space and velocity derivatives of $f_0$, and assuming a constant pressure along $z$: $n_e(z) k_B T_e(z) = P_e = {\rm constant}$, we obtain
$$\label{f1-basic}
f_1 (z, v, \mu) = - \frac{\mu v}{\nu} \left [ \left ( \frac{m_ev^2}{2 k_B T_e} - \frac{5}{2} \right )
\frac{1}{T_e} \, \frac{d T_e}{dz} + \frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \,\,\, ,$$
with
$$\label{dmumu-v}
\nu (v) = \frac{v}{\lambda_{\rm c}(v)} + \frac{v}{\lambda_{\rm T}(v)} = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{1+R(v/v_{\rm te})^{4-\alpha}}{(v/v_{\rm te})^3} \,\,\, .$$
It is convenient to introduce the normalization
$$\label{x-def}
x = \frac{v}{v_{\rm te}} \,\,\, ,$$
so that the zero-order Maxwellian distribution (Equation (\[maxwellian\])) at a given location (i.e., given values of $T_e$ and $v_{\rm te}$) can be written in the form
$$\label{f0}
f_0(x) = \pi^{-3/2} n_{e} \, v_{\rm te}^{-3} \, e^{-x^{2}} \,\,\, .$$
Further, Equation (\[dmumu-v\]) becomes
$$\label{dmumux3}
\nu = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{Rx^{4-\alpha}+1}{x^{3}} \,\,\, ,$$
and thus the expression (\[f1-basic\]) for $f_1$ becomes
$$\label{f1-x-mu}
f_1 = - \mu \lambda_{\rm ei} \, \frac{x^4}{Rx^{4-\alpha}+1} \, \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz} + \frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \,\,\, .$$
We can now compute the heat flux from
$$\label{heat-flux-def}
q_\parallel(z) = 2 \pi \, \int_0^\infty dv \, v^2 \int_{-1}^1 d\mu \, \mu \, \left ( \frac{m_e v^2}{2} \right ) \, v \, f_1 (z, v, \mu) \,\,\, ,$$
which yields
$$\begin{aligned}
\label{qparallel-mu-v-f1}
q_\parallel(z)
&=& 2\pi k_B T_e \, v_{\rm te}^4\int_0^\infty dx \, x^5 \int_{-1}^1 d\mu \, \mu \, f_1(z,x,\mu)
\cr
&=& - \frac{4}{3 \sqrt{\pi}} \, n_{e} \, k_B T_e \, v_{\rm te} \, \lambda_{\rm ei}
\int_0^\infty dx \, \frac{x^9}{Rx^{4-\alpha}+1} \, \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz}+\frac{eE_\parallel}{k_B T_e} \right ] \, e^{-x^2} \,\,\, .\end{aligned}$$
Writing this in the form
$$\label{qparallel-onsager-2}
q_\parallel = - \kappa_\parallel \, \frac{dT_e}{dz} - \alpha_\parallel \, E_\parallel \,\,\, ,$$
gives the thermoelectric coefficients
$$\label{kappa-parallel}
\kappa_{\parallel} = \frac{4}{3\sqrt{\pi}} \, n_e \, k_B \, v_{\rm te} \, \lambda_{\rm ei} \int_0^\infty \frac{x^9}{Rx^{4-\alpha}+1} \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx$$
and
$$\label{alpha-parallel}
\alpha_{\parallel} = \frac{4}{3\sqrt{\pi}} \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} \int_0^\infty \frac{x^9}{Rx^{4-\alpha}+1} \, e^{-x^2} \, dx \,\,\, ,$$
respectively.
Similarly, substituting for $f_1$ from Equation (\[f1-basic\]) in the expression for the parallel current density
$$\label{current-density-def}
j_\parallel(z) = - 2 \pi \, e \int_0^\infty dv \, v^2 \int_{-1}^1 d\mu \, \mu \, v \, f_1(z, v, \mu)$$
gives
$$\begin{aligned}
\label{jparallel-mu-v-f1}
j_\parallel(z)
&=& - 2 \pi e \, v_{\rm te}^4 \int dx \, x^3 \int_{-1}^1 d\mu \, \mu \, f_1(z,x,\mu)
\cr
&=& \frac{4}{3 \sqrt{\pi}} \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} \, \int_0^\infty dx \, \frac{x^7}{Rx^{4-\alpha}+1} \, \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz} + \frac{eE_\parallel}{k_B T_e} \right ] \, e^{-x^2} \,\,\, ,\end{aligned}$$
and writing this in the form
$$\label{jparallel-onsager-2}
j_\parallel = \beta_\parallel \, \frac{dT_e}{dz} + \sigma_\parallel \, E_\parallel$$
gives the identifications
$$\label{beta-parallel}
\beta_{\parallel} = \frac{4}{3\sqrt{\pi}} \, \frac{n_e \, e \, v_{\rm te} \, \lambda_{\rm ei}}{T_e} \int_{0}^{\infty} \frac{x^7}{Rx^{4-\alpha}+1} \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx$$
and
$$\label{sigma-parallel}
\sigma_{\parallel} = \frac{4}{3\sqrt{\pi}} \, \frac{n_e \, e^2 \, v_{\rm te} \, \lambda_{\rm ei}}{k_B T_e} \int_0^\infty \frac{x^7}{Rx^{4-\alpha}+1} \, e^{-x^2} \, dx \,\,\, .$$
Note that enforcing the current neutrality condition $j_\parallel=0$ in Equation (\[jparallel-onsager-2\]) shows that $E_\parallel = (-\beta_\parallel/\sigma_\parallel) \, dT_e/dz$. Substituting this condition into Equation (\[qparallel-onsager-2\]) implies that the thermal conductivity coefficient in a current neutralized environment is given by $\kappa^* = \kappa_\parallel - \alpha_\parallel \beta_\parallel/\sigma_\parallel$.
We now provide explicit expressions for these coefficients in the $R \ll 1$ (collision-dominated) and $R \gg 1$ (turbulence-dominated) limiting regimes.
Limit of small R
----------------
As expected, in the limit $R\ll 1$ we can replace the term $(Rx^{4-\alpha}+1)$ in the denominators of the integrands in expressions (\[kappa-parallel\]), (\[alpha-parallel\]), (\[beta-parallel\]), and (\[sigma-parallel\]) with unity, thus recovering the collisional [@1962pfig.book.....S] values. Noting that the integral
$$\int_0^\infty dx \, x^n \, e^{-x^2} = \frac{1}{2} \, \Gamma \left ( \frac{n + 1}{2} \right ) = \frac{1}{2} \, \left (\frac{n-1}{2} \right ) !$$
gives the purely collisional results
$$\label{kappa-sn}
\kappa_{\parallel} = \frac{2\Gamma(6)- 5\Gamma(5)}{3 \sqrt{\pi}} \, n_e k_B \, v_{\rm te} \, \lambda_{\rm ei} = \frac{40}{\sqrt{\pi}} \, n_e k_B \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, ;$$
$$\label{alpha-sn}
\alpha_{\parallel} = \frac{2 \, \Gamma(5)}{3 \sqrt{\pi}} \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} = \frac{16}{\sqrt{\pi}} \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, ;$$
$$\label{beta-sn}
\beta_{\parallel}=\frac{2\Gamma (5)-5\Gamma(4)}{3 \sqrt{\pi}} \, \frac{n_e \, e \, v_{\rm te} \, \lambda_{\rm ei}}{T_e}=\frac{6}{\sqrt{\pi}} \, \frac{n_e \, e \, v_{\rm te} \, \lambda_{\rm ei}} {T_e} \,\,\, ;$$
and
$$\label{sigma-sn}
\sigma_{\parallel} =\frac{2 \, \Gamma(4)}{3 \sqrt {\pi}} \frac{n_e e^2 \, v_{\rm te} \, \lambda_{\rm ei}}{k_B T_e} = \frac{4}{\sqrt {\pi}} \frac{n_e e^2 \, v_{\rm te} \, \lambda_{\rm ei}}{k_B T_e} \,\,\, .$$
Finally, in the zero-current regime, the effective thermal conductivity coefficient is
$$\label{kappa-star-sn}
\kappa^{*} \equiv \kappa_\parallel - \frac{\alpha_\parallel \beta_\parallel}{\sigma_\parallel} = \frac{16}{\sqrt{\pi}} \, n_e k_B \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, .$$
Limit of large R
----------------
On the other hand, when $R\gg 1$ we replace the term $(Rx^{4-\alpha}+1)$ in the denominators of the integrands in expressions (\[kappa-parallel\]), (\[alpha-parallel\]), (\[beta-parallel\]), and (\[sigma-parallel\]) with $Rx^{4-\alpha}$. This gives the following expressions:
$$\label{kappa-sn-rgg1}
\kappa_{\parallel} = \frac{2\Gamma(\frac{8+\alpha}{2})-5\Gamma(\frac{6+\alpha}{2})}{3 \sqrt{\pi}} \,\, \frac{n_e k_B \, v_{\rm te} \, \lambda_{\rm ei}}{R}\,\,\, ,$$
$$\label{alpha-sn-rgg1}
\alpha_{\parallel} = \frac{2 \, \Gamma(\frac{6+\alpha}{2})}{3 \sqrt{\pi} \, R} \, \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, ,$$
$$\label{beta-sn-rgg1}
\beta_{\parallel} = \frac{2\Gamma(\frac{6+\alpha}{2})-5\Gamma(\frac{4+\alpha}{2})}{3 \sqrt{\pi} \, R} \,\, \frac{n_e \, e \, v_{\rm te} \, \lambda_{\rm ei}}{T_e}\,\,\, ;$$
and
$$\label{sigma-sn-rgg1}
\sigma_{\parallel} = \frac{2 \, \Gamma(\frac{4+\alpha}{2})}{3 \sqrt {\pi} \, R} \,\, \frac{n_e e^2 \, v_{\rm te} \, \lambda_{\rm ei}}{k_B T_e}\,\,\, ;$$
The case $\alpha=0$ is of special interest as it corresponds (Equation (\[alpha-def\])) to a velocity-independent turbulent mean free path $\lambda_T$. In this case we have
$$\label{kappa-sn-rgg1alpha0}
\kappa_{\parallel}=\frac{2}{3\sqrt{\pi} \, R} \, n_e k_B \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, ,$$
$$\alpha_{\parallel}= \frac{4}{3 \sqrt{\pi} \, R} \, n_e \, e \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, ,$$
$$\label{beta-sn-rgg1alpha0}
\beta_{\parallel} = - \frac{1}{3 \sqrt{\pi} \, R} \, \frac{n_e \, e \, v_{\rm te} \, \lambda_{\rm ei}}{T_e} \,\,\, ;$$
and
$$\label{sigma-sn-rgg1alpha0}
\sigma_{\parallel} = \frac{2}{3 \sqrt {\pi} \, R} \, \frac{n_e e^2 \, v_{\rm te} \, \lambda_{\rm ei}}{k_B T_e} \,\,\, .$$
In a zero-current scenario, the effective thermal conductivity coefficient is
$$\label{kappa-star-sn-rgg1alpha0}
\kappa^{*} \equiv \kappa_\parallel - \frac{\alpha_\parallel \beta_\parallel}{\sigma_\parallel} = \frac{4}{3 \sqrt{\pi} \, R} \, n_e k_B \, v_{\rm te} \, \lambda_{\rm ei} \,\,\, .$$
As expected from the simple dimensional arguments of the previous Section, in the turbulence-dominated limit of large $R$ the transport coefficients are suppressed by a factor of $R$ with respect to the Spitzer values.
Figure \[fig:coefficients\] shows the full dependence on $R$ of all four thermoelectric coefficients for the case $\alpha=0$. We note from this Figure that in the turbulence-dominated high-$R$ limit, one cross-transport coefficient ($\beta_\parallel$) has not only changed in magnitude, but also in [*sign*]{}, compared to its collisional value (compare Equations (\[beta-sn\]) and (\[beta-sn-rgg1alpha0\])). Physically, this sign reversal is a consequence of the form of the Maxwellian distribution. In a uniform pressure environment,
$$\label{f0-T-dependence}
f_0(z,v) \sim \frac{n_e(z)}{[T_e(z)]^{3/2}} e^{-mv^2/2 k_B T_e(z)} = \left ( \frac {P_e}{k_B} \right ) \,
[T_e(z)]^{-5/2} \, e^{-mv^2/2 k_B T_e(z)} \,\,\, ,$$
from which we see that there are two contributions to the temperature gradient $dT_e/dz$, one involving the derivative of the normalization coefficient $T_e^{-5/2}$ and the other the derivative of the width of the Maxwellian velocity profile. The relative signs of these terms depend on the value of the speed $v$: increasing the temperature flattens the Gaussian velocity profile, leading to a general increase in $f_0$ at a high speeds but a general decrease at low speeds. As a consequence, the spatial derivative
$$\label{df0bdz}
\frac{\partial f_0(z,v)}{\partial z} = \frac{\partial f_0(z,v)}{\partial T_e} \, \frac{dT_e}{dz} \,\,\, ,$$
that appears in the expression (\[f1-general\]) for the lead anisotropic component $f_1(z,v,\mu)$ changes sign at some value of $v$. The overall sign of a particular transport coefficient (e.g., $\kappa_\parallel$, $\beta_\parallel$) depends on the relative contributions of the particular moments of $f_1(v,z,\mu)$ that appear[^2] in the expression for that coefficient. Higher-order moments of the velocity distribution are dominated by larger values of $v$, so that an increase in temperature generally leads to an increase in that moment, while lower-order moments place a greater weight on lower values of $v$, leading to a reduced increase (or even a decrease) in that moment. And, since (1) the moments of the velocity distribution involved in the calculation of $\kappa_\parallel$ and $\beta_\parallel$ depend on the value of $R$; and (2) the moments involved in the calculation of $\kappa_\parallel$ are generally higher than those involved in the calculation of $\beta_\parallel$, it follows that the signs of the various thermoelectric coefficients are not necessarily invariant with respect to the value of $R$.
Consider for definiteness the case of a negative temperature gradient, i.e., a temperature that decreases in the positive $z$ direction (the case of a positive temperature gradient is entirely similar). Also consider first the collisional (low-$R$) limit, for which the fourth-power dependence of the collision frequency $\nu$ causes all the moments involved to be quite high, specifically, the ninth and eleventh moments for $\kappa_\parallel$ (Equation (\[kappa-parallel\])) and the seventh and ninth moments for $\beta_\parallel$ (Equation (\[beta-parallel\])). These high velocity moments are all dominated by large values of $v$; thus, as we move toward increasing values of $z$ (i.e., toward regions of lower temperature), the predominant effect is a reduction in $f_0(z,v)$ with $z$: $\partial f_0(z,v)/\partial z < 0$. Both the heat flux and electron flow are predominantly carried by high-velocity electrons that move in the positive-$z$ direction to fill this relative void in $f_0$, and hence they are both oriented in the positive-$z$ direction, i.e., along the direction of decreasing temperature. Indeed, we see from Equation (\[f1-general\]) that the lead anisotropic term is aligned with the positive-$z$ direction: $f_1(z,v,\mu) \propto \mu$. The coefficients $\kappa_\parallel$ and $\beta_\parallel$ are therefore both positive (the former since $q_\parallel \sim -\kappa_\parallel \, dT_e/dz$ and the latter since the conventional current $j_\parallel \sim \beta_\parallel \, dT_e/dz$ flows in the opposite direction to the electron flow, i.e., in the negative-$z$ direction).
However, in the turbulence-dominated (high-$R$) limit, the velocity-distribution moments involved are all (four powers) lower because of the weaker velocity dependence of the collision frequency $\nu = \nu_T$ (compare Equation (\[alpha-def\]) \[with $\alpha = 0$\] and Equation (\[lambdac-def\])). Since the expression (\[kappa-parallel\]) for $\kappa_\parallel$ still involves relatively-high-order velocity moments (fifth and ninth), this reduction of all the moment orders is not sufficient to reverse the direction of the heat flux; it remains parallel to the direction of the negative temperature gradient. However, the considerably lower-order velocity moments (third and fifth) present in the expression (\[beta-parallel\]) for $\beta_\parallel$ are such that the value of the pertinent moments are now dominated by lower-velocity electrons. At these velocities a decrease in temperature causes an [*increase*]{} in $f_0(z,v)$. At the pertinent velocities, the spatial derivative of the isotropic zero-order term is therefore now positive: $\partial f_0(z,v)/\partial z > 0$, and so (Equation (\[f1-general\])) the lead anisotropic term now aligns in the negative-$z$ direction ($f_1(z,v,\mu) \propto - \mu$). The net result is that the predominant (low-velocity) electron flow is now in the negative-$z$ direction, i.e., in the direction of positive temperature gradient, antiparallel to the heat flux. The conventional current and the heat flux are now both in the positive $z$ direction, and so $\kappa_\parallel > 0$ and $\beta_\parallel < 0$.
![Transport coefficients values, relative to their collisional (Spitzer; $R=0$) limit as a function of $R$ for the case $\alpha = 0$ (turbulent mean free path $\lambda_T$ independent of velocity). The blue dashed line indicates negative values, which appear only for $\beta_\parallel$ in the high-$R$ limit; see explanation in the text.[]{data-label="fig:coefficients"}](f1.eps){width="0.56\linewidth"}
We remind the reader that all the above results have been derived under the simplifying assumption that the turbulent collision frequency $\nu_T$ is independent of pitch angle $\mu$, corresponding to isotropic scattering. While this has provided illustrative results on the impact of turbulent scattering on the various transport coefficients, in actual physical situations the scattering rate $\nu$ may depend on the pitch angle $\mu$ (and/or on the velocity $v$). This is indeed the case when angular scattering is produced by the presence of a spectrum of magnetic fluctuations in the plasma, next to be considered.
Transport reduction from pitch-angle scattering by transverse magnetic fluctuations {#dependent}
===================================================================================
Magnetostatic fluctuations $\delta \mathbf{B}_\perp$ perpendicular to the background magnetic field $B_0 \hat{\mathbf{z}}$ are a source of pitch-angle scattering in magnetized plasmas. For slab turbulence the pitch-angle diffusion coefficient takes the form
$$\label{stu-w}
D_{\mu\mu}^{T} = \frac{\pi}{2} (1-\mu^2) \, \Omega_{\rm ce} \,
\left . \frac{k_\parallel \, W(k_\parallel)}{B_0^2} \, \right \vert _{k_\parallel = \Omega_{\rm ce}/v_\parallel} \, \,\,\, ,$$
where $W(k_\parallel)$ is the spectral energy density of the magnetic fluctuations in wavenumber, i.e., $\int W(k_\parallel) \, dk_\parallel = (\delta B_\perp)^2$. The significance of the condition
$$\label{omegace}
\Omega_{\rm ce} = k_\parallel v_\parallel \equiv k_{\parallel} \, |\mu| \, v$$
in the expression (\[stu-w\]) is that scattering of electrons occurs as a result of gyro-resonance with zero-frequency magnetic modes during their cyclotronic orbits.
We shall consider the following form of the wavenumber spectrum:
$$W(k_{\parallel})=C(q) \, (\delta B_{\perp})^{2} \, \lambda_{B} \, \left [ 1+(k_\parallel \lambda_{B})^{2} \right ]^{-q} \,\,\, ,$$
where $C(q)$ is a normalization constant. Substituting this in Equation (\[stu-w\]) and using Equation (\[dmumudef\]) gives the corresponding turbulent scattering frequency:
$$\nu_{T} = \frac{2D^T_{\mu\mu}}{1-\mu^2} = \pi \, C(q) \, \Omega_{\rm ce} \,
\left(\frac{\delta B_{\perp}}{B_{0}}\right)^{2} \, \left(\frac{\Omega_{ce} \, \lambda_{B}}{|\mu| v}\right) \left[1+\left(\frac{\Omega_{ce}\lambda_{B}}{|\mu| v}\right)^{2}\right]^{-q} \,\,\, .$$
Notice the explicit dependence on $\mu$ which originates from the resonance condition (\[omegace\]). Introducing the normalized gyroradius (or rigidity parameter)
$$r_{\rm te} = \frac{v_{\rm te}}{\Omega_{\rm ce}\lambda_{B}} \,\,\, ,$$
we obtain, in the regime $r\ll 1$,
$$\nu_{T} = \pi \, C(q) \, \Omega_{\rm ce} \left( \frac{\delta B_{\perp}}{B_{0}} \right)^{2}
r_{\rm te}^{2q-1} \left (\frac{v}{v_{\rm te}} \right )^{2q-1} \, |\mu|^{2q-1} \,\,\, ,$$
which has the form (cf. Equation (\[nu\_t\_anisotropic\]))
$$\label{dmumu-t-mu}
\nu_{T} = \left (\frac{v_{\rm te}}{\lambda_{0}} \right ) \left (\frac{v}{v_{\rm te}} \right )^{1-\alpha} \, |\mu|^{\beta} \,\,\, ,$$
with
$$\label{alpha-beta-q}
1-\alpha=\beta=2q-1$$
and
$$\label{lamb}
\lambda_{0} =\frac{v_{\rm te}}{\pi \, C(q) \, \Omega_{\rm ce} \left( \frac{\delta B_{\perp}}{B_{0}} \right)^{2}
r_{\rm te}^{2q-1}}=\frac{1}{\pi \, C(q)} \, \lambda_{B} \, \left ( \frac{B_{0}}{\delta B_\perp} \right )^{2} \, r_{\rm te}^{2-2q}\,\,\, .$$
Note that the parameter $\lambda_{0}$ is *not* the turbulent mean free path. Rather, the turbulent mean free path is defined in terms of the parallel diffusion coefficient, which is itself a functional of the pitch-angle diffusion coefficient (or the scattering frequency):
$$\lambda_{T}\equiv \frac{3}{v} \, D_{\parallel}=\frac{3v}{8}\int _{-1}^{+1}d\mu \, \frac{(1-\mu^{2})^2}{D_{\mu\mu}^{T}}=
\frac{3v}{2}\int _{0}^{1}d\mu \, \frac{(1-\mu^{2})}{\nu_{T}}\,\,\, .$$
Substituting for $\nu_{T}$ from Equation (\[dmumu-t-mu\]), we obtain an expression for the turbulent mean free path:
$$\lambda_T = \lambda_{T0} \left ( \frac{v}{v_{\rm te} } \right)^{\alpha} = \lambda_{T0} \left ( \frac{v}{v_{\rm te} } \right)^{2 - 2q} \,\,\, ,$$
where $\lambda_{T0}$ is the turbulent mean free path at $v=v_{\rm te}$:
$$\begin{aligned}
\label{lambda_T}
\lambda_{T0} &=& \frac{3}{2 \pi \, C(q)} \, \lambda_{B} \, \left(\frac{B_{0}}{\delta B_{\perp}}\right)^{2} r_{\rm te}^{2-2q}\int _{0}^{1}d\mu \, (1-\mu^{2}) \, \mu^{1-2q} \cr
&=&\frac{3}{4 \pi \, C(q) (1-q)(2-q)} \,
\lambda_{B} \left(\frac{B_{0}}{\delta B_{\perp}}\right)^{2} r_{\rm te}^{2-2q} \,\,\, .\end{aligned}$$
For a typical spectrum, e.g., $2q=5/3$, $\lambda_{\rm T}\sim v^{2-2q}\sim v^{1/3}$, so that $\lambda_{\rm T}$ is only weakly dependent on velocity and hence on temperature ($\lambda_T \propto T_{e}^{1/6}$). Therefore, the case of a velocity-independent (or temperature-independent) mean free path (which corresponds precisely to the case $q=1$) deserves particular attention. However Equation (\[lambda\_T\]) shows that the turbulent mean free path formally diverges as $q \rightarrow 1$, which seems to suggest (Equation \[lambda-eff-c-t\])) that there is no reduction of transport below the collisional value in this case. In practice this divergence in $\lambda_T$ is avoided by considering the additional influence of collisions, with the result that the turbulent mean free path becomes finite and weakly (logarithmically) dependent on velocity [see below and the appendix in @2014ApJ...780..176K].
Returning to the result (\[df1dmu\]), which for an isobaric plasma can be written (cf. Equation (\[f1-basic\]))
$$\label{df1bdmu}
\frac{\partial f_1}{\partial \mu} = - \frac {v}{\nu (v)} \left [ \left ( \frac{m_e v^2}{2k_B T_e}-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz}+\frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \,\,\, ,$$
we now have
$$\label{dmumu-r-mu-x}
\nu(v) = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{1+R|\mu|^{2q-1}x^{2q+2}}{x^3} \,\,\, ,$$
where we have used the same normalization $x=v/v_{\rm te}$ and definition of the ratio $R = \lambda_{\rm ei}/\lambda_{0}$ as before. Substituting Equation (\[dmumu-r-mu-x\]) into Equation (\[df1bdmu\]), we obtain
$$\label{df1bdu-r-mu-x}
\frac{\partial f_{1}}{\partial \mu} = - \lambda_{\rm ei} \, \frac{x^4}{1+R|\mu|^{2q-1}x^{2q+2}} \, \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz} + \frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \,\,\, .$$
This is the generalization of Equation (\[f1-x-mu\]) to the case where $D^{T}_{\mu\mu}$ is proportional to $|\mu|^{2q-1}$ (Equations (\[dmumu-t-mu\]) and (\[alpha-beta-q\])).
Since, in the absence of collisions, the turbulent mean free path diverges for $q \geq 1$, the transport coefficients will also diverge for $q\geq1$ in the absence of collisions. We therefore focus on the physically-relevant $q < 1$ ($\alpha > 0$) cases. The singular case $q=1$ ($\alpha =0$) will be dealt with separately.
The case $q < 1$ {#smallq}
----------------
Except for the additional $\mu$ integral which produces a factor that depends on the value of the dimensionless spectral index $q$, there will be no difference in the scalings for large $R$ compared to those already given above for the case of isotropic scattering. Hence, while the necessary integrals can be evaluated in the large $R$ limit, it is simpler to replace the anisotropic scattering problem by an equivalent isotropic one, i.e., set
$$\label{nu-isotropic}
\nu(v) = \frac{v_{\rm te}}{\lambda_{\rm ei}} \, \frac{1+R x^{2q+2}}{x^3} \,\,\, ,$$
but with
$$R=\frac{\lambda_{\rm ei}}{\lambda_{T0}}$$
instead of $R=\lambda_{\rm ei}/{\lambda_{0}}$. This will lead to the previously given results of Section \[modified-spitzer\] with a somewhat different definition of $R$; these results are valid provided $q<1$, i.e., $\alpha>0$.
The case $q=1$ {#q1}
--------------
In the Lorentzian case $q=1$, corresponding to
$$\label{W-spectrum}
W(k_\parallel) = \frac{(\delta B_\perp)^2}{\pi} \, \frac{(1/\lambda_B)}{(1/\lambda_B)^2+k_\parallel^2} \,\,\, ,$$
we have
$$\label{nuT-Lorentzian}
\nu_T = |\mu| \left ( \frac{\delta B_\perp}{B_0} \right )^2 \frac{v}{\lambda_B} = \frac{|\mu| v}{\lambda_0} \,\,\, ,$$
so that
$$\label{Rm}
R = \frac{\lambda_{\rm ei}}{\lambda_{0}} = \left ( \frac{\lambda_{\rm ei}}{\lambda_B} \right ) \,
\left ( \frac{\delta B_\perp}{B_0} \right )^2 \simeq \frac{ 10^4 \, T_e^2 \, (\delta B_\perp/B_0)^2}{n_e \, \lambda_B} \,\,\, .$$
Equation (\[df1bdu-r-mu-x\]) can be analytically integrated over $\mu$ to give
$$\label{f1-r-mu-x}
f_1 = \begin{cases}
- \!\!\!\!\! &\frac{\lambda_{\rm ei}}{R} \, \ln (1+Rx^4\mu) \, \left [ \left ( x^2-\frac{5}{2} \right ) \frac{1}{T_e} \, \frac{dT_e}{dz} +
\frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \quad ; \quad \mu > 0 \cr
& \frac{\lambda_{\rm ei}}{R} \, \ln (1- Rx^4 \mu)\, \left [ \left ( x^2-\frac{5}{2} \right ) \frac{1}{T_e} \frac{dT_e}{dz} +
\frac{eE_\parallel}{k_B T_e} \right ] \, f_0 \quad ; \quad \mu < 0 \,\,\, .
\end{cases}$$
This result can now be substituted in the expressions for the normalized heat flux (cf. Equation (\[qparallel-mu-v-f1\])) and the current density (Equation (\[jparallel-mu-v-f1\])). Exploiting the hemispherical antisymmetry of Equation (\[f1-r-mu-x\]) in evaluating the integral over $\mu$, we obtain
$$\label{qparallel-x-mu-r-01}
q_{\parallel} = - \, \frac{4 n_e k_B T_e v_{\rm Te} \lambda_{\rm ei}}{\sqrt{\pi} \, R} \int_0^\infty dx \, x^5 \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz}+\frac{eE_\parallel}{k_B T_e} \right ] \, e^{-x^2} \, \int_0^1 d\mu \, \mu \ln ( 1+Rx^4 \mu ) \,\,\,$$
and
$$\label{jparallel-x-mu-r-01}
j_{\parallel} = \frac{4 n_e \, e \, v_{\rm Te} \lambda_{\rm ei}}{\sqrt{\pi} \, R} \int_0^\infty dx \, x^3 \left [ \left ( x^2-\frac{5}{2} \right )
\frac{1}{T_e} \, \frac{dT_e}{dz} + \frac{eE_\parallel}{k_B T_e} \right ] \, e^{-x^2} \, \int_0^1 d\mu \, \mu \, \ln ( 1+Rx^4 \mu ) \,\,\, .$$
Making the substitution $y=1+Rx^4 \mu$ and then integrating by parts, the integral over $\mu$ evaluates to
$$\label{integral-evaluation}
\int_0^1 d\mu \, \mu \ln ( 1+Rx^4 \mu ) = \frac{1}{2 (Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \,\,\, .$$
The values of the thermoelectric transport coefficients $\kappa_\parallel, \alpha_\parallel$, $\beta_\parallel$ and $\sigma_\parallel$ may now be obtained by substituting the expression (\[integral-evaluation\]) for the $\mu$ integral in the expressions (\[qparallel-x-mu-r-01\]) and (\[jparallel-x-mu-r-01\]). These transport coefficients are best represented in relation to their Spitzer ($R=0$) values. We find
$$\label{kappa-ratio-general}
\frac{\kappa_{\parallel}}{\kappa_{\parallel, S}} = \frac{3}{2R} \, \, \frac{\int_0^\infty \frac{1}{(Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \, x^5 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx}{\int_0^\infty x^9 \, \left ( x^2 - \frac{5}{2} \right ) \, e^{-x^2} \, dx} \,\,\, ;$$
$$\label{alpha-ratio-general}
\frac{\alpha_{\parallel}}{\alpha_{\parallel,S}} = \frac{3}{2R} \, \frac{\int_0^\infty \frac{1}{(Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \, x^5 \, e^{-x^2} \, dx}{\int_0^\infty x^9 \, e^{-x^2} \, dx} \,\,\, ;$$
$$\label{beta-ratio-general}
\frac{\beta_{\parallel}}{\beta_{\parallel,S}} = \frac{3}{2R} \, \frac{\int_0^\infty \frac{1}{(Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \, x^3 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx}{\int_0^\infty x^7 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx} \,\,\, ;$$
and
$$\label{sigma-ratio-general}
\frac{\sigma_{\parallel}}{\sigma_{\parallel,S}} = \frac{3}{2R} \, \frac{\int_0^\infty \frac{1}{(Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \, x^3 \, e^{-x^2} \, dx}{\int_0^\infty x^7 \, e^{-x^2} \, dx} \,\,\, .$$
For $R \ll 1$, we have a collision-dominated regime, and the results (\[kappa-ratio-general\]) through (\[sigma-ratio-general\]) should approach unity. Indeed, for $R \ll 1$, the logarithm may be expanded in a Taylor series $\ln (1+q) = q - q^2/2 + q^3/3 - \ldots$, giving
$$\label{int-r-small}
\frac{1}{2 (Rx^4)^2} \left \{ \left [ (Rx^4)^2 - 1 \right ] \ln (1 + Rx^4) + Rx^4 \left ( 1 - \frac{1}{2} Rx^4 \right ) \right \} \, {\overset{R \ll 1}\longrightarrow} \,\, \frac{1}{3} \, R x^4$$
in place of Equation (\[integral-evaluation\]). (This result may also be obtained quickly by expanding the logarithm to first order in the integral $\int_0^1 d\mu \, \mu \ln ( 1+Rx^4 \mu )$.) Using the limiting form in the numerators of (\[kappa-ratio-general\]) through (\[sigma-ratio-general\]) yields unity for all four ratios, as it should.
On the other hand, for the turbulence-dominated transport regime characterized by $R \gg 1$, we obtain
$$\label{kappa-ratio-rggg1}
\frac{\kappa_{\parallel}}{\kappa_{\parallel, S}} \, {\overset{R \gg 1}\longrightarrow} \,\, \frac{3 \int_0^\infty \, x^5 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^{2}} \, dx }{2 \int_0^\infty x^9 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx} \, \left ( \frac{\ln R}{R} \right ) = \frac{1}{40} \, \left ( \frac{\ln R}{R} \right ) \,\,\, ;$$
$$\label{alpha-ratio-rggg1}
\frac{\alpha_{\parallel}}{\alpha_{\parallel, S}} \, {\overset{R \gg 1}\longrightarrow} \,\, \frac{3 \int_0^\infty \, x^5 \, e^{-x^{2}} \, dx }{2 \int_0^\infty x^9 \, e^{-x^2} \, dx} \, \left ( \frac{\ln R}{R} \right ) = \frac{1}{8} \, \left ( \frac{\ln R}{R} \right ) \,\,\, ;$$
$$\label{beta-ratio-rggg1}
\frac{\beta_{\parallel}}{\beta_{\parallel, S}} \, {\overset{R \gg 1}\longrightarrow} \,\, \frac{3 \int_0^\infty \, x^3 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^{2}} \, dx }{2 \int_0^\infty x^7 \, \left ( x^2-\frac{5}{2} \right ) \, e^{-x^2} \, dx} \, \left ( \frac{\ln R}{R} \right ) = - \frac{1}{12} \, \left ( \frac{\ln R}{R} \right ) \,\,\, ;$$
and
$$\label{sigma-ratio-rggg1}
\frac{\sigma_{\parallel}}{\sigma_{\parallel, S}} \, {\overset{R \gg 1}\longrightarrow} \,\, \frac{3 \int_0^\infty \, x^3 \, e^{-x^{2}} \, dx }{2 \int_0^\infty x^7 \, e^{-x^2} \, dx} \, \left ( \frac{\ln R}{R} \right ) = \frac{1}{4} \, \left ( \frac{\ln R}{R} \right ) \,\,\, .$$
Note both the $\ln R/R$ dependence of these expressions and also the change of sign of $\beta_\parallel$ at large $R$ (which occurs for the same reasons given in the discussion at the end of Section \[modified-spitzer\]).
Application to solar flares {#application}
===========================
Heating and cooling of flare coronal plasma {#coronal-heating-cooling}
-------------------------------------------
A solar flare is ubiquitously characterized by enhanced emission in soft X-rays [e.g., @1969MNRAS.144..375C; @1970MNRAS.151..141C; @1982SoPh...78..107A]; indeed, the commonly-used GOES classification of flares is based on the soft X-ray flux in the $(1-8)$Å soft X-ray waveband. The plasma responsible for emitting these soft X-rays is generally located in the corona and has a temperature ${\, \lower3pt\hbox{$\sim$}\llap{\raise2pt\hbox{$>$}}\,}10^7$ K [e.g., @2011ApJ...727L..52R]. The [*in situ*]{} heating [@1983ApJ...266..383C; @2004ApJ...609..439F; @2007ApJ...659..750R] to these temperatures is generally believed to be due to a combination of ohmic heating associated with current dissipation in the primary site of magnetic reconnection, plus collisional heating by accelerated nonthermal particles, notably electrons. Ohmic heating by passage of the beam-neutralizing return current (see Section \[return-current\]) through the flaring corona may also play a role.
The heat energy transported by both nonthermal electrons and thermal conduction [@2003LNP...612..161H] to the chromosphere causes local heating and a corresponding increase in the emission measure of $10^7$ K gas, enhancing the overall soft X-ray emission. Further, the pressure gradients established by this rapid heating of chromospheric material through the region of radiative instability from $T\simeq 10^5$ K to $T \simeq 10^7$ K [@1969ApJ...157.1157C], drive a significant hydrodynamic response of the solar atmosphere [e.g., @1984ApJ...279..896N; @1989ApJ...341.1067M; @2005ApJ...630..573A], in particular an upward motion of soft-X-ray-emitting plasma into the corona, a process somewhat incorrectly, but nevertheless ubiquitously, termed “chromospheric evaporation.”
The hot $10^7$ K plasma thus created subsequently cools through radiation, conduction to the chromosphere, and flow of enthalpy. Estimating the pertinent cooling times shows that, under the assumption of classical, collisional [@1962pfig.book.....S] conductivity, thermal conduction generally dominates the cooling time at the highest temperatures. However, in many events the thermal plasma is sustained well beyond the duration of the impulsive hard X-ray burst (and hence heating by nonthermal electrons), for times much longer than the conductive cooling time. This has led to the suggestion [e.g., @1980sfsl.work..341M; @2012ApJ...759...71E] that energy is somehow injected, by unknown processes, into the corona after the impulsive phase has ceased. However, it is important to realize that this conclusion is based on thermal conductive losses governed by a purely collisional model of heat transport, which may not be valid if, as suggested above, turbulent processes play a significant role in the region of electron transport.
To illustrate, let us consider a flare volume of $V \simeq 10^{27}$ cm$^3$, with a plasma of density $n_e = 10^{10}$ cm$^{-3}$ embedded in a magnetic field $B \simeq 10^3$ Gauss. The magnetic energy density is $B^{2}/8\pi \simeq 10^{5}$ erg cm$^{-3}$ and the total available magnetic energy is $(B^{2}/8\pi) V \simeq 10^{32}$ erg, which is a good estimate of the amount of energy released by magnetic reconnection in a large flare [e.g., @2012ApJ...759...71E]. Taking the flare duration as $\tau \simeq 10^2$ s, this corresponds to a released power of $10^{30}$ erg s$^{-1}$. This energy is transformed into kinetic energy of the particles, thermal and non thermal. A fraction, say $10\%$, of this energy goes into the kinetic energy of non-thermal electrons [@2012ApJ...759...71E], giving a power $10^{29}$ erg s$^{-1}$ in accelerated electrons, a number close to that inferred from observations of hard X-ray emission in flares [e.g., @2011SSRv..159..107H]. This corresponds to a volumetric heating rate due to fast electrons $Q \simeq 100$ erg cm$^{-3}$ s$^{-1}$.
With these parameters established, we can now consider the heating (by fast electrons) and cooling (by conduction) of the soft X-ray emitting plasma, governed by the energy equation
$$\label{energy-balance}
\frac{\partial (n_e k_B T_e)}{\partial t} = - \frac{\partial q_\parallel}{\partial z} + Q \,\,\, .$$
As we have seen above, the conductive heat flux takes the form
$$\label{qparallel-energy}
q_{\parallel} = - \frac{2 n_e k_B \, (2 k_B T_e)^{1/2}}{m_e^{1/2}} \, \lambda \, \frac{dT_e}{dz} \,\,\, ,$$
where $\lambda$ is the mean free path, representing the combined effect of Coulomb collisions and turbulent scattering,
$$\lambda= \frac{\lambda_{\rm ei}}{1+R} \,\,\,,$$
and where the turbulent reduction factor
$$R =\frac{ \lambda_{ei}}{\lambda_0}=\frac{ 10^4 \, T_e^2 \, (\delta B_\perp/B_0)^2}{n_e \, \lambda_B} \,\,\, .$$
When $R\gg1$, i.e., when the turbulent parameter $\lambda_{0}$ is much smaller than the collisional mean free path $\lambda_{\rm ei}$ (corresponding to $ \lambda_{0}\ll 10^{4}T_{e}^{2}/n_{e}$), heat transport is regulated predominantly by turbulence. We call this a “turbulence-dominated regime,” although we again stress (cf. Section \[scattering-parameters\]) that the role of collisions in maintaining a near Maxwellian distribution of background electrons remains important. On the other hand, turbulence plays a negligible role when $R \ll 1$.
Let us first balance the terms on the right side of Equation (\[energy-balance\]) to obtain an estimate of the steady-state temperature in the presence of a heat source $Q$ balanced by thermal conduction:
$$2 n_e k_B \left ( \frac{2 k_B T_e}{m_e} \right )^{1/2} \lambda \, \left ( \frac{T_e}{L^2} \right ) \simeq Q \,\,\, ,$$
from which follows the scaling law
$$\label{scaling-law}
T_e \simeq \frac{m_e^{1/3}}{2 k_B} \, \left ( \frac{Q L^{2}}{n_e\lambda} \right )^{2/3} \,\,\, .$$
The vast majority of theoretical studies concerning the response of coronal loops to heating use the collisional [@1962pfig.book.....S] conductivity and corresponding mean free path $\lambda_{\rm ei}$; this includes the determination of equilibrium scaling laws and temperature distribution for active region loops and the modelling of evaporative cooling and enthalpy-based response to coronal heating [e.g., @1978ApJ...220.1137A; @1995ApJ...439.1034C; @2010LRSP....7....5R; @2012ApJ...758....5C]. When the mean free path is indeed close to its collisional value $\lambda_{ei}$ (Equation (\[lambda-ei\])), Equation (\[scaling-law\]) gives the scaling law appropriate to a collisional regime of transport [@1978ApJ...220..643R] :
$$\label{scaling-law-collisional}
T_e \simeq \frac{m_e^{1/7}}{2 k_B} \, \left ( 2 \pi e^4 \, \ln \Lambda \right )^{2/7} Q^{2/7} \, L^{4/7} \simeq 50 \, Q^{2/7} \, L^{4/7} \,\,\, .$$
Note that this is independent on the density $n_e$, and that this scaling follows straightforwardly from the familiar balancing relation $Q \propto T_e^{7/2}/L^2$ appropriate to a collisional environment. Substituting the values $Q \simeq 100$ erg cm$^{-3}$ s$^{-1}$ and $L \simeq 10^9$ cm, we obtain
$$\label{scaling-law-collisional-result}
T_e \simeq 3 \times 10^7 \, {\rm K} \,\,\, ,$$
a value nicely consistent with soft X-ray emission.
On the other hand, returning to Equation (\[scaling-law\]) with $R\gg1$, we now obtain the fundamentally different scaling law for the equilibrium temperature in a turbulence dominated regime:
$$\label{scaling-law-turbulent}
T_e \simeq \frac{m_e^{1/3}}{2 k_B} \, \left ( \frac{Q}{n_e} \right )^{2/3}L^{4/3}\lambda_{B}^{-2/3}
\left (\frac{\delta B_{\perp}}{B_{0}} \right)^{4/3} \,\,\, ,$$
which now depends on the density $n_e$. With a magnetic field perturbation ratio $\delta B_\perp /B_0 \simeq 0.1$ and a magnetic correlation length $\lambda_B \simeq 10^6$ cm ($\lambda_0 \simeq 10^8$ cm) gives the significantly larger temperature
$$\label{scaling-law-turbulence-result}
T_e \simeq 1 \times 10^8 \, {\rm K} \,\,\, .$$
Hence, for a given heating rate $Q$ and loop properties $n_e$ and $L$, the turbulent suppression of heat transport associated with large values of $R$ leads to a higher steady-state temperature than that obtained by using Spitzer conductivity. Plasma at temperatures $\sim$$10^8$ K ($\simeq 10$ keV) can make a meaningful contribution to the [*hard*]{} X-ray emission from the flare. Since thermal conduction is inhibited, possibly also by other collective plasma processes [@1979ApJ...228..592B; @1980ApJ...242..799S], such $10^8$ K temperatures will likely be confined rather than extending along the entire magnetic loop. For a given heating rate $Q$, Equations (\[scaling-law-collisional\]) and (\[scaling-law-turbulent\]) give quite different dependencies of the flaring coronal temperature $T_e$ on the loop length $L$ ($L^{4/7}$ and $L^{4/3}$, respectively), a result that should be observationally testable.
We note that the steady state temperature $T_{e}=3\times 10^{7}$ K in Equation (\[scaling-law-collisional-result\]) above was obtained by assuming a collisional transport regime. Such an assumption is valid only under the dual conditions that both the Knudsen number ${\rm Kn} = \lambda_{\rm ei}/L_T$ and the turbulent reduction factor $R$ are $\ll 1$. Consistency therefore demands that we evaluate the validity of these two assumptions. We find that at these temperatures (and densities $n_e \sim 10^{10}$ cm$^{-3}$), the collisional mean free path $\lambda_{\rm ei}$ is actually of the order of the loop length $L \simeq 10^9$ cm; thus the Knudsen number is of order unity and so the use of a collision-related expression for the heat flux is somewhat questionable. In such conditions the heat flux is instead determined [e.g.. @1979ApJ...228..592B] by a flux-limited value equal to a fraction of the free-streaming limit (Equation (\[qparallel\])), leading to a generally higher equilibrium temperature and conductive cooling time than is appropriate for the collisional case. With regard to the second assumption, the main thrust of the present paper is that, whether or not the Knudsen number is small, the heat flux may be further limited (by a factor $\simeq R$) by the presence of collisionless pitch-angle scattering. Thus, if the turbulent mean free path is substantially less than the collisional mean free path (or, for Knudsen numbers of order unity or more, the length of the flaring loop), then the much larger temperatures ($T_{e}\sim 10^{8}$ K; Equation (\[scaling-law-turbulence-result\])) appropriate to a turbulence-dominated regime apply.
We now turn to a consideration of the role of conduction in cooling the coronal plasma after the energy input $Q$ has ceased. Balancing the heating and conductive cooling in Equation (\[energy-balance\])), and using the expression (\[qparallel-energy\]), we obtain an expression for the cooling time-scale $\tau_{c}$:
$$\label{cooling-time}
\tau_{c} \simeq \frac{m_e^{1/2}}{(2 k_B T_e)^{1/2}} \, \frac{L^2}{\lambda} =
\left ( \frac{L}{v_{\rm te}} \right ) \, \left ( \frac{L}{\lambda} \right ) = \left ( \frac{L}{v_{\rm te}} \right ) \, \left ( \frac{L}{\lambda_{\rm ei}} \right ) \, (1+R) \,\,\, .$$
The cooling time $\tau_{c}$ is thus the free-streaming transport time-scale $L/v_{\rm te}$ of thermal electrons multiplied by the inverse of the Knudsen number ${\rm Kn} = \lambda/L$. For a temperature $T_{e} \simeq 3 \times 10^7$ K (Equation (\[scaling-law-collisional-result\])), the electron thermal speed $v_{\rm Te} \simeq 3 \times 10^9$ cm s$^{-1}$, so that for a loop length $L \simeq 10^9$ cm, the free-streaming escape time is $L/v_{\rm te} \simeq$ 0.3 s. The collisional mean free path $\lambda_{\rm ei}\simeq 10^9$ cm and hence $L/\lambda_{ei}\sim 1$, which when $R \ll 1$, yields a cooling time of the order of the free-streaming time scale. A smaller initial temperature $T_{e} \simeq 10^7$ K has the collisional mean free path decreased by an order of magnitude giving a cooling time $\tau_{c}\sim 3s$. These time scales are much shorter than the duration of the soft X-ray emission, which has led to the realization that the coronal plasma will cool very rapidly after the cessation of the energy input term $Q$, to the point where some form of post-impulsive-phase energy input to the corona is needed to sustain the soft X-ray emission for the observed times ${\, \lower3pt\hbox{$\sim$}\llap{\raise2pt\hbox{$>$}}\,}100$ s [see, e.g., @1980sfsl.work..341M; @2012ApJ...759...71E]. However, in the presence of turbulence the cooling time $\tau_{\rm c}$ is further enhanced by a factor $(1+R)$, and so cooling by thermal conduction parallel to the guiding magnetic field can thus be significantly inhibited. This increase in the cooling time significantly reduces the previously-assumed requirement [@1980sfsl.work..341M; @2012ApJ...759...71E] for post-impulsive phase heating of the coronal plasma.
Before moving on, we parenthetically note that similar issues regarding thermal conductivity (and its suppression) in stochastic magnetic fields arise in the context of galaxy cluster formation and in the theory of cooling flows .
Return current effects {#return-current}
----------------------
The very significant electrical currents associated with the injection of non-thermal electrons necessitate a current-neutralizing return current, set up by a combination of electrostatic and inductive processes, the relative role of which has been a matter of some debate . However, irrespective of the detailed physics responsible for establishing this return current, driving it through the finite resistivity of the ambient medium requires a local electric field ${\cal E}_\parallel = j_\parallel/\sigma_\parallel$, which in turn causes both an Ohmic heating rate $Q_{\rm rc} = j_\parallel \, {\cal E}_\parallel = j_\parallel^2/\sigma_\parallel$ and an additional energy loss rate $| dE/dt | = e \, {\cal E}_\parallel \, v = j_\parallel \, {\cal E}_\parallel/n_e = j_\parallel^2/n_e \, \sigma_\parallel$ for each of the accelerated electrons.
Now, the transport of [*non-thermal*]{} electrons is dominated by non-diffusive cold-target energy losses [see, e.g., @1971SoPh...18..489B; @1973SoPh...31..143B; @1978ApJ...224..241E], and hence we expect that the transport of such electrons, and hence the current density $j_\parallel$ that they carry, is largely unaffected by collisionless pitch-angle scattering. (Although @2014ApJ...780..176K have shown that the direct beam current $j_\parallel$ is also reduced somewhat due to the presence of pitch-angle scattering, the reduction factor is not as large as the transport coefficient reduction factors $R$ considered here, so that we may assume that the current density $j_\parallel$ associated with the injected electrons is essentially the same as in the purely collisional case.) Any change in ohmic energy losses is therefore driven primarily by changes in the parallel electrical conductivity $\sigma_\parallel$. Reducing the value of $\sigma_\parallel$ through turbulence results in a greater rate of Ohmic heating $Q_{\rm rc}$ (and hence higher coronal heating rates) and also a greater energy loss rate $|dE/dt|$ for the accelerated electrons.
This enhancement of the return-current electron energy loss rates affects the heating rate as a function of position [@1980ApJ...235.1055E] and hence the hydrodynamic response of the atmosphere [e.g., @1984ApJ...279..896N; @1989ApJ...341.1067M]. It also reduces the amount of energy precipitating into the chromospheric footpoints, thus possibly accounting for the “gentle” evaporation observed by, e.g., @1988ApJ...329..456Z. Enhanced return current energy losses also result in a more effective confinement of hard-X-ray-producing electrons in the corona, which may offer an alternative explanation for loop-top coronal sources .
Estimates of the ratio of return current heating to collisional energy loss in the flaring corona show that, for moderately large flares they are comparable [see Figure 3 of @1980ApJ...235.1055E]. The same figure shows that the ratio of return current heating to collisional energy loss in the chromosphere, where most of the electron heating occurs, can be up to several percent. Thus, enhancing the return current heating/energy loss rate by even an order of magnitude through turbulent modification of the electrical conductivity $\sigma_\parallel$ and could possibly transform the flaring corona into a return-current-dominated regime . This has very significant implications, ranging from the spatial distribution of hard X-ray emission and electron heating, to the total number of accelerated electrons required to produce a given hard X-ray intensity .
A more dominant role for return current losses in the energy loss rate for accelerated electrons has a possibly even more interesting effect. Since the energy loss rate for an individual electron $| dE/dt | = e \, {\cal E}_\parallel \, v = e \, v \, j_\parallel/\sigma_\parallel$, which is proportional to the injected current $j_\parallel$ and hence the electron injection rate, and since the total hard X-ray yield is proportional to the injection rate divided by the energy loss rate [@1988ApJ...331..554B], it follows that the hard X-ray yield in a return-current-loss dominated regime [*is independent of the injected number of electrons*]{} [@1980ApJ...235.1055E]. Such a possible saturation of hard X-ray flux with increasing flare intensity has been reported by @2007ApJ...666.1268A.
Summary and conclusions {#summary}
=======================
Motivated by observations suggesting the presence of magnetic fluctuations in flaring loops and also suggesting that turbulent pitch-angle scattering plays a significant role in the transport of energy by both thermal and non-thermal electrons in solar flares, we have derived formulae for the thermal and electrical conductivities in the presence of both collisions and magnetic turbulence.
The enhanced electron confinement effected by the addition of collisionless pitch-angle scattering can reduce the [*thermal*]{} conductivity of the corona, thus decreasing thermal conductive losses and so increasing coronal temperatures compared to those in a model with collisionally-dominated transport. This may explain the localization of coronal X-ray sources in the apex of the loop [see, e.g., @2011SSRv..159..107H for a review]. It also increases the cooling time for the flare-heated coronal plasma, possibly alleviating the need for post-impulsive-phase heating by unidentified processes [@1980sfsl.work..341M; @2012ApJ...759...71E]. Finally, it means that the corona becomes more of a “warm” target in the calculation of the energy loss rate of accelerated electrons, which has an impact on the relationship between the source-integrated electron spectrum and the injected spectrum [@1988ApJ...331..554B; @2003ApJ...595L.115B; @2003ApJ...595L.119E; @2011SSRv..159..301K; @2015ApJ...809...35K] and hence on the overall energetics associated with accelerated electrons [@1986NASACP...2439..505D; @1997JGR...10214631M; @2003ApJ...595L..97H; @2004JGRA..10910104E; @2005JGRA..11011103E; @2012ApJ...759...71E].
The suppressed value of the [*electrical*]{} conductivity may significantly increase the importance of ohmic heating, both in the thermodynamics of the flare-heated atmosphere and in the propagation of the accelerated electrons themselves. In particular, because the inclusion of return current energy losses affects the “bremsstrahlung efficiency” (energy of hard X-rays produced per electron energy injected in the corona), it may significantly alter the injected electron flux required to produce given hard X-ray flux, with further attendant implications for the overall role of accelerated electrons in flare energetics.
Because of these important implications for quantitative details of the impulsive phase of solar flares and even for its overall viability , we urge workers in the field to consider such anomalous transport effects in their modeling of particle transport, thermal conduction, and the electrodynamics of solar flares.
This work is partially supported by a STFC consolidated grant. Financial support by the European Commission through the “Radiosun” (PEOPLE-2011-IRSES-295272) is gratefully acknowledged. AGE was supported by grant NNX10AT78G from NASA’s Goddard Space Flight Center.
[^1]: This approximation is, for the Coulomb interaction, equivalent to the Lorentz gas approximation which has immobile heavy hard spheres as scattering agents, as in the Drude-Lorentz model of electric conductivity, hence the name “Lorentz plasma” [e.g., @1964PhFl....7..407K], see also [@1966JETP...23..145R] for the analogy between low-frequency electrostatic turbulence and the Lorentz gaz
[^2]: Note that only one term arises in evaluating the velocity-space derivative $\partial f_0(z,v)/\partial v$ that prefixes the electric field $E_\parallel$ term in the expression for $f_1(z,v,\mu)$. Thus only one velocity-moment term appears in each of the expressions for $\alpha_\parallel$ and $\sigma_\parallel$, and the signs of these two thermoelectric coefficients are therefore fixed.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this survey, we present five different proofs for the transcendence of Kempner’s number, defined by the infinite series $\sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas and methods that are used for proving deeper results. We outline proofs for some of these results and also point out references where the reader can find all the details.'
---
\[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Question]{}
\[thm\][Remark]{}
Boris Adamczewski\
CNRS, Université de Lyon, Université Lyon 1\
Institut Camille Jordan\
43 boulevard du 11 novembre 1918\
69622 Villeurbanne Cedex\
France\
<[email protected]>\
.2 in
[*À Jean-Paul Allouche, pour son soixantième anniversaire.*]{}
[*Le seul véritable voyage, le seul bain de Jouvence,\
ce ne serait pas d’aller vers de nouveaux paysages,\
mais d’avoir d’autres yeux...*]{}
[Marcel Proust, [*À la recherche du temps perdu*]{}]{}
Introduction
============
Proving that a given real number is transcendental is usually an extremely difficult task. Even for classical constants like $e$ and $\pi$, the proofs are by no means easy, and most mathematicians would be happy with a single proof of the transcendence of $e+\pi$ or $\zeta(3)$. In contrast, this survey will focus on the simple series $$\kappa := \sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$$ that can be easily proved to be transcendental. The first proof is due to Kempner [@Kempner] in 1916 and, in honor of this result, we refer to $\kappa$ as the [*Kempner number*]{}[^1]. If the transcendence of $\kappa$ is not a real issue, our aim is instead to look at the many faces of $\kappa$, which will lead us to give five different proofs of this fact. This must be (at least for the author) some kind of record, even if we do not claim this list of proofs to be exhaustive. In particular, we will not discuss Kempner’s original proof. Beyond the transcendence of $\kappa$, the different proofs we give all offer the opportunity to mention some interesting ideas and methods that are used for proving deeper results. We outline proofs for some of these results and also point out references where the reader can find all the details.
The outline of the paper is as follows. In Section \[section: knight\], we start this survey with a totally elementary proof of the transcendence of the Kempner number, based on a digital approach. Quite surprisingly, a digital approach very much in the same spirit has a more striking consequence concerning the problem of finding good lower bounds for the number of non-zero digits among the first $N$ digits of the binary expansion of algebraic irrational numbers. In Section \[section: mahler\], we give a second proof that relies on Mahler’s method. We also take the opportunity to discuss a little-known application of this method to transcendence in positive characteristic. Our third proof is a consequence of a $p$-adic version of Roth’s theorem due to Ridout. It is given in Section \[section: roth\]. More advanced consequences of the Thue–Siegel–Roth–Schmidt method are then outlined. In Section \[section: folding\], we give a description of the continued fraction expansion of $\kappa$ which turns out to have interesting consequences. We present two of them, one concerning a question of Mahler about the Cantor set and the other the failure of Roth’s theorem in positive characteristic. Our last two proofs rely on such a description and the Schmidt subspace theorem. They are given, respectively, in Sections \[sec: cf1\] and \[sec: cf2\]. The first one uses the fact that $\kappa$ can be well approximated by a familly of quadratic numbers of a special type, while the second one uses the fact that $\kappa$ and $\kappa^2$ have very good rational approximations with the same denominators. Both proofs give rise to deeper results that are described briefly.
Throughout this paper, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote, respectively, the floor and the ceiling of the real number $x$. We also use the classical notation $f(n)\ll g(n)$ (or equivalently $g(n)\gg f(n)$), which means that there exists a positive real number $c$, independent of $n$, such that $f(n)< cg(n)$ for all sufficiently large integers $n$.
An ocean of zeros {#section: knight}
=================
We start this survey with a totally elementary proof of the transcendence of the Kempner number, due to Knight [@Kn91]. This proof, which is based on a digital approach, is also reproduced in the book of Allouche and Shallit [@AS Chap. 13].
Set $$f(x) := \sum_{n=0}^{\infty} x^{2^n} ,$$ so that $\kappa = f(1/2)$. For every integer $i\geq 0$, we let $a(n,i)$ denote the coefficient of $x^n$ in the formal power series expansion of $f(x)^i$. Thus $a(n,i)$ is equal to the number of ways that $n$ can be written as a sum of $i$ powers of $2$, where different orderings are counted as distinct. For instance, $a(5,3)=3$ since $$5 = 1+2+2=2+1+2=2+2+1 .$$ Note that for positive integers $n$ and $i$, we clearly have $$\label{eq: ank}
a(n,i) \leq (1+\log_2 n)^{i} .$$ The expression $$\kappa^{i} = f(1/2)^{i} = \sum_{n=0}^{\infty} \frac{a(n,i)}{2^n}$$ can be though of as a “fake binary expansion" of $\kappa^{i}$ in which carries have not been yet performed.
Let us assume, to get a contradiction, that $\kappa$ is an algebraic number. Then there exist integers $a_0,\ldots,a_d$, with $a_d>0$, such that $$a_0 + a_1 \kappa+\cdots + a_d\kappa^d =0 .$$ Moving all the negative coefficients to the right-hand side, we obtain an equation of the form $$\label{eq: fake}
a_{i_1}\kappa^{i_1} + \cdots + a_{i_r}\kappa^{i_r} + a_d\kappa^{d} = b_{j_1}\kappa^{j_1} + \cdots + b_{j_s}\kappa^{j_s} ,$$ where $r+s=d$, $0\leq i_1<\cdots <i_r<d$, $0\leq j_1<\cdots <j_s$, and coefficients on both sides are nonnegative.
Let $m$ be a positive integer and set $N := (2^{d}-1)2^{m}$, so that the binary expansion of $N$ is given by $$(N)_2 = \underbrace{1\cdots 1}_{d}\; \underbrace{0\cdots 0}_{m} .$$ Then for every integer $n$ in the interval $I := [N - (2^{m-1}-1) , N + 2^{m}-1]$ and every integer $i$, $0\leq i\leq d$, we have $$a(n,i) = \left\{
\begin{array}{ll}
d!, & \mbox{if } n=N \mbox{ and } i=d; \\
0,& \mbox{otherwise.}
\end{array}
\right.$$ Indeed, every $n\not=N\in I$ has more than $d$ nonzero digits in its binary expansion, while $N$ has exactly $d$ nonzero digits.
Now looking at Equality (\[eq: fake\]) as an equality between two fake binary numbers, we observe that
- on the right-hand side, all fake digits with position in $I$ are zero (an ocean of zeros),
- on the left-hand side, all fake digits with position in $I$ are zero except for the one in position $N$ that is equal to $a_dd!$ (an island).
Note that $d$ is fixed, but we can choose $m$ as large as we want. Performing the carries on the left-hand side of (\[eq: fake\]) for sufficiently large $m$, we see that the fake digit $a_dd!$ will produce some nonzero binary digits in a small (independent of $m$) neighborhood of the position $N$. On the other hand, the upper bound (\[eq: ank\]) ensures that, for sufficiently large $m$, carries on the right-hand side of (\[eq: fake\]) will never reach this neighborhood of the position $N$. By uniqueness of the binary expansion, Equality (\[eq: fake\]) is thus impossible. This provides a contradiction.
Beyond Knight’s proof
---------------------
Unlike $\kappa$, which is a number whose binary expansion contains absolute oceans of zeros, it is expected that all algebraic irrational real numbers have essentially random binary expansions (see the discussion in Section \[section: roth\]). As a consequence, if $\xi$ is an algebraic irrational number and if $\mathcal P(\xi,2,N)$ denotes the number of $1$’s among the first $N$ digits of the binary expansion of $\xi$, we should have $$\mathcal P(\xi,2,N) \sim \frac{N}{2} \cdot$$ Such a result seems to be out of reach of current approaches, and to find good lower bounds for $\mathcal P(\xi,2,N)$ remains a challenging problem.
A natural (and naive) approach to study this question can be roughly described as follows: if the binary expansion of $\xi$ contains too many zeros among its first digits, then some partial sums of its binary expansion should provide very good rational approximations to $\xi$; but on the other hand, we know that algebraic irrationals cannot be too well approximated by rationals. More concretely, we can argue as follows. Let $\xi:= \sum_{i\geq 0}1/2^{n_i}$ be a binary algebraic number. Then there are integers $p_k$ such that $$\displaystyle\sum_{i=0}^k \frac{1}{2^{n_i}}= \frac{p_k}{2^{n_k}} \;\;\mbox{ and }\;\;
\left \vert \xi - \frac{p_k}{2^{n_k}} \right\vert < \frac{2}{2^{n_{k+1}}} \cdot$$ On the other hand, since $\xi$ is algebraic, given a positive $\varepsilon$, Ridout’s theorem (see Section \[section: roth\]) implies that $$\left\vert \xi - \frac{p_k}{2^{n_k}} \right\vert > \frac{1}{2^{(1+\varepsilon)n_k}} ,$$ for every sufficiently large integer $k$. This gives that $n_{k+1} < (1 + \varepsilon) n_k+1$ for such $k$. Hence, for any positive number $c$, we have $$\label{ri}
\mathcal P(\xi,2,N) > c \log N ,$$ for every sufficiently large $N$.
Quite surprisingly, a digital approach very much in the same spirit as Knight’s proof of the transcendence of $\kappa$ led Bailey, J. M. Borwein, Crandall, and Pomerance [@BBCP04] to obtain the following significant improvement of (\[ri\]).
Let $\xi$ be an algebraic real number of degree $d\geq 2$. Then there exists an explicit positive number $c$ such that $$\mathcal P(\xi,2,N) > cN^{1/d} ,$$ for every sufficiently large $N$.
We do not give all the details, for which we refer the reader to [@BBCP04]. Let $\xi$ be an algebraic number of degree $d\geq 2$, for which we assume that $$\label{eq: H}
\mathcal P(\xi,2,N) < c N^{1/d} ,$$ for some positive number $c$. Let $a_0,\ldots,a_d$, $a_d>0$ such that $$a_0+a_1\xi +\cdots +a_d\xi^d=0 .$$ Let $\sum_{i\geq 0} 1/2^{n_i}$ denote the binary expansion of $\xi$ and set $f(x) := \sum_{i\geq 0} x^{n_i}$. We also let $a(n,i)$ denote the coefficient of $x^n$ in the power series expansion of $f(x)^i$. Without loss of generality we can assume that $n_0=0$. This assumption is important, in fact, for it ensures that $$a(n,d-1)=0 \implies a(n,i)=0, \mbox{ for every }i, 0\leq i\leq d-1 .$$ Set $T_i(R):=\sum_{m\geq 1} a(R+m,i)/2^m$ and $T(R):=\sum_{i=0}^d a_iT_i(R)$. A fundamental remark is that $T(R)\in\mathbb Z$. Let $N$ be a positive integer and set $K:=\lceil 2d\log N\rceil$. Our aim is now to estimate the quantity $$\sum_{R=0}^{N-K} \vert T(R)\vert .$$
[*Upper bound.*]{} We first note that $$\label{eq: ineq}
a(n,i)\leq {n+i-1\choose i-1} \mbox{ and }
\sum_{R=0}^Na(R,i) \leq \mathcal P(\xi,2,N)^{i} .$$ Using these inequalities, it is possible to show that $$\begin{aligned}
\
\sum_{R=0}^{N-K} T_i(R) & = & \sum_{m=1}^{\infty} 2^{-m} \sum_{R=0}^{N-K} a(R+m,i)\nonumber \\
&<& \sum_{R=0}^{N} a(R,i) + 2^{-K} \sum_{R=K}^N T_i(R) \nonumber \\
& \leq & \mathcal P(\xi,2,N)^i+1 ,\nonumber \end{aligned}$$ for $N$ sufficiently large. We thus obtain that $$\begin{aligned}
\label{eq: maj}
\sum_{R=0}^{N-K} \vert T(R)\vert & \leq & \sum_{i=1}^d \vert a_k\vert \left(\mathcal P(\xi,2,N)^{i}+1\right) \nonumber \\
&\leq & a_dc^dN + O(N^{1-1/d}) .\end{aligned}$$
[*Lower bound.*]{} We first infer from (\[eq: H\]) and (\[eq: ineq\]) that $$\mbox{Card} \left\{ R \in [0,N] \mid a(R,d-1)>0\right\} < c^{d-1}N^{1-1/d} .$$ Let $0=R_1<R_2<\cdots <R_M$ denote the elements of this set, so that $M<c^{d-1}N^{1-1/d}$. Set also $R_{M+1}:=N$. Then $$\sum_{i=1}^M(R_{i+1}-R_i) = N .$$ Let $\delta >0$ and set $$\mathcal I := \left\{ i \in [0,M] \mid R_{i+1}-R_i \geq \frac{\delta}{3}c^{1-d}N^{1/d} \right\} .$$ Then we have $$\label{eq: min1}
\sum_{i\in \mathcal I} \left( R_{i+1} - R_i\right)\geq \left(1-\frac{\delta}{3}\right) N .$$ Now let $i\in \mathcal I$. Note that Roth’s theorem (see Section \[section: roth\]) allows us to control the size of blocks of consecutive zeros that may occur in the binary expansion of $\xi$. Concretely, it ensures the existence of an integer $$j_i\in\left( \frac{1}{2+\delta/2}(R_{i+1}-R_i-d\log N), (R_{i+1}-R_i-d\log N)\right)$$ such that $a(j_i,1)>0$. Thus $a(R_i+j_i,d)>0$ since by assumption $n_0=0$, and then a short computation gives that $T(R_i+j_i-1)>0$.
By definition, $a(R,d-1)=0$ for every $R\in (R_{i},R_{i+1})$ and thus $a(R,i)=0$ for every $R\in (R_i,R_{i+1})$ and every $i\in[0,d-1]$. For such integers $R$, a simple computation gives $$T(R-1) = \frac{1}{2}T(R) +\frac{1}{2} a_da(R,d)$$ and thus $T(R)>0$ implies $T(R-1)>0$. Applying this argument successively to $R$ equal to $R_i+j_i-1,R_i+j_i-2,\ldots, R_i+1$, we finally obtain that $T(R)>0$ for every integer $R\in [R_i,R_i +j_i)$. The number of integers $R\in [0,N]$ such that $T(R)>0$ is thus at least equal to $$\sum_{i\in\mathcal I} \frac{1}{2+\delta/2}\left( R_{i+1}-R_i-d\log N\right) ,$$ which, by (\[eq: min1\]), is at least equal to $(1/2 -\delta/3)N$ for sufficiently large $N$. Since $T(R)\in \mathbb Z$, we get that $$\label{eq: min2}
\sum_{R=0}^{N-K} \vert T(R)\vert \geq \left(\frac{1}{2}-\frac{\delta}{3}\right)N ,$$ for sufficiently large $N$.
[*Conclusion.*]{} For sufficiently large $N$, Inequalities (\[eq: maj\]) and (\[eq: min2\]) are incompatible as soon as $c\leq ((2+\delta)a_d)^{-1/d}$. Thus, choosing $\delta$ sufficiently small, this proves the theorem for any choice of $c$ such that $c<(2a_d)^{-1/d}$.
We end this section with a few comments on Theorem BBCP.
- It is amusing to note that replacing Roth’s theorem by Ridout’s theorem in this proof only produces a minor improvement: the constant $c$ can be replaced by a slightly larger one (namely by any $c< a_d^{-1/d}$).
- A deficiency of Theorem BBCP is that it is not effective: it does not give an explicit integer $N$ above which the lower bound holds. This comes from the well-known fact that Roth’s theorem is itself ineffective. The authors of [@AdFa] show that one can replace Roth’s theorem by the much weaker Liouville inequality to derive an effective version of Theorem BBCP. This version is actually slightly weaker, because the constant $c$ is replaced by a smaller constant, but the proof becomes both totally elementary and effective.
- Last but not least: Theorem BBCP immediately implies the transcendence of the number $$\sum_{n=0}^{\infty} \frac{1}{2^{\left\lfloor n^{\log\log n}\right\rfloor}} ,$$ for which no other proof seems to be known!
Functional equations {#section: mahler}
====================
Our second proof of the transcendence of $\kappa$ follows a classical approach due to Mahler. In a series of three papers [@Mah29; @Mah30A; @Mah30B] published in 1929 and 1930, Mahler initiated a totally new direction in transcendence theory. [*Mahler’s method*]{} aims to prove transcendence and algebraic independence of values at algebraic points of locally analytic functions satisfying certain type of functional equations. In its original form, it concerns equations of the form $$f(x^k) = R(x,f(x)) ,$$ where $R(x,y)$ denotes a bivariate rational function with coefficients in a number field. In our case, we consider the function $$f(x) := \sum_{n=0}^{\infty} x^{2^n} ,$$ and we will use the fact that it is analytic in the open unit disc and satisfies the following basic functional equation: $$\label{eq: fe}
f(x^2) = f(x) - x .$$
Note that we will in fact prove much more than the transcendence of $\kappa=f(1/2)$, for we will obtain the transcendence of $f(\alpha)$ for every nonzero algebraic number $\alpha$ in the open unit disc. This is a typical advantage when using Mahler’s method. Before proceeding with the proof we need to recall a few preliminary results.
[*Preliminary step 1.*]{} The very first step of Mahler’s method consists in showing that the function $f(x)$ is transcendental over the field of rational function $\mathbb C(x)$. There are actually several ways to do that. Instead of giving an elementary but [*ad hoc*]{} proof, we prefer to give the following general statement that turns out to be useful in this area.
Let $(a_n)_{n\geq 0}$ be an aperiodic sequence with values in a finite subset of $\mathbb Z$. Then $f(x)=\sum_{n\geq 0}a_nx^n$ is transcendental over $\mathbb C(x)$.
Note that $f(x)\in\mathbb Z[[x]]$ has radius of convergence one and the classical theorem of Pólya–Carlson[^2] thus implies that $f(x)$ is either rational or transcendental. Furthermore, since the coefficients of $f(x)$ take only finitely many distinct values and form an aperiodic sequence, we see that $f(x)$ cannot be a rational function.
[*Preliminary step 2.*]{} We will also need to use Liouville’s inequality as well as basic estimates about [*height functions*]{}. There are, of course, several notions of heights. The most convenient works with the absolute logarithmic Weil height that will be denoted by $h$. We refer the reader to the monograph of Waldschmidt [@Wa_book Chap. 3] for an excellent introduction to heights and in particular for a definition of $h$. Here we just recall a few basic properties of $h$ that will be used in the sequel. All are proved in [@Wa_book Chap. 3]. For every integer $n$ and every pair of algebraic numbers $\alpha$ and $\beta$, we have $$\label{eq: height1}
h(\alpha^n)=\vert n\vert h(\alpha)$$ and $$\label{eq: height1bis}
h(\alpha+\beta) \leq h(\alpha)+h(\beta) +\log 2 .$$ More generally, if $P(X,Y)\in\mathbb Z[X,Y]\setminus\{0\}$ then $$\label{eq: height2}
h(P(\alpha,\beta)) \leq \log L(P) + (\deg_XP)h(\alpha) + (\deg_Y P)h(\beta) ,$$ where $L(P)$ denote the length of $P$, which is classically defined as the sum of the absolute values of the coefficients of $P$. We also recall Liouville’s inequality: $$\label{eq: liouville}
\log\vert \alpha\vert \geq - d h(\alpha) ,$$ for every nonzero algebraic number $\alpha$ of degree at most $d$.
We are now ready to give our second proof of transcendence for $\kappa$.
Given a positive integer $N$, we choose a nonzero bivariate polynomial $P_N\in \mathbb Z[X,Y]$ whose degree in both $X$ and $Y$ is at most $N$, and such that the order of vanishing at $x=0$ of the formal power series $$A_N(x) := P_N(x,f(x))$$ is at least equal to $N^2$. Note that looking for such a polynomial amounts to solving a homogeneous linear system over $\mathbb Q$ with $N^2$ equations and $(N+1)^2$ unknowns, which is of course always possible. The fact that $A_N(x)$ has a large order of vanishing at $x=0$ ensures that $A_N$ takes very small values around the origin. More concretely, for every complex number $z$, $0\leq \vert z\vert <1/2$, we have $$\label{eq: anmaj}
\vert A_n(z) \vert \leq c(N) \vert z\vert^{N^2} ,$$ for some positive $c(N)$ that only depends on $N$.
Now we pick an algebraic number $\alpha$, $0< \vert \alpha\vert <1$, and we assume that $f(\alpha)$ is also algebraic. Let $L$ denote a number field that contains both $\alpha$ and $f(\alpha)$ and let $d:=[L:\mathbb Q]$ be the degree of this extension. The functional equation (\[eq: fe\]) implies the following for every positive integer $n$: $$A_N(\alpha^{2^{n}}) = P_N(\alpha^{2^{n}},f(\alpha^{2^{n}})) = P_N\left(\alpha^{2^{n}},f(\alpha)- \sum_{k=0}^{n-1} \alpha^{2^k}\right) \in L .$$ Thus $A_N(\alpha^{2^{n}})$ is always an algebraic number of degree at most $d$. Furthermore, we claim that $A_N(\alpha^{2^{n}})\not=0$ for all sufficiently large $n$. Indeed, the function $A_N(x)$ is analytic in the open unit disc and it is nonzero because $f(x)$ is transcendental over $\mathbb C(x)$, hence the identity theorem applies.
Now, using (\[eq: height1\]), (\[eq: height1bis\]) and (\[eq: height2\]), we obtain the following upper bound for the height of $A_N(\alpha^{2^n})$: $$\begin{array}{lll}
h(A_N(\alpha^{2^n})) & = &h(P_N(\alpha^{2^{n}},f(\alpha^{2^{n}})))\\ \\
& \leq & \log L(P_N) + Nh(\alpha^{2^n}) + Nh(f(\alpha^{2^{n}})) \\ \\
& = & \log L(P_N) + 2^nNh(\alpha) + Nh(f(\alpha) - \sum_{k=0}^{n-1} \alpha^{2^k}) \\ \\
& \leq & \log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2 .\\ \\
\end{array}$$ From now on, we assume that $n$ is sufficiently large to ensure that $A_N(\alpha^{2^{n}})$ is nonzero and that $\vert \alpha^{2^n}\vert < 1/2$. Since $A_N(\alpha^{2^n})$ is a nonzero algebraic number of degree at most $d$, Liouville’s inequality (\[eq: liouville\]) implies that $$\log \vert A_N(\alpha^{2^n})\vert \geq - d \left(\log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2\right) .$$ On the other hand, since $\vert \alpha^{2^n}\vert < 1/2$, Inequality (\[eq: anmaj\]) gives that $$\log \vert A_N(\alpha^{2^n})\vert \leq \log c(N) + 2^nN^2\log \vert \alpha\vert .$$ We thus deduce that $$\log c(N) + 2^nN^2\log \vert \alpha\vert \geq - d \left(\log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2\right) .$$ Dividing both sides by $2^n$ and letting $n$ tend to infinity, we obtain $$N \leq \frac{2dh(\alpha)}{\vert \log \vert\alpha\vert \vert} \cdot$$ Since $N$ can be chosen arbitrarily large independently of the choice of $\alpha$, this provides a contradiction and concludes the proof.
Beyond Mahler’s proof {#allouche}
---------------------
Mahler’s method has, by now, become a classical chapter in transcendence theory. As observed by Mahler himself, his approach allows one to deal with functions of several variables and systems of functional equations as well. It also leads to algebraic independence results, transcendence measures, measures of algebraic independence, and so forth. Mahler’s method was later developed by various authors, including Becker, Kubota, Loxton and van der Poorten, Masser, Nishioka, and Töpfer, among others. It is now known to apply to a variety of numbers defined by their decimal expansion, their continued fraction expansion, or as infinite products. For these classical aspects of Mahler’s theory, we refer the reader to the monograph of Ku. Nishioka [@Ni_liv] and the references therein.
We end this section by pointing out another feature of Mahler’s method that is unfortunately less well known. A major deficiency of Mahler’s method is that, in contrast with the Siegel $E$- and $G$-functions, there is not a single classical transcendental constant that is known to be the value at an algebraic point of an analytic function solution to a Mahler-type functional equation. Roughly, this means that the most interesting complex numbers for number theorists seemingly remain beyond the scope of Mahler’s method. However, a remarkable discovery of Denis is that Mahler’s method can be applied to prove transcendence and algebraic independence results involving [*periods of $t$-modules*]{}, which are variants of the more classical periods of abelian varieties, in the framework of the arithmetic of function fields of positive characteristic. For a detailed discussion on this topic, we refer the reader to the recent survey by Pellarin [@Pel2], and also [@Pel1]. Unfortunately, we cannot begin to do justice here to this interesting topic. We must be content to give only a hint about the proof of the transcendence of an analogue of $\pi$ using Mahler’s method, and we hope that the interested reader will look for more in [@Pel1; @Pel2].
Let $p$ be a prime number and $q=p^{e}$ be an integer power of $p$ with $e$ positive. We let $\mathbb F_q$ denote the finite field of $q$ elements, $\mathbb F_q[t]$ the ring of polynomials with coefficients in $\mathbb F_q$, and $\mathbb F_q(t)$ the field of rational functions. We define an absolute value on $\mathbb F_q[t]$ by $\vert P\vert = q^{\deg_t P}$ so that $\vert t\vert =q$. This absolute value naturally extends to $\mathbb F_q(t)$. We let $\mathbb F_q((1/t))$ denote the completion of $\mathbb F_q(t)$ for this absolute value and let $C$ denote the completion of the algebraic closure of $\mathbb F_q((1/t))$ for the unique extension of our absolute value to the algebraic closure of $\mathbb F_q((1/t))$. Roughly, this allows to replace the natural inclusions $$\mathbb Z\subset \mathbb Q\subset \mathbb R\subset \mathbb C$$ by the following ones $$F_q[t]\subset F_q(t)\subset F_q((1/t))\subset C .$$ The field $C$ is a good analogue for $\mathbb C$ and allows one to use some tools from complex analysis such as the identity theorem. In this setting, the formal power series $$\Pi := \prod_{n=1}^{\infty} \frac{1}{1- t^{1-q^n}} \in \mathbb F_q((1/t))\subset C$$ can be thought of as an analogue of the number $\pi$. To be more precise, the Puiseux series $$\widetilde{\Pi} = t(-t)^{1/(q-1)} \prod_{n=1}^{\infty} \frac{1}{1- t^{1-q^n}} \in C$$ is a fundamental period of Carlitz’s module and, in this respect, it appears to be a reasonable analogue for $2i\pi$. Of course, proving the transcendence of either $\Pi$ or $\widetilde{\Pi}$ over $\mathbb F_p(t)$ remains the same. As discovered by Denis [@Denis], it is possible to deform the infinite product given in our definition of $\Pi$, in order to obtain the following “analytic function" $$f_{\Pi}(x) := \prod_{n=1}^{\infty} \frac{1}{1- tx^{q^n}}$$ which converges for all $x\in C$ such that $\vert x\vert <1$. A remarkable property is that the function $f_{\Pi}(x)$ satisfies the following Mahler-type functional equation: $$f_{\Pi}(x^q) = \frac{f_{\Pi}(x)}{(1-tx^q)} \cdot$$ As the principle of Mahler’s method also applies in this framework, one can prove along the same lines as in the proof we just gave for the transcendence of $\kappa$ that $f_{\Pi}$ takes transcendental values at every nonzero algebraic point in the open unit disc of $C$. Considering the rational point $1/t$, we obtain the transcendence of $\Pi = f_{\Pi}(1/t)$.
Note that there are many other proofs of the transcendence of $\Pi$. The first is due to Wade [@Wa] in 1941. Other proofs were then given by Yu [@Yu] using the theory of Drinfeld modules, by Allouche [@All90] using automata theory and Christol’s theorem, and by De Mathan [@DM] using tools from Diophantine approximation.
$p$-adic rational approximation {#section: roth}
===============================
The first transcendence proof that graduate students in mathematics usually meet concerns the so–called Liouville number $${\mathcal L}:= \sum_{n=1}^{\infty} \frac{1}{b^{n!}} \cdot$$ This series is converging so quickly that partial sums $$\frac{p_n}{q_n} := \sum_{k=1}^{n}\frac{1}{b^{k!}}$$ provide infinitely many extremely good rational approximations to $\mathcal L$, namely $$\left\vert {\mathcal L} - \frac{p_n}{q_n} \right\vert
< \frac{2}{q_n^{n+1}} \cdot$$ In view of the classical Liouville inequality [@Li], these approximations prevent $\mathcal L$ from being algebraic. Since $\kappa$ is also defined by a lacunary series that converges very fast, it is tempting to try to use a similar approach. However, we will see that this requires much more sophisticated tools.
Liouville’s inequality is actually enough to prove the transcendence for series such as $\sum_{i=0}^{\infty} 1/2^{n_i}$, where $\limsup (n_{i+1}/n_i)=+\infty$, but it does not apply if $n_i$ has only an exponential growth like $n_i=2^{i}$, $n_i=3^{i}$ or $n_i=F_i$ (the $i$th Fibonacci number). In the case where $\limsup (n_{i+1}/n_i)>2$, we can use Roth’s theorem [@Ro55].
Let $\xi$ be a real algebraic number and $\varepsilon$ be a positive real number. Then the inequality $$\left\vert \xi - \frac{p}{ q} \right\vert < \frac{1}{q^{2 + \varepsilon}}$$ has only a finite number of rational solutions $p/q$.
For instance, the transcendence of the real number $\xi := \sum_{i=0}^{\infty} 1/2^{3^{i}}$ is now a direct consequence of the inequality $$0<\left\vert \xi - \frac{p_n}{q_n} \right\vert
< \frac{2}{q_n^3} ,$$ where $p_n/q_n := \sum_{i=0}^n 1/2^{3^{i}}$. However, the same trick does not apply to $\kappa$, for we get that $$\left\vert \kappa - \frac{p_n}{q_n} \right\vert
\gg \frac{1}{q_n^2} ,$$ if $p_n/q_n := \sum_{i=0}^n 1/2^{2^{i}}$.
The transcendence of $\kappa$ actually requires the following $p$-adic extension of Roth’s theorem due to Ridout [@Ri57]. For every prime number $\ell$, we let $\vert \cdot \vert_\ell$ denote the $\ell$-adic absolute value normalized such that $\vert \ell \vert_\ell = \ell^{-1}$.
Let $\xi$ be an algebraic number and $\varepsilon$ be a positive real number. Let $S$ be a finite set of distinct prime numbers. Then the inequality $$\left( \prod_{\ell \in S} \vert p \vert_\ell \cdot \vert q\vert_\ell \right) \cdot
\left\vert \xi - \frac{p}{q} \right\vert < \frac{1}{q^{2+\varepsilon}}$$ has only a finite number of rational solutions $p/q$.
With Ridout’s theorem in hand, the transcendence of $\kappa$ can be easily deduced: we just have to take into account that the denominators of our rational approximations are powers of $2$.
Let $n$ be a positive integer and set $$\rho_n := \sum_{i=1}^{n}\frac{1}{2^{2^{i}}}.$$ Then there exists an integer $p_n$ such that $\rho_n = p_n/q_n$ with $q_n = 2^{2^n}$. Observe that $$\left\vert \kappa - \frac{p_n}{q_n} \right\vert < \frac{2}{2^{2^{n+1}}}
= \frac{2}{(q_n)^{2}},$$ and let $S=\{2\}$. Then, an easy computation gives that $$\vert q_n \vert_2 \cdot \vert p_n\vert_2 \cdot
\left\vert \kappa - \frac{p_n}{q_n}\right\vert < \frac{2}{(q_n)^{3}} \cdot$$ Applying Ridout’s theorem, we get that $\kappa$ is transcendental.
Of course there is no mystery, the difficulty in this proof is hidden in the proof of Ridout’s theorem.
Beyond Roth’s theorem
---------------------
The Schmidt subspace theorem [@Schmidt80] provides a formidable multidimensional generalization of Roth’s theorem. We state below a simplified version of the $p$-adic subspace theorem due to Schlickewei [@Sch77], which turns out to be very useful for proving transcendence of numbers defined by their base-$b$ expansion or by their continued fraction expansion. Note that our last two proofs of the transcendence of $\kappa$, given in Sections \[sec: cf1\] and \[sec: cf2\], both rely on the subspace theorem. Several recent applications of this theorem can also be found in [@Bilu].
We recall that a [*linear form*]{} (in $m$ variables) is a homogeneous polynomial (in $m$ variables) of degree $1$.
Let $m\ge 2$ be an integer and $\varepsilon$ be a positive real number. Let $S$ be a finite set of distinct prime numbers. Let $L_1, \ldots , L_m$ be $m$ linearly independent linear forms in $m$ variables with real algebraic coefficients. Then the set of solutions ${\bf x} = (x_1, \ldots, x_m)$ in $\mathbb Z^m$ to the inequality $$\left(\prod_{i=1}^m\prod_{\ell \in S} \vert x_i\vert_\ell \right) \cdot \prod_{i=1}^m
{\vert L_{i} ({\bf x}) \vert } \leq
(\max\{|x_1|, \ldots , |x_m|\})^{-\varepsilon}$$ lies in finitely many proper subspaces of $\mathbb Q^m$.
Let us first see how the subspace theorem implies Roth’s theorem. Let $\xi$ be a real algebraic number and $\varepsilon$ be a positive real number. Consider the two independent linear forms $\xi X - Y$ and $X$. The subspace theorem implies that all the integer solutions $(p, q)$ to $$\label{Ch7:equation:rothlin}
\vert q \vert \cdot \vert q \xi - p\vert < \vert q\vert^{-\varepsilon}$$ are contained in a finite union of proper subspaces of $\mathbb Q^2$. There thus is a finite set of lines $x_1 X + y_1 Y = 0,\ \ldots ,\
x_t X + y_t Y = 0$ such that, every solution $(p, q)\in\mathbb Z^2$ to (\[Ch7:equation:rothlin\]), belongs to one of these lines. This means that the set of rational solutions $p/q$ to $\left\vert \xi - p/q \right\vert < q^{-2-\varepsilon}$ is finite, which is Roth’s theorem.
### A theorem of Corvaja and Zannier
Let us return to the transcendence of $\kappa$. Given an integer $b\geq 2$ and letting $S$ denote the set of prime divisors of $b$, it is clear that the same proof also gives the transcendence of $\sum_{n=0}^{\infty}1/b^{2^n}$. However, if we try to replace $b$ by a rational or an algebraic number, we may encounter new difficulties. As a good exercise, the reader can convince himself that the proof will still work with $b={5 \over 2}$ or $b={{17} \over 4}$, but not with $b={3 \over 2}$ or $b={5 \over 4}$. Corvaja and Zannier [@CZ] make clever use of the subspace theorem that allows them to overcome the problem in all cases. Among other results, they proved the following nice theorem.
Let $(n_i)_{i\geq 0}$ be a sequence of positive integers such that $\liminf n_{i+1}/n_i>1$ and let $\alpha$, $0<\vert \alpha\vert<1$, be an algebraic number. Then the number $$\sum_{i=0}^{\infty} \alpha^{n_i}$$ is transcendental.
Of course, we recover the fact, already proved in Section \[section: mahler\] by Mahler’s method, that the function $f(x)= \sum_{n=0}^{\infty} x^{2^n}$ takes transcendental values at every nonzero algebraic point in the open unit disc. The proof of Theorem CZ actually requires an extension of the $p$-adic subspace theorem to number fields (the version we gave is sufficient for rational points). We also take the opportunity to mention that the main result of [@Ad04] is actually a consequence of Theorem 4 in [@CZ].
In order to explain the idea of Corvaja and Zannier we somewhat oversimplify the situation by considering only the example of $f({4 \over 5})$. We refer the reader to [@CZ] for a complete proof. We assume that $f({4 \over 5})$ is algebraic and we aim at deriving a contradiction. A simple computation gives that $$\left\vert f\left({4 \over 5}\right) - \sum_{k=0}^n \left(\frac{4}{5}\right)^{2^k} \right\vert < 2 \left(\frac{4}{5}\right)^{2^{n+1}} ,$$ for every nonnegative integer $n$. This inequality can obviously be rephrased as $$\left\vert f\left({4 \over 5}\right) - \sum_{k=0}^n \left(\frac{4}{5}\right)^{2^k} - \left(\frac{4}{5}\right)^{2^{n+1}} - \left(\frac{4}{5}\right)^{2^{n+2} }\right\vert
< 2 \left(\frac{4}{5}\right)^{2^{n+3}} ,$$ but the subspace theorem will now take care of the fact that the last two terms on the left-hand side are $S$-units (for $S=\{2,5\}$). Multiplying by $5^{2^{n+2}}$, we obtain that $$\left\vert 5^{2^{n+2}}f\left({4 \over 5}\right) - 5^{2^{n+2}-2^n}p_n - 4^{2^{n+1}}5^{2^{n+1}} - 4^{2^{n+2}}\right\vert < 2 \left(\frac{4}{5}\right)^{2^{n+3}}5^{2^{n+2}} ,$$ for some integer $p_n$. Consider the following four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{ll}
L_1 (X_1,X_2,X_3,X_4) = & f({4 \over 5})X_1-X_2-X_3-X_4, \\
L_2 (X_1,X_2,X_3,X_4) = & X_1, \\
L_3 (X_1,X_2,X_3,X_4) = & X_3 , \\
L_4(X_1,X_2,X_3,X_4)=& X_4.
\end{array}$$ For every integer $n \geq 1$, consider the integer quadruple $${\bf x}_n = (x_1^{(n)},x_2^{(n)},x_3^{(n)},x_4^{(n)}):=\left( 5^{2^{n+2}} , 5^{2^{n+2}-2^n}p_n , 4^{2^{n+1}}5^{2^{n+1}} , 4^{2^{n+2}} \right) .$$ Note that $\Vert {\bf x}_n\Vert_{\infty}\leq 5\cdot 5^{2^{n+2}}$. Set also $S=\{2,5\}$. Then a simple computation shows that $$\left(\prod_{i=1}^4\prod_{\ell \in S} \vert x_i^{(n)}\vert_\ell \right) \cdot \prod_{i=1}^4
{\vert L_{i} ({\bf x}_n) \vert } \leq 2 \left(\frac{4^8}{5^7}\right)^{2^{n}} <
\Vert {\bf x}_n\Vert_{\infty}^{-\varepsilon} ,$$ for some $\varepsilon>0$. We then infer from the subspace theorem that all points ${\bf x}_n$ lie in a finite number of proper subspaces of $\mathbb Q^4$. Thus, there exist a nonzero integer quadruple $(x,y,z,t)$ and an infinite set of distinct positive integers ${\mathcal N}$ such that $$\label{eq: plan}
5^{2^{n+2}} x + 5^{2^{n+2}-2^n}p_n y + 4^{2^{n+1}}5^{2^{n+1}} z + 4^{2^{n+2}} t = 0,$$ for every $n$ in ${\mathcal N}$. Dividing (\[eq: plan\]) by $ 5^{2^{n+2}}$ and letting $n$ tend to infinity along ${\mathcal N}$, we get that $$x + \kappa y=0 .$$ Since $\kappa$ is clearly irrational, this implies that $x=y=0$. But then Equality (\[eq: plan\]) becomes $$4^{2^{n+1}}5^{2^{n+1}} z = -4^{2^{n+2}} t ,$$ which is impossible for large $n\in\mathcal N$ unless $z=t=0$ (look at, for instance, the $5$-adic absolute value). This proves that $x=y=z=t=0$, a contradiction.
Note that the proof of the transcendence of $f(\alpha)$, for every algebraic number $\alpha$ with $0<\vert \alpha\vert <1$, actually requires the use of the subspace theorem with an arbitrary large number of variables (depending on $\alpha$). For instance, we need $14$ variables to prove the transcendence of $f(2012/2013)$.
### The decimal expansion of algebraic numbers
The decimal expansion of real numbers such as $\sqrt 2$, $\pi$, and $e$ appears to be quite mysterious and, for a long time, has baffled mathematicians. After the pioneering work of É. Borel [@Borel1; @Borel2], most mathematicians expect that all algebraic irrational numbers are [*normal numbers*]{}, even if this conjecture currently seems to be out of reach. Recall that a real number is normal if for every integer $b\geq 2$ and every positive integer $n$, each one of the $b^n$ blocks of digits of length $n$ occurs in its base-$b$ expansion with the same frequency. We end this section by pointing out an application of the $p$-adic subspace theorem related to this problem.
Let $\xi$ be a real number and $b\geq 2$ be a positive integer. Let $(a_n)_{n\geq -k}$ denote the base-$b$ expansion of $\xi$, that is, $$\xi= \displaystyle\sum_{n\geq -k} \frac{a_n}{b^n} = a_{-k}\cdots a_{-1}a_0{\scriptscriptstyle \bullet} a_1a_2\cdots .$$ Following Morse and Hedlund [@HM], we define the [*complexity function*]{} of $\xi$ with respect to the base $b$ as the function that associates with each positive integer $n$ the positive integer $$p(\xi,b,n) := \mbox{Card} \{(a_j,a_{j+1},\ldots, a_{j+n-1}), \; j \geq 1\}.$$ A normal number thus has the maximum possible complexity in every integer base, that is, $p(\xi,b,n) = b^n$ for every positive integer $n$ and every integer $b\geq 2$. One usually expects such a high complexity for numbers like $\sqrt 2$, $\pi$, and $e$. Ferenczi and Mauduit [@FM] gave the first lower bound for the complexity of all algebraic irrational numbers by means of Ridout’s theorem. More recently, Adamczewski and Bugeaud [@AdBuAnnals] use the subspace theorem to obtain the following significant improvement of their result.
Let $b \geq 2$ be an integer and $\xi$ be an algebraic irrational number. Then $$\label{bfm}
\lim_{n\to \infty} \frac{p(\xi,b,n)}{ n} = + \infty .$$
Note that Adamczewski [@Ad10] obtains a weaker lower bound for some transcendental numbers involving the exponential function. For a more complete discussion concerning the complexity of the base-$b$ expansion of algebraic numbers, we refer the reader to [@Ad10; @AdBuAnnals; @AdBuCant; @Miw09].
We only outline the main idea for proving Theorem AB1 and refer the reader to [@AdBuAnnals] or [@AdBuCant] for more details. Let $\xi$ be an algebraic number and let us assume that $$\label{eq: comp}
\liminf_{n\to \infty} \frac{p(\xi,b,n)}{ n} < +\infty,$$ for some integer $b\geq 2$. Our goal is thus to prove that $\xi$ is rational. Without loss of generality, we can assume that $0<\xi<1$.
Our assumption implies that the number of distinct blocks of digits of length $n$ in the base-$b$ expansion of $\xi$ is quite small (at least for infinitely many integers $n$). Thus, at least some of these blocks of digits have to reoccur frequently, which forces the early occurrence of some repetitive patterns in the base-$b$ expansion of $\xi$. This rough idea can be formalized as follows. We first recall some notation from combinatorics on words. Let $V=v_1\cdots v_r$ be a finite word. We let $\vert V\vert=r$ denote the length of $V$. For any positive integer $k$, we write $V^k$ for the word $$\overbrace{V\cdots V}_{\mbox{$k$ times}} .$$ More generally, for any positive real number $w$, $V^w$ denotes the word $V^{\lfloor w \rfloor}V'$, where $V'$ is the prefix of $V$ of length $\left\lceil(w-\lfloor w\rfloor)\vert V\vert\right\rceil$. With this notation, one can show that the assumption (\[eq: comp\]) ensures the existence of a real number $w>1$ and of two infinite sequences of finite words $(U_n)_{n\geq 1}$ and $(V_n)_{n\geq 1}$ such that the base-$b$ expansion of $\xi$ begins with the block of digits $0{\scriptscriptstyle \bullet} U_nV_n^w$ for every positive integer $n$. Furthermore, if we set $r_n := \vert U_n \vert$ and $s_n := \vert V_n\vert$, we have that $s_n$ tends to infinity with $n$ and there exists a positive number $c$ such that $r_n/s_n <c$ for every $n\geq 1$.
This combinatorial property has the following Diophantine translation. For every positive integer $n \ge 1$, $\xi$ has to be close to the rational number with ultimately periodic base-$b$ expansion $$0{\scriptscriptstyle \bullet} U_nV_nV_nV_n\cdots .$$ Precisely, one can show the existence of an integer $p_n$ such that $$\label{Ch7:equation:pj}
\left\vert \xi - \frac{p_n}{ b^{r_n} (b^{s_n} - 1)} \right\vert
\ll \frac{1 }{b^{r_n + w s_n}} \cdot$$
Consider the following three linearly independent linear forms with real algebraic coefficients: $$\begin{array}{ll}
L_1 (X_1, X_2, X_3) = & \xi X_1 - \xi X_2 - X_3, \\
L_2 (X_1, X_2, X_3) = & X_1, \\
L_3 (X_1, X_2, X_3) = & X_2.
\end{array}$$ Evaluating them at the integer points $${\bf x}_n =(x_1^{(n)},x_2^{(n)},x_3^{(n)}):= (b^{r_n+s_n}, b^{r_n}, p_n) ,$$ we easily obtain that $$\left(\prod_{i=1}^3\prod_{p \in S} \vert x_i^{(n)}\vert_p \right) \cdot
\prod_{i=1}^3 {\vert L_i ({\bf x}_n) \vert } \ll
\left( \max\{b^{r_n + s_n}, b^{r_n}, p_n\} \right)^{-\varepsilon},$$ where $\varepsilon := (\omega-1)/2(c+1) >0$ and $S$ denotes the set of prime divisors of $b$. We then infer from the subspace theorem that all points ${\bf x}_n$ belong to a finite number of proper subspaces of $\mathbb Q^3$. There thus exist a nonzero integer triple $(x,y,z)$ and an infinite set of distinct positive integers ${\mathcal N}$ such that $$\label{Ch7:equation:plan}
x b^{r_n+s_n} + y b^{r_n} + z p_n = 0 ,$$ for every $n$ in ${\mathcal N}$. Dividing (\[Ch7:equation:plan\]) by $b^{r_n+s_n}$ and letting $n$ tend to infinity along ${\mathcal N}$, we get that $$x+\xi z = 0 ,$$ as $s_n$ tends to infinity. Since $(x,y,z)$ is a nonzero vector, this implies that $\xi$ is a rational number. This ends the proof.
Interlude: from base-$b$ expansions to continued fractions {#section: folding}
==========================================================
It is usually very difficult to extract any information about the continued fraction expansion of a given irrational real number from its decimal or binary expansion and vice versa. For instance, $\sqrt 2$, $e$, and $\tan 1$ all have a very simple continued fraction expansion, while they are expected to be normal and thus should have essentially random expansions in all integer bases. In this section, we shall give an exception to this rule: our favorite binary number $\kappa$ has a predictable continued fraction expansion that enjoys remarkable properties involving both repetitive and symmetric patterns (see Theorem Sh1 below). Our last two proofs of transcendence for $\kappa$, given in Sections \[sec: cf1\] and \[sec: cf2\], both rely on Theorem Sh1.
For an introduction to continued fractions, the reader is referred to standard books such as Perron[@Perron], Khintchine [@Kh], or Hardy and Wright [@HaWr]. We will use the classical notation for finite or infinite continued fractions $$\frac{p}{q}
= a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}
= [a_0, a_1,\cdots, a_n]$$ resp., $$\xi =
a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ddots+\cfrac{1}
{a_n + \cfrac{1}{\ddots}}}}} = [a_0, a_1,\cdots, a_n, \cdots]$$ where $p/q$ is a positive rational number (resp. $\xi$ is a positive irrational real number), $n$ is a nonnegative integer, $a_0$ is a nonnegative integer, and the $a_i$’s are positive integers for $i \geq 1$. Note that we allow $a_n=1$ in the first equality. If $A=a_1a_2\cdots$ denotes a finite or an infinite word whose letters $a_i$ are positive integers, then the expression $[0,A]$ stands for the finite or infinite continued fraction $[0,a_1,a_2,\ldots]$. Also, if $A=a_1a_2\cdots a_n$ is a finite word, we let $A^R:=a_na_{n-1}\cdots a_1$ denote the reversal of $A$. As in the previous section, we use $\vert A\vert$ to denote the length of the finite word $A$.
The following elementary result was first discovered by Mendès France [@MF][^3].
Let $c, a_0, a_1, \ldots,a_n$ be positive integers. Let $p_n/q_n := [a_0, a_1, \cdots, a_n]$. Then $$\label{fold}
\frac{p_n}{q_n} + \frac{(-1)^n}{c q_n^2}
= [a_0, a_1, a_2, \cdots, a_n, c, -a_n, -a_{n-1}, \cdots, -a_1].$$
For a proof of the folding lemma, see, for instance, [@AS p. 183]. In Equality (\[fold\]) negative partial quotients occur. However, we have two simple rules that permit to get rid of these forbidden partial quotients: $$\label{eq: rule1}
[\ldots,a,0,b,\ldots] = [\ldots,a+b,\ldots]$$ and $$\label{eq: rule2}
[\ldots,a, -b_1, \cdots, -b_r] = [,\ldots,a-1,1,b_1-1,b_2,\ldots,b_r] .$$
As first discovered independently by Shallit [@Sh1; @Sh2] and Kmošek [@Km], the folding lemma can be used to describe the continued fraction expansion of some numbers having a lacunary expansion in an integer base, such as $\kappa$. Following Theorem 11 in [@Sh1], we give now a complete description of the continued fraction expansion of $2\kappa$. The choice of $2\kappa$ instead of $\kappa$ is justified by obtaining a nicer formula.
Let $A_1:=1112111111$ and $B_1:=11121111$. For every positive integer $n$, let us define the finite words $A_{n+1}$ and $B_{n+1}$ as follows: $$A_{n+1} = A_n12 (B_n)^R$$ and $$B_{n+1} \mbox{ is the prefix of } A_{n+1} \mbox{ with length }\vert A_{n+1}\vert -2 .$$ Then the sequence of words $A_n$ converges to an infinite word $$A_{\infty} = 111 2 111111 12 1111 2 111 12 \cdots$$ and $$2\kappa = [1,A_{\infty}] =[1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,\ldots] .$$
In particular, the partial quotients of $2\kappa$ take only the values $1$ and $2$. This shows that $\kappa$ is badly approximable by rationals and [*a fortiori*]{} that the transcendence of $\kappa$ is beyond the scope of Roth’s theorem (the $p$-adic version of Roth’s theorem was thus really needed in Section \[section: roth\]).
First note that by the definition of $A_{n+1}$, the word $A_n$ is a prefix of $A_{n+1}$, for every nonnegative integer $n$, which implies that the sequence of finite words converges (for the usual topology on words) to an infinite word $A_{\infty}$.
For every integer $n\geq 0$, we set $$\frac{P_n}{Q_n} := \sum_{k=0}^{n} \frac{2}{2^{2^k}} \cdot$$ We argue by induction to prove that $$P_n/Q_n = [1,A_{n-2}] ,$$ for every integer $n\geq 3$. We first note that $P_3/Q_3=[1,1,1,1,2,1,1,1,1,1,1]=[1,A_1]$. Let $n\geq 3$ be an integer and let us assume that $P_n/Q_n=[1,A_{n-2}]$. By the definition of $P_n/Q_n$, we have that $$\label{eq: pnqn}
\frac{P_{n+1}}{Q_{n+1}} = \frac{P_n}{Q_n} + \frac{1}{2Q_n^2} \cdot$$ Furthermore, an easy induction shows that for every integer $k\geq 1$, $\vert A_k\vert$ is even and $A_k$ ends with $11$ so that $$\label{eq: 11}
(A_k)^R = 11 (B_k)^R .$$ Since $\vert A_{n-2}\vert$ is even, we can apply the folding Lemma and we infer from Equalities (\[eq: pnqn\]) and (\[eq: 11\]), and from the transformation rules (\[eq: rule1\]) and (\[eq: rule2\]) that $$\begin{array}{lll}
P_{n+1}/Q_{n+1} &=& [1,A_{n-2},2,- (A_{n-2})^R] \\ \\
&=& [1,A_{n-2},1,1, 0,1,(B_{n-2})^R]\\ \\
&=& [1,A_{n-2},1,2, (B_{n-2})^R] \\ \\
&=& [1,A_{n-1}] .
\end{array}$$ This proves that $P_n/Q_n=[1,A_{n-2}]$ for every $n\geq 3$. Since the sequence $(P_n/Q_n)_{n\geq 0}$ converges to $2\kappa$ and $A_n$ is always a prefix of $A_{n+1}$, we obtain that $2\kappa=[1,A_{\infty}]$, as desired.
Two applications
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In the second part of the paper [@Sh1], Shallit [@Sh2] extends his construction and obtained the following general result[^4].
Let $b\geq 2$ and $n_0\geq 0$ be integers and let $(c_n)_{n\geq 0}$ be a sequence of positive integers such that $c_{n+1}\geq 2c_n$, for every integer $n\geq n_0$. Set $d_n:=c_{n+1}-2c_n$ and $$S_b(n) := \sum_{k=0}^n \frac{1}{b^{c_k}} \cdot$$ If $n\geq n_0$ and $S_b(n)=[a_0,a_1,\ldots,a_r]$, with $r$ even, then $$S_b(n+1) = [a_0,a_1,\ldots,a_r,b^{d_n}-1,1,a_r-1,a_{r-1},\ldots,a_1] .$$
This result turns out to have interesting consequences, two of which are recalled below.
### A question of Mahler about the Cantor set
Mahler [@Mah84] asked the following question: how close can irrational numbers in the Cantor set be approximated by rational numbers? We recall that the [*irrationality exponent*]{} of an irrational real number $\xi$, denoted by $\mu(\xi)$, is defined as the supremum of the real numbers $\mu$ for which the inequality $$\left\vert \xi - \frac{p}{ q} \right\vert < \frac{1}{q^{\mu}}$$ has infinitely many rational solutions $p/q$. Mahler’s question may thus be interpreted as follows: are there elements in the Cantor set with any prescribed irrationality exponent?
This question was first answered positively by Levesley, Salp and Velani [@LSV] by means of tools from metric number theory. A direct consequence of Shallit’s result is that one can also simply answer Mahler’s question by providing explicit example of numbers in the Cantor set with any prescribed irrationality exponent. We briefly outline how to prove this result and refer the reader to [@Bu08] for more details. Some refinements along the same lines can also be found in [@Bu08].
Let $\tau\geq 2$ be a real number. Note first that the number $$\xi_{\tau} := 2 \sum_{n= 1}^{\infty} \frac{1}{3^{\lfloor \tau^n\rfloor}}$$ clearly belongs to the Cantor set. Furthermore, the partial sums of $\xi_{\tau}$ provide infinitely many good rational approximations which ensure that $\mu(\xi_{\tau})\geq \tau$. When $\tau\geq (3+\sqrt 5)/2$, a classical approach based on triangles inequalities allows to show that $\mu(\xi_{\tau})\leq \tau$. However, the method fails when $\tau$ satisfies $2\leq \tau<(3+\sqrt 5)/2$.
In order to overcome this difficulty, we can use repeatedly Theorem Sh2 with $b=3$ and $c_n=\lfloor \tau^n\rfloor$ to obtain the continued fraction expansion of $\xi_{\tau}/2$. Set $\xi_{\tau}/2:=[0,b_1,b_2,\ldots]$ and let $s_n$ denote the denominator of the $n$th convergent to $\xi_{\tau}/2$. If $\tau=2$, we see that the partial quotients $b_n$ are bounded, which implies $\mu(\xi_2)=\mu(\xi_2/2)=2$, as desired. We can thus assume that $\tau >2$. Let us recall that once we know the continued fraction of an irrational number $\xi$, it becomes easy to deduce its irrationality exponent. Indeed, if $\xi=[a_0,a_1,\ldots]$, it is well-known that $$\label{eq: mes}
\mu(\xi) = 2 + \limsup_{n\to\infty} \frac{\ln a_{n+1}}{\ln q_n} ,$$ where $p_n/q_n$ denotes the $n$th convergent to $\xi$. Equality (\[eq: mes\]) is actually a direct consequence of the inequality $$\frac{1}{(2+a_{n+1})q_n^2}<\left\vert \xi - \frac{p_n}{q_n} \right\vert < \frac{1}{a_{n+1}q_n^2 }$$ and the fact that the convergents provide the best rational approximations (see, for instance, [@Kh Chapter 6]). When $\tau >2$, the formula given in Theorem Sh2 shows that the large partial quotients[^5] of $\xi_{\tau}/2$ are precisely those equal to $3^{d_n}-1$ which occur first at some positions, say $r_n+1$. But then Theorem Sh2 implies that $s_{r_n}$ is the denominator of $\sum_{k=1}^n1/3^{\lfloor \tau^k\rfloor}$, that is $s_{r_n}= 3^{\lfloor \tau^n\rfloor}$. A simple computation thus shows that $$\limsup_{n\to\infty} \frac{\ln b_{n+1}}{\ln s_n} = \limsup_{n\to\infty} \frac{\ln (3^{d_n}-1)}{\ln 3^{\lfloor \tau^n\rfloor}}= \tau-2 ,$$ since $d_n=\lfloor 3^{\tau^{n+1}}\rfloor - 2 \lfloor 3^{\tau^n}\rfloor$. Then we infer from Equality (\[eq: mes\]) that $\mu(\xi_{\tau})= \mu(\xi_{\tau}/2)= \tau$, as desired.
### The failure of Roth’s theorem in positive characteristic
We consider now Diophantine approximation in positive characteristic. Let $\mathbb F_p((1/t))$ be the field of Laurent series with coefficients in the finite field $\mathbb F_p$, endowed with the natural absolute value $\vert \cdot \vert$ defined at the end of Section \[section: mahler\]. In this setting, the approximation of real numbers by rationals is naturally replaced by the approximation of Laurent series by rational functions. In analogy with the real case, we define the irrationality exponent of $f(t) \in\mathbb F_p((1/t))$, denoted by $\mu(f)$, as the supremum of the real number $\mu$ for which the inequality $$\left\vert f(t) - \frac{P(t)}{Q(t)} \right\vert < \frac{1}{\deg Q^{\mu}}$$ has infinitely many rational solutions $P(t)/Q(t)$.
It is well-known that Roth’s theorem fails in this framework. Indeed, Mahler [@Mah49] remarked that it is even not possible to improve Liouville’s bound for the power series $$f(t): = \sum_{n=0}^{\infty}t^{-p^n} \in\mathbb F_p[[1/t]]$$ is algebraic over $\mathbb F_p(t)$ with degree $p$, while $\mu(f)=p$. Osgood [@Os] and then Lasjaunias and de Mathan [@LaDeM] obtained an improvement of the Liouville bound (namely the Thue bound) for a large class of algebraic functions. However not much is known about the irrationality exponent of algebraic functions in $\mathbb F_p((1/t))$. For instance, it seems that we do not know whether $\mu(f)=2$ for almost every[^6] algebraic Laurent series in $\mathbb F_p((1/t))$. We also do not know what the set $\mathcal E$ of possible values taken by $\mu(f)$ is precisely when $f$ runs over the algebraic Laurent series.
In this direction, we mention that it is possible to use an analogue of Theorem Sh2 for power series with coefficients in a finite field (the proof of which is identical). Thakur [@Th99] uses such a result in order to exhibit explicit power series $f(t)\in\mathbb F_p[[1/t]]$ with any prescribed irrationality measure $\nu\geq 2$, with $\nu$ rational. In other words, this proves that $\mathbb Q_{\geq 2}\subset \mathcal E$, where $\mathbb Q_{\geq 2}:=\mathbb Q\cap [2,+\infty)$. These power series are defined as linear combinations of Mahler-type series which shows that they are algebraic, while the analogue of Theorem Sh2 allows us to describe their continued fraction expansion and thus to easily compute the value of $\mu(f)$, as previously. Note that this result can also be obtained by considering only continued fractions, as shown independently by Thakur [@Th99] and Schmidt [@Schmidt00]. It is expected, but not yet proved, that $\mathcal E=\mathbb Q_{\geq 2}$. For a recent survey about these questions, we refer the reader to [@Th09].
Approximation by quadratic numbers {#sec: cf1}
==================================
A famous consequence of the subspace theorem provides a natural analogue of Roth’s theorem in which rational approximations are replaced by quadratic ones. More precisely, if $\xi$ is an algebraic number of degree at least $3$ and $\varepsilon$ is a positive real number, then the inequality $$\label{eq: sch}
\left\vert \xi - \alpha\right\vert < \frac{1}{H(\alpha)^{3+\varepsilon}} ,$$ has only finitely many quadratic solutions $\alpha$. Here $H(\alpha)$ denotes the (naive) height of $\alpha$, that is, the maximum of the modulus of the coefficients of its minimal polynomial.
In this section, we give our fourth proof of transcendence for $\kappa$ which is obtained as a consequence of Theorem Sh1 (see Section \[section: folding\]) and Theorem AB2 stated below. We observe that some repetitive patterns occur in the continued fraction expansion of $2\kappa$ and then we use them to find infinitely many good quadratic approximations $\alpha_n$ to $2\kappa$. However, a more careful analysis would show that $$\left\vert 2\kappa - \alpha_n \right\vert \gg \frac{1}{H(\alpha_n)^3} ,$$ so that we cannot directly apply (\[eq: sch\]). Fortunately, the subspace theorem offers a lot of freedom and adding some information about the minimal polynomial of our approximations finally allows us to conclude.
We keep the notation from Sections \[section: roth\] and \[section: folding\]. Let ${\bf a}=a_1a_2\cdots$ be an infinite word and $w\geq 1$ be a real number. We say that ${\bf a}$ satisfies Condition $(*)_w$ if there exists a sequence of finite words $(V_n)_{n \ge 1}$ such that the following hold.
- For any $n \ge 1$, the word $V_n^w$ is a prefix of the word ${\bf a}$.
- The sequence $(\vert V_n\vert)_{n \ge 1}$ is increasing.
The following result is a special instance of Theorem 1 in [@AdBuActa].
Let $(a_n)_{n \ge 1}$ be a bounded sequence of positive integers such that ${\bf a}=a_1a_2\cdots$ satisfies Condition $(*)_w$ for some real number $w>1$. Then the real number $$\xi:= [0, a_1, a_2, \ldots]$$ is either quadratic or transcendental.
We only outline the main idea of the proof and refer the reader to [@AdBuActa] for more details.
Assume that the parameter $w > 1$ is fixed, as well as the sequence $(V_n)_{n \ge 1}$ occurring in the definition of Condition $(*)_w$. Set also $s_n:=\vert V_n\vert$. We want to prove that the real number $$\xi:= [0, a_1, a_2, \ldots]$$ is either quadratic or transcendental. We assume that $\alpha$ is algebraic of degree at least three and we aim at deriving a contradiction.
Let $p_n/q_n$ denote the $n$th convergent to $\xi$. The key fact for the proof is the observation that $\xi$ has infinitely many good quadratic approximations obtained by truncating its continued fraction expansion and completing by periodicity. Let $n$ be a positive integer and let us define the quadratic number $\alpha_n$ as having the following purely periodic continued fraction expansion: $$\alpha_n:= [0, V_nV_nV_n \cdots ] .$$ Then $$\left\vert \xi -\alpha_n\right\vert < \frac{1}{q_{\lfloor ws_n\rfloor^2}} ,$$ since by assumption the first $\lfloor ws_n\rfloor$ partial quotients of $\xi$ and $\alpha_n$ are the same. Now observe that $\alpha_n$ is a root of the quadratic polynomial $$P_n (X) := q_{s_n-1} X^2 + (q_{s_n} - p_{s_n-1}) X - p_{s_n} .$$ By Rolle’s theorem, we have $$\vert P_n (\xi)\vert = \vert P_n (\xi) - P_n (\alpha_n)\vert
\ll q_{s_n} |\xi - \alpha_n| \ll \frac{q_{s_n} }{ q_{\lfloor w s_n\rfloor}^2} \cdot$$ Furthermore, by the theory of continued fractions we also have $$\vert q_{s_n} \xi - p_{s_n}\vert \le \frac{1}{q_{s_n}} \cdot$$ Consider the four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{lll}
L_1(X_1, X_2, X_3, X_4)& = & \xi^2 X_2 + \xi (X_1 - X_4) - X_3, \\
L_2(X_1, X_2, X_3, X_4) &= & \xi X_1 - X_3, \\
L_3(X_1, X_2, X_3, X_4) &= & X_1, \\
L_4(X_1, X_2, X_3, X_4) &= & X_2.
\end{array}$$ Evaluating them on the integer quadruple $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, a simple computation using continuants shows that $$\prod_{1 \le j \le 4} |L_j (q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})|
\ll q_{s_n}^2 q_{\lfloor w s_n\rfloor}^{-2} < \frac{1}{q_{s_n}^{\varepsilon}} ,$$ for some positive number $\varepsilon$, when $n$ is large enough. We can thus apply the subspace theorem. We obtain that all the integer points $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, $n\in\mathbb N$, belong to a finite number of proper subspaces of $\mathbb Q^4$. After some work, it can be shown that this is possible only if $\xi$ is quadratic, a contradiction.
We are now ready to give a new proof of transcendence for $\kappa$.
Shallit [@Sh1] proves that the continued fraction of $\kappa$ is not ultimately periodic. Here we include, for the sake of completeness, a similar proof that the word $A_{\infty}$ is not ultimately periodic. Set $A_{\infty}:=a_1a_2\cdots$. We argue by contradiction assuming that $A_{\infty}$ is ultimately periodic. There thus exist two positive integers $r$ and $n_0$ such that $$\label{eq: per}
a_{n+jr} = a_{n} ,$$ for every $n\geq n_0$ and $j\geq 1$.
For every positive integer $i$, set $k_i:= \vert A_i\vert = 10\cdot 2^{i-1}$. Let us fix a positive integer $n$ such that $k_n \geq r+ n_0$. Theorem Sh1 implies that $A_{\infty}$ begins with $$A_{n+1} = a_1\cdots a_{k_n-2}a_{k_n-1}a_{k_n} 12 a_{k_n-2}\cdots a_1$$ and since $A_n$ always ends with $11$ (see (\[eq: 11\])), we obtain that $$\label{eq: 1112}
A_{n+1}=a_1\cdots a_{k_n-2} 11 12 a_{k_n-2}\cdots a_1 .$$ We thus have $$\label{eq: sym}
a_{k_n+x+1} = a_{k_n-x} ,$$ for every integer $x$ with $2\leq x \leq k_n-1$. Since the word $1112$ occurs infinitely often in $A_{\infty}$, we see that $r\geq 4$. We can thus choose $x=r-2$ in (\[eq: sym\]), which gives that $$a_{k_n+r-1} = a_{k_n-r+2} .$$ Then (\[eq: per\]) implies that $$a_{k_n+r-1} = a_{k_n-1}$$ and $$a_{k_n-r+2} =a_{k_n +2} .$$ Finally, we get that $$a_{k_n-1} = a_{k_n +2} ,$$ that is $$1=2 ,$$ a contradiction. Hence, $A_{\infty}$ is not ultimately periodic.
Now, set $$\xi := \frac{4\kappa-3}{2-2\kappa} \cdot$$ Clearly it is enough to prove that $\xi$ is transcendental. By Theorem Sh1, we have that $$2\kappa =[1,A_{\infty}] = [1,a_1,a_2,\ldots] $$ and a simple computation shows that $$\xi = [0,a_3,a_4,\cdots] .$$ Let us show that the infinite word $a_3a_3\cdots$ satisfies Condition $(*)_{1+3/10}$. Let $n$ be a positive integer. Using the definition of $A_{n+2}$, we infer from (\[eq: 1112\]) that $$A_{n+2} = a_1\cdots a_{k_n-2} 1112 a_{k_n-2}\cdots a_1 12 a_3a_4\cdots a_{k_n-2} 2 111a_{k_n-2}\cdots a_1$$ Setting $V_n := a_3a_4\cdots a_{k_n-2} 1112 a_{k_n-2}\cdots a_112$, we obtain that $A_{n+2}$ begins with $$a_1a_2 V_n^{1+ (k_n-4)/2k_n} .$$ Since $k_n\geq 10$, we thus deduce that $A_{\infty}$ begins with $a_1a_2V_n^{1+3/10}$, for every integer $n\geq 1$. This shows that the infinite word $a_3a_4\cdots$ satisfies Condition $(*)_{1+3/10}$. Applying Theorem AB2, we obtain that $\xi$ is either quadratic or transcendental. However, we infer from Lagrange’s theorem that $\xi$ cannot be quadratic, for we just have shown that the word $a_3a_4\cdots$ is not ultimately periodic. Thus $\xi$ is transcendental, concluding the proof.
Beyond this proof
-----------------
Very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. From numerical evidence and a belief that these numbers behave like most of the numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. Apparently, Khintchine [@Kh] was the first to consider this question (see [@All00; @Sha; @MIW] for surveys including a discussion of this problem). Although almost nothing has been proved yet in this direction, Lang [@LangSMF; @Lang] made some more general speculations, including the fact that algebraic numbers of degree at least three should behave like most of the numbers with respect to the Gauss–Khintchine–Kuzmin–Lévy laws. This conjectural picture is thus very similar to the one encountered in Section \[section: roth\] regarding the expansion of algebraic irrational numbers in integer bases.
As a first step, it is worth proving that real numbers with a “too simple" continued fraction expansion are either quadratic or transcendental. Of course, the term “simple" can lead to many interpretations. It may, for instance, denote real numbers whose continued fraction expansion can be produced in a simple algorithmical, dynamical, combinatorial or arithmetical way. In any case, it is reasonable to expect that such expansions should be close to be periodic. In this direction, there is a long tradition of using an excess of periodicity to prove the transcendence of some continued fractions (see, for instance, [@ADQZ; @Baker62; @Bax; @Dav1; @Dav2; @Dav3; @Maillet; @Quef1; @Quef2]). Adamczewski and Bugeaud [@AdBuActa] use the freedom offered by the subspace theorem to prove some combinatorial transcendence criteria, which provide significant improvements of those previously obtained in [@ADQZ; @Dav1; @Dav2; @Quef1; @Quef2]. Some transcendence measures for such continued fractions are also recently given in [@AdBuJEMS; @Bu3], following the general approach developed in [@AdBuPLMS] (see also [@AdBuCrelle] for similar results related to integer base expansions).
Theorem AB2 gives the transcendence of non-quadratic real numbers $\xi$ whose continued fraction expansion begins with arbitrarily long blocks of partial quotients of the form $V_n^{1+\varepsilon}$, for some positive $\varepsilon$. The key fact in the proof to apply the subspace theorem is to see that the linear form $$\xi^2 X_2 + \xi (X_1 - X_4) - X_3$$ takes small values at the integer quadruples $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, where $s_n=\vert V_n\vert$ and where $p_n/q_n$ denotes the $n$th convergent to $\xi$. This idea can be naturally generalized to the important case where the repetitive patterns do not occur at the very beginning of the expansion. Indeed, if the continued fraction expansion of $\xi$ begins with a block of partial quotients of the form $U_nV_n^{1+\varepsilon}$, then $\xi$ is well approximated by the quadratic number $\alpha'_n:=[0,U_nV_nV_nV_n\cdots]$. As shown in [@AdBuActa], when dealing with these more general patterns, we can argue similarly, but we have now to estimate the linear form $$L(X_1,X_2,X_3,X_4):= \xi^2X_1-\xi(X_2+X_3)+X_4$$ at the integer quadruples $$\begin{array}{c}
{\bf x}_n:= (q_{r_n-1}q_{r_n+s_n}-q_{r_n}q_{r_n+s_n}, q_{r_n-1}p_{r_n+s_n}-q_{r_n}p_{r_n+s_n-1}, \hspace{3cm}\\
\hspace{2cm}p_{r_n-1}q_{r_n+s_n}-p_{r_n}q_{r_n+s_n-1}, p_{r_n-1}p_{r_n+s_n}-p_{r_n}p_{r_n+s_n-1}) ,
\end{array}$$ where $r_n:=\vert U_n\vert$, $s_n:=\vert V_n\vert$, and $p_n/q_n$ denotes the $n$th convergent to $\xi$. In a recent paper, Bugeaud [@Bu12] remarks that the quantity $\vert L({\bf x}_n)\vert$ was overestimated in the proof given in [@AdBuActa][^7]. Taking into account this observation, the method developed in [@AdBuActa] has much striking consequences than those initially announced. As an illustration, we now have that the continued fraction expansion of an algebraic number of degree at least $3$ cannot be generated by a finite automaton.
Simultaneous approximation by rational numbers {#sec: cf2}
==============================================
Another classical feature of the subspace theorem is that it can also deal with simultaneous approximation of several real numbers by rationals. Our last proof of the transcendence of $\kappa$ relies on this principle. We use the occurrences of some symmetric patterns in the continued fraction expansion of $2\kappa-1$ to find good simultaneous rational approximations of $2\kappa-1$ and $(2\kappa-1)^2$.
We keep the notation of Sections \[section: folding\] and \[sec: cf1\]. Set $\xi:=(2\kappa-1)$. Clearly it is enough to prove that $\xi$ is transcendental. We argue by contradiction assuming that $\xi$ is algebraic.
By Theorem Sh1, we have that $\xi=[0,A_{\infty}]= [0,a_1,a_2,\ldots]$. We let $p_n/q_n$ denote the $n$th convergent to $\xi$. We also set as previously $k_n:=\vert A_n\vert$. By the theory of continued fraction, we have $$\left\vert \xi - \frac{ p_{k_n-1} }{ q_{k_n-1} } \right\vert < \frac{1}{q_{k_n-1}^2}
\mbox{ and }
\left\vert \xi - \frac{ p_{k_n} }{ q_{k_n} } \right\vert < \frac{1}{q_{k_n}^2} \cdot$$ Let us also recall the so-called mirror formula (see, for instance, [@AdAl]): $$\label{eq: mirror}
\frac{q_{n-1}}{q_{n}} =[0,a_n,\ldots,a_1] .$$ Since $A_{n} = B_{n-1} 1112 (B_{n-1})^R$, we have $$\frac{q_{k_n-1}}{q_{k_n}} =[0,B_{n-1},2,1,1,1, (B_{n-1})^R]$$ and a simple computation using continuants shows that[^8] $$\left\vert \xi - \frac{q_{k_n-1}}{q_{k_n}} \right\vert \ll \frac{1}{q_{k_n}}
\mbox{ and }
\left\vert \frac{p_{k_n}}{q_{k_n}} - \frac{q_{k_n-1}}{q_{k_n}} \right\vert \ll \frac{1}{q_{k_n}} \cdot$$ Then we have $$\begin{array}{lll}
\left\vert \xi^2 - \frac{ p_{k_n-1} }{ q_{k_n} } \right\vert &=&
\left\vert \xi^2 - \frac{ p_{k_n-1} }{ q_{k_n-1} } \cdot \frac{ q_{k_n-1} }{ q_{k_n} } \right\vert \\ \\
&=& \left\vert \left (\xi-\frac{ p_{k_n-1} }{ q_{k_n-1} }\right)\left(\xi+\frac{ q_{k_n-1} }{ q_{k_n} } \right)
+ \xi\left(\frac{ p_{k_n-1} }{ q_{k_n-1} }- \frac{ q_{k_n-1} }{ q_{k_n} } \right)\right\vert \\ \\
& \ll& q_{k_n}^{-1} \cdot
\end{array}$$
Consider the four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{lll}
L_1(X_1, X_2, X_3, X_4)& = & \xi^2 X_1 - X_4, \\
L_2(X_1, X_2, X_3, X_4) &= & \xi X_1 - X_3, \\
L_3(X_1, X_2, X_3, X_4) &= & \xi X_2-X_4, \\
L_4(X_1, X_2, X_3, X_4) &= & X_2.
\end{array}$$ Evaluating them on the integer quadruple $(q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})$, our previous estimates implies that $$\prod_{1 \le j \le 4} \left\vert L_j (q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})\right\vert
\ll \frac{1}{q_{k_n}} \cdot$$ The subspace theorem thus implies that all the integer points $(q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})$, $n\geq 1$, belong to a finite number of proper subspaces of $\mathbb Q^4$. Thus, there exists a nonzero integer quadruple $(x,y,z,t)$ such that $$q_{k_n}x+ q_{k_n-1}y+ p_{k_n}z+ p_{k_n-1}t = 0 ,$$ for all $n$ in an infinite set of positive integers $\mathcal N$. Dividing by $q_{k_n}$ and letting $n$ tend to infinity along $\mathcal N$, we obtain that $$x+ \xi(y+z) + \xi^2 t =0 .$$ Since $A_{\infty}$ is not ultimately periodic (see Section \[sec: cf1\]), it follows from Lagrange’s theorem that $\xi$ is not quadratic and thus $x=t=(y+z)=0$. This gives that $q_{k_n-1} = p_{k_n}$ for every $n\in\mathcal N$. But $$\frac{q_{k_n-1} }{q_{k_n}} = [0,B_n,2,1,1,1, (B_n)^R]
\not= [0,B_n,1,1,1,2, (B_n)^R] = \frac{p_{k_n} }{q_{k_n}} ,$$ a contradiction.
Beyond this proof
-----------------
As already mentioned in Section \[sec: cf1\], there is a long tradition in using an excess of periodicity to prove the transcendence of some continued fractions. The fact that occurrences of symmetric patterns can actually give rise to transcendence statements is more surprising. The connection between palindromes[^9] in continued fractions and simultaneous approximation of a number and its square is reminiscent of works of Roy about extremal numbers [@Roy03bis; @Roy03; @Roy04] (see also [@Fi]). Inspired by this original discovery of Roy, Adamczewski and Bugeaud [@AdBuFourier] use the subspace theorem and the mirror formula (\[eq: mirror\]) to establish several combinatorial transcendence criteria for continued fractions involving symmetric patterns (see also [@Bu12] for a recent improvement of one of these criteria). The same authors [@AdBuJEMS] also give some transcendence measures for such continued fractions. As a simple illustration, they prove in [@AdBuFourier] that if the continued fraction of a real number begins with arbitrarily large palindromes, then this number is either quadratic or transcendental.
It is amusing to mention that in contrast there are only very partial results about the transcendence of numbers whose decimal expansion involves symmetric patterns (see [@AdBuIMRN]). In particular, it is not known whether a real number whose decimal expansion begins with arbitrarily large palindromes is either rational or transcendental.
Beyond the study of extremal numbers and transcendence results, Adamczewski and Bugeaud [@AdBuJLMS] use continued fractions with symmetric patterns to provide explicit examples for the famous Littlewood conjecture on simultaneous Diophantine approximation.
Acknowledgments
===============
This work was supported by the project Hamot, ANR 2010 BLAN-0115-01. The idea to make this survey came up with a talk I gave for the conference *Diophantine Analysis and Related Fields* that was held in Tokyo in March 2009. I would like to take the opportunity to thank the organizers of this conference and especially Noriko Hirata-Kohno. I would also like to thank the anonymous referees for their useful comments. My interest for transcendence grew quickly after I followed some lectures by Jean-Paul Allouche when I was a Ph.D. student. I am deeply indebted to Jean-Paul for giving such stimulating lectures!
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[^1]: The number $\kappa$ is sometimes erroneously called the Fredholm number (see, for instance, the discussion in [@Sha]).
[^2]: Note that this argument could also be replaced by the use of two important results from automata theory: the Cobham and Christol theorems (see [@AS] and also Section \[allouche\] for another use of Christol’s theorem).
[^3]: The folding lemma is an avatar of the so–called mirror formula, another very useful elementary identity for continued fractions, which is the object of the survey [@AdAl]. Many references to work related to these two identities can be found in [@AdAl].
[^4]: Köhler [@Ko] also obtains independently almost the same result after he studied [@Sh1]. It is also worth mentioning that this result was somewhat anticipated, although written in a rather different form, by Scott and Wall in 1940 [@SW].
[^5]: That is those which are larger than all previous ones.
[^6]: This could mean something like, among algebraic Laurent series $f$ of degree and height at most $M$, the proportion of those with $\mu(f)=2$ tends to one as $M$ tends to infinity.
[^7]: The authors of [@AdBuActa] use that $\vert L({\bf x}_n)\vert \ll q_{r_n} q_{r_n+s_n} q_{ r_n \lfloor (1+\varepsilon) s_n \rfloor }^{-2}$ while it is actually elementary to see that $\vert L({\bf x}_n)\vert \ll q_{r_n}^{-1} q_{r_n+s_n} q_{ r_n \lfloor (1+\varepsilon) s_n \rfloor }^{-2}$.
[^8]: Indeed, the theory of continuants implies that the denominator of the rational number $[0,B_{n-1}]$, say $r_n$, grows roughly $\sqrt{q_{k_n}}$ and thus $r_n^{-2}$ essentially behaves like $q_{k_n}^{-1}$. For more details about continuants, see, for instance, [@AdBuCant Sec.8.2], [@AdBuCrelle1 Sec. 5], or [@AdBuJLMS Sec.3].
[^9]: Recall that a palindrome is a word $W=a_1\cdots a_n$ that is equal to its reversal $W^R=a_n\cdots a_1$. Thus a palindrome may be considered as a perfect symmetric pattern.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Large Synoptic Survey Telescope (LSST) will explore the entire southern sky over 10 years starting in 2022 with unprecedented depth and time sampling in six filters, *ugrizy*. Artificial power on the scale of the 3.5deg LSST field-of-view will contaminate measurements of baryonic acoustic oscillations (BAO), which fall at the same angular scale at redshift $z \sim 1$. Using the HEALPix framework, we demonstrate the impact of an “un-dithered” survey, in which 17% of each LSST field-of-view is overlapped by neighboring observations, generating a honeycomb pattern of strongly varying survey depth and significant artificial power on BAO angular scales. We find that adopting large dithers (i.e., telescope pointing offsets) of amplitude close to the LSST field-of-view radius reduces artificial structure in the galaxy distribution by a factor of $\sim$10. We propose an observing strategy utilizing large dithers within the main survey and minimal dithers for the LSST Deep Drilling Fields. We show that applying various magnitude cutoffs can further increase survey uniformity. We find that a magnitude cut of $r < 27.3$ removes significant spurious power from the angular power spectrum with a minimal reduction in the total number of observed galaxies over the ten-year LSST run. We also determine the effectiveness of the observing strategy for Type Ia SNe and predict that the main survey will contribute $\sim$100,000 Type Ia SNe. We propose a concentrated survey where LSST observes one-third of its main survey area each year, increasing the number of main survey Type Ia SNe by a factor of $\sim$1.5, while still enabling the successful pursuit of other science drivers.'
author:
- |
Christopher M. Carroll, Eric Gawiser, Peter L. Kurczynski, Rachel A. Bailey,\
Rahul Biswas, David Cinabro, Saurabh W. Jha, R. Lynne Jones,\
K. Simon Krughoff, Aneesa Sonawalla, W. Michael Wood-Vasey,\
for the LSST Dark Energy Science Collaboration Dept. of Physics and Astronomy, Dartmouth College,\
6127 Wilder Laboratory, Hanover, NH 03755-3528, USA;\
Dept. of Physics and Astronomy, Rutgers Univ.,\
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA;\
Argonne National Laboratory,\
9700 S. Cass Avenue, Argonne, IL 60439, USA;\
Dept. of Physics and Astronomy, Wayne State Univ.,\
42. W. Warren Ave., Detroit, MI 48202, USA;\
Astronomy Dept., Univ. of Washington,\
3910 15th Ave NE, Seattle, WA 98195-1580, USA;\
Dept.of Astronomy and Astrophysics, Univ. of Chicago,\
5801 South Ellis Avenue, Chicago, Illinois 60637, USA;\
Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PITT PACC),\
Physics and Astronomy Dept., Univ. of Pittsburgh, Pittsburgh PA, 15260, USA
title: |
Improving the LSST dithering pattern\
and cadence for dark energy studies
---
INTRODUCTION {#sec:intro}
============
The Large Synoptic Survey Telescope (LSST) will observe $\sim$20,000deg$^2$ of the sky over 10 years and will make major contributions to astronomy.[@SciBook] The main science drivers of LSST include detection of near-Earth objects, the creation of detailed surveys of the Solar System and the Milky Way, enhancing the search for faint transient and variable phenomena, and probing the nature of dark energy and dark matter. LSST will provide four probes of dark matter and dark energy via (1) weak lensing cosmic shear (WL) of galaxies, (2) baryonic acoustic oscillations (BAO) present in the power spectrum of galaxy clustering, (3) the evolution of the mass function of galaxy clusters, and (4) the distance-redshift relationship to Type Ia SNe.
LSST was designed to satisfy the majority of its science goals in one data set (main survey) that will include 90% of the observing time. The remaining telescope time is then split between observing the LSST Deep Drilling Fields (DDFs) and other Òmini-surveys.Ó The wide range of science objectives leads to competing demands in the design of the main survey. These competing demands can be understood by considering the frequency of visits (cadence) and the exposure time (depth) dedicated to each part of the sky. In general, studies of time-variable phenomena such as Type Ia SNe prefer high cadence at the expense of uniform depth, while studies that emphasize the detection of spatial correlations, such as BAO in galaxy clustering, prefer uniform depth at the expense of high cadence.
Both cadence and uniformity of depth are impacted by the standard observing technique of taking exposures at small excursions away from nominal survey positions (dithering). A survey design that seeks uniform coverage on the sky inevitably produces areas of overlap between adjacent nominal survey positions. In LSST, these overlap regions are important because they constitute a significant sky-fraction of the survey that has higher cadence than non-overlap regions. By smoothing over areas of unequal coverage, dithering improves uniformity and changes the cadence in these overlap regions.
Our theory of modern cosmology relies heavily upon the cosmological principle, which holds that the universe when viewed from a large enough scale appears homogenous and isotropic from any vantage point. The discovery of an accelerating expanding universe drives us to test this cosmological principle via supernovae studies.[@Riess1998] If the universe is indeed isotropic, then the distance-redshift relation traced by light curves of Type Ia SNe observed in different directions on the sky should be identical. However, if the redshift-distance relation varies with position on the sky, the cosmological principle would not hold, and we would be in need of a new or modified theory of cosmic evolution. By using WL and BAO in addition to Type Ia SNe, we can probe for dark energy anisotropy and place tight constraints on the dark energy equation of state.[@Ivezic2011]
The intent of our work is to optimize the effectiveness of the LSST observing strategy for dark energy studies while maintaining uniformity in the main survey and limiting any adverse affects on other areas of study. Although the primary focus of our work is Type Ia SNe and survey uniformity, it is worth noting that many of the same considerations taken here also enter when optimizing for studies of weak gravitational lensing and galaxy clusters.[@Jee2011; @Morrison2012]
METHODOLOGY AND ANALYSIS GOALS
==============================
Operations Simulator {#sec:opsim}
--------------------
The Operations Simulator (OpSim) was developed in order to verify that LSST can meet all of its required science goals given a specific survey design.[@SciBook] OpSim constructs a detailed ten-year simulated LSST run, providing information on each observation including position on the sky, time and filter, weather conditions, and more. Such detailed models of the LSST ten-year survey are vital for estimating the final co-added depth of the stacked images in each filter as a function of sky position. From this we can determine the overall uniformity of the survey and detect any noticeable patterns or effects that may propagate through our analysis leading to systematic errors. In addition, we can determine the total number and cadence of observations per area of the sky, which is fundamental in determining the efficiency of the LSST observing strategy for dark energy studies as well as the other science drivers. OpSim also accounts for telescope and camera parameters such as slew time, filter exchanges, and scheduled maintenance. These simulations model seeing and weather conditions from on site measurements to predict results as close to actual telescope data as possible.
The LSST original baseline cadence strategy tiles the Southern Hemisphere of the sky with hexagons. Inscribing the hexagon within the roughly circular LSST 3.5deg field-of-view (FOV) produces doubly observed areas within an LSST FOV caused by overlaps with neighboring observations. For our analysis we use OpSim run 2.168 which includes 10 Deep Drilling Fields (DDFs). Each DDF consists of a single fixed LSST FOV which is intended for long-term, rapid observations and is planned to be visited more often than the canonical main survey cadence, resulting in increased sensitivity to faint objects. The LSST DDFs will provide rich data sets due to increased time sampling meant for observing extremely faint, distant objects. Our analysis of OpSim excludes these DDFs and focuses specifically on the LSST main survey.
An additional observing strategy is implemented by modifying the OpSim output. This modification is used to post-process the simulation to create large dithers ($\sim$0.5 FOV) that offset the LSST pointing centers on return visits. The dithering pattern that we utilize appears in the output database for OpSim but has not yet been implemented in the Simulator itself. This pattern uses a lattice of points that fills a hexagon inscribed in the LSST FOV. Each night, the next vertex is chosen as an offset for all telescope pointings versus the center of the hexagon, until all 217 points in the hexagon have been utilized and the pattern begins anew. By the end of the 10 year survey, due to the randomness in which nights a given field is observed, the pattern of dithering vectors applied in each field is close to random and includes shifts as large as the radius of the LSST FOV. These large dithers help to increase survey uniformity by reducing the sharp contrast of overlap regions from their pointing centers. LSST has always planned to utilize some form of dithering to cover the small gaps between CCDs on repeated visits, but the CCD gaps are so small compared to the FOV that this minimal dithering pattern is well approximated by not dithering at all in the current OpSim code, and we will henceforth refer to this as the “un-dithered" survey. By instead utilizing large dithers we hope to create the most uniform survey possible with negligible negative effects on other science drivers.
Analyzing OpSim output with HEALPix {#sec:model}
-----------------------------------
The need to model overlap regions prompted us to utilize the Hierarchical Equal Area isoLatitude Pixelization[^1] (HEALPix) software package[@Gorski2005]. HEALPix provides us with a tool for creating uniform tessellation of a sphere with a variable number of equal area pixels. By increasing the number of pixels, and thereby decreasing the area of each pixel, we fill an LSST FOV with an adequate number of pixels to highlight the overlap regions produced by neighboring observations. The total number of pixels is set by the resolution parameter $N_{\mathrm{side}}$ according to the formula $N_{\mathrm{pix}} = 12 \times N_{\mathrm{side}}^2$. Throughout our body of work we have chosen the resolution parameter as $N_{\mathrm{side}}$ = 128, which yields $N_{\mathrm{pix}}$ = 196608, approximately 50 HEALPix pixels per LSST FOV. Each individual HEALPix pixel covers $\sim$0.21deg$^2$ and this allows us to make sky maps of OpSim metadata and to observe the overlap regions. Figures \[fig:num\_obs\] and \[fig:num\_obs\_dith\] show our findings in all filters *ugrizy* of the main survey and illustrates how large dithers affect the survey, blending the overlap regions and creating a noticeably more uniform survey. Further analysis of the effects of these dithers on survey uniformity will be discussed in Section \[sec:results\].
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![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](u_band_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](g_band_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](r_band_p361.png "fig:"){height="3cm"}
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![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](i_band_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](z_band_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\] Total number of observations in the OpSim un-dithered survey in all bands, *ugrizy*.](y_band_p361.png "fig:"){height="3cm"}
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![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](u_band_dith_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](g_band_dith_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](r_band_dith_p361.png "fig:"){height="3cm"}
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![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](i_band_dith_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](z_band_dith_p361.png "fig:"){height="3cm"} ![\[fig:num\_obs\_dith\] Total number of observations in the OpSim dithered survey in all bands, *ugrizy*.](y_band_dith_p361.png "fig:"){height="3cm"}
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Cadence for Type Ia SNe {#sec:cadence}
-----------------------
Most science drivers benefit from an increased number of observations, but to optimize LSST for dark energy studies with Type Ia SNe we also need to increase the cadence of these observations. For Type Ia SNe, light curves require frequent multi-color observations to act as calibrated standard candles, allowing us to probe the farthest edges of the observable universe. We set out to determine the number of HEALPix pixels on the sky with suitable cadence for cosmological studies of Type Ia SNe. Calibrating Type Ia SNe requires an even distribution of observations at $\sim$2-day cadence to capture light curve peaks. A secondary requirement for accurate calibration is observations in multiple bands. With ample filter exchanges, we consider any 60-day period which contains 60 or more observations to be a good observing ÒseasonÓ with adequate cadence for cosmological studies using Type Ia SNe. We seek to determine the total number of good observing seasons in the 10-year main survey.
RESULTS {#sec:results}
=======
Survey depth and estimated galaxy counts {#sec:gal_count}
----------------------------------------
Among the numerous outputs provided by OpSim are simulated estimates of the sky brightness and 5-$\sigma$ limiting magnitude for each observation. In our work we use the modified 5-$\sigma$ limiting magnitude output of OpSim, where the 5-$\sigma_{\mathrm{mod}}$ limiting magnitude is calculated by using a V sky brightness model adjusted in each bandpass for moon phase, differences in atmospheric light scatter, and twilight hours to best simulate a real point source detection depth for each visit. We then calculate the stacked 5-$\sigma$ limiting magnitude to estimate the final co-added depth in each HEALPix pixel for that filter; from this we can estimate the number of galaxies we project LSST to find in each pixel. We calculate the co-added depth as $$\label{eq:mag_depth}
\operatorname{5-\sigma}_{\mathrm{stack}} = 1.25\log \sum_{i} 10^{(0.8 \times \operatorname{5-\sigma}_{\mathrm{mod},i})},$$ assuming that individual exposures are combined with optimal S/N weighting.[@Gawiser2006] Taking the median value in each filter we compare our 5-$\sigma_{\mathrm{stack}}$ values to the LSST science requirements[@LSSTReq] for co-added depth per filter in Table \[tab:mag\_depth\]. Using the power-law fit for r-band number counts from MUSYC[@Gawiser2006] we determine the estimated number of galaxies to be detected per HEALPix pixel by integrating over all magnitudes brighter than the co-added survey depth in each pixel $$\label{eq:numgal}
N_{\mathrm{gal}} = \int_{-\infty}^{\operatorname{5-\sigma}_{\mathrm{stack}}} 10^{-3.52} \times 10^{0.34m} \,dm.$$
[cccc]{}
------------------------------------------------------------------------
Filter & Median depth (un-dithered) & Median depth (dithered) & SRD Stretch Goal\
------------------------------------------------------------------------
*u* & 26.5 & 26.5 & 26.1\
------------------------------------------------------------------------
*g* & 27.1 & 27.2 & 27.4\
------------------------------------------------------------------------
*r* & 27.3 & 27.4 & 27.5\
------------------------------------------------------------------------
*i* & 26.7 & 26.8 & 26.8\
------------------------------------------------------------------------
*z* & 25.8 & 25.9 & 26.1\
------------------------------------------------------------------------
*y* & 24.6 & 24.7 & 24.9\
We modified this integral to correct for incompleteness near the 5-$\sigma$ survey limit as described in Fleming et al. 1995[@Fleming1995]. Table \[tab:mag\_cuts\] shows our results on galaxy detection in *r* band. Implementing the proposed large dithers causes a slight increase in the total number of detected galaxies over the course of the survey. There are uncertainties in this process, which translate to uncertainties in the window function estimated in this way. We estimate this window function via the angular power spectrum (Section \[sec:power\_spec\]). Here we assume that the less power in the survey window function, the smaller the uncertainties.
Galaxy angular power spectrum {#sec:power_spec}
-----------------------------
Characterizing the overall uniformity requires analyzing the spherical harmonic transform of our galaxy detection sky maps. We determine the deviation from survey average ($\Delta N/N_{ave}$) in each pixel and use the HEALPix routine *ianafast* to determine the galaxy angular power spectrum shown in Figure \[fig:power\_spec\]. To ensure a robust, unbiased galaxy sample, we must minimize deviations in the overall co-added survey depth, which creates a more uniform detection limit. From the galaxy angular power spectrum we evaluate the effects that large dithers have on the uniformity of the survey. Comparing the two angular power spectra, it is clear that adding large dithers to the output of OpSim significantly reduces artificial structure in the galaxy distribution. This benefit provided by the implementation of large dithers is crucial for studies of dark energy: note that $\ell \sim 150$ corresponds to an angular diameter of $\sim$$1^\circ$, close to the BAO angular scale at $z \sim 1$. The additional low-$\ell$ power introduced by the large dithers results from the long border of reduced depth that can be seen in Figure \[fig:num\_obs\_dith\]. This could be reduced with a careful trimming of that border.
![\[fig:power\_spec\] Galaxy angular power spectrum (*left*): un-dithered survey, (*right*): with large dithers](full_ps.png){height="6cm"}
Magnitude cutoffs {#sec:mag_cuts}
-----------------
Further uniformity can be achieved by introducing uniform magnitude cutoffs that smooth variations in the final co-added depth of the heterogeneous survey. Figure \[fig:mag\_cuts\] presents our results for several magnitude cutoffs in the OpSim main survey and their corresponding angular power spectra. Each successive cutoff decreases the total number of galaxies detected in the overall survey, but we find these losses to be mild for cutoffs as bright as $r = 27.3$ (see Table \[tab:mag\_cuts\]). We find a magnitude cutoff of 27.3 to remove sufficient artificial structure in the galaxy angular power spectrum without unnecessary reduction of the LSST galaxy sample size. A more detailed optimization using magnitude cutoffs is the topic of ongoing work.
[ccc]{}
------------------------------------------------------------------------
Mag. Cut & Num. of Galaxies (un-dithered) & Num. of Galaxies (dithered)\
------------------------------------------------------------------------
none & 1.41$\times10^{10}$ & 1.46$\times10^{10}$\
------------------------------------------------------------------------
27.80 & 1.41$\times10^{10}$ & 1.46$\times10^{10}$\
------------------------------------------------------------------------
27.50 & 1.37$\times10^{10}$ & 1.45$\times10^{10}$\
------------------------------------------------------------------------
27.30 & 1.29$\times10^{10}$ & 1.35$\times10^{10}$\
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27.02 & 1.07$\times10^{10}$ & 1.11$\times10^{10}$\
It is important to note that the most precise probe of dark energy planned for LSST is the joint BAO+WL analysis of bright, $\mathrm{S/N} > 20$, $i < 25.3$ galaxies.[@SciBook] Our magnitude cut of $r < 27.3$ is substantially more inclusive than this criterion for typical $r - i = 1$.
![\[fig:mag\_cuts\] Galaxy angular power spectra with magnitude cutoffs in (*top*): the un-dithered survey, (*bottom*): with large dithers. The lack of power in the upper right panels results from an oversimplified model for incompleteness. Added low-$\ell$ power in the bottom row comes from reduced depth in survey borders.](magcut_ps.png){height="7cm"}
Cadence results {#sec:cad_results}
---------------
For each HEALPix pixel, we determine the number of good observing seasons over the ten-year survey that meet our cadence criteria described in Section \[sec:cadence\]. Using a conservative limit for supernovae detection rate of 1 SN per square degree per month[@Graur2014], we estimate the total number of supernovae detected over the entire ten-year survey with cadence acceptable for cosmological studies. Our results are presented in Figure \[fig:sn\_count\]. It is important to note that these results are strictly from the main survey and are independent of DDFs, which will contribute an additional $\sim$10,000 high-quality Type Ia SN light curves. We subtract one-third of our SNe detection to account for phase coverage and other cadence issues.
It is also important to note that the main focus of our work is not to determine an absolute number of SNe detected over the length of the survey, rather to show the relative increase in detection via the proposed dithering pattern. We compare the results of OpSim with and without dithers to determine the effects of the overlap regions on supernova count. Large dithers like those proposed for LSST appear to slightly increase the overall number of observations with adequate cadence for cosmological SNe studies. It should be noted that the results presented on supernovae capture are made on the basis of number of visits regardless of filter. Proper calibration of a supernova light curve requires observations in multiple bands.
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![\[fig:sn\_count\] Total number of good observing seasons in (*left*): the un-dithered survey, (*right*): with large dithers. The total SNe detection counts have been adjusted for edge effects and other cadence issues.](undith_season.png "fig:"){height="5cm"} ![\[fig:sn\_count\] Total number of good observing seasons in (*left*): the un-dithered survey, (*right*): with large dithers. The total SNe detection counts have been adjusted for edge effects and other cadence issues.](dith_season.png "fig:"){height="5cm"}
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The “cadence output” can be translated into the input for the supernova analysis package SNANA [@SNANA]. Here each pixel is used to make an entry in a SNANA simulation library. The repeated observations of a pixel are time ordered and their observation conditions are taken from the LSST OpSim output to build up a simulation library entry. The new feature here is that this pixelated “cadence output” naturally takes into account LSST field overlaps which had previously been ignored in SNANA LSST supernova simulations. The set of simulation library entries can then be used to used to simulate the observations of various sorts of supernovae (Type Ia and/or core collapse) with various assumptions about dithering to optimize the survey strategy for cosmology with Type Ia SNe.
DISCUSSION
==========
Time-restricted survey area {#sec:conc_survey}
---------------------------
We propose a drastic change to the scheduled operation of LSST in order to increase cadence and to improve Type Ia SNe capture for dark energy studies. The observational strategy present in OpSim 2.168 covers $\sim$20,000deg$^2$ of the Southern Hemisphere of the sky, with a typical gap between visit pairs of 3 nights. Our analysis in Section \[sec:cad\_results\] on the efficiency of LSST to observe Type Ia SNe for cosmological studies has given us hope of a modest population with which to refine our understanding of dark energy. To increase our supernova sample means to increase the cadence of observations of these eventsnamely the number of good seasons defined in Section \[sec:cadence\]. We propose a time-dependent restriction on the observing strategy to concentrate on one-third of the main survey each year over nine years.
Using the analysis methods described in this paper, we determine the cadence of observations for a mock concentrated survey based on OpSim 2.168 where all observations for each year of operation are redistributed by declination. We keep the first year of the default survey untouched and move through each consecutive year, keeping the observations fixed in right ascension and randomly redistribute their declination within the corresponding year’s declination bin. Our “survey" results on the improved cadence and capture of Type Ia SNe using this method are compared to the OpSim run in Table \[tab:conc\_survey\]. We find the concentrated survey produces a factor of $\sim$1.5 increase in Type Ia SNe over the default un-dithered and dithered strategy, and we suspect that the detailed SNANA simulations underway will reveal a larger improvement.
[ccccc]{}
------------------------------------------------------------------------
& Default survey (un-dithered) & Default survey (dithered) & Concentrated Survey\
------------------------------------------------------------------------
SNe counts & 1.10$\times10^5$ & 1.15$\times10^5$ & 1.70$\times10^5$\
------------------------------------------------------------------------
Improvement Factor & 1 & 1.05 & 1.55 &\
It should be noted that our coarse approach to a concentrated survey creates artifacts in the survey, such as undesired observations towards the Galactic center. We subtract these false observations from our SNe count for the concentrated survey. Our calculations are therefore missing observations present within the OpSim survey. A full OpSim run would be needed in order to determine more accurate estimates.
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![\[fig:conc\_cad\] Total number of good observing seasons within the concentrated survey. Undesirable observations within the Galactic plane have been removed. The dividing lines in declination between our three equal area regions show up as horizontal lines of reduced cadence.](rc_undith.png "fig:"){height="5cm"}
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CONCLUSION
==========
The astronomical community will benefit from the information LSST collects over its ten-year survey. It is imperative that this survey be optimized to produce maximum results. With a multitude of science drivers, a balance must be struck between competing demands for cadence and survey uniformity. We find each of these survey characteristics to be affected by dithering. Large dithering improves survey uniformity by a factor of $\sim$10 while also modestly improving the cadence over a larger area. Uniform magnitude cutoffs further improve survey uniformity, with a magnitude cut of $r < 27.3$ being roughly optimal to reduce spurious power without a significant loss of statistics. We find that a concentrated survey design, whereby large regions of the sky are emphasized together and uniformity over the entire survey area is achieved at the end of each three years, can significantly improve the yield of Type Ia supernovae. This study illustrates the need for more in-depth analyses of the effects large dithers have on each of the science objectives. Variations on our proposed dithering strategy and the effect on Type Ia SN cadence and survey uniformity is the subject of ongoing work. We caution the reader that our estimates of the yield of Type Ia supernovae are highly approximate and result from an analysis of the total number of LSST observations without regard to which filter those observations were made in. Improved estimates will result from ongoing detailed SNANA simulations of the resulting quality of SN Ia light curves.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We thank Tony Tyson and Beth Willman for their detailed LSST Publications Board review of this article, as well as Pat Burchat and Jeff Newman for helpful comments. This research was supported by grants from the National Science Foundation, AST-1055919, and the Department of Energy, DE-SC0011636.
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[^1]: http://healpix.jpl.nasa.gov/
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The goal of an algorithm substitution attack (ASA), also called a subversion attack (SA), is to replace an honest implementation of a cryptographic tool by a subverted one which allows to leak private information while generating output indistinguishable from the honest output. Bellare, Paterson, and Rogaway provided at CRYPTO ’14 a formal security model to capture this kind of attacks and constructed practically implementable ASAs against a large class of *symmetric encryption schemes*. At CCS’15, Ateniese, Magri, and Venturi extended this model to allow the attackers to work in a fully-adaptive and continuous fashion and proposed subversion attacks against *digital signature schemes*. Both papers also showed the impossibility of ASAs in cases where the cryptographic tools are deterministic. Also at CCS’15, Bellare, Jaeger, and Kane strengthened the original model and proposed a universal ASA against sufficiently random encryption schemes. In this paper we analyze ASAs from the perspective of steganography – the well known concept of hiding the presence of secret messages in legal communications. While a close connection between ASAs and steganography is known, this lacks a rigorous treatment. We consider the common computational model for secret-key steganography and prove that successful ASAs correspond to secure stegosystems on certain channels and vice versa. This formal proof allows us to conclude that ASAs are stegosystems and to “rediscover” several results concerning ASAs known in the steganographic literature.'
author:
- Sebastian Berndt
- Maciej Liśkiewicz
title: |
Algorithm Substitution Attacks from a\
Steganographic Perspective
---
Introduction
============
The publication of secret internal documents of the NSA by Edward Snowden (see [e.g.]{}[@ball2013revealed; @greenwald2014no; @perlroth2013nsa]) allowed the cryptographic community a unique insight into some well-kept secrets of one of the world’s largest security agency. Two conclusions may be drawn from these reveals:
- On the one hand, even a large organization such as the NSA seems not to be able to break well established implementations of cryptographic primitives such as RSA or AES.
- On the other hand, the documents clearly show that the NSA develops methods and techniques to *circumvent* the well established security notions by [e.g.]{}manipulating standardization processes ([e.g.]{}issues surrounding the number generator `Dual_EC_DRBG` [@checkoway2014dual; @schneier2007did; @shumow2007back]) or reason about metadata.
This confirms that the security guarantees provided by the cryptographic community are sound, but also indicates that some security definitions are too narrow to evade all possible attacks, including (non-)intentional improper handling of theoretically sound cryptographic protocols. A very realistic attack which goes beyond the common framework is a modification of an appropriate implementation of a secure protocol. The modified implementation should remain indistinguishable from a truthful one and its aim is to allow leakage of secret information during subsequent runs of the subverted protocol. Attacks of this kind are known in the literature [@young1997kleptography; @young1996dark; @bellare2015asa; @bellare2014asa; @ateniese2015sig; @russell2016cliptography] and an overview on this topic is given in the current survey [@schneier2015survey] by @schneier2015survey
A powerful class of such attacks that we will focus on – coined *secretly embedded trapdoor with universal protection (SETUP) attacks* – was presented over twenty years ago by Young and Yung in the *kleptographic* model framework [@young1996dark; @young1997kleptography]. The model is meant to capture a situation where an adversary (or “big brother” as we shall occasionally say) has the opportunity to implement (and, indeed, “mis-implement” or subvert) a basic cryptographic tool. The difficulty in detecting such an attack is based on the hardness of program verification. By using *closed source* software, the user must trust the developers that their implementation of cryptographic primitives is truthful and does not contain any backdoors. This is especially true for hardware-based cryptography [@bellare2014asa]. But it is difficult to verify this property. Even if the software is *open source* – the source code is publicly available – the sheer complexity of cryptographic implementations allows only very specialized experts to be able to judge these implementations. Two of the most prominent bugs of the widely spread cryptographic library `OpenSSL`[^1] – the *Heartbleed bug* and Debian’s faulty implementation of the pseudorandom number generator – remained undiscovered for more than two years [@schneier2015survey].
Inspired by Snowden’s reveals, the recent developments reignited the interest in these kind of attacks. @bellare2014asa named them *algorithm substitution attacks (ASA)* and showed several attacks on certain symmetric encryption schemes [@bellare2014asa]. Note that they defined a very weak model, where the only goal of the attacker was to distinguish between two ciphertexts, but mostly used a stronger scenario with the aim to recover the encryption key. @degabriele2015asa criticized the model of [@bellare2014asa] by pointing out the results crucially rely on the fact that a subverted encryption algorithm always needs to produce valid ciphertexts (the *decryptability assumption*) and proposed a refined security notion [@degabriele2015asa]. The model of algorithm substitution attacks introduced in [@bellare2014asa] was extended to signature schemes by @ateniese2015sig in [@ateniese2015sig]. Simultaneously, @bellare2015asa [@bellare2015asa] strengthened the result of [@bellare2014asa] by enforcing that the attack needs to be stateless.
In this paper we thoroughly analyze (general) ASAs from the *steganographic* point of view. The principle goal of steganography is to hide information in unsuspicious communication such that no observer can distinguish between normal documents and documents that carry additional information. Modern steganography was first made popular due to the prisoners’ problem by Simmons [@simmons1984prisoners] but, interestingly, the model was inspired by detecting the risk of ASAs during development of the SALT2 treaty between the Soviet Union and the United States in the late seventies [@simmons1998history]. This sheds some light on the inherent relationship between these two frameworks which is well known in the literature (see [e.g.]{}[@young1996dark; @young1997kleptography; @russell2016destroying]). A related result showing that so called *decoy password vaults* are very closely related to stegosystems on a certain kind of channels was presented by @pasquini2017decoy in [@pasquini2017decoy].
Our main achievement is providing a strict relationship between secure algorithm substitution attacks and the common computational model for secret-key steganography. Particularly, we prove that successful ASAs correspond to secure stegosystems on certain channels and vice versa. This formal proof allows us to conclude that ASAs are stegosystems and to “rediscover” results of [@bellare2014asa; @bellare2015asa; @ateniese2015sig] concerning ASAs.
The computational model for steganography used in this paper was first presented by Hopper, Langford, and von Ahn [@hopper2002provably; @hopper2009provably] and independently proposed by Katzenbeisser and Petitcolas [@katzenbeisser2002defining]. A *stegosystem* consists of an encoder and a decoder sharing a key. The encoder’s goal is to *embed* a secret message into a sequence of documents which are send via a public *communication channel* ${\mathcal{C}}$ monitored by an adversary (often called the *warden* due to the prisoners problem of Simmons [@simmons1984prisoners]). The warden wants to distinguish documents that carry no secret information from those sent by the encoder. If all polynomial-time (in the security parameter $\kappa$) wardens fail to distinguish these cases, we say that the stegosystem is *secure*. If the decoder is able to reconstruct the secret message from the sequence send by the encoder, the system is called *reliable*.
Our Results {#our-results .unnumbered}
-----------
We first investigate algorithm substitution attacks against symmetric encryption schemes in the framework by @bellare2015asa [@bellare2015asa]. We model encryption schemes as steganographic channels in appropriate way which allows to relate algorithm substitution attacks with steganographic systems and vice versa. This leads to the following result.
\[thm:stego:asa:iff:on:ses\] Assume that $\operatorname{\mathsf{SES}}$ is a symmetric encryption scheme. Then there exists an indistinguishable and reliable algorithm substitution attack against $\operatorname{\mathsf{SES}}$ if and only if there exists a secure and reliable stegosystem on the channel determined by $\operatorname{\mathsf{SES}}$.
The proof of the theorem is constructive in the sense that we give an explicit construction of an algorithm substitution attack against $\operatorname{\mathsf{SES}}$ from a stegosystem and vice versa. As conclusion we provide a generic ASA against *every* symmetric encryption scheme $\operatorname{\mathsf{SES}}$ whose insecurity is negligible if, roughly speaking, $\operatorname{\mathsf{SES}}$ has sufficiently large min-entropy. Our algorithm against $\operatorname{\mathsf{SES}}$ achieves almost the same performance as the construction of @bellare2015asa (see Theorem 4.1 and Theorem 4.2 in [@bellare2015asa] and also our discussion in Section \[Sec:ASA:against:encrypted:as:stego\]).
Next, we generalize our construction and show a generic algorithm substitution attack $\operatorname{\mathsf{ASA}}$ against any (polynomial-time) randomized algorithm $\operatorname{\mathsf{R}}$ which, with hardwired secret $s$, takes inputs $x$ and generates outputs $y$. Algorithm $\operatorname{\mathsf{ASA}}$, using a hidden hardwired random key ${\textit{ak}}$, returns upon the secret $s$ the sequence $\tilde{y}_1,\tilde{y}_2,\ldots$ such that the output is indistinguishable from $\operatorname{\mathsf{R}}(s,x_1), \operatorname{\mathsf{R}}(s,x_2),\ldots$ and $\tilde{y}_1,\tilde{y}_2,\ldots$ embeds the secret $s$. From this result we conclude:
\[thm:generic-attack\] There exists a generic algorithm substitution attack $\operatorname{\mathsf{ASA}}$ that allows an undetectable subversion of any cryptographic primitive of sufficiently large min-entropy.
\[thm:min-entropy\] Let $\Pi$ be a cryptographic primitive consisting with algorithms $(\Pi.A_{1},\Pi.A_{2},\ldots,$ $ \Pi.A_{r})$ such that $\{A_{i}\mid i\in
I\}$ for some $I\subseteq \{1,\ldots,r\}$ are deterministic. Then there is no ASA on $\Pi$ which subverts only algorithms $\{A_{i}\mid i\in I\}$.
As a corollary we obtain the result of @ateniese2015sig (Theorem 1 in [@ateniese2015sig]) that for every *coin-injective* signature scheme, there is a successful algorithm substitution attack of negligible insecurity. Moreover we get (Theorem 2 in [@ateniese2015sig]) that for every *coin-extractable* signature scheme, there is a successful and secure ASA. We can conclude also (Theorem 3 in [@ateniese2015sig]) that *unique signature schemes* are resistant to ASAs fulfilling the *verifiability condition*. Roughly speaking the last property means that each message has exactly one signature and the ASA can only produce valid signatures.
We furthermore introduce the concept of *universal ASAs* that can be used without a detailed description of the implementation of the underlying cryptographic primitive and note that almost all known ASAs belong to this class. Based upon this definition, we prove the following upper bound on the information that can be embedded into a single ciphertext:
\[thm:universal:bound\] No universal ASA is able to embed more than $\mathcal{O}(1)\cdot \log(\kappa)$ bits of information into a single ciphertext.
The paper is organized as follows. Section \[sec:prelim\] contains the basic preliminaries and notations that we use throughout this work, Section \[sec:asa\] presents the formal definitions of algorithm substitution attacks, and Section \[sec:stego\] gives the necessary background on steganography. In order to relate ASAs and steganography, we make use of an appropriate channel for symmetric encryption schemes defined in Section \[sec:channel\]. The proof of Theorem \[thm:stego:asa:iff:on:ses\] is given in Section \[Sec:ASA:against:encrypted:as:stego\], where one direction is contained in Theorem \[thm:asa:on:ses:impl:stego\] and the other direction is given as Theorem \[thm:ses:on:ses:impl:stego\]. We generalize our results to arbitrary randomized algorithms in Section \[sec:general\]. Combining the positive results of Theorem \[thm:generic-attack:against:R\] with the generic stegosystem provided by Theorem \[thm:rejsam:secure\] allows us to conclude Theorem \[thm:generic-attack\]. The negative results of Theorem \[thm:no-attack:against:R\] directly give Theorem \[thm:min-entropy\]. Finally, Section \[sec:bound\] defines universal ASAs and contains the upper bound on the transmission rate of these ASAs via a sequence of lemmata that results in Corollary \[cor:bound\] implying Theorem \[thm:universal:bound\].
Basic Preliminaries and Notations {#sec:prelim}
=================================
We use the following standard notations. A function $f\colon {\mathbb{N}}\to {\mathbb{N}}$ is *negligible*, if for all $c\in {\mathbb{N}}$, there is an $n_{0}\in {\mathbb{N}}$ such that $f(n) < n^{-c}$ for all $n\geq n_{0}$. The set of all strings of length $n$ on an alphabet $\Sigma$ is denoted by $\Sigma^{n}$ and the set of all strings of length at most $n$ is denoted by $\Sigma^{\leq
n}:=\cup_{i=0}^{n}\Sigma^{i}$. If $S$ is a set, $x\gets S$ denotes the uniform random assignment of an element of $S$ to $x$. If $\mathsf{A}$ is a randomized algorithm, $x\gets \mathsf{A}$ denotes the random assignment (with regard to the internal randomness of $\mathsf{A}$) of the output of $\mathsf{A}$ to $x$. The *min-entropy* measures the amount of randomness of a probability distribution $D$ and is defined as ${H_{\infty}}(D)=\inf_{x\in \operatorname{supp}(D)}\{-\log \Pr_{D}(x)\}$, where $\operatorname{supp}(D)$ is the support of $D$. Moreover, PPTM stands for probabilistic polynomial-time Turing machine.
A *symmetric encryption scheme* $\operatorname{\mathsf{SES}}$ is a triple of probabilistic polynomial-time algorithms $({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Dec}}})$ with parameters ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ describing the length of the encrypted message and ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)$ describing the length of a generated cipher message. The algorithms have the following properties:
- The *key generator* ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}$ produces upon input $1^{\kappa}$ a key $k$ with $|k|=\kappa$.
- The *encryption algorithm* ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}$ takes as input the key $k$ and a message $m\in \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$ of length ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ and produces a *ciphertext* $c\in \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)}$ of length ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)$.
- The *decryption algorithm* ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Dec}}}$ takes as input the key $k$ and a ciphertext $c\in \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)}$ and produces a message $m'\in
\{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$.
If the context is clear, we also write $\operatorname{\mathsf{Gen}}$, $\operatorname{\mathsf{Enc}}$, $\operatorname{\mathsf{Dec}}$, $\operatorname{\mathsf{ml}}$ and $\operatorname{\mathsf{cl}}$ without the prefix $\operatorname{\mathsf{SES}}$. We say that $(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$ is *reliable*, if $\operatorname{\mathsf{Dec}}(k,\operatorname{\mathsf{Enc}}(k,m))=m$ for all $k$ and all $m$.
An *cpa-attacker* $\operatorname{\mathsf{A}}$ against a symmetric encryption scheme is a PPTM that mounts *chosen-plaintext-attacks (cpa)*: It is given a challenging oracle $\operatorname{CH}$ that either equals $\operatorname{\mathsf{Enc}}_{k}$ for a randomly generated key $k$ or produces random bitstrings of length $\operatorname{\mathsf{cl}}(\kappa)$. For an integer $\lambda$, let $\operatorname{RAND}(\lambda)$ be an algorithm that returns uniformly distributed bitstrings of length $\lambda$. The goal of $\operatorname{\mathsf{A}}$ is to distinguish between those settings. Formally, this is defined via the following experiment named $\operatorname{\mathsf{CPA-Dist}}$:
attacker $\operatorname{\mathsf{A}}$, symmetric encryption scheme $\operatorname{\mathsf{SES}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$
$k\gets \operatorname{\mathsf{Gen}}(1^{\kappa})$; $b\gets \{0,1\}$ $b' \gets \operatorname{\mathsf{A}}^{\operatorname{CH}}(1^{\kappa})$
$b=0$ [**then**]{} [ **return** $\operatorname{\mathsf{Enc}}(k,m)$]{}
A symmetric encryption scheme $\operatorname{\mathsf{SES}}$ is *cpa-secure* if for every attacker $\operatorname{\mathsf{A}}$ there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
&\operatorname{\mathbf{Adv}}^{\operatorname{cpa}}_{\operatorname{\mathsf{SES}}}(\kappa) :=
|\Pr[\operatorname{\mathsf{CPA-Dist}}_{\operatorname{\mathsf{A}},\operatorname{\mathsf{SES}}}(\kappa)=\textsf{true}] - 1/2|\leq
\operatorname{\mathsf{negl}}(\kappa).\end{aligned}$$ The maximal advantage of any attacker against $\operatorname{\mathsf{SES}}$ is called the *insecurity* of $\operatorname{\mathsf{SES}}$ and is defined as $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{cpa}}_{\operatorname{\mathsf{SES}}}(\kappa) := \max_{\operatorname{\mathsf{A}}}\{\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{A}},\operatorname{\mathsf{SES}}}^{\operatorname{cpa}}(\kappa)\}.\end{aligned}$$
For a $\operatorname{\mathsf{SES}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$ we will assume that it has nontrivial randomization measured by the min-entropy ${H_{\infty}}(\operatorname{\mathsf{SES}})$ of ciphertexts that is defined via $$2^{-{H_{\infty}}(\operatorname{\mathsf{SES}})} = \max_{k,m,c} \Pr[{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m)=c].$$
For two numbers $\ell,\ell'\in {\mathbb{N}}$, denote the *set of all function* from $\{0,1\}^{\ell}$ to $\{0,1\}^{\ell'}$ by $\operatorname{\mathsf{Fun}}(\ell,\ell')$. Clearly, in order to specify a random element of $\operatorname{\mathsf{Fun}}(\ell,\ell')$, one needs $2^{\ell}\times \ell'$ bits and we can thus not use completely random functions in an efficient setting. Therefore we will use efficient functions that are indistinguishable from completely random functions. A *pseudorandom function* is a pair of PPTMs $\operatorname{\mathsf{F}}=({\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}},{\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Gen}}})$ such that ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Gen}}}$ upon input $1^{\kappa}$ produces a key $k\in \{0,1\}^{\kappa}$. The keyed function ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}$ takes the key $k\gets {\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$ and a bitstring $x$ of length ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{in}}}(\kappa)$ and produces a string ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{k}(x)$ of length ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{out}}}(\kappa)$. An attacker, called *distinguisher* $\operatorname{\mathsf{Dist}}$, is a PPTM that upon input $1^{\kappa}$ gets oracle access to a function that either equals ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{k}$ for a randomly chosen key $k$ or is a completely random function $f$. The goal of $\operatorname{\mathsf{Dist}}$ is to distinguish between those cases. A pseudorandom function $\operatorname{\mathsf{F}}$ is secure if for every distinguisher $\operatorname{\mathsf{Dist}}$ there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
&\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Dist}},\operatorname{\mathsf{F}}}^{\operatorname{prf}}(\kappa) \ := \\
& \quad \quad \left| \Pr[\operatorname{\mathsf{Dist}}^{{\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{k}}(1^{\kappa})=1] -
\Pr[\operatorname{\mathsf{Dist}}^{f}(1^{\kappa})=1]\right| \leq \operatorname{\mathsf{negl}}(\kappa),\end{aligned}$$ where $k\gets {\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$ and $f\gets
\operatorname{\mathsf{Fun}}({\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{in}}}(\kappa),{\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{out}}}(\kappa))$. If $\operatorname{\mathsf{Dist}}$ outputs $1$, this means that the distinguisher $\operatorname{\mathsf{Dist}}$ believes that he deals with a truly random function.
As usual, the maximal advantage of any distinguisher against $\operatorname{\mathsf{F}}$ is called the *prf-insecurity* $\operatorname{\mathbf{InSec}}_{\operatorname{\mathsf{F}}}^{\operatorname{prf}}(\kappa)$ and defined as $$\begin{aligned}
\operatorname{\mathbf{InSec}}_{\operatorname{\mathsf{F}}}^{\operatorname{prf}}(\kappa) := \max_{\operatorname{\mathsf{Dist}}}\{\operatorname{\mathbf{Adv}}^{\operatorname{prf}}_{\operatorname{\mathsf{Dist}},\operatorname{\mathsf{F}}}(\kappa)\}.\end{aligned}$$
Algorithm Substitution Attacks against Encryption Schemes {#sec:asa}
=========================================================
While it is certainly very useful for an attacker to be able to reconstruct the key, one can also consider situations, where the extractor should be able to extract different information from the ciphertexts or signatures. We will thus generalize the algorithm substitution attacks described in the literature to the setting, where the substituted algorithm also takes a message ${\textit{am}}$ as argument and the goal of the extractor is to derive this message from the produced ciphertext. By always setting ${\textit{am}}:= k$, this is the setting described by @bellare2015asa in [@bellare2015asa]. We thus strengthen the model of [@bellare2014asa] and [@bellare2015asa] in this sense.
Below we give in detail our definitions based upon the model proposed by Bellare et al. in [@bellare2015asa]. If the substitution attack is *stateful*, we allow the distinguisher that tries to identify the attack to also choose this state and observe the internal state of the attack. Every algorithm substitution attack thus needs to be *stateless*, as in the model of Bellare et al. in [@bellare2015asa]. Note that this is a stronger requirement than in [@bellare2014asa] and [@ateniese2015sig], as those works also allowed stateful attacks.
In our setting an *algorithm substitution attack* against a symmetric encryption scheme $\operatorname{\mathsf{SES}}=({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Dec}}})$ is a triple of PPTMs $$\begin{array}{rcl}
\operatorname{\mathsf{ASA}}&=&({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Ext}}}) \\
\end{array}$$ with parameter ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ for the *message length* – the length of the attacker message – and the following functionality.
- The *key generator* ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}$ produces upon input $1^{\kappa}$ an attacker key ${\textit{ak}}$ of length $\kappa$.
- The *encryption algorithm* ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$ takes an attacker key ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))$, attacker message ${\textit{am}}$ such that ${\textit{am}}\in \{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, an encryption key $k\in \operatorname{supp}({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}$ $(1^{\kappa}))$, an encryption message $m\in \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, and a state $\sigma\in \{0,1\}^{*}$ and produces a ciphertext $c$ of length ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)$ and a new state $\sigma'$.
- The *extraction algorithm* ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Ext}}}$ takes as input an attacker key ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))$ and $\ell =
{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$ a ciphertext $c_{1},\ldots,c_{\ell}$ with $c_{i}\in
\{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{cl}}}(\kappa)}$ and produces an attacker message ${\textit{am}}'$.
An algorithm substitution attack needs (a) to be indistinguishable from the symmetric encryption scheme and (b) should be able to reliably extract the message ${\textit{am}}$ of length ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ from the ciphertexts. Due to information-theoretic reasons, it might be impossible to embed the attacker message ${\textit{am}}$ into a single ciphertext: If ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}$ uses $10$ bits of randomness, at most $10$ bits from ${\textit{am}}$ can be reliably embedded into a ciphertext. Hence, the algorithm substitution attack needs to produce more than one ciphertext in this case. For message $m_{1},\ldots,m_{\ell}$, the complete output, denoted as ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{\ell}({\textit{ak}},{\textit{am}},k,m_{1},\ldots,m_{\ell})$ is defined as follows:
1. $\sigma = \varnothing$
2. [**for**]{} $j=1$ to $\ell$ [**do**]{} $(c_{j},\sigma)\gets {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}({\textit{ak}},{\textit{am}},k,m_{j},\sigma)$
3. [**return**]{} $c_{1},\ldots,c_{\ell}$
To formally define the probability that the extractor is able to reliably extract ${\textit{am}}$ from the given ciphertexts $c_{1},\ldots,c_{\ell}$, we define its *reliability*[^2] as $1-\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)$, where the *unreliability* $\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}$ is given as $$\begin{aligned}
&\max \{
\Pr[{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Ext}}}({\textit{ak}},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{\ell}({\textit{ak}},{\textit{am}},k,m_{1},\ldots,m_{\ell}))\neq
{\textit{am}}]\},\end{aligned}$$ with the maximum taken over all ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})),
{\textit{am}}\in\{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, and $m_{i}\in\{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$. The algorithm is *successful*, if there is negligible function $\operatorname{\mathsf{negl}}$ with $\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) \leq \operatorname{\mathsf{negl}}(\kappa).$
The indistinguishability of an ASA is defined as follows. Call a *watchdog* $\operatorname{\mathsf{Watch}}$ a PPTM that tries to distinguish the output of the attacker encryption algorithm ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$ from the original encryption algorithm $\operatorname{\mathsf{Enc}}$. The indistinguishability is defined via the game named $\operatorname{\mathsf{ASA-Dist}}$:
watchdog $\operatorname{\mathsf{Watch}}$, algorithm substitution attack $\operatorname{\mathsf{ASA}}=({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}, {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}},$ $ {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Ext}}})$, and encryption scheme $\operatorname{\mathsf{SES}}=({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}},{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Dec}}})$
${\textit{ak}}\gets {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$; $b\gets \{0,1\}$ $b' \gets \operatorname{\mathsf{Watch}}^{\operatorname{CH}}(1^{\kappa})$ $b=0$ [**then**]{} $c\gets {\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m)$ $(c,\sigma) \gets {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}({\textit{ak}},{\textit{am}},k,m,\sigma)$
An algorithm substitution attack $\operatorname{\mathsf{ASA}}$ is called *indistinguishable* from the symmetric encryption scheme $\operatorname{\mathsf{SES}}$, if for every watchdog $\operatorname{\mathsf{Watch}}$, there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
&\operatorname{\mathbf{Adv}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{Watch}}, \operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) \
:=\\ &\quad \quad |\Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)=\textsf{true}] -1/2|
\leq \ \operatorname{\mathsf{negl}}(\kappa).\end{aligned}$$
The maximal advantage of any watchdog distinguishing $\operatorname{\mathsf{ASA}}$ from $\operatorname{\mathsf{SES}}$ is called the *indistinguishability* or *insecurity* of $\operatorname{\mathsf{ASA}}$ and is defined as $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) := \max_{\operatorname{\mathsf{Watch}}}\{\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}^{\operatorname{enc-watch}}(\kappa)\}.\end{aligned}$$
In [@bellare2014asa], Bellare et al. proposed a (stateless) construction $\operatorname{\mathsf{ASA}}$ against all symmetric encryption schemes $\operatorname{\mathsf{SES}}$. They prove in Theorem 3 that if $\operatorname{\mathsf{SES}}$ is a randomized, stateless, coin-injective symmetric encryption scheme with randomness-length $r$ and if the ASA uses a PRF $\operatorname{\mathsf{F}}$ then for a watchdog $\operatorname{\mathsf{Watch}}$ that makes $q$ queries to its $\operatorname{CH}$ oracle we can construct an adversary $\operatorname{\mathsf{A}}$ such that $ \operatorname{\mathbf{Adv}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{Watch}}, \operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) \le q/2^{2^r} +
\operatorname{\mathbf{Adv}}^{\operatorname{prf}}_{\operatorname{\mathsf{A}},\operatorname{\mathsf{F}}}(\kappa)$, where $\operatorname{\mathsf{A}}$ makes $q$ oracle queries and its running time is that of $\operatorname{\mathsf{Watch}}$.
Bellare et al. conclude that as long as their scheme uses a non-trivial amount of randomness, for example $r \ge 7$ bits resulting $2^r \ge 128$, Theorem 3 implies that the subversion is undetectable.
Backgrounds of Steganography {#sec:stego}
============================
The definitions of the basic steganography concepts presented in this section are essentially those of [@hopper2009provably] and [@dedic2009upper].
In order to define undetectable hidden communication, we need to introduce a notion of *unsuspicious* communication. We do this via the notion of a *channel* ${\mathcal{C}}$. A channel ${\mathcal{C}}$ on the alphabet $\Sigma$ with maximal document length ${{{\mathcal{C}}\,\!.\!\,n}}$ is a function that maps a string of previously send elements $h\in (\Sigma^{\leq {{{\mathcal{C}}\,\!.\!\,n}}})^{*}$ – the *history* – to a probability distribution upon $\Sigma^{\leq {{{\mathcal{C}}\,\!.\!\,n}}}$. We denote this probability distribution by ${\mathcal{C}}_{h}$. The elements of $\Sigma^{\leq {{{\mathcal{C}}\,\!.\!\,n}}}$ are called *documents*. As usually, we will assume that the sequences of documents are efficiently prefix-free recognizable.
A *stegosystem* $\operatorname{\mathsf{S}}$ on a family of channels $\bm{{\mathcal{C}}}=\{{\mathcal{C}}^{\kappa}\}_{\kappa\in {\mathbb{N}}}$ is a triple of probabilistic polynomial-time (according to the security parameter $\kappa$) algorithms: $$\begin{array}{rcl}
\operatorname{\mathsf{S}}&=&({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}},{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Dec}}})
\end{array}$$ with parameters ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ describing the *message length* of the subliminal (hidden, or attacker) message and ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$ describing the length of a generated sequence of stego documents to embed the whole hidden message. The algorithms have the following functionality:
- The *key generator* ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}}$ takes the unary presentation of an integer $\kappa$ – the *security parameter* – and outputs a key (we will call it an attacker key) ${\textit{ak}}\in \{0,1\}^{\kappa}$ of length $\kappa$.
- The *stegoencoder* ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}$ takes as input the key ${\textit{ak}}$, the attacker (or hidden) message ${\textit{am}}\in
\{0,1\}^{{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, a history $h$, and a state $\sigma$ and outputs a document $d$ from ${\mathcal{C}}^{\kappa}$ such that ${\textit{am}}$ is (partially) embedded in this document and a new state. In order to produce the document, ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}$ also has sampling access to ${\mathcal{C}}^{\kappa}_{h}$. We denote this by writing ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}^{{\mathcal{C}}}({\textit{ak}},{\textit{am}},h,\sigma)$.
- The *(history-ignorant) stegodecoder* ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Dec}}}$ takes as input the key ${\textit{ak}}$ and $\ell={\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$ documents $d_{1},\ldots,d_{\ell}$ and outputs a message ${\textit{am}}'$. A history-ignorant stegodecoder thus has no knowledge of previously sent documents. The stegodecoders of nearly all known systems are history-ignorant.
To improve readability, if the stegosystem is clear from the context, we will omit the prefix $\operatorname{\mathsf{S}}$. If $\bm{{\mathcal{C}}}=\{{\mathcal{C}}^{\kappa}\}_{\kappa\in \mathbb{N}}$ is a family of channels, the *min-entropy* of ${H_{\infty}}(\bm{{\mathcal{C}}},\kappa)$ is defined as ${H_{\infty}}(\bm{{\mathcal{C}}},\kappa) = \min_{h\in
\Sigma^{*}}\{{H_{\infty}}({\mathcal{C}}^{\kappa}_{h})\}$. In order to be useful, the stegodecoder should *reliably* decode the embedded message from the sequence of documents. As in the setting of algorithm substitution attack, the complete output of $\ell$ documents of the stegosystem for the history $h$ on the subliminal message ${\textit{am}}$ of length ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ is denoted as ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}^{\ell,{\mathcal{C}}}({\textit{ak}},{\textit{am}},h)$ and is defined as follows.
1. $\sigma = \varnothing$
2. [**for**]{} $j=1$ to $\ell$ [**do**]{}
3. $(d_{j},\sigma)\gets{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}^{{\mathcal{C}}}({\textit{ak}},{\textit{am}},h,\sigma)$; $h = h \mid\mid d_{j}$
4. [**return**]{} $d_{1},\ldots,d_{\ell}$
The *unreliability* $\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)$ of the stegosystem $\operatorname{\mathsf{S}}$ on the channel family $\{{\mathcal{C}}^{\kappa}\}_{\kappa\in {\mathbb{N}}}$ with security parameter $\kappa$ is defined as $$\begin{aligned}
& \operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) := \\
& \max_{{\textit{ak}}, {\textit{am}}} \max_{h}
\{\Pr[{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Dec}}}({\textit{ak}},{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}^{\ell,{\mathcal{C}}}}({\textit{ak}},{\textit{am}},h))\neq {\textit{am}}]\},\end{aligned}$$ where the maximum is taken over all ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})),
{\textit{am}}\in \{0,1\}^{{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, and $h\in(\Sigma^{n(\kappa)})^{*}$. If there is a negligible function $\operatorname{\mathsf{negl}}$ such that $\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\leq \operatorname{\mathsf{negl}}(\kappa)$, we say that $\operatorname{\mathsf{S}}$ is *reliable* on ${\mathcal{C}}$. Furthermore, the *reboot-reliability* of the stegosystem $\operatorname{\mathsf{S}}$ is defined as $$\begin{aligned}
& \operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) := \\
&\max_{{\textit{ak}}, {\textit{am}}} \max_{\tau}\max_{ h_1,\ldots,h_{\tau}}\max_{\ell_1,\ldots,\ell_{\tau}} \{
\Pr[{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Dec}}}({\textit{ak}},
d_1,d_2,\ldots,d_{\ell}
)\neq {\textit{am}}]\} \end{aligned}$$ where the maxima are taken over all ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))$, ${\textit{am}}\in \{0,1\}^{{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, all positive integers $\tau\le \ell$, all histories $h_{1},\ldots,h_{\tau}$, and all positive integers $\ell_1,\ldots,\ell_{\tau}$ such that $\ell_1 + \ldots + \ell_{\tau}=\ell$. The documents $d_{1},\ldots, d_{\ell}$ are the concatenated output of the runs $$\begin{aligned}
{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}^{\ell_1,{\mathcal{C}}}}({\textit{ak}},{\textit{am}},h_1)\mid\mid
\ldots \mid\mid
{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}^{\ell_\tau,{\mathcal{C}}}}({\textit{ak}},{\textit{am}},h_\tau). \end{aligned}$$ We say that the stegosystem $\operatorname{\mathsf{S}}$ is *reboot-reliable* if $\operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)$ is bounded from above by a negligible function. This corresponds to a situation where the stegoencoder is restarted $\tau$ times, each time with the history $h_i$, and is allowed to generate $\ell_i$ documents. Note that reboot-reliability is a strictly stronger requirement than reliability and we can thus conclude $$\begin{aligned}
\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\leq \operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa).\end{aligned}$$
To define the security of a stegosystem, we first specify the abilities of an attacker: A *warden* $\operatorname{\mathsf{Ward}}$ is a probabilistic polynomial-time algorithm that will have access to a *challenge oracle* $\operatorname{CH}$. This challenge oracle can be called with a message ${\textit{am}}$ and a history $h$ and is either equal to ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}^{{\mathcal{C}}}({\textit{ak}},{\textit{am}},h,\sigma)$ for a key ${\textit{ak}}\gets {\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$ or equal to random documents of the *channel*.
The goal of the warden is to distinguish between those oracles. It also has access to samples of the channel ${\mathcal{C}}^{\kappa}_{h}$ for a freely chosen history $h$. Formally, the *chosen-hiddentext-attack-advantage* is defined via the following game $\operatorname{\mathsf{SS-CHA-Dist}}$:
warden $\operatorname{\mathsf{Ward}}$, stegosystem $\operatorname{\mathsf{S}}$, channel ${\mathcal{C}}$
${\textit{ak}}\gets {\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$ $b\gets \{0,1\}$ $b' \gets \operatorname{\mathsf{Ward}}^{\operatorname{CH},{\mathcal{C}}}(1^{\kappa})$ $b=0$ [**then**]{} $d \gets {\mathcal{C}}^{\kappa}_{h}$ $(d,\sigma)\gets {\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{Enc}}}({\textit{ak}},{\textit{am}},h,\sigma)$
A stegosystem $\operatorname{\mathsf{S}}$ is called *secure against chosen-hiddentext attacks* if for every warden $\operatorname{\mathsf{Ward}}$, there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}}, \operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
&:= |\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)=\textsf{true}] -1/2| \\
& \leq \ \operatorname{\mathsf{negl}}(\kappa).\end{aligned}$$
The maximal advantage of any warden against $\operatorname{\mathsf{S}}$ is the *insecurity* $\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)$ and defined as $\max_{\operatorname{\mathsf{Ward}}}\{\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\}$.
A very common technique in the design of secure stegosystems called *rejection sampling* goes back to an idea of Anderson, presented in [@anderson1996limits]. The basic concept is that the stegoencoder samples from the channel until he finds a document that already encodes the hiddentext. This was first used by Cachin in [@Cachin2004] to construct a secure stegosystem in the information-theoretic sense.
In the following, let $\operatorname{\mathsf{F}}$ be pseudorandom function that maps input strings of length ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{in}}}(\kappa)$ (documents) to strings of length ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{out}}}(\kappa)=\log(\operatorname{\mathsf{ml}}(\kappa))+1$ (message parts). To simplify notation, we treat the output of ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{k}$ as a pair $(b,j)$ with $|b|=1$ and $|j|=\log(\operatorname{\mathsf{ml}}(\kappa))$. The encoder of the *rejection sampling stegosystem*, which we denote as $\operatorname{\mathsf{RejSam}}^{\operatorname{\mathsf{F}}}$, is defined as follows:
key ${\textit{ak}}$, message ${\textit{am}}$, history $h$, state $\sigma$
$i := 0$; $d \gets {\mathcal{C}}_{h}$ $i := i+1$ $(b,j) := {\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{{\textit{ak}}}(d)$
The key generator ${\operatorname{\mathsf{RejSam}}^{\operatorname{\mathsf{F}}}\!.\!\operatorname{\mathsf{Gen}}}$ is equal to ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Gen}}}$ and the decoder derives ${\textit{am}}$, as long as its input documents contain every bit ${\textit{am}}[j]$, by applying ${\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{{\textit{ak}}}$ to these documents. Below we present the description of the decoder. Note that the stegosystem is stateless.
key ${\textit{ak}}$, documents $d_{1},\ldots,d_{{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}$
let ${\textit{am}}_{j} := \bot$ $(b,j) := {\operatorname{\mathsf{F}}\!.\!\operatorname{\mathsf{Eval}}}_{k}(d_{i})$ let ${\textit{am}}_{j} := b$
In [@hopper2009provably], Hopper et al. were the first to prove the security of this stegosystem in the complexity-theoretic model. Their argument was simplified by Dedi[ć]{} et al. in [@dedic2009upper] and by Backes and Cachin in [@backes2005public]. The version given here is based upon the stateless construction of Dedi[ć]{} et al. and also uses the idea of Bellare et al. in [@bellare2015asa] to apply the *coupon collector’s problem* to completely get rid of the state by randomly choosing an index to embed.
The analysis of the coupon collector’s problem shows that by sending $\operatorname{\mathsf{ml}}(\kappa)\cdot (\ln \operatorname{\mathsf{ml}}(\kappa)+\beta)$ documents – for an appropriate value $\beta$ – one only introduces a term $\exp(-\beta)$ into the unreliability (see [e.g.]{}[@mitzenmacher2005probability] for a proof of this fact), which can be made negligible by setting $\beta\geq \operatorname{\mathsf{ml}}(\kappa)-\ln(\operatorname{\mathsf{ml}}(\kappa))$. The output length on messages of length $\operatorname{\mathsf{ml}}(\kappa)$ will thus be bounded by $\operatorname{\mathsf{ml}}(\kappa)^{2}$.
The security of this system directly follows from the analysis of Dedi[ć]{} et al. in [@dedic2009upper]:
\[thm:rejsam:secure\] For every polynomial $\operatorname{\mathsf{ml}}(\kappa)$, there exists a universal history-ignorant stegosystem $\operatorname{\mathsf{S}}=\operatorname{\mathsf{RejSam}}^{\operatorname{\mathsf{F}}}$ with security parameter $\kappa$ and $s\ge 1$ such that for every channel ${\mathcal{C}}^{\kappa}$ we have
- ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)=\operatorname{\mathsf{ml}}(\kappa)$,
- $\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) \leq
\mathcal{O}(\operatorname{\mathsf{ml}}(\kappa)^{4}\cdot
2^{-{H_{\infty}}({\mathcal{C}}^{\kappa})}+\operatorname{\mathsf{ml}}(\kappa)^{2}\cdot
\exp(-s))+\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}},{\mathcal{C}}}(\kappa)$, and
- $\operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) \leq
\operatorname{\mathsf{ml}}(\kappa)^{2}(2\cdot
\exp(-2^{{H_{\infty}}({\mathcal{C}}^{\kappa})-3}) +
\exp(-2^{-2} s))+
\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}},{\mathcal{C}}}(\kappa)$.
The notation $\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}},{\mathcal{C}}}(\kappa)$ indicates the insecurity of the pseudorandom function $\operatorname{\mathsf{F}}$ *relative to the channel ${\mathcal{C}}$*. Informally, this means that the attacker against $\operatorname{\mathsf{F}}$ also has sampling access to ${\mathcal{C}}$ (for a formal definition, see [@dedic2009upper]). For an *efficiently sampleable* channel ${\mathcal{C}}$ ([i.e.]{}one that can be simulated by a PPTM), it clearly holds that $\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}},{\mathcal{C}}}(\kappa)=\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}}}(\kappa)$. All channels used in this work are efficiently sampleable and we will thus omit the index ${\mathcal{C}}$ from the term $\operatorname{\mathbf{InSec}}$.
Encryption Schemes as Steganographic Channels {#sec:channel}
=============================================
Let $\operatorname{\mathsf{SES}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$ be a symmetric encryption scheme that encodes messages of length $\operatorname{\mathsf{ml}}(\kappa)$ into ciphertexts of length $\operatorname{\mathsf{cl}}(\kappa) \geq \operatorname{\mathsf{ml}}(\kappa)$ and let $\ell$ be a polynomial of $\kappa$. For $\operatorname{\mathsf{SES}}$ we define a channel family, named ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)$, indexed with parameter $\kappa\in {\mathbb{N}}$, where the documents will correspond to the input of generalized algorithm substitution attack against encryption schemes. The essential idea behind the definition of the channel ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)$ is that for all $k\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$ and every sequence of messages $m_{1},m_{2},\ldots, m_{\ell(\kappa)}$, with $m_{i}\in\{0,1\}^{\operatorname{\mathsf{ml}}(\kappa)}$, for the history $$h= k \mid\mid m_{1} \mid\mid m_{2} \mid\mid \ldots \mid\mid m_{\ell(\kappa)}$$ the distribution of the sequences of documents $$\begin{aligned}
c_{1} \mid\mid c_{2} \mid\mid \ldots \mid\mid c_{\ell(\kappa)}
\end{aligned}$$ generated by the channel is exactly the same as the distribution for $$\begin{aligned}
\operatorname{\mathsf{Enc}}(k,m_{1}) \mid\mid \operatorname{\mathsf{Enc}}(k,m_{2}) \mid\mid \ldots \mid\mid \operatorname{\mathsf{Enc}}(k,m_{\ell(\kappa)}).
\end{aligned}$$ To give a formal definition of $\{{\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)\}_{\kappa\in {\mathbb{N}}}$ we need to specify the probability distributions for any history $h$. Thus, we define the family, on the alphabet $\{0,1\}$, as follows.
For the empty history $h=\varnothing$, define $$\begin{aligned}
{\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)_{\varnothing} \end{aligned}$$ as the distribution of all keys generated by $\operatorname{\mathsf{Gen}}(1^{\kappa})$. For a key $k\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$ and a (possibly empty) sequence of messages $m_{1},m_{2},\ldots, m_{r}$, with $m_{i}\in \{0,1\}^{\operatorname{\mathsf{ml}}(\kappa)}$ and $0\leq r\leq \ell(\kappa)-1$, the distribution $$\begin{aligned}
{\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)_{k\mid\mid m_{1}\mid\mid
m_{2}\mid\mid \ldots \mid\mid m_{r}}
\end{aligned}$$ is the uniform distribution on all messages $m_{r+1}\in
\{0,1\}^{\operatorname{\mathsf{ml}}(\kappa)}$. For $k\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$, a sequence of messages $m_{1},m_{2},\ldots,$ $ m_{\ell(\kappa)}$ with $m_{i}\in
\{0,1\}^{\operatorname{\mathsf{ml}}(\kappa)}$, and a (possibly empty) sequence of ciphertexts $c_{1},\ldots,c_{r}$, with $c_{i}\in \operatorname{supp}(\operatorname{\mathsf{Enc}}(k,m_{((i-1)\bmod
\ell(\kappa))+1})),$ the distribution $$\begin{aligned}
{\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)_{k\mid\mid m_{1}\mid\mid
m_{2}\mid\mid \ldots \mid\mid m_{\ell(\kappa)}\mid\mid
c_{1}\mid\mid c_{2}\mid\mid \ldots \mid\mid
\ldots \mid\mid c_{r}}
\end{aligned}$$ is the distribution of $\operatorname{\mathsf{Enc}}(k,m_{(r \bmod \ell(\kappa))+1})$.
ASAs against Encryption in the Steganographic Model {#Sec:ASA:against:encrypted:as:stego}
===================================================
The main message of our paper is that algorithm substitution attacks against a primitive $\Pi$ are equivalent to the use of steganography on a corresponding channel ${\mathcal{C}}_{\Pi}$ determined by the protocol $\Pi$. Focusing on symmetric encryption schemes as a common cryptographic primitive, we will show in this section exemplary proofs for the general relations between ASAs and steganography.
In the previous section we showed a formal specification of the family of communication channels ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)$ determined by a symmetric encryption scheme $\operatorname{\mathsf{SES}}$. We will now prove that a secure and reliable stegosystem on ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)$ implies the existence of an indistinguishable and successful algorithm substitution attack on $\operatorname{\mathsf{SES}}$. On the other hand, we will also show that the existence of an indistinguishable and successful algorithm substitution attack on $\operatorname{\mathsf{SES}}$ implies a secure and reliable stegosystem on ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(\ell)$.
As a consequence we get a construction of an ASA against any encryption scheme using a generic stegosystem like [e.g.]{}this proposed by Dedi[ć]{} et al. [@dedic2009upper]. Thus, we can conclude Theorem 1 and Theorem 3 proposed by Bellare et al. in [@bellare2014asa] that there exist indistinguishable and successful ASAs against encryption schemes. Moreover we obtain Theorem 4 in [@bellare2014asa] which says that an ASA is impossible for unique ciphertext symmetric encryption schemes.
Steganography implies ASAs
--------------------------
\[thm:asa:on:ses:impl:stego\] Assume $\operatorname{\mathsf{SES}}$ is a symmetric encryption scheme and let $\operatorname{\mathsf{S}}$ be a stegosystem on the channel ${\mathcal{C}}:={\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa))$ determined by $\operatorname{\mathsf{SES}}$. Then there exists an algorithm substitution attack $\operatorname{\mathsf{ASA}}$ against $\operatorname{\mathsf{SES}}$ of indistinguishability, resp. reliability such that: $$\begin{array}{rcl}
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) &\le& \operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
\quad \text{and} \\
\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) &=& \operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa).
\end{array}$$
Let $\operatorname{\mathsf{SES}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$ be a symmetric encryption scheme and $\operatorname{\mathsf{S}}=(\operatorname{\mathsf{SGen}},\operatorname{\mathsf{SEnc}},\operatorname{\mathsf{SDec}})$ be a stegosystem on the channel ${\mathcal{C}}$. To simplify notation, let $\ell=\ell(\kappa):={\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$. We will construct the algorithm substitution attack $\operatorname{\mathsf{ASA}}=(\operatorname{\mathsf{AGen}},\operatorname{\mathsf{AEnc}},$ $\operatorname{\mathsf{AExt}})$ on $\operatorname{\mathsf{SES}}$ from the stegosystem $\operatorname{\mathsf{S}}$ and show the indistinguishability and success of $\operatorname{\mathsf{ASA}}$ depending on security and reliability of $\operatorname{\mathsf{S}}$. The components of the $\operatorname{\mathsf{ASA}}$ are defined as follows.
The key generator $\operatorname{\mathsf{AGen}}$ just simulates $\operatorname{\mathsf{SGen}}$ – the key generator of the stegosystem. It will output the attack key ${\textit{ak}}$. The encoding algorithm $\operatorname{\mathsf{AEnc}}$ on input ${\textit{ak}}\in
\operatorname{supp}(\operatorname{\mathsf{AGen}}(1^{\kappa}))$, ${\textit{am}}\in
\{0,1\}^{{\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, $k\in
\operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$, and $m\in
\{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$ simulates $\operatorname{\mathsf{SEnc}}$ on channel ${\mathcal{C}}$ with input key ${\textit{ak}}$, the message ${\textit{am}}$ and the history $h=k\mid\mid m^{\ell}$, where $m^{\ell}$ is the string of length $\ell\cdot |m|$ containing $\ell$ copies of $m$. Whenever $\operatorname{\mathsf{SEnc}}$ makes a query to its channel oracle, algorithm $\operatorname{\mathsf{AEnc}}$ uses $\operatorname{\mathsf{Enc}}$ on input $k$ and $m$ to produce a corresponding ciphertext and sends it to $\operatorname{\mathsf{SEnc}}$. The encoder $\operatorname{\mathsf{AEnc}}$ then outputs the document $d$ generated by $\operatorname{\mathsf{SEnc}}$. Finally, the extraction algorithm $\operatorname{\mathsf{AExt}}$ on input ${\textit{ak}}\in \operatorname{supp}(\operatorname{\mathsf{AGen}}(1^{\kappa}))$ and documents $d_{1},\ldots,d_{\ell}$ just simulates $\operatorname{\mathsf{SDec}}$ on the same inputs.
As one can see from the definitions, $\operatorname{\mathsf{ASA}}$ is a generalized algorithm substitution attack against $\operatorname{\mathsf{SES}}$. We will now prove that it is indistinguishable from $\operatorname{\mathsf{SES}}$ and that it is successful.
We prove first indistinguishability of the system. Let $\operatorname{\mathsf{Watch}}$ be a watchdog against the above $\operatorname{\mathsf{ASA}}$ with maximal advantage, i.e. $$\begin{aligned}
\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}^{\operatorname{enc-watch}}(\kappa) = \operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa),
\end{aligned}$$ where $\operatorname{\mathbf{Adv}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{Watch}}, \operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)$ is equal to the success probability that $\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)=\textsf{true}$. We will now construct a warden $\operatorname{\mathsf{Ward}}$ from $\operatorname{\mathsf{Watch}}$ such that $$\begin{aligned}
\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
=
\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}^{\operatorname{enc-watch}}(\kappa). \end{aligned}$$ Thus, we will get that $$\label{ineq:insecasa:insecstego}
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)
\le
\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa).$$ The warden $\operatorname{\mathsf{Ward}}$ on input $1^{\kappa}$ just simulates the watchdog $\operatorname{\mathsf{Watch}}$ and gives the same output as $\operatorname{\mathsf{Watch}}$ at the end of the simulation. Whenever the watchdog makes a query on input ${\textit{am}}$, $k$, and $m$ to its challenging oracle (that is either equal to $\operatorname{\mathsf{SES}}$’s encryption algorithm $\operatorname{\mathsf{Enc}}(k,m)$ or to $\operatorname{\mathsf{ASA}}$’s encryption $\operatorname{\mathsf{AEnc}}({\textit{ak}},{\textit{am}},k,m,\sigma)$ for ${\textit{ak}}\gets \operatorname{\mathsf{AGen}}(1^{\kappa})$), the warden $\operatorname{\mathsf{Ward}}$ queries its own challenging oracle with message ${\textit{am}}$, state $\sigma$ and history $h=k\mid\mid m^{\ell}$. Note that the challenging oracle of $\operatorname{\mathsf{Ward}}$ is either equal to the channel ${\mathcal{C}}$ or to $\operatorname{\mathsf{SEnc}}({\textit{ak}},{\textit{am}},h,\sigma)$ for ${\textit{ak}}\gets \operatorname{\mathsf{SGen}}(1^{\kappa})$.
If the challenging oracle of $\operatorname{\mathsf{Ward}}$ is equal to the steganographic encoding $\operatorname{\mathsf{SEnc}}({\textit{ak}},{\textit{am}},h,\sigma)$ ([i.e.]{}the bit $b$ in $\operatorname{\mathsf{SS-CHA-Dist}}$ equals $1$, denoted by $\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\langle b=1\rangle$), the answer of $\operatorname{\mathsf{Ward}}$ is the same as the output of the $\operatorname{\mathsf{Watch}}$ in case it queries the ASA’s encoding algorithm $\operatorname{\mathsf{AEnc}}({\textit{ak}},{\textit{am}},k,m)$ by construction. Thus, $$\begin{aligned}
&\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\langle b=1\rangle=\textsf{true}]\\
&= \ \Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)\langle b=1\rangle=\textsf{true}].\end{aligned}$$
If the challenging oracle of $\operatorname{\mathsf{Ward}}$ is equal to the channel (the bit $b$ in $\operatorname{\mathsf{SS-CHA-Dist}}$ equals $0$), by the definition of the channel ${\mathcal{C}}$ for the symmetric encryption scheme $\operatorname{\mathsf{SES}}$, the answer of the challenging oracle is equal to the output of $\operatorname{\mathsf{Enc}}(k,m)$. Hence, $$\begin{aligned}
&\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\langle b=0\rangle=\textsf{true}]\\
& = \
\Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)\langle b=0\rangle=\textsf{true}].\end{aligned}$$ We thus have $$\begin{array}{rcl}
\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
& = &
|\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)=\textsf{true}] - 1/2| \\
& = & |\Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)=\textsf{true} -1/2| \\
& = &
\operatorname{\mathbf{Adv}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{Watch}}, \operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)
\end{array}$$ which completes the proof of .
We still need to prove that $\operatorname{\mathsf{AExt}}$ is reliably able to extract the attacker message ${\textit{am}}$ from the ciphertext. But, as $\operatorname{\mathsf{AExt}}=\operatorname{\mathsf{SDec}}$, the reboot-reliability of $\operatorname{\mathsf{SDec}}$ directly implies that $\operatorname{\mathsf{AExt}}$ is successful with probability of $1-\operatorname{\mathsf{negl}}(\kappa)$.
By combining Theorem \[thm:asa:on:ses:impl:stego\] and Theorem \[thm:rejsam:secure\], we can conclude the following corollary.
\[coroll:thm:ipl:bellare\] For every symmetric encryption scheme $\operatorname{\mathsf{SES}}$, there exists an algorithm subsection attack $\operatorname{\mathsf{ASA}}$ with message length $\operatorname{\mathsf{ml}}(\kappa)$ and parameter $s\ge 1$ such that $$\begin{array}{rcl}
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)&\leq&
\mathcal{O}(\operatorname{\mathsf{ml}}(\kappa)^{4}\cdot 2^{-{H_{\infty}}({\mathcal{C}}^{\kappa})})+\\
&&\mathcal{O}(\operatorname{\mathsf{ml}}(\kappa)^{2}\cdot
\exp(-s))+
\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}}}(\kappa),\\[2mm]
\operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) &\leq&
2 \operatorname{\mathsf{ml}}(\kappa)^{2}\cdot
\exp(-2^{{H_{\infty}}({\mathcal{C}}^{\kappa})-3}) +\\
&& \operatorname{\mathsf{ml}}(\kappa)^{2}\cdot \exp(-2^{-2} s)+
\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}}}(\kappa)
\end{array}$$ where ${\mathcal{C}}:={\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa))$
One can compare this corollary to the construction used in the proof of Theorem 4.1 and Theorem 4.2 in [@bellare2015asa]. We can see that our generic algorithm substitution attack gets almost the same bounds for insecurity and for unreliability.
Note that the protocols in [@bellare2015asa; @ateniese2015sig] and our generic protocol of Corollary \[coroll:thm:ipl:bellare\] have a very bad rate: $
\frac{\operatorname{\mathsf{ml}}}{\operatorname{\mathsf{ml}}\cdot (\ln \operatorname{\mathsf{ml}}+\beta)} = 1/(\ln \operatorname{\mathsf{ml}}+\beta)
$ for an appropriate value $\beta$. One can easily modify the above constructions such that instead of one bit $b$ of a message ${\textit{am}}$ we embed a block of $\log(\operatorname{\mathsf{ml}})$ bits per ciphertext. This improves the rate to $
\frac{\log \operatorname{\mathsf{ml}}}{\ln(\operatorname{\mathsf{ml}}) - \ln\log(\operatorname{\mathsf{ml}}) +\beta} = \Theta(1).
$
ASAs imply Steganography
------------------------
\[thm:ses:on:ses:impl:stego\] Assume $\operatorname{\mathsf{SES}}$ is a symmetric encryption scheme and let $\operatorname{\mathsf{ASA}}$ be an algorithm substitution attack against $\operatorname{\mathsf{SES}}$ of output length ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$. Then there exists a stegosystem $\operatorname{\mathsf{S}}$ with the output length ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)=2\cdot {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)+1$ on the channel ${\mathcal{C}}:={\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa))$ determined by $\operatorname{\mathsf{SES}}$ such that $\operatorname{\mathsf{S}}$’s insecurity, resp. its reliability satisfy $$\begin{array}{rcl}
\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) &\le& \operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)
\quad \text{and} \\
\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) &=& \operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa) .
\end{array}$$
Let $\operatorname{\mathsf{SES}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Enc}},\operatorname{\mathsf{Dec}})$ be a symmetric encryption scheme and $\operatorname{\mathsf{ASA}}=(\operatorname{\mathsf{AGen}},\operatorname{\mathsf{AEnc}},\operatorname{\mathsf{AExt}})$ be an algorithm substitution attack against $\operatorname{\mathsf{SES}}$. To simplify notation, let $\ell={\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)$. We construct the stegosystem $\operatorname{\mathsf{S}}=(\operatorname{\mathsf{SGen}},\operatorname{\mathsf{SEnc}},\operatorname{\mathsf{SDec}})$ on ${\mathcal{C}}$ out of the $\operatorname{\mathsf{ASA}}$.
The key generation algorithm $\operatorname{\mathsf{SGen}}$ simply simulates $\operatorname{\mathsf{AGen}}$. It will output the key ${\textit{ak}}$. To encode a message ${\textit{am}}$ using the key ${\textit{ak}}$, the stegoencoding algorithm $\operatorname{\mathsf{SEnc}}$ generates for any history $h$ a sequence of ${\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)=2\ell+1$ documents such that the last $\ell$ documents embed the message ${\textit{am}}$. To describe the algorithm we need to distinguish between different given histories $h$.
In this case, $\operatorname{\mathsf{SEnc}}$ chooses a random key $k\gets {\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$ using the generation algorithm of $\operatorname{\mathsf{SES}}$ and outputs $k$.
Encoder $\operatorname{\mathsf{SEnc}}$ samples a random message $m_{r+1}$ and outputs it.
The stego-encoder $\operatorname{\mathsf{SEnc}}$ simulates $\operatorname{\mathsf{AEnc}}({\textit{ak}},{\textit{am}},k,m_{(r+1) \bmod \ell +1})$ and outputs the generated ciphertext.
Note that by construction, in any case the last $\ell$ documents generated by $\operatorname{\mathsf{SEnc}}^{2\ell+1}$ embed the message ${\textit{am}}$ in the same way as done by $\operatorname{\mathsf{ASA}}^{\ell}$.
If the decoder $\operatorname{\mathsf{SDec}}$ is given documents $d_{1},\ldots,d_{2\ell+1}$, we output $\operatorname{\mathsf{AExt}}({\textit{ak}},d_{\ell+2},\ldots,d_{2\ell+1})$.
As one can see from the definitions, the decoding algorithm of $\operatorname{\mathsf{S}}$ is history-ignorant. We will prove that on the channel ${\mathcal{C}}={\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{SES}}}(2\ell+1)$ the security and reliability of the stegosystem $\operatorname{\mathsf{S}}$ satisfy the stated conditions.
We first analyze the security of the system. Let $\operatorname{\mathsf{Ward}}$ be a warden against $\operatorname{\mathsf{S}}$ on ${\mathcal{C}}$ with maximal advantage, i.e. $$\begin{aligned}
\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
=
\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa),
\end{aligned}$$ where $ \operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
= \Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)=\textsf{true}].
$ We will construct a watchdog $\operatorname{\mathsf{Watch}}$ against the algorithm substitution attack $\operatorname{\mathsf{ASA}}$ with the same advantage as $\operatorname{\mathsf{Ward}}$: $$\begin{aligned}
\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}^{\operatorname{enc-watch}}(\kappa) =
\operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa).
\end{aligned}$$ This will prove that $$\label{ineq:insecstego:insecasa}
\operatorname{\mathbf{InSec}}^{\operatorname{cha}}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
\le
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa).$$
The watchdog $\operatorname{\mathsf{Watch}}$ on input $1^{\kappa}$ simply simulates the warden $\operatorname{\mathsf{Ward}}$. Whenever the warden $\operatorname{\mathsf{Ward}}$ makes a query to its channel oracle ${\mathcal{C}}$ with a history $h$, the watchdog $\operatorname{\mathsf{Watch}}$ simulates the oracle response as follows:
- If $h=\varnothing$, the watchdog uses $\operatorname{\mathsf{Gen}}(1^{\kappa})$ to construct a key $k$ and returns $k$ to the warden.
- If $h=k\mid\mid m_{1}\mid\mid \ldots \mid\mid
m_{r}$ with $r < \ell$, the watchdog uniformly chooses a message $m_{r+1}$ from $\{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$ and outputs $m_{r+1}$.
- If $h=k\mid\mid m_{1}\mid\mid \ldots\mid\mid
m_{\ell}\mid\mid c_{1}\mid\mid \ldots\mid\mid c_{r}$ with $r\geq 0$, the watchdog computes $c_{r+1}\gets \operatorname{\mathsf{Enc}}(k,m_{((r+1)\bmod \ell)+1})$ and outputs $c_{r+1}$.
Clearly, this simulates the channel distribution ${\mathcal{C}}$ perfectly. If the warden queries its challenge oracle ${\operatorname{\mathsf{Ward}}\!.\!\operatorname{CH}}$ with chosen message ${\textit{am}}$, state $\sigma$, and history $h$ (that is either equivalent to sampling from ${\mathcal{C}}_{h}$ or to calling $\operatorname{\mathsf{SEnc}}({\textit{ak}},{\textit{am}},h,\sigma)$), the watchdog simulates the response of the oracle ${\operatorname{\mathsf{Ward}}\!.\!\operatorname{CH}}$ as follows:
- If $h=\varnothing$ then $\operatorname{\mathsf{Watch}}$ chooses a random key $k\gets \operatorname{\mathsf{Gen}}(1^{\kappa})$ and outputs it.
- If $h=k\mid\mid m_1\mid\mid m_2\mid\mid \ldots \mid\mid
m_r$ for $0\leq r\leq \ell-1$ then $\operatorname{\mathsf{Watch}}$ samples a random message $m$ and outputs it.
- If $h=k\mid\mid m_{1}\mid\mid m{_2}\mid\mid \ldots
\mid\mid m_{\ell}\mid\mid c_{1}\mid\mid \ldots \mid\mid
c_{r}$ with $r\geq 0$ then $\operatorname{\mathsf{Watch}}$ queries its own oracle on $k$ and $m_{((r+1)\bmod \ell)+1}$.
If ${\operatorname{\mathsf{Watch}}\!.\!\operatorname{CH}}$ is equal to $\operatorname{\mathsf{Enc}}$ of $\operatorname{\mathsf{SES}}$ (the bit $b$ in $\operatorname{\mathsf{ASA-Dist}}$ is set to $0$) the corresponding answer is identically distributed to a sample of the channel ${\mathcal{C}}$. Hence, $$\begin{aligned}
&\Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)\langle
b=0\rangle=\textsf{true}]=\\
&\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\langle b=0\rangle=\textsf{true}].
\end{aligned}$$ On the other hand, if ${\operatorname{\mathsf{Watch}}\!.\!\operatorname{CH}}$ is equal to $\operatorname{\mathsf{AEnc}}$ (the bit $b$ in $\operatorname{\mathsf{ASA-Dist}}$ is set to $1$), the corresponding answer is identically distributed to $\operatorname{\mathsf{SEnc}}({\textit{ak}},{\textit{am}},h,\sigma)$ and thus $$\begin{aligned}
& \Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)\langle b=1\rangle=\textsf{true}]=\\
&\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)\langle b=1\rangle=\textsf{true}].
\end{aligned}$$ We thus have $$\begin{array}{ll}
& \operatorname{\mathbf{Adv}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)
\ =\\
&\quad\quad |\Pr[\operatorname{\mathsf{ASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)=\textsf{true}] - 1/2| \ =\\
&\quad\quad |\Pr[\operatorname{\mathsf{SS-CHA-Dist}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)=\textsf{true}] - 1/2| \ =\\
&\quad\quad \operatorname{\mathbf{Adv}}^{\operatorname{cha}}_{\operatorname{\mathsf{Ward}},\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa)
\end{array}$$ which proves .
The reliability of $\operatorname{\mathsf{S}}$ is the same as the success probability of $\operatorname{\mathsf{ASA}}$ since $\operatorname{\mathsf{SDec}}$ simply simulates $\operatorname{\mathsf{AExt}}$.
By using the fact that channels with min-entropy $0$ can not be used for steganography (see [e.g.]{}Theorem 6 in [@hopper2009provably]) and observing that channels corresponding to deterministic encryption schemes have min-entropy $0$, we can conclude the following corollary:
For all deterministic encryption schemes $\operatorname{\mathsf{SES}}$ and all algorithm substitution attacks $\operatorname{\mathsf{ASA}}$ against $\operatorname{\mathsf{SES}}$: $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)\geq 1.\end{aligned}$$
Note that this exactly Theorem 4 in [@bellare2014asa].
General Results {#sec:general}
===============
Let $\operatorname{\mathsf{R}}$ be a polynomial-time randomized algorithm with hardwired secret $s$ which takes inputs $x$ and generates outputs $y$. The general task of an algorithm substitution attack against $\operatorname{\mathsf{R}}$ is to construct a subverted algorithm $\operatorname{\mathsf{AR}}_{{\textit{ak}}}$ which using a hidden hardwired random key ${\textit{ak}}$ outputs on the secret $s$ in the sequence of calls $\operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_1), \operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_2),\ldots$ a sequence such that
the output $\operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_1), \operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_2),\ldots$ is indistinguishable from $\operatorname{\mathsf{R}}(s,x_1), \operatorname{\mathsf{R}}(s,x_2),\ldots$ and
$\operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_1), \operatorname{\mathsf{AR}}_{\textit{ak}}(s,x_2),\ldots$ embeds the secret $s$.
In our setting we model the attack on $\operatorname{\mathsf{R}}$ as a stegosystem on a channel determined by $\operatorname{\mathsf{R}}$ and define such a channel.
ASA against a Randomized Algorithm
----------------------------------
In this section we give formal definitions for algorithm substitution attack $\operatorname{\mathsf{AR}}$, its advantage $\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{AR}},\operatorname{\mathsf{R}}}$, etc. Formally, an *algorithm substitution attack against $\operatorname{\mathsf{R}}$* is a triple of efficient algorithms $\operatorname{\mathsf{ASA}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{AR}},\operatorname{\mathsf{Ext}})$, where $\operatorname{\mathsf{Gen}}$ generates the key ${\textit{ak}}$, the algorithm $\operatorname{\mathsf{AR}}$ takes the key ${\textit{ak}}$, a secret $s$ and all inputs $x_{1},x_{2},\ldots$ to $\operatorname{\mathsf{R}}$ and the extractor $\operatorname{\mathsf{Ext}}$ tries to extract $s$ from the outputs of $\operatorname{\mathsf{AR}}$ with the help of ${\textit{ak}}$ (but without knowing $x_{1},x_{2},\ldots$). Similarly to the setting for encryption schemes, $\operatorname{\mathsf{ASA}}$ is called *indistinguishable*, if every PPTM $\operatorname{\mathsf{Watch}}$ – the *watchdog* – is not able to distinguish between $\operatorname{\mathsf{AR}}_{{\textit{ak}}}(s,x_{1}),\operatorname{\mathsf{AR}}_{{\textit{ak}}}(s,x_{2}),\ldots$ and $\operatorname{\mathsf{R}}(x_{1}),\operatorname{\mathsf{R}}(x_{2}),\ldots$ even if he is allowed to choose $s$ and all $x_{i}$. This is defined via the game $\operatorname{\mathsf{RASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}$ defined analogously to $\operatorname{\mathsf{ASA-Dist}}$. The maximal advantage of any watchdog distinguishing $\operatorname{\mathsf{ASA}}$ from $\operatorname{\mathsf{R}}$ is called the *insecurity* or indistinguishability of $\operatorname{\mathsf{ASA}}$ and is formally defined as $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{asa}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa) := \max_{\operatorname{\mathsf{Watch}}}\{\operatorname{\mathbf{Adv}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}^{\operatorname{asa}}(\kappa)\},\end{aligned}$$ where $$\begin{aligned}
&\operatorname{\mathbf{Adv}}^{\operatorname{asa}}_{\operatorname{\mathsf{Watch}}, \operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa)
:= \\
&\quad\quad |\Pr[\operatorname{\mathsf{RASA-Dist}}_{\operatorname{\mathsf{Watch}},\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa)=\textsf{true}] - 1/2|.\end{aligned}$$
The *unreliability* of $\operatorname{\mathsf{ASA}}$ is also defined like before: $$\begin{aligned}
&\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa) \ :=\\ &\max
\{
\Pr[{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{AExt}}}({\textit{ak}},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{AR}}}({\textit{ak}},{\textit{am}},x_{1},\ldots,x_{\ell}))\neq
{\textit{am}}]\},\end{aligned}$$ where the maximum is taken over all ${\textit{ak}}\in \operatorname{supp}({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})), \allowbreak
{\textit{am}}\in\{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, and $x_{1},\ldots,x_{\ell}$ being inputs to $\operatorname{\mathsf{R}}$.
Known examples which fit into this setting include [e.g.]{}the subversion-resilient signature schemes presented in the work of Ateniese et al. [@ateniese2015sig].
Channel determined by a Randomized Algorithm
--------------------------------------------
Let $\operatorname{\mathsf{R}}$ be a polynomial-time randomized algorithm with parameter $\kappa$. We assume that the secret $s$ is generated by $\operatorname{\mathsf{Gen}}$ and the inputs $x$ to $\operatorname{\mathsf{R}}$ are generated by the randomized polynomial-time algorithm $\operatorname{\mathsf{GenInput}}$, associated with $\operatorname{\mathsf{R}}$ (which may be chosen adversarially as shown in the definition above). Let $\ell$ be a polynomial of $\kappa$. For $\operatorname{\mathsf{R}}$ we define a channel family, named ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)$, indexed with parameter $\kappa\in {\mathbb{N}}$, with documents which correspond to the input of $\operatorname{\mathsf{AR}}$. The essential idea behind the definition of the channel ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)$ is that for all $s\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$ and every sequence of inputs $x_{1},x_{2},\ldots, x_{\ell(\kappa)}$, with $x_{i}\in\operatorname{supp}(\operatorname{\mathsf{GenInput}}(1^{\kappa}))$, for the history $$h= s \mid\mid x_{1} \mid\mid x_{2} \mid\mid \ldots \mid\mid x_{\ell(\kappa)}$$ the distribution of the sequences of documents $$\begin{aligned}
y_{1} \mid\mid y_{2} \mid\mid \ldots \mid\mid y_{\ell(\kappa)}
\end{aligned}$$ generated by the channel is exactly the same as the distribution for $$\begin{aligned}
\operatorname{\mathsf{R}}(s,x_{1}) \mid\mid \operatorname{\mathsf{R}}(s,x_{2}) \mid\mid \ldots \mid\mid \operatorname{\mathsf{R}}(s,x_{\ell(\kappa)}).
\end{aligned}$$ To give a formal definition of $\{{\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)\}_{\kappa\in {\mathbb{N}}}$ we need to specify the probability distributions for any history $h$. Thus, we define the family, on the alphabet $\{0,1\}$, as follows: For empty history $h=\varnothing$, we define ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)_{\varnothing}$ as the distribution on all possible keys generated by $\operatorname{\mathsf{Gen}}(1^{\kappa})$. For $s\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$ and a (possibly empty) sequence inputs $x_{1},x_{2},\ldots, x_{r}$ with $x_{i}\in \operatorname{supp}(\operatorname{\mathsf{GenInput}}(1^{\kappa}))$ and $0\leq r\leq \ell(\kappa)-1$, the distribution ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)_{s\mid\mid x_{1}\mid\mid
x_{2}\mid\mid \ldots \mid\mid x_{r}}$ is the distribution on inputs $x_{r+1}\gets \operatorname{\mathsf{GenInput}}(1^{\kappa})$. For $s\in \operatorname{supp}(\operatorname{\mathsf{Gen}}(1^{\kappa}))$, a sequence of inputs $x_{1},x_{2},\ldots, x_{\ell(\kappa)}$ with $x_{i}\in\operatorname{supp}(\operatorname{\mathsf{GenInput}}(1^{\kappa}))$, and a (possibly empty) sequence of $\operatorname{\mathsf{R}}$’s outputs $y_{1},\ldots,y_{r}$ with $y_{i}\in \operatorname{supp}(\operatorname{\mathsf{R}}(s,x_{((i-1)\bmod
\ell(\kappa))+1}))$, the probability distribution of ${\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}(\ell)_{s\mid\mid x_{1}\mid\mid
x_{2}\mid\mid \ldots \mid\mid x_{\ell(\kappa)}\mid\mid
y_{1}\mid\mid y_{2}\mid\mid \ldots \mid\mid
\ldots \mid\mid y_{r}}$ is the probability distribution of $\operatorname{\mathsf{R}}(s,x_{(r \bmod \ell(\kappa))+1})$.
Results
-------
The theorems proved in the previous section can simply be generalized by using our general construction of the channel ${\mathcal{C}}^{k}_{\operatorname{\mathsf{R}}}(\ell)$ for the randomized algorithm $\operatorname{\mathsf{R}}$ and the generic stegosystem $\operatorname{\mathsf{RejSam}}^{\operatorname{\mathsf{F}}}$ provided by Theorem \[thm:rejsam:secure\].
\[thm:generic-attack:against:R\] For every randomized algorithm $\operatorname{\mathsf{R}}$, there exists a generic algorithm substitution attack $\operatorname{\mathsf{ASA}}$ against $\operatorname{\mathsf{R}}$ such that $$\begin{array}{rcl}
\operatorname{\mathbf{InSec}}^{\operatorname{\operatorname{\mathsf{ASA}}}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa)&\leq&
\mathcal{O}(\operatorname{\mathsf{ml}}(\kappa)^{4}\cdot 2^{-{H_{\infty}}({\mathcal{C}}^{\kappa})})+\\
&&\mathcal{O}(\operatorname{\mathsf{ml}}(\kappa)^{2}\cdot
\exp(-s))+
\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}}}(\kappa),\\[2mm]
\operatorname{\mathbf{UnRel}}^{\star}_{\operatorname{\mathsf{S}},{\mathcal{C}}}(\kappa) &\leq&
2 \operatorname{\mathsf{ml}}(\kappa)^{2}\cdot
\exp(-2^{{H_{\infty}}({\mathcal{C}}^{\kappa})-3}) +\\
&& \operatorname{\mathsf{ml}}(\kappa)^{2}\cdot \exp(-2^{-2} s)+
\operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\operatorname{\mathsf{F}}}(\kappa)
\end{array}$$ where ${\mathcal{C}}:={\mathcal{C}}^{\kappa}_{\operatorname{\mathsf{R}}}({\operatorname{\mathsf{S}}\!.\!\operatorname{\mathsf{ol}}}(\kappa))$.
\[thm:no-attack:against:R\] For all deterministic algorithms $\operatorname{\mathsf{R}}$ and all algorithm substitution attacks $\operatorname{\mathsf{ASA}}$ against $\operatorname{\mathsf{R}}$: $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{asa}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{R}}}(\kappa) = 1.\end{aligned}$$
Theorem \[thm:generic-attack\] is thus just a consequence of Theorem \[thm:generic-attack:against:R\] and Theorem \[thm:min-entropy\] is just a consequence of Theorem \[thm:no-attack:against:R\].
These general results also imply several other results from the literature, for example on signature schemes. Ateniese et al. [@ateniese2015sig] study algorithm substitution attacks[^3] on *signature schemes* $\operatorname{\mathsf{SIG}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Sign}},\operatorname{\mathsf{Vrfy}})$, where
- The *key generator* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}}$ produces upon input $1^{\kappa}$ a pair $({\textit{pk}},{\textit{sk}})$ of keys with $|{\textit{pk}}|=|{\textit{sk}}|=\kappa$. We call ${\textit{pk}}$ the *public key* and ${\textit{sk}}$ the *secret key*.
- The *signing algorithm* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}}$ takes as input the secret key ${\textit{sk}}$ and a message $m\in \{0,1\}^{{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{ml}}}}$ of length ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ and produces a signature $\sigma\in \{0,1\}^{{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{sl}}}(\kappa)}$ of length ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{sl}}}(\kappa)$.
- The *verifying algorithm* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Vrfy}}}$ takes as input the public key ${\textit{pk}}$, the message $m$ and a signature $\sigma$ and outputs a bit $b$.
On the positive side (from the view of an algorithm substitution attack) they show that all randomized *coin-injective* schemes and all *coin-extractable* schemes have ASA. A randomized algorithm $A$ is *coin-injective*, if the function $f_{A}(x,\rho)=A(x;\rho)$ (where $\rho$ denotes the random coins used by $A$) is injective and *coin-extractable* if there is another randomized algorithm $B$ such that $\Pr[B(A(x;\rho))=\rho] \geq 1-\operatorname{\mathsf{negl}}$ for a negligible function $\operatorname{\mathsf{negl}}$. They prove the following theorems:
For every coin-injective signature scheme $\operatorname{\mathsf{SIG}}$, there is a successful algorithm substitution attack $\operatorname{\mathsf{ASA}}$ and a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{asa}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SIG}}}(\kappa)\leq \operatorname{\mathbf{InSec}}^{\operatorname{prf}}_{\mathsf{F}}(\kappa)+\operatorname{\mathsf{negl}}(\kappa)
\end{aligned}$$ for a pseudorandom function $\mathsf{F}$.
For every coin-extractable signature scheme $\operatorname{\mathsf{SIG}}$, there is a successful algorithm substitution attack $\operatorname{\mathsf{ASA}}$ and a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{asa}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SIG}}}(\kappa)\leq \operatorname{\mathsf{negl}}(\kappa).
\end{aligned}$$
Both of these results are easily implied by Theorem \[thm:generic-attack:against:R\].
On the negative side (from the view of an algorithm substitution attack), they show that *unique signature schemes* are resistant to ASAs fulfilling the *verifiability condition*. Informally this means that (a) each message has exactly on signature (for a fixed key-pair) and (b) each signature produced by the ASA must be valid.
For all unique signature schemes $\operatorname{\mathsf{SIG}}$ and all algorithm substitution attacks $\operatorname{\mathsf{ASA}}$ against them that fulfill the verifiability condition, there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
\operatorname{\mathbf{InSec}}^{\operatorname{asa}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SIG}}}(\kappa)\geq 1-\operatorname{\mathsf{negl}}(\kappa).
\end{aligned}$$
As unique signature schemes do not provide enough min-entropy for a stegosystem, this results follows from Theorem \[thm:min-entropy\].
A Lower Bound for Universal ASA {#sec:bound}
===============================
A setting similar to steganography, where *universal* stegosystems exist, that can be used for *any* channel of sufficiently large min-entropy, would be quite useful for attackers that plan to launch algorithm substitution attacks. Such a system would allow them to attack any symmetric encryption scheme *without* knowing the internal specification of the encryption algorithm. A closer look at the results in [@bellare2014asa; @bellare2015asa; @ateniese2015sig] reveals that their attacks do indeed go without internal knowledge of the used encryption algorithm. They only manipulate the random coins used in the encryption process. Note that ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m;r)$ (where $r$ denotes the random coins used by $\operatorname{\mathsf{Enc}}$) is a deterministic function, as ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}$ is a PPTM.
We thus define a *universal algorithm substitution attack* as a triple of PPTMs such that for every symmetric encryption scheme $\operatorname{\mathsf{SES}}$, the triple $$\begin{aligned}
\operatorname{\mathsf{ASA}}^{\operatorname{\mathsf{SES}}}=({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(\cdot,\cdot; \cdot)},{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Ext}}}) \end{aligned}$$ is an ASA against $\operatorname{\mathsf{SES}}$. Hence, ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$ has only oracle access to the encryption algorithm ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}$ of the encryption scheme: It may thus choose arbitrary values $k$, $m$, and $r$ and receives a ciphertext $$\begin{aligned}
c\gets {\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m; r) \end{aligned}$$ without having a complete description of the encryption schemes.
As noted above, all attacks in [@bellare2014asa; @bellare2015asa; @ateniese2015sig] are universal and @bellare2015asa explicitly state in their work [@bellare2015asa] that their ASA works against any encryption scheme of sufficiently large min-entropy. We also remark that the rejection sampling ASA presented earlier is universal.
For a universal algorithm substitution attack $\operatorname{\mathsf{ASA}}$ and a symmetric encryption scheme $\operatorname{\mathsf{SES}}$, let ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}},\kappa,{\textit{ak}},{\textit{am}},k,m_j,\sigma)$ be the expected number of oracle calls that a single call of the substitution encoder ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(\cdot,\cdot;\cdot)}({\textit{ak}},{\textit{am}},k,m_j,\sigma)$ makes to its encryption oracle ${\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}$. We then define $$\begin{aligned}
&{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}},\kappa)=\\
&\max_{\substack{{\textit{ak}}\in\operatorname{supp}({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})),\\ {\textit{am}}\in \{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)},\\ k\in \operatorname{supp}({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})),\\ m\in \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)},\\ \sigma\in \{0,1\}^{*}}} \{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}},\kappa,{\textit{ak}},{\textit{am}},k,m_j,\sigma)\}.\end{aligned}$$ For a family $\mathcal{F}$ of encryption schemes, let ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\mathcal{F},\kappa)$ be the maximal value of ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}},\kappa)$ for $\operatorname{\mathsf{SES}}\in \mathcal{F}$.
In the steganographic setting, @dedic2009upper showed in [@dedic2009upper] that (under the cryptographic assumption that one-way functions exist) no universal stegosystem can embed more than $\mathcal{O}(1)\cdot \log(\kappa)$ bits per document and thus proved that the rejection sampling based systems have optimal rate. The needed ingredients of this proof are summarized by two key lemmas based on Lemma 12 and Lemma 13 in [@Berndt2016].
\[lem:secure\_asa\] Let $\operatorname{\mathsf{ASA}}$ be a algorithm substitution attack for the symmetric encryption scheme $\operatorname{\mathsf{SES}}$ such that $\operatorname{\mathsf{ASA}}$ is secure against $\operatorname{\mathsf{SES}}$. Then for all integers $\kappa\in {\mathbb{N}}$, messages $m\in \{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, ciphertexts $c_{1},c_{2},\ldots,c_{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}\gets
{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}({\textit{ak}},{\textit{am}},k,m,\sigma)$ and all positions $i\in
\{1,\ldots,{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)\}$: $$\begin{aligned}
\Pr_{{\textit{ak}}\gets {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})}[c_{i}\not\in \operatorname{supp}({\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m))]
\leq \operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa).\end{aligned}$$
\[lem:reliable\_asa\] Let $\operatorname{\mathsf{ASA}}$ be a universal and reliable algorithm substitution attack against the symmetric encryption scheme $\operatorname{\mathsf{SES}}$. Then for every $\kappa$, the probability that the encoder ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$ produces a ciphertext, which was not provided by the encryption oracle, is at least $$\begin{aligned}
1-\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}}(\kappa)-\frac{({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)\cdot {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}},\kappa))^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}}{2^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}}.
\end{aligned}$$
We will now show how one can modify an existing symmetric encryption scheme $\operatorname{\mathsf{SES}}$ with the help of a signature scheme $\operatorname{\mathsf{SIG}}$ into a family of encryption schemes such that no universal ASA can achieve a super-logarithmic rate on all of these encryption schemes. The construction is very similar to the construction used in [@Berndt2016].
A *signature scheme* $\operatorname{\mathsf{SIG}}=({\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}},{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}},{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Vrfy}}})$ is a triple of probabilistic polynomial-time algorithms with the following properties:
- The *key generator* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}}$ produces upon input $1^{\kappa}$ a pair $({\textit{pk}},{\textit{sk}})$ of keys with $|{\textit{pk}}|=|{\textit{sk}}|=\kappa$. We call ${\textit{pk}}$ the *public key* and ${\textit{sk}}$ the *secret key*.
- The *signing algorithm* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}}$ takes as input the secret key ${\textit{sk}}$ and a message $m\in \{0,1\}^{{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{ml}}}}$ of length ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)$ and produces a signature $\sigma\in \{0,1\}^{{\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{sl}}}(\kappa)}$ of length ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{sl}}}(\kappa)$.
- The *verifying algorithm* ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Vrfy}}}$ takes as input the public key ${\textit{pk}}$, the message $m$ and a signature $\sigma$ and outputs a bit $b$.
We say that $(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Sign}},\operatorname{\mathsf{Vrfy}})$ is *reliable*, if $\operatorname{\mathsf{Vrfy}}({\textit{pk}},m,\operatorname{\mathsf{Sign}}({\textit{sk}},m))=1$ for all ${\textit{pk}}$, ${\textit{sk}}$ and $m$.
A *forger* $\operatorname{\mathsf{Fo}}$ is a probabilistic polynomial time algorithm that upon input ${\textit{pk}}$ and oracle access to $\operatorname{\mathsf{Sign}}_{{\textit{sk}}}$ tries to produce a pair $(m,\sigma)$ such that $\operatorname{\mathsf{Vrfy}}_{{\textit{pk}}}(m,\sigma)=1$. Formally, this is defined via the following experiment $\operatorname{\mathsf{Sig-Forge}}$:
Forger $\operatorname{\mathsf{Fo}}$, Signature Scheme $\operatorname{\mathsf{SIG}}=(\operatorname{\mathsf{Gen}},\operatorname{\mathsf{Sign}},\operatorname{\mathsf{Vrfy}})$ length $\kappa$
$({\textit{pk}},{\textit{sk}}) \gets \operatorname{\mathsf{Gen}}(1^{\kappa})$ $(m,\sigma) \gets \operatorname{\mathsf{Fo}}^{\operatorname{\mathsf{Sign}}_{{\textit{sk}}}}({\textit{pk}})$ Let $Q$ be the set of messages given to $\operatorname{\mathsf{Sign}}_{{\textit{sk}}}$ by $\operatorname{\mathsf{Fo}}$
A signature scheme $\operatorname{\mathsf{SIG}}$ is called *existentially unforgeable*, if for every forger $\operatorname{\mathsf{Fo}}$, there is a negligible function $\operatorname{\mathsf{negl}}$ such that $$\begin{aligned}
\operatorname{\mathbf{Adv}}^{\operatorname{sig}}_{\operatorname{\mathsf{Fo}},\operatorname{\mathsf{SIG}}}(\kappa) :=
\Pr[\operatorname{\mathsf{Sig-Forge}}_{\operatorname{\mathsf{Fo}},\operatorname{\mathsf{SIG}}}(\kappa)=1]\leq \operatorname{\mathsf{negl}}(\kappa).\end{aligned}$$
The maximal advantage of any forger against $\operatorname{\mathsf{SIG}}$ is called the *insecurity* of $\operatorname{\mathsf{SIG}}$ and is defined as $$\begin{aligned}
\operatorname{\mathbf{InSec}}_{\operatorname{\mathsf{SIG}}}^{\operatorname{sig}}(\kappa) := \max_{\operatorname{\mathsf{Fo}}}\{\operatorname{\mathbf{Adv}}^{\operatorname{sig}}_{\operatorname{\mathsf{Fo}},\operatorname{\mathsf{SIG}}}(\kappa)\}.\end{aligned}$$
For $({\textit{pk}},{\textit{sk}})\in
\operatorname{supp}({\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))$, let $\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}$ be the encryption scheme with
- ${\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Gen}}}={\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}$, [i.e.]{}the key generation algorithm remains the same.
- The encryption algorithm ${\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Enc}}}$ is given as:
key $k$, message $m$
$c \gets {\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m)$ $\gets {\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}}({\textit{sk}},c)$
- Similarly, the decryption algorithm ${\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Dec}}}$ is given as:
key $k$, ciphertext $(c,\sigma)$
By using this family $$\begin{aligned}
\mathcal{F}(\operatorname{\mathsf{SES}},\operatorname{\mathsf{SIG}})=\{\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\}_{({\textit{pk}},{\textit{sk}})\in \operatorname{supp}({\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))},\end{aligned}$$
we can derive the following upper bound on the rate of each universal ASA:
Let $\operatorname{\mathsf{SES}}$ be a symmetric encryption scheme, $\operatorname{\mathsf{SIG}}$ be a signature scheme and $\mathcal{F}=\mathcal{F}(\operatorname{\mathsf{SES}},\operatorname{\mathsf{SIG}})$ be defined as above. For every universal algorithm substitution attack $\operatorname{\mathsf{ASA}}$ against $\operatorname{\mathsf{SES}}$, there exist a forger $\operatorname{\mathsf{Fo}}$ on $\operatorname{\mathsf{SIG}}$ with advantage at least $$\begin{aligned}
1-\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\mathcal{F}}(\kappa)-\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\mathcal{F}}(\kappa)-\varphi(\operatorname{\mathsf{ASA}},\kappa)
\end{aligned}$$ for every $\kappa$, where $$\begin{aligned}
\varphi(\operatorname{\mathsf{ASA}},\kappa)=\frac{({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)\cdot {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\mathcal{F},\kappa))^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}}{2^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}}.
\end{aligned}$$
The proof is analogue to the proof of [@Berndt2016 Theorem 13].
Fix $\kappa\in {\mathbb{N}}$ and $({\textit{pk}},{\textit{sk}})\in \operatorname{supp}({\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa}))$. We will now construct an forger on $\operatorname{\mathsf{SIG}}$ with the help of the algorithm substitution attacker $\operatorname{\mathsf{ASA}}$. Choose a random attacker message ${\textit{am}}^{*}\gets
\{0,1\}^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$, a random attacker key ${\textit{ak}}^{*}\gets {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$, a random message $m^{*}\gets \{0,1\}^{{\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}$ and a random key $k^{*}\gets {\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Gen}}}(1^{\kappa})$.
The forger now simulates the run of the algorithm substitution attack ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{{\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Enc}}}(\cdot,\cdot;\cdot)}({\textit{ak}}^{*},{\textit{am}}^{*},k^{*},m^{*})$ against the symmetric encryption scheme $\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}$. Whenever ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$ makes an access $(k,m;r)$ to its encryption oracle, the forger computes $c={\operatorname{\mathsf{SES}}\!.\!\operatorname{\mathsf{Enc}}}(k,m;r)$ and uses its signing oracle ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}}_{{\textit{sk}}}$ upon $c$. This returns a valid signature $\sigma$ for $c$ and the forger returns $(c,\sigma)$ to ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$. This simulation hence yields the same result as ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{{\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Enc}}}(\cdot,\cdot;\cdot)}({\textit{ak}}^{*},{\textit{am}}^{*},k^{*},m^{*})$. Denote the first document produced by the run of the algorithm substitution attack ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}^{{\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Enc}}}(\cdot,\cdot;\cdot)}({\textit{ak}}^{*},{\textit{am}}^{*},k^{*},m^{*})$ as $(\widehat{c},\widehat{\sigma})$. By , the probability that the pair $(\widehat{c},\widehat{\sigma})$ does not belong to to the support $\operatorname{supp}({\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}\!.\!\operatorname{\mathsf{Enc}}}(k,m))$ ([i.e.]{}it is no valid ciphertext-signature pair) is bounded by $\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)$. Furthermore, implies that the probability that $(\widehat{c},\widehat{\sigma})$ is equal to any $(c,\sigma)$ which was given to the ASA is at most $\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)+\varphi(\operatorname{\mathsf{ASA}},\kappa)$. We can thus conclude that with probability $$\begin{aligned}
1-&\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)-\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)-\\
&\frac{({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)\cdot {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}},\kappa))^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}}{2^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}},
\end{aligned}$$ the ciphertext-signature pair $(\widehat{c},\widehat{\sigma})$ is a valid ciphertext-signature pair and was not produced by the oracle ${\operatorname{\mathsf{SIG}}\!.\!\operatorname{\mathsf{Sign}}}_{{\textit{sk}}}$ The advantage of the forger against the signature scheme $\operatorname{\mathsf{SIG}}$ is thus at least $$\begin{aligned}
1-&\operatorname{\mathbf{InSec}}^{\operatorname{enc-watch}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)-\operatorname{\mathbf{UnRel}}_{\operatorname{\mathsf{ASA}},\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}}}(\kappa)-\\
&\frac{({\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)\cdot {\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{query}}}(\operatorname{\mathsf{SES}}_{{\textit{pk}},{\textit{sk}}},\kappa))^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ol}}}(\kappa)}}{2^{{\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{ml}}}(\kappa)}},
\end{aligned}$$ The running time of the forger is polynomial in $\kappa$ due to the polynomial running time of ${\operatorname{\mathsf{ASA}}\!.\!\operatorname{\mathsf{Enc}}}$.
This allows us to conclude the following corollary bounding the number of bits embeddable into a single ciphertext by a universal algorithm substitution attack.
\[cor:bound\] There is no universal algorithm substitution attack that embeds more than $\mathcal{O}(1)\cdot \log(\kappa)$ bits per ciphertext (unless one-way functions do not exist).
Conclusions
===========
In this work, we proved that ASAs in the strong undetectability model of Bellare, Jaeger and Kane [@bellare2015asa] are a special case of stegosystems on a certain kind of channels described by symmetric encryption schemes. This gives a rigorous proof of the well-known connection between steganography and algorithm substitution attacks. We make use of this relationship to show that a wide range of results on ASAs are already present in the steganographic literature. Inspired by this connection, we define *universal ASAs* that work with no knowledge on the internal implementation of the symmetric encryption schemes and thus work for *all* such encryption schemes with sufficiently large min-entropy. As almost all known ASAs are universal, we investigate their rate – the number of embedded bits per ciphertext – and prove a logarithmic upper bound of this rate.
[00]{}
\#1 \#1[[DOI:]{}0[\#1]{} ]{} \#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{} \#1[\#1]{}
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[^1]: <https://www.openssl.org/>
[^2]: In [@bellare2015asa], this is called the *key recovery security*.
[^3]: To be more precise, their attacks only replace the signing algorithm $\operatorname{\mathsf{Sign}}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A neutron scattering investigation of the magnetoelectric coupling in PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$ (PFN) has been undertaken. Ferroelectric order occurs below 400 K, as evidenced by the softening with temperature and subsequent recovery of the zone center transverse optic phonon mode energy ($\hbar \Omega_{0}$). Over the same temperature range, magnetic correlations become resolution limited on a terahertz energy scale. In contrast to the behavior of nonmagnetic disordered ferroelectrics (namely Pb(Mg,Zn)$_{1/3}$Nb$_{2/3}$O$_{3}$), we report the observation of a strong deviation from linearity in the temperature dependence of $(\hbar \Omega_{0})^{2}$. This deviation is compensated by a corresponding change in the energy scale of the magnetic excitations, as probed through the first moment of the inelastic response. The coupling between the short-range ferroelectric and antiferromagnetic correlations is consistent with calculations showing that the ferroelectricity is driven by the displacement of the body centered iron site, illustrating the multiferroic nature of magnetic lead based relaxors in the dynamical regime.'
author:
- 'C. Stock'
- 'S.R. Dunsiger'
- 'R. A. Mole'
- 'X. Li'
- 'H. Luo'
title: 'Coupled magnetic and ferroelectric excitations in PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$'
---
*Introduction* – Creating materials with a strong coupling between ferroelectric and magnetic order has been a central goal of research in $3d$ transition metal compounds [@Cheong07:6; @Kimura03:426; @Scott12:222]. Fundamentally, this phenomenon is startling, since typically ferroelectricity results from relative shifts of negative and positive ions with closed electronic configurations, while magnetism is related to ordering of spins of electrons in incomplete ionic shells. While multiferroicity has now been reported in a variety of compounds, as typified by compounds like BiFeO$_{3}$ and BiMnO$_{3}$, coupling between the magnetic and polar orders is often weak and occurs on widely disparate temperature scales [@Santos02:122]. Magnon-phonon interactions are typically also very weak in improper multiferroics like TbMnO$_{3}$, where only tiny shifts of $\sim$ 0.1 meV in phonon frequencies due to magnetic ordering have been measured [@Rovillain10:81]. By contrast, we demonstrate a strong intrinsic coupling in a disordered ferroelectric between short range ferroelectric and magnetic orders by measuring the magnetic and lattice fluctuations. Recently, the multiferroic nature of disordered systems has attracted considerable attention and the correlation between short range ferroelectric and antiferromagnetic orders has been reported for several different systems [@r1; @r2; @r3]. For example, in the magnetic relaxor ${(1-x)}$BiFeO$_{3}$-$x$BaTiO$_3$, elastic neutron diffraction studies suggest the polar nanoregions are identical to short range magnetic nanodomains [@r3].
Some of the most studied nonmagnetic ferroelectric compounds are the lead based relaxors with the chemical form PbBO$_{3}$, where the B site is a mixture between two different ions [@Park97:82; @Ye98:81; @Bokov06:41]. PbMg$_{1/3}$Nb$_{2/3}$O$_{3}$ (PMN) and PbZn$_{1/3}$Nb$_{2/3}$O$_{3}$ (PZN) are prototypical relaxors in which the random occupancy of the B site results in a suppression of a classical ferroelectric phase transition, replaced by short range ferroelectric order. These polar nanoregions manifest as a strong temperature dependent neutron and x-ray diffuse scattering cross-section which has a significant dynamic component. The latter component tracks the frequency dependence of the dielectric constant [@Stock10:81; @Hirota02:65; @Xu04:70; @Stock07:76; @Gehring09:79]. A dynamic signature of ferroelectricity in these compounds is a soft and transverse optic phonon mode located at the Brillouin zone center. The dielectric constant is related to the soft phonon energy ($\hbar \Omega_{0}$) via the Lyndane-Sachs-Teller relation- $1/\epsilon \propto (\hbar \Omega_{0})^{2}$ [@Lyddane41:59]. PbTiO$_{3}$ displays a conventional soft ferroelectric mode, where $(\hbar \Omega)^{2}$ softens to a minimum energy at the structural transition and then recovers linearly below the phase transition [@Shirane70:2; @Kempa06:79]. Even though a long-range ferroelectric ground state is absent in relaxors PMN and PZN, a soft zone center transverse optic mode is present and recovers at the same temperature where static short range polar correlations are onset [@Wakimoto02:65; @Stock04:69; @Cowley11:60].
The disordered compound PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$ (PFN) is also based on a perovskite structure with the B site a mixture of nonmagnetic Nb$^{5+}$ and magnetic Fe$^{3+}$ ($d^{5}$, $S$=5/2) ions. Two structural transitions are reported, the cubic unit cell transforming to tetragonal ($a/c=0.9985$) at 376 K, followed by a monoclinic structure ($\beta$=89.94$^{\circ}$) below 355 K [@Lampis99:11]. The dielectric constant is peaked near 375 K, however is broad in temperature as well as frequency dependent, indicative of short range ferroelectric order [@Majumder06:99]. A weak cusp at $\sim 150$ K and history dependence below $\sim 20$ K has been observed in measurements of the bulk magnetization [@bokov; @blinc; @bhat]. Neutron diffraction measurements have observed short-range three dimensional antiferromagnetic correlations peaked at a wave vector $\vec{Q}$=(1/2,1/2,1/2) coexisting with a magnetic Bragg peak indicative of a long-range magnetically ordered component. The presence of two distinct lineshapes for the magnetic diffraction has led to models involving two phases defined by different Fe$^{3+}$ clustering sizes [@Kleemann10:105].
Motivated by dielectric measurements suggestive of a coupling between the magnetic and ferroelectric orders (Ref. ) we present a neutron scattering study of the lattice and magnetic fluctuations in PFN performed at the PUMA spectrometer (FRM2 reactor) on a 1 cm$^{3}$ sample. We show that the temperature dependent lattice dynamics are dominated by a soft transverse optic mode $(\hbar\Omega)$ measured near the nuclear zone center. Although $(\hbar\Omega)^{2}$ recovers, it deviates from the linear response observed in other ferroelectrics (namely PbTiO$_{3}$, PMN, and PZN). An investigation of the first moment of the magnetic inelastic response suggests that this is compensated by a corresponding change in the energy scale of magnetic excitations, indicative of magnetoelectric coupling [*in the dynamical regime*]{}. These measurements are complementary to Raman scattering studies [@correa; @garcia], where the zone center phonon response is also explored, typically at higher characteristic energies. While Correa [*et al.*]{} [@correa] report anomalous shifts with sublattice magnetization of the Fe-O phonon mode frequency centered around 87 meV, the shifts are very small ($<3$ cm$^{-1}$ or 0.37 meV). The so called $F_{2g}$ mode associated with Pb localization around 65 cm$^{-1}$ (8 meV) is completely unaffected by the antiferromagnetic transition observed in their sample. We show that the magnetoelastic effects observed using inelastic neutron scattering in the low energy regime are much more dramatic. More generally, using Raman techniques, the assignment of phonon modes in the cubic state of the relaxor ferroelectric is controversial [@correa], making neutron scattering invaluable.
![\[mag\_elastic\] a) the elastic ($\hbar \omega$=0) magnetic intensity as a function of temperature at $\vec{Q}$=(1/2,1/2,1/2). b) the inverse of the correlation length as a function of temperature derived from the constant energy scans displayed in panel e). The static magnetization is shown in panels $c)$ and $d)$ under the application of a 100 Gauss field along the \[100\] axis.](mag_elastic.eps){width="8.2cm"}
$\textit{Static magnetic properties}$ – Figure \[mag\_elastic\] plots the static magnetic properties of PFN measured through the use of elastic neutron scattering (with an energy resolution $\delta E$=1.25 meV=0.30 THz) and static bulk magnetization. Panel $a)$ shows a plot of the elastic magnetic intensity as a function of temperature at $\vec{Q}$=(1/2,1/2,1/2), located at the peak of the magnetic intensity (see panel $e$). It shows only a smooth growth of intensity and no anomaly indicative of a well defined phase transition. The increase in elastic magnetic intensity is mimicked by the static magnetization (panel $c)$ which shows a deviation from Curie-Weiss behavior over a similar temperature range. The combination of the half-integer position in momentum and its broad lineshape indicates the origin of the scattering is from clusters of Fe$^{3+}$ ions. A Lorentzian squared lineshape was used to model the data, motivated by random fields [@Birgeneau83:28]. Such random fields are thought to be the underlying cause of the avoided long range magnetic ordering, which would be characterized by a Bragg peak. Similar ideas have been proposed to understand the ferroelectric properties of disordered PMN and PZN [@Westphal92:68; @Fisch03:67]. Given the large vertical resolution on the PUMA spectrometer (Ref. ), we integrated over the vertical direction analytically, giving the ${3/2}$ power to the momentum dependence shown below. The magnetic correlation lengths (panel $b$) were thus extracted from fits to a resolution convolution of the following lineshape (examples illustrated in $e$): $$\begin{aligned}
I(\vec{Q})=C {(\gamma r_{0})^2 \over 4} g^{2} f^2(Q) m^{2} e^{-\langle u^{2} \rangle Q^{2}} {V^{*} {\xi^{3}/\pi^{2}} \over [1+(|\vec{Q}-\vec{Q}_{0}|\xi)^{2}]^{3/2}},
\nonumber\end{aligned}$$ where $C$ is the calibration constant, $(\gamma r_{0})^{2}/4$ is 73 mbarns sr$^{-1}$, $g=2$ is the Land[é]{} factor, $f^{2}(Q)$ is the Fe$^{3+}$ magnetic form factor, $m$ is the magnetic moment size, $V^{*}$ is the volume of the Brillouin zone, $\xi$ the correlation length, $e^{-\langle u^{2} \rangle Q^{2}}$ is the Debye-Waller factor, and $a$ is the lattice constant. Performing scans along all high symmetry directions (\[111\], \[110\], and \[001\]), the magnetic correlation length is found to be spatially isotropic within error. As demonstrated in Fig. \[mag\_elastic\] $b)$, this quantity never diverges, but saturates at the small value of $\xi$=17Å with an inflection point around $\sim$ 80 K. The inflection point in the temperature dependent correlation length is mimicked by the static magnetization presented in Fig. \[mag\_elastic\] $d)$, albeit at much lower temperatures of $\sim$ 25 K. The divergence between field-cooled (FC) and zero-field cooled (ZFC) magnetization reflects the development of a spin-glass-like state at low temperatures and is close to the $\sim$ 20 K anomaly observed using muon spin relaxation [@Rotaru09:79]. While there is apparent sample dependence evidenced by the different values quoted in the literature for this divergence in the magnetization (27.6 K [@Kumar08:93],$\sim$20 K [@Rotaru09:79], and 10.6 K [@Kleemann10:105]), as pointed out in Ref. , the exact value may be strongly technique and field dependent. The differing onset temperatures of spin glass-like ordering produced by different techniques (neutrons - 80 K compared to static magnetization - 25 K) which probe different timescales is similar to results published on canonical spin glasses and frustrated magnets, where the magnetic correlations are dominated by slow fluctuations over a broad frequency range, sampled with differing energy resolutions [@Stock10:105; @Murani78:41].
The high temperature fit of the static magnetization yields a Curie constant of $k_{B}\Theta_{CW}$=-240 $\pm$ 10 K which is a measure of the average Fe$^{3+}$-Fe$^{3+}$ exchange coupling. The negative sign indicates antiferromagnetic coupling between the spins. Within the mean-field approximation $k_{B}\Theta_{CW}=-{1\over3} z J S(S+1)$ (with $z$=3 neighbors and $S={5\over2}$) gives an estimate of the average $J=2.4 \pm 0.4$ meV $\sim$ 28 K. This implies a magnetic band width of the order of 10 meV ($\sim 2SJ$). The energy scale of the magnetic coupling is consistent with the presence of strong low energy magnetic spectral weight at high temperatures, within the spectrometer resolution (Fig. \[mag\_elastic\]$e$).
![\[phonon\] a) the phonon dispersion for both the T$_{1}$ and T$_{2}$ modes. b)-d) representative constant-Q scans taken at $\vec{Q}$=(2.15, 1.85, 0) e) constant-Q scans at the zone center at 150 K (filled circles) and 3 K (open circles) f) the frequency squared of the soft transverse optic mode as a function of wavevector squared near the zone center.](figure_phonons.eps){width="8.2cm"}
$\textit{Ferroelectric properties}$ – The lattice dynamics are similar in structure and energy scale to PMN and PbTiO$_{3}$ [@Swainson09:79; @Tomeno06:73; @Hlinka06:73]. The phonons are described by a transverse optic mode which is gapped at the nuclear zone center and a lower energy acoustic mode which is gapless. The dispersion near the nuclear zone center is shown in Fig. \[phonon\] $a)$ and illustrative constant-$\vec{Q}$ scans are shown in panels $b-d)$. The constant-$Q$ scans show there is little change in frequency in the low-energy acoustic mode. However, the higher energy optic mode gradually hardens as the temperature is decreased, as expected for the recovery from a structural transition. In a similar manner to the case of PMN, we observed the optic mode is over damped in energy near the nuclear zone center, plotted in panel $e)$, where a very broad and unresolvable peak was observed at 150 K. A more well defined peak at $\sim$ 11 meV is observed at 3 K [@Stock12:86]. To extract the zone center soft mode energy, we rely on the optic mode frequencies at finite $q$ away from the zone center and fit the results to $(\hbar \Omega(q))^{2}=(\hbar \Omega_{\circ})^{2}+\alpha q^{2}$. $\Omega_{\circ}$ and $\alpha$ are respectively the optic mode frequency at the zone center and a temperature independent measure of the curvature near the zone center. Representative results from this analysis are plotted in panel $f)$ and where the zone center frequency could be measured, good agreement was observed. This method has been applied before to the relaxors and found to be in good agreement with zone center scans, as well as Raman data [@Cao08:78; @Shirane70:2].
$\textit{Magnetic dynamics}$ – The temperature dependent magnetic dynamics are illustrated in Fig. \[mag\_inelastic\] through constant $\vec{Q}$=(1/2,1/2,1/2) scans (panels $a-c$) and constant $\hbar\omega$= 2 meV scans (panels $d-f$). The fluctuations are dominated by a peak at $\vec{Q}$=(1/2, 1/2, 1/2) which is both momentum and energy broadened, characteristic of short range and glass-like dynamics. The dynamic magnetic response is not dispersive and is overdamped for all temperatures investigated. Constant $\vec{Q}$=(1/2,1/2,1/2) scans were fit to a resolution convolved damped harmonic oscillator as described in the supplementary information.
![\[mag\_inelastic\] $a-c)$ are constant-$\vec{Q}$=(1/2,1/2,1/2) scans at several temperatures. The solid curves represent a fit to the simple harmonic oscillator described in the text. $d-f)$ are constant $\hbar \omega$=2 meV scans at the sample temperatures. The resolution full-widths are represented by the solid bars.](figure_mag_fluctuations.eps){width="9.0cm"}
![\[param\] $a)$ the soft transverse optic phonon frequency squared as a function of temperature for both PFN and PMN. $b)$ and $c)$ illustrates the magnetic fitting parameters described in the text. $b)$ shows the line width as a function of temperature and $c)$ the susceptibility. $d)$ the zeroeth moment for the static and dynamic components on the THz timescale. $e)$ shows the change in the first moment with temperature.](param_figure.eps){width="8.0cm"}
The temperature variation of $(\hbar\Omega_{0})^{2}$ (proportional to $1/\epsilon$), the timescale of the magnetic fluctuations ($2\Gamma \sim 1/\tau$), and the susceptibility ($\chi_{0}$) at $\vec{Q}$=(1/2, 1/2, 1/2) are illustrated in Fig. \[param\] $a-c)$. The energy of the soft optic mode in PMN is also plotted in Fig. \[param\] $a)$, showing a linear recovery down to low temperatures [@Stock05:74; @Wakimoto02:65]. For comparison, the data from PMN has been scaled by a factor of 1.4 to agree with PFN at 400 K. While at temperatures above $\sim$ 100 K the zone center energy tracks the measured response in prototypical relaxors (such as PMN), a significant deviation from the linear recovery is observed at low temperatures.
The linewidth of the magnetic fluctuations measured at $\vec{Q}$=(1/2,1/2,1/2) is plotted in Fig. \[param\] $b)$ and illustrates a linear decrease towards saturation at $\sim$ 80 K. The dashed line is given by $2\Gamma=k_{B}T$, demonstrating that the energy scale of the magnetic fluctuations at high temperatures is set by $k_{B}T$. Figure \[param\] $c)$ illustrates the susceptibility $\chi_{0}$ derived from fits shown in Fig. \[mag\_inelastic\]. The dashed line is a high temperature fit to $1/(T-\Lambda)$ with a deviation at $\sim$ 80 K - the same temperature as the inflection point in the elastic correlation length (Fig. \[mag\_elastic\] $b$) and the saturation of the inelastic linewidth (Fig. \[param\] $b$).
Given the broad nature of the magnetic response in momentum and energy of our data and the ambiguity over its functional form, it is important to ensure all the spectral weight is accounted for. In general, the total integrated magnetic intensity over all momentum and energy transfers is a conserved quantity satisfying the zeroeth moment sum rule. We calculated the zeroeth moment sum (Fig. \[param\] $d$) by integrating the magnetic intensity over the $Q$ and $E$ range presented in Fig.\[mag\_inelastic\]. The data were normalized against the known cross section of an acoustic phonon. The dashed line is the expected value based on a $S={5\over2}$ moment for Fe$^{3+}$, indicating the experiment indeed probed all of the magnetic scattering. The filled points are the total spectral weight summing over the elastic and inelastic channels, while the open circles represent the inelastic component. The difference at low temperatures corresponds to an estimate of the total spectral weight measured to be static, or within resolution limits (0.3 THz). The decrease of the spectral weight with increasing temperature indicates the magnetic response extends to higher energies not directly probed.
The deviation of $(\hbar\Omega)^{2}$ from linearity (Fig. \[param\]$a$) differs from nonmagnetic counterparts PMN and PZN and is suggestive of a coupling to another energy scale. Given the magnetic response is strongly damped (Fig. \[mag\_inelastic\] ), we have investigated the magnetic energy scale through a study of the first moment $\langle \mathcal{H} \rangle$ calculated from the Hohenberg Brinkman (Ref. ) sum rule. The sum rule is a general result for isotropic correlations and is applicable to the case where no sharp peak is observed in a constant $Q$ scan: $$\begin{aligned}
\label{first_moment} \langle \mathcal{H} \rangle=-{3\over4}{{\int_{-\infty}^{\infty}dE (E\chi''(Q,E))}\over{(1-\cos(\vec{Q}\cdot\vec{d}))}}.
\nonumber\end{aligned}$$ The integration was performed at $\vec{Q}$=(1/2,1/2,1/2) and $d$ is the distance between nearest neighbors. The change in the first moment with temperature is plotted in Fig. \[param\] $e)$, showing a substantial reduction with decreasing temperature. The change is of order the expected change in energy between the soft optic phonon in PFN and PMN, indicating a coupling between the two orders.
$\textit{Summary and Conclusions}$ – Our results illustrate that in the presence of short-range magnetic order, the soft phonon dynamics are directly altered from the linear recovery in $(\hbar \Omega)^{2}$ observed in classic and disordered ferroelectrics. The first moment, a measure of the magnetic energy scale, illustrates the change in energy is taken up by the magnetic spin terms demonstrating coupling between the two orders. Such a coupling may be expected given the local bonding environment in PFN. The ferroelectric order in compounds of the form ABO$_{3}$ has been found to be determined by the condensation of predominately the Last and Slater phonon modes, with significant contributions from shifts in the A and O sites [@Harada70:A26]. In the fully ordered case of PFN, the exchange interaction between two Fe$^{3+}$ ions would involve orbitals from Pb and also O. Therefore, the hybridization associated with ferroelectric order would be expected to alter the exchange pathways coupling magnetic ions. Such a scenario has been suggested to exist in fully site ordered EuTiO$_{3}$ [@Katsufuji01:64]. An alternate explanation is proposed as a result of calculations (Ref. ) which suggest that the ferroelectricity in PFN predominately originates from the displacement of the Fe$^{3+}$ site, the ratio of the displacement of Fe$^{3+}$ to Nb$^{5+}$ being greater than 10. Such a large difference in displacement could provide a route for explaining the strong coupling between the two orders observed here.
Strong magnetoelectric coupling appears to be favored in disordered systems where the symmetry contraints of the lattice are relaxed, as in compounds like pervoskite (Sr,Mn)TiO$_{3}$ and nonperovskite (Ni,Mn)TiO$_{3}$ [@r1; @r2]. However, enhanced dielectric constants have been reported previously in a number of candidate materials for multiferroicity, which were later shown to arise from nanoscale disorder [@zhu07; @ruff12]. Using a microscopic technique robust against such extrinsic effects, we have found evidence of coupling between the short range magnetic and ferroelectric orders in the archetypal ferroelectric PFN. The temperature dependence of $(\hbar \Omega_{0})^{2}$ associated with the soft transverse optic mode, sensitive to ferroelectric correlations, deviates strongly from the linear recovery observed in classic ferroelectrics as well as prototypical nonmagnetic relaxors Pb(Mg,Zn)$_{1/3}$Nb$_{2/3}$O$_{3}$. We are grateful for funding from EU-NMI3, the Carnegie Trust for the Universities of Scotland, STFC, and Deutsche Forschungsgemeinschaft Grant TRR 80.
Appendix
========
Here we present supplementary information regarding the experimental details, spectrometer calibration, and sum rules used in the main text. Supplementary data regarding the momentum dependence of the magnetic scattering are also presented. The data demonstrate the absence of well-defined spin waves and show that the magnetic excitations are represented by strongly overdamped fluctuations characteristic of the short range magnetic order.
Experimental details for the neutron scattering and susceptibility measurements
-------------------------------------------------------------------------------
Neutron scattering measurements were performed on the PUMA thermal triple-axis spectrometer located at the FRM2 reactor (Garching, Germany). Two sets of measurements were performed with the 1 cm$^{3}$ (with lattice constant $a$=4.01 Å) sample to measure both the magnetic and lattice fluctuations. To measure the phonon dispersion curves and temperature dependence, the sample was oriented in the (HK0) scattering plane. The magnetic scattering was investigated with the sample mounted in the (HHL) plane. In both sets of measurements, the sample was cooled in a closed cycle refrigerator. A PG(002) vertically focused monochromator was used to select an incident energy E$_{i}$ and the final energy was fixed at E$_{f}$=14.8 meV using a PG(002) flat analyzer crystal. The energy transfer was then defined as $\hbar \omega$=E$_{i}$-E$_{f}$. To measure the phonon curves, it was desirable to obtain a high count rate at the expense of momentum resolution and therefore horizontal focusing was used on both the monochromator and the analyser. The horizontal focusing, on both the incident and scattered sides was removed for studies of the static and fluctuating magnetic response. Higher order contamination was reduced through the use of a pyrolytic graphite filter in the scattered beam. The counting time was determined by a low efficiency monitor placed in the incident beam and was corrected for variable contamination by higher order scattering from the monochromator using the same calibration described elsewhere. [@Stock04:69]
Magnetization measurements were performed using a Quantum Design Materials Properties Measurement System (MPMS) on a small 7.0 mg piece of PFN taken from the same crystal growth batch. A field of 100 Gauss was applied along the $a$ axis and measurements under field cooled (FC) and zero-field cooled conditions were performed.
Spectrometer calibration constant derived from acoustic phonons
---------------------------------------------------------------
To calculate the zeroeth and first moments of the magnetic scattering, we have put the magnetic intensities on an absolute scale. The calibration constant for the experiment was obtained by measuring a low-energy acoustic phonon. In the long-wavelength (low $q$) limit, it can be assumed that we are in the hydrodynamic regime where only the center of mass is moving and the structure factor for the acoustic phonon will match that of the nearby nuclear Bragg peak. In the setup used on PUMA with an incident beam monitor with an efficiency $\propto {1\over{\sqrt{E}}}$, the measured energy integrated intensity takes the form $$\begin{aligned}
I(\vec{Q})=A\left({\hbar \over {2 \omega_{0}} } \right) [1+n(\omega_{0})]|F_{N}|^{2} {{Q^{2} \cos^{2}(\beta)} \over {M}}e^{-2W}.\end{aligned}$$ where $A$ is the spectrometer calibration constant, $\hbar \omega_{0}$ is the acoustic phonon frequency, $[1+n(\omega_{0})]$ is the Bose factor, $|F_{N}|^{2}$ the structure factor of the nuclear Bragg peak, $M$ the mass of the unit cell, and $e^{-2W} \sim 1$ is the Debye Waller factor. For the measured magnetic scattering in the (HHL) scattering plane, we have used an acoustic phonon measured at $\vec{Q}$=(0.15,0.15,2) and T=300 K as a reference with a horizontally flat monochromator and analyzer.
Lineshape- inelastic magnetic response
--------------------------------------
To describe the broad overdamped lineshape characterizing the magnetic dynamics, a relaxational form determined by a single energy scale $\Gamma\propto1/\tau$ described by $\chi''(Q,E)\propto \chi_{0}(Q)E\Gamma/(E^{2}+\Gamma^{2})$ was initially fit to the constant $Q$ scans shown in Fig. 3 of the main text. Using this form for the magnetic dynamics, $\chi_{0}$ is related to the real part of the susceptibility. While this line shape described the data well at high temperatures, it failed to fit the response below $\sim$ 100 K. To correct this, following Ref. , we fit the following damped harmonic oscillator lineshape to all temperatures. $$\begin{aligned}
\label{SHO} S(E)= {\chi_{0}} [n(E)+1] \times \\
\left( {{1}\over {[1+
{{\left(E- E_{0}\right)^{2}}\over {\Gamma^{2}}} }]} - {{1}\over {[1+
{{\left(E+ E_{0}\right)^{2}}\over {\Gamma^{2}}} }]} \right),
\nonumber\end{aligned}$$ where $\chi_{0}$ is a measure of the strength of the magnetic scattering, $[n(E)+1]$ is the thermal population (or Bose) factor, $\hbar \Omega$ the mode position, and $\Gamma$ is the half-width. To account for elastic scattering from static (defined by our resolution width) correlations, we have included a Gaussian in the fit centered at the elastic energy position. The parameter E$_{0}$ was found to be temperature independent with E$_{0}$=0.5 $\pm$ 0.2 meV and can be physically interpreted as a magnetic anisotropy energy scale.
![\[dispersion\] The momentum dependence of the scattering at T=3 K. $a-b)$ illustrate constant energy scans at E=6 and 2 meV. $c-e)$ show constant-Q scans taken at positions close to $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$)](dispersion.eps){width="8.2cm"}
Zeroeth moment sum rule for magnetic scattering
-----------------------------------------------
The total integrated magnetic intensity over all momentum and energy transfers is a conserved quantity satisfying the zeroeth moment sum rule. Accounting for the orientation factor in magnetic neutron scattering and the fact that there is ${1 \over 2}$ a Fe$^{3+}$ site per unit cell, the integral over $S(\vec{Q},E)$ is, $$\begin{aligned}
\langle \mu_{eff}^{2} \rangle = \int dE \int d^{3}Q S(\vec{Q},E)=...\nonumber \\
{1\over \pi} \int dE \int d^{3}Q [1+n(E)]\chi''(\vec{Q},E)=... \nonumber \\
{2 \over 3} g^{2} \mu_{B}^{2} S(S+1) \times {1 \over 2}.\end{aligned}$$ Setting $S={5 \over 2}$ gives a total expected integral of 11.7 $\mu_{B}^{2}$. This is in agreement with the total integrated moment discussed in the main text.
$\vec{Q}$-E dependence
----------------------
The zeroeth and first moment analysis outlined in the main text relies on a knowledge of the momentum dependence of the magnetic scattering with energy transfer. Fig. \[dispersion\] illustrates the momentum dependence of the magnetic scattering through both constant energy (panels $a-b)$ and also constant-Q scans (panels $c-e)$). The constant energy scans have been fitted to a Gaussian centered at $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$) and show only a single central peak. The constant-Q scans also do not display any sign of spin waves or dispersion of the magnetic excitations. The first moment analysis has used the fact that the excitations are peaked only near $\vec{Q}$=(${1\over 2}, {1\over 2}, {1\over 2}$) and this is substantiated by the results.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein-Uhlenbeck semigroup ${\mathrm{e}}^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and applying the powers $L^N$ of the Ornstein-Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by [@Portal2014], making all relevant quantities explicit.'
address: |
Delft Institute of Applied Mathematics, Delft University of Technology\
P.O. Box 5031\
2600 GA Delft\
The Netherlands\
[email protected]
author:
- Jonas Teuwen
bibliography:
- 'library.bib'
title: 'On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroup'
---
Introduction
============
Much effort [@Harboure2000; @Kemppainen2015; @Maas2010b; @MaasNeervenPortal2011; @MauceriMeda2007; @MauceriMeda2012; @Muckenhaupt; @Pineda2008; @Portal2014; @Sjogren1997; @Teuwen2015] has gone into developing the harmonic analysis of the [*Ornstein-Uhlenbeck operator*]{} $$\label{eq:OU-operator}
L := \frac12 \Delta - \langle x, \nabla \rangle.$$ On the space $L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)$, where $\gamma$ is the Gaussian measure $$\label{eq:Gaussian-measure}
{\mathrm{d}}\gamma(x) := \pi^{-d/2} {\mathrm{e}}^{-|x|^2} {\mathrm{d}}{x},$$ this operator can be viewed as the Gaussian counterpart of the Laplace operator $\Delta$. Indeed, one has $L = -\nabla^*\nabla$, where $\nabla$ is the the usual gradient and the $\nabla^*$ is its adjoint in $L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)$. It is a classical fact that the semigroup operators $e^{tL}$, $t>0$, are integral operators of the form $${\mathrm{e}}^{tL} u(\cdot) = \int_{{\mathbf{R}}^d} M_t(\cdot, y) u(y) {\,\mathrm{d}}\gamma(y),$$ where $M_t$ is the so-called [*Mehler kernel*]{} [@Sjogren1997] $$\label{eq:Mehler-kernel-oneformula}
\begin{aligned}
M_t(x, y)
& = \frac{\exp\Bigl(\displaystyle-\frac{|{\mathrm{e}}^{-t}x - y|^2}{1 - {\mathrm{e}}^{-2t}}
\Bigr)}{(1 - {\mathrm{e}}^{-2t})^{d/2}} {\mathrm{e}}^{|y|^2}
\end{aligned}$$ (see [@Teuwen2015] for a representation of $M_t$ which makes the symmetry in $x$ and $y$ explicit).
Developing a Hardy space theory for $L$ is the subject of active current research [@Portal2014; @MauceriMeda2007]. In this theory the derivatives $(d^k/dt^k)e^{t L} = L^N e^{tL}$ play an important role. The aim of the present paper is to derive closed form expressions for the integral kernels of these derivatives, that is, to determine explicitly the kernels $M_t^N$ such that we have the identity $$\label{eq:Mehler}
L^N{\mathrm{e}}^{tL} u(\cdot) = \int_{{\mathbf{R}}^d} M_t^N(\cdot, y) u(y) {\,\mathrm{d}}\gamma(y).$$ Direct application of the derivatives $d^N/dt^N$ to the Mehler kernel yields expressions which become intractible even for small values of $N$. Our approach will be to expand the Mehler kernel in terms of the $L^2$-normalised Hermite polynomials and then to apply $L^N$ to it, thus exploiting the fact that the Hermite polynomials are eigenfunctions for $L$.
As an application of our main result, which is proved in section \[sec:mainresult\] after developing some preliminary material in the sections \[sec:prelim\]-\[sec:Weyl\], we shall give a direct proof for the kernel bounds of [@Portal2014] in section \[sec:application\].
Preliminaries {#sec:prelim}
=============
In this preliminary section we collect some standard properties of Hermite polynomials and their relationship with the Ornstein-Uhlenbeck operator. Most of this material is classical and can be found in [@Sjogren1997; @MR1215939].
Hermite polynomials
-------------------
The [*Hermite polynomials*]{} $H_n$, $n\ge 0$, are defined by Rodrigues’s formula $$\label{eq:Hermite-Rodrigues}
H_n(x) := (-1)^n {\mathrm{e}}^{x^2} \partial_x^n {\mathrm{e}}^{-x^2}.$$ Their $L^2$-normalizations, $$h_n := \frac{H_n}{\sqrt{2^{n} n!}}$$ form an orthonormal basis for $L^2({\mathbf{R}},{\mathrm{d}}\gamma)$. We shall use the fact that the generating function for the Hermite polynomials is given by $$\label{eq:Generating-function-identity} \sum_{n = 0}^\infty \frac{H_n(x)}{n!} t^n ={\mathrm{e}}^{2 tx - t^2}.$$ The relationship with the Ornstein-Uhlenbeck operator is encoded in the eigenvalue identity $L H_n = -n H_n$, from which it follows that for all $t \ge 0$ we have ${\mathrm{e}}^{tL} H_n =
{\mathrm{e}}^{-t n} H_n$. From this one quickly deduces that the Mehler kernel is given by $$\label{eq:Mehler_kernel_non_comp}
M_t(x, y) := \sum_{n = 0}^\infty {\mathrm{e}}^{-t n} h_n(x)h_n(y).$$ We will need two further identities for the Hermite polynomials which can be found, e.g., in [@NIST:DLMF Chapter 18]: the integral representation $$\label{eq:Hermite-integral}
H_n(x) = \frac{(-2i)^n {\mathrm{e}}^{x^2}}{\sqrt\pi} \int_{-\infty}^\infty
\xi^n {\mathrm{e}}^{2ix \xi} {\mathrm{e}}^{-\xi^2} {\,\mathrm{d}}\xi$$ and the “binomial” identity $$\label{eq:Hermite-binomial-type}
H_n(x + y) = \sum_{k = 0}^n \binom{n}{k} (2y)^{n - k} H_k(x) .$$
Hermite polynomials in several variables
----------------------------------------
The Hermite polynomials on ${\mathbf{R}}^d$ are defined, for multiindices $\alpha = (\alpha_1, \dots, \alpha_d)\in{\mathbf{N}}^d$ by the tensor extensions $$\label{eq:Hermite-Rodrigues-Rd}
H_\alpha(x) := \prod_{n = 1}^{d} h_{\alpha_i}(x_i),$$ for $x=(x_1,\dots,x_d)$ in ${\mathbf{R}}^d$. The normalized Hermite polynomials $$\label{eq:Hermite-normalized}
h_\alpha := \frac{H_\alpha}{\sqrt{2^{|\alpha|} \alpha!}}, \ \text{ where } \alpha! = \alpha_1! \cdot \dots \cdot \alpha_d!$$ form an orthonormal basis in ${{L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)}}$, and we have the eigenvalue identity $$\label{eq:OU-Hermite-action}
LH_{\alpha} = -|\alpha| H_\alpha, \ \text{ where } |\alpha| = \alpha_1 + \dots + \alpha_d.$$ If we consider the action of $L^N {\mathrm{e}}^{tL}$ on a Hermite polynomial $h_\alpha$, through the multinomial theorem applied to $|\alpha|^k$ we get (writing $L_d$ for the operator $L$ in dimension $d$ and $L_1$ for the operator $L$ in dimension $1$) $$\label{eq:reduction-to-d1}
\begin{aligned} L_d^N {\mathrm{e}}^{tL} & h_\alpha(x) = |\alpha|^N {\mathrm{e}}^{-t|\alpha|}
h_{\alpha_1}(x_1) \cdot \hdots \cdot
h_{\alpha_d}(x_d)\\
&= \sum_{|n| = N} \binom{N}{n_1, n_2, \hdots, n_d} \alpha_1^{n_1}
\cdot \hdots \cdot \alpha_d^{n_d} {\mathrm{e}}^{-t\alpha_1}\cdot\hdots\cdot {\mathrm{e}}^{\alpha_d}
h_{\alpha_1}(x_1) \cdot \hdots\cdot
h_{\alpha_d}(x_d)
\\ & = \sum_{|n| = N} \binom{N}{n_1, n_2, \hdots, n_d} L_1^{n_1}
{\mathrm{e}}^{tL_1} h_{\alpha_1}(x_1) \cdot \hdots \cdot L_1^{n_d}
{\mathrm{e}}^{tL_1} h_{\alpha_d}(x_d).
\end{aligned}$$ This implies that we can reduce the question of computing the $d$-dimensional version of the integral kernel to the one-dimension one.
A combinatorial lemma {#sec:setup}
=====================
From now on we concentrate on the Ornstein-Uhlenbeck operator $L$ in one dimension, i.e., in $L^2({\mathbf{R}},{\mathrm{d}}\gamma)$. We are going to follow the approach of [@Sjogren1997]. Recalling the identity $L h_n = -n h_n$, we will apply $L^N$ to the generating function of the Hermite polynomials . A problem which immediately occurs is that $\Delta$ and $\langle x, \nabla \rangle$ do not commute, and because of this we cannot use a standard binomial formula to evaluate $L^N$. Instead, we note that $$L g = -t \partial_t g.$$ In particular this implies that $$L^N g = (-1)^k D_N g,$$ where $$\label{eq:Differential-operator-generated}
D_N := \underbrace{t \partial_t \circ t \partial_t \circ \dots \circ t \partial_t}_{\text{$k$ times}} = (t\partial_t)^k.$$ The following lemma will be very useful.
\[lem:Lk-powers-expanded\] We have $$\label{eq:Expanded-Differentiatial-operator-generated}
D_N = \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n \partial_t^n, $$ where ${\genfrac{\{}{\}}{0pt}{}{N}{n}}$ are the Stirling numbers of the second kind.
The Stirling numbers of the second kind are quite well-known combinatorial objects. For the sake of completeness we will state their definition and recall some relevant properties below. For more information we refer the reader to [@LINT]. The related Stirling numbers of the first kind will not be needed here.
We begin by recalling the definition of [*falling factorial*]{} $$\label{eq:falling-factorial}
(j)_n := j(j - 1)\dots (j - n + 1) = \frac{j!}{(j - n)!},$$ for non-negative integers $k \geq n$.
\[def:stirling-numbers-second-kind\] For non-negative integers $N \ge n$, the number [*Stirling number of the second kind*]{} ${\genfrac{\{}{\}}{0pt}{}{N}{n}}$ is defined as the number of partitions of an $N$-set into $n$ non-empty subsets.
These numbers satisfy the recursion identity $$\label{eq:Stirling-numbers-second-kind-recursion}
{\genfrac{\{}{\}}{0pt}{}{N}{n}} = N {\genfrac{\{}{\}}{0pt}{}{N - 1}{n}} + {\genfrac{\{}{\}}{0pt}{}{N - 1}{n - 1}}.$$ For all non-negative integers $j$ and $k$ one has the generating function identity $$\label{eq:Stirling-numbers-second-kind-generating-function-1}
j^N = \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} (j)_n.$$
Weyl Polynomials {#sec:Weyl}
================
Before turning to the proof of lemma \[lem:Lk-powers-expanded\], let us already mention that it only depends on the commutator identity $[t,\partial_t] = -1$. This brings us to the observation that *Weyl polynomials* provide the natural habitat for our expressions. Rougly speaking, a Weyl polynomial is a polynomial in two non-commuting variables $x$ and $y$ which satisfy the commutator identity $[x, y] = -1$. This is made more precise in the following definition.
The [*Weyl algebra*]{} over a field ${\mathbf{F}}$ of characteristic zero is the ring ${\mathbf{F}}\langle x,y\rangle$ of all polynomials of the form $p(x,y) = \sum_{m=0}^M \sum_{n=0}^N c_{mn} x^m y^n$ with coefficients $c_{mn}\in{\mathbf{F}}$ in two noncommuting variables $x$ and $y$ which satisfy the commutator identity $$[x,y]:=xy-yx = -1.$$
We now have the following abstract version of lemma \[lem:Lk-powers-expanded\]:
\[lem:Weyl-expansion\] In the Weyl algebra ${\mathbf{F}}\langle x, y\rangle$ we have the identity $$\label{eq:Weyl-base-thm}
(x y)^m = \sum_{i = 1}^m {\genfrac{\{}{\}}{0pt}{}{m}{i}} x^i y^i,$$ where ${\genfrac{\{}{\}}{0pt}{}{m}{i}}$ are the Stirling numbers of the second kind.
As a preparation for the proof of lemma \[lem:Weyl-expansion\] we make a couple of easy computations. If we set $D := xy$, then $$\begin{aligned}
D x^m &= x^m D + m x^m\label{eq:Weyl-push-1},\\
D y^m &= y^m D - m y^m\label{eq:Weyl-push-2}.\end{aligned}$$ This can be shown by induction on $m$. For instance, note that $$D x^m = x(D + 1)x^{m - 1} = x D x^{m - 1} + x^m.$$ If we take this a bit further and have $p \in {\mathbf{F}}[D]$, then $$\begin{aligned}
\label{eq:Weyl-poly-1}
p(D) x^m &= x^m p(D + m),\\
\label{eq:Weyl-poly-2}
p(D) y^m &= y^m p(D - m).\end{aligned}$$ The $m$-th powers, $m\ge 1$, of $x$ and $y$ satisfy $$\begin{aligned}
\label{eq:Weyl-xmym}
x^m y^m &= \prod_{i = 0}^{m - 1} (D - i),\\
\label{eq:Weyl-ymxm}
y^m x^m &= \prod_{i = 1}^m (D + i).\end{aligned}$$ This can be seen using induction: $$x^{m + 1} y^{m + 1} = x^m D y^m \overset{\eqref{eq:Weyl-poly-2}}{=} x^m y^m (D - m)$$ and $$y^{m + 1} x^{m + 1} = y^m (D + 1) x^m = y^m D x^m + y^m x^m \overset{\eqref{eq:Weyl-poly-1}}{=} y^m x^m (D + (m + 1)).$$
The [*weighted degree*]{} of a monomial $x^m y^n\in {\mathbf{F}}\langle x,y\rangle$ is the integer $m-n$. A polynomial in ${\mathbf{F}}\langle x,y \rangle$ is said to be [*homogeneous of weighted degree $j$*]{} if all its constituting monomials have weighted degree $j$.
Left multiplication by $xy$ is [*homogeneity preserving*]{}, i.e., for all $j\in{\mathbf{Z}}$ it maps the set of homogeneous monomials of weighted degree $j$ into itself. To prove this, first consider a monomial $x^m y^n$ of dweighted egree $j = m-n$. Then, $$(xy) x^m y^n \mathrel{\overset{\eqref{eq:Weyl-push-1}}{=}} (x^m (xy) + m x^m)y^n = x^m(xy)y^n +mx^m y^n = x^{m + 1} y^{n + 1} + m x^m y^n,$$ and we see that weighted degree of homogeneity is indeed preserved. The general case follows immediately. Through we conclude that left multiplication $x^k y^k$ is homogeneity preserving as well, for all non-negative integers $k$. We claim that left multiplication by $x^i y^j$ is homogeneity preserving only if $i = j$. To see this note that $$y x^i y^j = x^i y^{i + j} + i
x^{i - 1} y^j$$ from which we can deduce that $$x^m y^M x^n y^N = x^{n + m} y^{N + M} + \text{ lower order terms}.$$ From which the claim follows.
Finally, a polynomial is homogeneity preserving if and only if all of its constituting monomials have this property. If this were not to be the case we could look at the highest-order non-homogeneous term and note from above $x^m y^M x^n y^N$ would give terms of a lower order in the polynomial expansion which cannot cancel as they have different powers of $x$ or $y$.
It follows from these observations that $$\label{eq:Weyl-G0}
F_0 := \biggl\{\sum_{n=0}^N c_{n} x^n y^n \ \bigg| \ N\in{\mathbf{N}}, \, c_1,\dots,c_N\in{\mathbf{F}}\biggr\}$$ is precisely the set of *homogeneity preserving polynomials* in ${\mathbf{F}}\langle
x, y\rangle$.
Now everything is in place to give the proof of lemma \[lem:Weyl-expansion\].
As $(xy)^k$ is homogeneity preserving, we infer that there are coefficients $a_i^k$ in ${\mathbf{F}}$ such that $$\label{eq:Weyl-lemma-intermediate-step-wanted-form}
(xy)^k = \sum_{i = 0}^k a_i^k x^i y^i.$$ We will apply $x^j$ to the right on both sides of and derive an expression for the $a_i^k$. First note that gives $$(xy)^k x^j = x^j (xy + j)^k,$$ and together with gives $$x^i y^i x^j \overset{\eqref{eq:Weyl-xmym}}{=}
\prod_{\ell = 0}^{i - 1} (xy - \ell) x^j \overset{\eqref{eq:Weyl-poly-1}}{=} x^j \prod_{\ell = 0}^{i - 1} (xy - \ell + j).$$ Hence, to find the coefficients $a_i^k$ it is sufficient to consider $$(xy + j)^k = \sum_{i = 0}^k a_n^k \prod_{\ell = 0}^{i - 1}(xy - \ell + j).$$ Comparing the constant terms on both the left-hand side and right-hand side, we find $$\label{eq:Weyl-result-generating-function}
j^k = \sum_{i = 0}^k a_i^k \prod_{\ell = 0}^{i - 1} (j - \ell) = \sum_{i = 0}^k a_i^k (j)_i,$$ where $(j)_i$ is the falling factorial as in . Comparing with the generating function of the Stirling numbers of the second kind ${\genfrac{\{}{\}}{0pt}{}{k}{i}}$ as given in , we see that $a_i^k = {\genfrac{\{}{\}}{0pt}{}{k}{i}}$. This concludes the proof of lemma \[lem:Weyl-expansion\].
The integral kernel of $L^N {\mathrm{e}}^{t L}$ {#sec:mainresult}
===============================================
As mentioned before, as a first step we would like to apply $D_N$ to the generating function $g(x,t) := {\mathrm{e}}^{-2tx + t^2 } = {\mathrm{e}}^{-(x - t)^2 + x^2}$ for the Hermite polynomials . We first compute the action of $\partial_t^N$ on the generating function.
We have $$\label{eq:derivatives-generating-function-Hermite}
\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = {\mathrm{e}}^{-(x - t)^2 + x^2} H_N(x - t).$$
We first note that, $$\begin{aligned}
\partial_t {\mathrm{e}}^{-(x - t)^2} &= 2(x - t) {\mathrm{e}}^{-(x - t)^2} = - \partial_x {\mathrm{e}}^{-(x - t)^2}.\end{aligned}$$ Using this we get $$\begin{aligned}
\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} &= {\mathrm{e}}^{x^2} \partial_t^N {\mathrm{e}}^{-(x -
t)^2}\\
&= {\mathrm{e}}^{x^2} \partial_t^{N - 1} \partial_t {\mathrm{e}}^{-(x - t)^2}\\
&= -{\mathrm{e}}^{x^2} \partial_t^{N - 1} \partial_x {\mathrm{e}}^{-(x - t)^2}\\
&= (-)^2{\mathrm{e}}^{x^2} \partial_t^{N - 2} \partial_x^2 {\mathrm{e}}^{-(x - t)^2}\\
&= \dots\\
&= (-1)^N {\mathrm{e}}^{x^2} \partial_x^N {\mathrm{e}}^{-(x - t)^2}.\end{aligned}$$ By a change of variables, $$\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} {\mathrm{e}}^{(x - t)^2} \partial_y^N {\mathrm{e}}^{-y^2} \Bigr|_{y = x - t}.$$ Hence, by , $$\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = {\mathrm{e}}^{-(x - t)^2 + x^2} H_N(x - t).$$
\[lem:power-Ornstein-Uhlenbeck-generating-Hermite\] For all $x\in{\mathbf{R}}$ and $t>0$ we have $$\label{eq:power-Ornstein-Uhlenbeck-generating-Hermite}
L^N {\mathrm{e}}^{-(x - t)^2 + x^2} = (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n H_n(x - t).$$
This is now easy to prove. Recalling that $L = -t \partial_t$ and using , we get $$\begin{aligned}
L^N {\mathrm{e}}^{-(x - t)^2 + x^2} &= D_N {\mathrm{e}}^{-(x - t)^2 + x^2}\\
&= (-1)^N \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n \partial_t^n {\mathrm{e}}^{-(x - t)^2 + x^2}\\
&= (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n H_n(x - t).\end{aligned}$$
Our next theorem is the main result of this paper and provides an explicit expression for the integral kernel of $L^N e^{t L}$.
\[th:integral-kernel\] Let $L$ be the Ornstein-Uhlenbeck operator on $L^2({\mathbf{R}}^d,{\mathrm{d}}\gamma)$, let $t>0$, and let $N\ge 0$ be an integer. The integral kernel $M_t^N$ of $L^N {\mathrm{e}}^{tL}$ is given by $$\label{eq:Mehler-kernel-of-powers}
\begin{split}
M_t^N(x, y) &= M_t(x, y) \sum_{|n| = N} \binom{N}{n_1, \dots, n_d} \prod_{i = 1}^d
\sum_{m_i = 0}^{n_i} \sum_{\ell_i = 0}^{m_i} 2^{-m_i}
{\genfrac{\{}{\}}{0pt}{}{n_i}{m_i}} \binom{m_i}{\ell_i}\\
&\quad \times \biggl(-\frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}} \biggr)^{2m_i - \ell_i}
H_{\ell_i}(x_i) H_{2m_i - \ell_i}\biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\biggr).
\end{split}$$
We first prove the result for $d = 1$. Pulling $L^N$ through the integral expression for $e^{t L}$ involving the Mehler kernel, we must find a suitable expression for the kernel $M_t^N(\cdot, y) = L^N
M_t(\cdot, y)$. Using and the normalization of $H_m$ in we get $$\begin{aligned}
M_t^N(x, y) &= L^N \sum_{m = 0}^\infty \frac{{\mathrm{e}}^{-t m}}{m!}\frac1{2^m} H_m(x) H_m(y)\\
&\overset{\eqref{eq:Hermite-integral}}{=} L^N \sum_{m = 0}^\infty \frac{{\mathrm{e}}^{-t m}}{m!}\frac1{2^m} H_m(x) \frac{(-2i)^m}{\sqrt{\pi}} {\mathrm{e}}^{y^2} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} \xi^m {\mathrm{e}}^{2 i y \xi} {\,\mathrm{d}}\xi\\
&= L^N \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2 i y \xi} \sum_{m = 0}^\infty \frac1{m!} H_m(x) (-i \xi {\mathrm{e}}^{-t})^m{\,\mathrm{d}}\xi\\
&\overset{\eqref{eq:Generating-function-identity}}{=} L^N
\frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2
i y \xi} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2 + x^2} {\,\mathrm{d}}\xi.\end{aligned}$$ The operator $L^N$ is applied with respect to $x$ here, so by lemma \[lem:power-Ornstein-Uhlenbeck-generating-Hermite\] we get $$\begin{aligned}
M_t^N(x, y) &= \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}}
\int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2 i y \xi} L^N {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2 + x^2} {\,\mathrm{d}}\xi\\
&\overset{\eqref{eq:power-Ornstein-Uhlenbeck-generating-Hermite}}{=}(-1)^N
\frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \sum_{m = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{m}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i y \xi} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i
\xi {\mathrm{e}}^{-t})^m H_m(y + i \xi {\mathrm{e}}^{-t}) {\mathrm{e}}^{-\xi^2} {\,\mathrm{d}}\xi,\end{aligned}$$ where in last line we have used the analytic continuation of the algebraic identity . Similarly we can expand $H_m(y + i \xi {\mathrm{e}}^{-t})$ using . This gives $$H_m(y + i \xi {\mathrm{e}}^{-t}) = \sum_{\ell = 0}^m \binom{m}{\ell} H_\ell(y)
(2 i \xi {\mathrm{e}}^{-t})^{m - \ell},$$ so that $M_t^N$ can be written as $$\label{eq:M_k-integral}
(-1)^N \frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \sum_{m = 0}^N \sum_{\ell = 0}^m {\genfrac{\{}{\}}{0pt}{}{N}{m}} \binom{m}{\ell} H_\ell(y) 2^{m - \ell} \int_{-\infty}^\infty {\mathrm{e}}^{2 i y \xi - \xi^2} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i \xi {\mathrm{e}}^{-t})^{2m - \ell} {\,\mathrm{d}}\xi.$$ If we set $M = 2m - \ell$, this reduces our task to computing the integral $$\begin{aligned}
\notag
\frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty &{\mathrm{e}}^{2 i y \xi - \xi^2} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i \xi {\mathrm{e}}^{-t})^M {\,\mathrm{d}}{\xi}\\
\label{eq:Hermite-integral-derivation-1}
&= \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{2 i (x {\mathrm{e}}^{-t} - y) \xi} {\mathrm{e}}^{-(1 - {\mathrm{e}}^{-2 t}) \xi^2} (i \xi {\mathrm{e}}^{-t})^M
{\,\mathrm{d}}{\xi}.\end{aligned}$$ To make the computation less convolved, let us set $$\alpha_t := \sqrt{1 - {\mathrm{e}}^{-2t}}, \text{ and, } \beta_t(x, y) := \frac{x {\mathrm{e}}^{-t} - y}{\sqrt{1 - {\mathrm{e}}^{-2t}}}.$$ This allows us to write the exponential in the integral as $${\mathrm{e}}^{2 i (x{\mathrm{e}}^{-t} - y) \xi} {\mathrm{e}}^{-(1 - {\mathrm{e}}^{-2 t}) \xi^2} = {\mathrm{e}}^{2 i \alpha_t \beta_t(x,y) \xi} {\mathrm{e}}^{-\alpha_t^2 \xi^2}.$$ This reduces the problem, after the substitution $\alpha_t \xi \to \xi$, to computing the integral $$\frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \frac{i^M {\mathrm{e}}^{-M t}}{\alpha_t^{M + 1}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i \beta_t(x, y) \xi} {\mathrm{e}}^{-\xi^2} \xi^M {\,\mathrm{d}}{\xi}.$$ The final integral is an integral expression for the Hermite polynomials , so $$\begin{aligned}
\ & \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \frac{i^M {\mathrm{e}}^{-M t}}{\alpha_t^{M + 1}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i \beta_t(x,y) \xi} {\mathrm{e}}^{-\xi^2} \xi^M {\,\mathrm{d}}{\xi}
\\ & \qquad \overset{\eqref{eq:Hermite-integral}}{=} {\mathrm{e}}^{y^2-\beta_t(x,y)^2}
\frac1{\alpha_t^{M + 1}} \frac{(-1)^M {\mathrm{e}}^{-M t}}{2^M} H_M(\beta_t(x,y)).\end{aligned}$$ Next we note that $\exp(y^2 - \beta_t(x,y)^2) \alpha_t^{-1} = M_t$, the Mehler kernel from . Hence, $$\label{eq:Mehler-kernel-Lk-intermediate-1}
\begin{split}
M_t^N(x, y) &= M_t(x, y) \sum_{m = 0}^N \sum_{\ell = 0}^m
\binom{m}{\ell} {\genfrac{\{}{\}}{0pt}{}{N}{m}}
\biggl(-\frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}} \biggr)^{2m - \ell}
2^{-m}\\
&\quad \times H_\ell(x) H_{2m - \ell}\biggl(\frac{x {\mathrm{e}}^{-t} - y}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\biggr).
\end{split}$$ Applying we get the result in $d$ dimensions.
An application {#sec:application}
==============
As an application of our main result, in this section we give an alternative proof of the bounds on the kernels $K$ and $\tilde K$ of [@Portal2014] (see the definition below), making the dependence on the parameters more explicit. These kernels play a central role in the study of the Hardy space $H^1({\mathbf{R}}^d,{\mathrm{d}}\gamma)$ in [@Portal2014], where the standard Calderón reproducing formula is replaced by $$u = C \int_0^\infty (t^2 L)^{N + 1} {\mathrm{e}}^{\frac{t^2}{\alpha} L} u \frac{{\,\mathrm{d}}{t}}{t} + \int_{{\mathbf{R}}^d} u {\,\mathrm{d}}\gamma,$$ where $C$ is a suitable constant only depending on $N$ and $\alpha$ (this can be seen by letting $u$ be a Hermite polynomial). The kernels $K$ and $\tilde K$ then occur in several decompositions, and the estimates below allow them to be related to classical results about the Mehler kernel.
We define the kernels $K$ and $\tilde K$ by $$\begin{aligned}
\int_{{\mathbf{R}}^d} K_{t^2, N, \alpha}(x, y) u(y) {\,\mathrm{d}}\gamma(y) &= (t^2 L)^N
{\mathrm{e}}^{\frac{t^2}{\alpha}
L} u(x),\\
\int_{{\mathbf{R}}^d} \tilde{K}_{t^2, N, \alpha, j}(x, y) u(y) {\,\mathrm{d}}\gamma(y) &= (t^2 L)^N
{\mathrm{e}}^{\frac{t^2}{\alpha}
L} t \partial_{x_j}^* u(x).
\end{aligned}$$
Note that the operators on the right-hand sides are indeed given by integral kernels: the first is a scaled version of the operator we have already been studying, and a duality argument implies that the second is given by the integral kernel $$\tilde{K}_{t^2, N, \alpha,j}(x, y) =
t \partial_{x_j} K_{t^2, N, \alpha}(x, y).$$ Thus, both kernels are given as appropriate derivatives of the Mehler kernel.
We begin with a technical lemma which is a rephrased version of [@Portal2014 Lemma 3.4]. One should take note that we define the kernels with respect to the Gaussian measure whereas, [@Portal2014] defines these with respect to the Lebesgue measure.
\[lem:Mehler-alpha-efficient1\] For all $\alpha > 1$ and all $t>0$ and $x, y$ in ${\mathbf{R}}^d$ we have $$\label{eq:Mehler-alpha-efficient2}
\frac{|{\mathrm{e}}^{-\frac{t}{\alpha}}x - y|^2}{1 - e^{-2\frac{t}{\alpha}}}
\geq\frac{\alpha}2 {\mathrm{e}}^{-2t} \frac{|{\mathrm{e}}^{-t}x - y|^2}{1 -
{\mathrm{e}}^{-2t}} - \frac{t^2 \min{(|x|^2, |y|^2)}}{1 - {\mathrm{e}}^{-2\frac{t}{\alpha}}}.$$ Additionally, we have $$\label{eq:Mehler-alpha-efficient3}
\alpha {\mathrm{e}}^{-2t} \leq \frac{1 - {\mathrm{e}}^{-2t}}{1 - {\mathrm{e}}^{-2\frac{t}{\alpha}}} \leq \alpha.$$
Let $N$ be a positive integer, $0 < t < T$. The for all large enough $\alpha>1$ we have
1. If $t |x| \leq C$, then $$\displaystyle |K_{t^2, \alpha, N}(x, y)| \lesssim_{T, N}
\alpha \exp(\frac{\alpha}2C^2)
M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8{\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr).$$
2. If $t |x| \leq C$, then $$\displaystyle |\tilde{K}_{t^2, \alpha, N,j}(x, y)|
\lesssim_{T, N} \alpha \exp(\frac{\alpha}2C^2) M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr).$$
For $K_{t^2, \alpha, N}$, we use Theorem \[th:integral-kernel\] to obtain, after taking absolute values, $$\begin{aligned}
|K_{t^2, \alpha, N}(x, y)| &\leq M_{\frac{t^2}{\alpha}}(x, y) \sum_{|k| = N} \binom{N}{n_1, \dots, n_d}
\prod_{i = 1}^d t^{2k_i }\sum_{\ell_i = 0}^{n_i} \sum_{m_i = 0}^{m_i} 2^{-m_i}
\binom{m_i}{\ell_i} {\genfrac{\{}{\}}{0pt}{}{n_i}{m_i}}\\
&\quad \times \biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{2m_i - \ell_i}
| H_{\ell_i}(x_i)| \biggl| H_{2m_i - \ell_i}\biggl(\frac{x_i {\mathrm{e}}^{-\frac{t^2}{\alpha}} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr) \biggr |.\end{aligned}$$ Recalling that $\ell_1 + \dots + \ell_d \leq N$, using the assumptions $t \leq T$ and $t |x| \leq
C$ we can bound $t^{2k_i} |H_{\ell_i}(x)|$ by considering the highest order term to obtain $$t^{2k_i} |H_{\ell_i}(x)| \lesssim_{C,N,T} 1.$$ Using we proceed by looking at $$\begin{aligned}
M_{\frac{t^2}{\alpha}}(x, y) &= M_{t^2}(x, y) \bigg(\frac{{1 - {\mathrm{e}}^{-2t^2}}}{{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\bigg)^{1/2}\exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr)\\
&\le \alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \Bigg[\exp\biggl(-\frac12\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr) \bigg]^2.
$$ We can now bound the final Hermite polynomial in the expression of the kernel. Setting $M_i = 2m_i - \ell_i$ we get $$\begin{aligned}
\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{M_i}
\biggl|H_{M_i}\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i}{\sqrt{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)\biggr| &\lesssim_N
\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1
- {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{M_i}
\biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{\sqrt{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i}\\
&\le \biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i}.\end{aligned}$$ Also, $$\begin{aligned}
\biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i} \exp\biggl(-\frac1{2}\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x_i - y_i|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr) \lesssim 1.\end{aligned}$$ Putting these estimates together, using Lemma \[lem:Mehler-alpha-efficient1\], and taking $\alpha>1$ so large that $$1-\frac{\alpha}{4 {\mathrm{e}}^{2T}} \le -\frac{\alpha}{8 {\mathrm{e}}^{2T}} \ \hbox{ and } \
1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}} \ge \frac{t^2}{\alpha},$$ we obtain $$\begin{aligned}
|K_{t^2, \alpha, N}(x, y)| & \lesssim_{T, N}\alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac12\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr)\\
&\leq \alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac{\alpha}{4 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac12 \frac{t^4 |x|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}} \biggr)\\
&= \alpha M_{t^2}(x, y) \exp\biggl(\Big(1-\frac{\alpha}{4 {\mathrm{e}}^{2T}}\Big) \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac12 \frac{t^4 |x|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}} \biggr)\\
&\leq \alpha M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac\alpha2 t^2 |x|^2 \biggr)
\\ &\leq \alpha \exp\biggl(\frac\alpha2 C^2 \biggr) M_{t^2}(x, y)
\exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr),\end{aligned}$$ using $t|x| \leq C$ in the last step.
For the bound on $\tilde{K}$ we consider $$\begin{aligned}
t \partial_{x_i} \biggl[H_{\ell_i}(x_i) H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr) \biggr] &= t H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr) \partial_{x_i} H_\ell(x_i)\\
&\quad + H_{\ell_i}(x_i) t \partial_{x_i} H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr).\end{aligned}$$ So, as the first term on the right-hand side just decreases in degree we look at $$\begin{aligned}
t \partial_{x_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}
\biggr)^{m_i} = m_i \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}
\biggr)^{m_i - 1} t \frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\end{aligned}$$ The last term is bounded as $t \downarrow 0$, and the rest of the proof is as before.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was partially supported by NWO-VICI grant 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).
The author wishes to thank Alex Amenta and Mikko Kemppainen for inspiring discussions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the gravity dual to a class of three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on a biaxially squashed three-sphere, with a non-trivial background gauge field. This is described by a 1/2 BPS Euclidean solution of four-dimensional $\mathcal{N}=2$ gauged supergravity, consisting of a Taub-NUT-AdS metric with a non-trivial instanton for the graviphoton field. The holographic free energy of this solution agrees precisely with the large $N$ limit of the free energy obtained from the localized partition function of a class of Chern-Simons quiver gauge theories. We also discuss a different supersymmetric solution, whose boundary is a biaxially squashed Lens space $S^3/{\mathbb{Z}}_2$ with a topologically non-trivial background gauge field. This metric is of Eguchi-Hanson-AdS type, although it is not Einstein, and has a single unit of gauge field flux through the $S^2$ cycle.'
---
2.3 cm
[**The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere**]{}
2 cm
[Dario Martelli$^1$ and James Sparks$^2$\
]{}
$^1$*Department of Mathematics, King’s College, London,\
The Strand, London WC2R 2LS, United Kingdom\
*
0.8cm
$^2$*Mathematical Institute, University of Oxford,\
24-29 St Giles’, Oxford OX1 3LB, United Kingdom\
*
2 cm
Introduction
============
Supersymmetric gauge theories on compact curved backgrounds are interesting for various reasons. For example, supersymmetry may be combined with localization techniques, allowing one to perform a variety of exact computations in strongly coupled field theories. The authors of [@Hama:2011ea] presented a construction of ${\cal N}=2$ supersymmetric gauge theories in three dimensions in the background of a $U(1)\times U(1)$-invariant squashed three-sphere and R-symmetry gauge field. The gravity dual of this construction was recently given in [@Martelli:2011fu]. It consists of a 1/4 BPS Euclidean solution of four-dimensional $\mathcal{N}=2$ gauged supergravity, which in turn may be uplifted to a supersymmetric solution of eleven-dimensional supergravity. In particular, the bulk metric in [@Martelli:2011fu] is simply AdS$_4$, and the graviphoton field is an instanton with (anti)-self-dual field strength. The asymptotic metric and gauge field then reduce to the background considered in [@Hama:2011ea].
The purpose of this letter is to present the gravity dual to a different field theory construction, obtained recently in [@Imamura:2011wg]. In this reference the authors have constructed three-dimensional ${\cal N}=2$ supersymmetric gauge theories in the background of the $SU(2)\times U(1)$-invariant squashed three-sphere (which we refer to as *biaxially* squashed) and a non-trivial background $U(1)$ gauge field, and have computed the corresponding partition functions using localization. Differently from a similar construction discussed briefly in [@Hama:2011ea], this partition function depends non-trivially on the squashing parameter. As we will see, the gravity dual to this set-up will have some distinct features with respect to the solution in [@Martelli:2011fu]. In particular, the metric is not simply AdS$_4$, although it will again be an Einstein metric, and there is a self-dual graviphoton.
The plan of the rest of this paper is as follows. In section \[revsect\] we review the construction of [@Imamura:2011wg]. In section \[nuts\] we discuss the gravity dual. In section \[bolty\] we describe a different supersymmetric solution, consisting of a non-Einstein metric and a non-instantonic graviphoton field. Section \[discussione\] concludes.
Supersymmetric gauge theories on the biaxially squashed $S^3$ {#revsect}
=============================================================
In the construction of [@Imamura:2011wg] the metric on the three-sphere is, up to an irrelevant overall factor, given by s\^2\_3 & = & \_1\^2 + \_2\^2 + \_3\^2 , \[fametric\] where $\sigma_i$ are the standard $SU(2)$ left-invariant one-forms on $S^3$, defined as ${\mathrm{i}}\sigma_i \tau_i = -2 \mathtt{g}^{-1}{\mathrm{d}}\mathtt{g}$, where $\tau_i$ denote the Pauli matrices and $\mathtt{g}\in SU(2)$. The background $U(1)$ gauge field reads A\^[(3)]{} & = & \_3 , \[newgau\] and the spinors in the supersymmetry transformations obey the equation (setting the radius $r=2$ in [@Imamura:2011wg]) \_\^[(3)]{}- \_- A\^[(3)]{}\_\_\^& = & 0 , \[newks\] where $\nabla_\alpha^{(3)}$, $\alpha=1,2,3$, is the spinor covariant derivative constructed from the metric (\[fametric\]), and $\gamma_\alpha$ generate $\mathrm{Cliff}(3,0)$. There are *two* linearly independent solutions to (\[newks\]), transforming as a doublet under $SU(2)$, whose explicit form is given in [@Imamura:2011wg]. This will be important for identifying the gravity dual.
In [@Imamura:2011wg] the authors constructed Chern-Simons, Yang-Mills, and matter Lagrangians for the ${\cal N}=2$ vector multiplets $V=(\mathscr{A}_\alpha,\sigma,\lambda,D)$ and chiral multiplets $\Phi = (\phi,\psi,F)$, in the background of the metric (\[fametric\]) and R-symmetry gauge field (\[newgau\]). These are invariant under a set of supersymmetry transformations, provided the spinorial parameters obey the equation (\[newks\]). The supersymmetric completion of the Chern-Simons Lagrangian contains new terms, in addition to those appearing in flat space, proportional to $\sigma^2$ and $\sigma A^{(3)}\wedge {\mathrm{d}}\mathscr{A}$ (*cf*. eq. (32) of [@Imamura:2011wg]). The Yang-Mills and matter Lagrangians are total supersymmetry variations (*cf*. eq. (31) of [@Imamura:2011wg]) and therefore can be used to compute the partition function using localization. In particular, the partition function localizes on supersymmetric configurations obeying \_ = D = 0 , = u = , \[newloc\] with the matter fields all being zero. Notice that although $D=0$, the Chern-Simons Lagrangian is non-zero because of the new term proportional to $\sigma^2$, and therefore it contributes classically to the localized partition function, as in previous constructions. The Yang-Mills and matter terms contribute one-loop determinants from the Gaussian integration about the the classical solutions (\[newloc\]). The final partition function may be expressed again in terms of double sine functions $s_b(z)$, and for a $U(N)$ gauge theory at Chern-Simons level $k\in{\mathbb{Z}}$ reads \[partition\] Z & = & \_ u ( u\^2) , where $b=(1+{\mathrm{i}}\sqrt{v^2-1})/v$. The exponential term is the classical contribution from the Chern-Simons Lagrangian, evaluated on (\[newloc\]); the numerator is the one-loop vector multiplet determinant and involves a product over the roots $\alpha$ of the gauge group $G$; while the denominator is the one-loop matter determinant and involves a product over chiral fields of R-charge $\Delta_a$ in representations $\mathcal{R}_a$, with $\rho$ running over weights in the weight-space decomposition of $\mathcal{R}_a$. Following [@Martelli:2011qj], one can easily extract the large $N$ behaviour of this partition function for a class of non-chiral ${\cal N}=2$ quiver Chern-Simons-matter theories. The calculation was done in [@Imamura:2011wg], and the result is that the leading contribution to the free energy (defined as ${\mathcal{F}}=-\log Z$) is given by \_v & = & \_[v=1]{} , \[newfree\] and thus depends very simply on the squashing parameter $v$. In the next section we will present the supergravity dual to this construction, in particular showing that the holographic free energy precisely agrees with the field theory result (\[newfree\]).
The gravity dual {#nuts}
================
As anticipated in [@Martelli:2011fu], we will show that the gravity dual to the set-up described in the previous section is a supersymmetric solution of $d=4$, $\mathcal{N}=2$ gauged supergravity. In *Lorentzian* signature, the bosonic part of the action is given by \[4dSUGRA\] S\_ &=& \^4x . Here $R$ denotes the Ricci scalar of the four-dimensional metric $g_{\mu\nu}$, and the cosmological constant is given by $\Lambda=-3g^2$. The graviphoton is an Abelian gauge field $A^L$ with field strength $F^L={\mathrm{d}}A^L$; here the superscript $L$ emphasizes that this is a Lorentzian signature object. A solution to the equations of motion derived from (\[4dSUGRA\]) is supersymmetric if there is a non-trivial spinor $\epsilon$ satisfying the Killing spinor equation \[LKSE\] &=& 0 . Here $\Gamma_\mu$, $\mu=0,1,2,3$, generate the Clifford algebra $\mathrm{Cliff}(1,3)$, so $\{\Gamma_\mu,\Gamma_\nu\}=2g_{\mu\nu}$.
Since the background of [@Imamura:2011wg] preserves half of the maximal supersymmetry in three dimensions, we should seek a *1/2 BPS Euclidean* solution of $d=4$, $\mathcal{N}=2$ gauged supergravity, whose metric has as conformal boundary the biaxially squashed metric on $S^3$ (\[fametric\]), and whose background $U(1)$ gauge field restricted to this asymptotic boundary reduces to (\[newgau\]). This very strongly suggests that the appropriate solution is a Euclideanized version of the 1/2 BPS Reissner-Nordström-Taub-NUT-AdS solution discussed in [@AlonsoAlberca:2000cs].
We will first present this Euclidean solution, and then discuss the Wick rotation that leads to it. The metric reads s\^2\_4 &= & r\^2 + (r\^2-[s]{}\^2)(\_1\^2+\_2\^2) +\_3\^2 , \[TNAdS\] where (r) & = & ([s]{}- r)\^2 , \[finalome\] and ${s}$ is the NUT parameter.[^1] The $SU(2)$ left-invariant one-forms $\sigma_i$ may be written in terms of angular variables as \_1 + \_2 = \^[-]{} ( + ) ,\_3 = + . The graviphoton field is A & = & s \_3 . \[gaugefield\] In the orthonormal frame \[frame\] e\_1 &=& \_1 , e\_2 = \_2 ,\
e\_3 & = & 2s \_3 , e\_4 = r , the curvature may be written as F &=& A = - (e\_[12]{}+e\_[34]{}) . \[instantfield\] Thus the gauge field is an *instanton*, as in the solution discussed in [@Martelli:2011fu]. In particular, with our choice of orientation the curvature is self-dual, and the on-shell gauge field action is finite. Since the stress-energy tensor of an instanton vanishes, the metric (\[TNAdS\]) is accordingly an Einstein metric. However, differently from the solution in [@Martelli:2011fu], one can check that this metric is *not* locally AdS$_4$. It is in fact a Euclidean version of the well-known Taub-NUT-AdS metric, with a special value of the mass parameter. This metric is locally asymptotically AdS$_4$, and therefore it can be interpreted holographically [@Chamblin:1998pz]. Notice that for $|s|\leq 1/(2g)$ the gauge field (\[gaugefield\]) is *real*, while for $|s|> 1/(2g)$ it is *purely imaginary*; the intermediate case with $|s|=1/(2g)$ has vanishing gauge field instanton and the metric reduces to Euclidean AdS$_4$.
For large $r$ the metric becomes \[approxmetric\] s\^2\_4 & & + r\^2 , while to leading order the gauge field reduces to \[A3\] A & & A\^[(3)]{} s \_3 . We see that the conformal boundary may be identified *precisely* with the metric (\[fametric\]), and the background gauge field with (\[newgau\]), by setting $s = \tfrac{1}{2gv}$. Recall here that in order to uplift to eleven-dimensional supergravity one should also set $g=1$ [@Martelli:2011fu]. Notice that when $|v|=1$ the boundary metric reduces to the round metric on $S^3$, and the background gauge field vanishes. Correspondingly, in the bulk the instanton field vanishes, and the metric becomes AdS$_4$.
Wick rotation and regularity {#wick-rotation-and-regularity .unnumbered}
----------------------------
Let us discuss briefly how this solution was obtained. The reader not interested in these details may safely jump to the discussion of the Killing spinors and the holographic free energy.
As we are interested in a 1/2 BPS solution, we may begin by appropriately Wick rotating the solution (2.1), (2.4) of [@AlonsoAlberca:2000cs]. We take their parameter $\aleph=+1$, so as to obtain a biaxially squashed $S^3$ as constant $r$ surface. The Wick rotation may then be taken to be $t\rightarrow {\mathrm{i}}\tau$, $N\rightarrow {\mathrm{i}}s$, $Q\rightarrow {\mathrm{i}}Q$, together with a change in sign of the metric. This leads to the following metric and gauge field s\^2\_4 &=& r\^2 + (r\^2-[s]{}\^2)(\^2+\^2\^2) + (+ 2 [s]{})\^2 ,\
\[intermedi\] A\^L &=& + , where (r) &=& g\^2 (r\^2-[s]{}\^2)\^2 + (1-4g\^2[s]{}\^2)(r\^2+[s]{}\^2)-2Mr + (P\^2-Q\^2) . \[genom\] This depends on the parameters $s,g,M,P,Q$. Notice we have kept a Lorentzian superscript on $A^L$ in (\[intermedi\]) – the reason for this will become clear momentarily.
For the 1/2 BPS solution of interest, the Euclideanized BPS equations of [@AlonsoAlberca:2000cs] imply that M\^2 &=& (1-4g\^2[s]{}\^2) ,\
[s]{}\^2 P(1-4g\^2[s]{}\^2) &=& sMQ - P(P\^2-Q\^2) ,\[BPS\] and the corresponding 1/2 BPS solution then depends on only *two* parameters. We take these to be $s$ and $Q$, with P & =& [s]{} ,M = -Q , \[jimllfixit\] then solving (\[BPS\]). The factors of ${\mathrm{i}}$ in (\[jimllfixit\]) may look problematic, but there are (at least) two different ways of obtaining real solutions. We require $s$ and $M$ to be real in order that the metric in (\[intermedi\]) is real. If $|s| \leq 1/(2g)$ then $P$ and $Q$ will be purely imaginary, and we may write $P={\mathrm{i}}p$, $Q=-{\mathrm{i}}q$ to obtain the *real* gauge field \[Wiki\] A && -A\^L = + .
Redefining $\tau = 2{s}\psi$, in terms of standard Euler angles $(\theta,\varphi,\psi)$ notice that the metric (\[intermedi\]) takes the form presented in (\[TNAdS\]), albeit with a more general form of the function $\Omega (r)$, given by (\[genom\]) and (\[jimllfixit\]). That (\[TNAdS\]) has only one free parameter $s$, and not the two we have above, follows from imposing *regularity* of the Euclidean metric. At any fixed $r>s$ that is not a root of $\Omega(r)$, we obtain a smooth biaxially squashed $S^3$ metric. In order to obtain a complete metric, the space must “close off” at the largest root $r_0$ of $\Omega(r)$, so that $\Omega(r_0)=0$. More precisely, if $r_0>s$ this should be a single root, while if $r_0=s$ the metric will be regular only if $r_0=s$ is a double root of $\Omega(r)$. We shall return to the former case in section \[bolty\], here focussing on the case $r_0=s$. The condition $\Omega(r_0=s)=0$ immediately fixes q &=& -[s]{} , so that now (see also [@Emparan:1999pm]) p &=& [s]{} = -q , M = [s]{}(1-4g\^2[s]{}\^2) . It is then in fact automatic that $r={s}$ is a double root of $\Omega$.
In conclusion, we end up with the metric (\[TNAdS\]), with $\Omega (r)$ given in (\[finalome\]), and gauge field (\[gaugefield\]). The gauge field is manifestly non-singular and one can check that the metric indeed smoothly closes off at $r={s}$, giving the topology $M_4={\mathbb{R}}^4$.
Killing spinors {#killing-spinors .unnumbered}
---------------
In this subsection we briefly discuss the supersymmetry of the Euclidean solution (\[TNAdS\]), (\[gaugefield\]), in particular reproducing the three-dimensional spinor equation (\[newks\]) asymptotically.
In Lorentzian signature the Killing spinor equation is (\[LKSE\]). However, in Wick rotating we have introduced a factor of ${\mathrm{i}}$ into the gauge field in (\[Wiki\]), so that $A^L={\mathrm{i}}A$. Thus the appropriate Killing spinor equation to solve in this case is \[funnyKSE\] &=& 0 . This possibility of Wick rotating the gauge field (or not) was also discussed in [@Dunajski:2010uv]. In particular, the authors of [@Dunajski:2010uv] pointed out that any Euclidean solution with a real gauge field that solves (\[funnyKSE\]) will automatically be 1/2 BPS. The reason is simple: if $\epsilon$ solves (\[funnyKSE\]), then so does its conjugate $\epsilon^c$. We shall see this explicitly below.
We introduce the following representation for the generators of Cliff$(4,0)$ \_4 & =&
0 & \_2\
-\_2 & 0
, \_ =
0 & \_\
\_& 0
, where $\alpha \in 1,2,3$, $\tau_\alpha$ are the Pauli matrices, and hats denote tangent space quantities. Decomposing the Dirac spinor $\epsilon$ into positive and negative chirality parts as &=& (
[c]{} \_+\
\_-
) , where $\epsilon_\pm$ are two-component spinors, it is then straightforward, but tedious, to verify that in the orthonormal frame (\[frame\]) \[spinors\] \_+ &=& (
[c]{} (r)\_+\
\^\*(r)\_-
) , \_- = (
[c]{} \^\*(r)\_+\
(r)\_-
) , is the general solution to the $\mu=r$ component of (\[funnyKSE\]), where $\chi_\pm$ are independent of $r$ and we have defined (r) & & (g(r+[s]{})-)\^[1/2]{} . If we now define the charge conjugate spinor $\epsilon^c \equiv B\epsilon^*$, where $B$ is the charge conjugation matrix defined in [@Martelli:2011fu], then it is straightforward to see that taking the conjugate $\epsilon\rightarrow \epsilon^c$ simply maps $\chi_+\rightarrow -\chi_-^*$, $\chi_-\rightarrow\chi_+^*$.
Let us analyze the large $r$ asymptotics of the Killing spinor equation (\[funnyKSE\]), and its solutions (\[spinors\]). We begin by expanding \_+ &=& r\^[1/2]{} ,\
\[spinorexpand\] \_- &=& r\^[1/2]{} , where we have defined the $r$-independent two-component spinor \[chispinor\] & & (
[c]{}\_+\
\_-
) . We then write the asymptotic expansion of the metric as s\^2\_4 &=& + ,\
\[threemetric\] s\^2\_3 && g\^2 . It is then straightforward to extract the coefficient of $r^{1/2}$ in the Killing spinor equation (\[funnyKSE\]). One finds that the positive and negative chirality projections lead to the *same* equation for $\chi$, namely \[3dkse\] \_\^[(3)]{}+ gA\^[(3)]{}\_- \_- \_\_3 & = & 0 , where $\nabla^{(3)}$ denotes the spin connection for the three-metric (\[threemetric\]), and $A^{(3)}$ is defined in (\[A3\]). Using the explicit form for $A^{(3)}$ in (\[A3\]), the identity $\gamma_\alpha\gamma_\beta=\gamma_{\alpha\beta}
+g^{(3)}_{\alpha\beta}$, and recalling that $s=1/(2gv)$, $g=1$, we precisely obtain the spinor equation (\[newks\]). Finally, one can verify that the $d=4$ spinors (\[spinors\]), with $\chi$ satisfying (\[3dkse\]), do indeed solve (\[funnyKSE\]).
The holographic free energy {#the-holographic-free-energy .unnumbered}
---------------------------
The holographic free energy of the Taub-NUT-AdS solution was discussed in [@Emparan:1999pm], but of course in this latter reference there was no instanton field, which is crucial for supersymmetry. The calculation proceeds essentially as in section 2.5 of [@Martelli:2011fu], except for the following caveat. The integrability condition for the Killing spinor equation (\[funnyKSE\]) gives the equations of motion following from the action S\_ &=& -\^4x(R + 6g\^2 + F\^2) , which has *opposite* (relative) sign for the gauge field term compared with (\[4dSUGRA\]) (see also [@Dunajski:2010uv]). This is clear from the fact that our equation (\[funnyKSE\]) was obtained from the Lorentzian form of the equation by sending $A\to {\mathrm{i}}A$. It is therefore natural to expect that in the computation of the holographic free energy we have to evaluate the action $S_{\mathrm{Euclidean}}$ on shell.
Setting $g=1$ and cutting off the space at $r=R$, the bulk gravity contribution is given by I\^\_ &=& \^4 x = - + . Denoting by $R[\gamma]$ the scalar curvature of the boundary metric, and by $K$ the trace of its second fundamental form, the combined gravitational boundary terms I\_\^+I\_\^& = & \^3x ( 2 + R \[\] - K ) \[gravbdry\] have the following asymptotic expansion I\_\^+I\_\^& = & - + + + [O]{}(1/R) , where in particular notice there is a non-zero *finite* contribution. The instanton action is[^2] I\^F\_ &=& -\^4 x F\_F\^ = - . Therefore the total on-shell action $S_{\mathrm{Euclidean}}$, obtained after removing the cut-off ($R \to \infty$), is given by I & = & I\^\_ + I\_\^+I\_\^+ I\^F\_ = . Since the round sphere result[^3] is ${s}=1/2$, we thus see that I\_[s]{} & =& = (2[s]{})\^2 I\_[s=1/2]{} , which since $v=1/(2s)$ precisely agrees with the field theory result (\[newfree\]).
A supersymmetric Eguchi-Hanson-AdS solution {#bolty}
===========================================
In the previous section the Taub-NUT-AdS solution existed for both $1-4g^2{s}^2\geq 0$ and $1-4g^2{s}^2\leq 0$, with the sign determining whether the gauge field is real or purely imaginary, in a fixed choice of Wick rotation. However, in this section we consider a different solution which exists only when $1-4g^2{s}^2\leq 0$, or equivalently $|s|\geq 1/(2g)$. In this case the Euclidean supersymmetry equation takes the same form as the Lorentzian equation (\[LKSE\]), namely &=& 0 . \[KSE2\] We will show that there is a one-parameter family of regular solutions in this class, of topology $M_4=T^*S^2$, for which there are Killing spinors solving (\[KSE2\]).
When $|s|\geq 1/(2g)$ we may rewrite (\[jimllfixit\]) as \[realBPS\] P &=& -[s]{} , M = Q , which are now *real*. Again setting $\tau=2{s}\psi$, the metric takes the form given in (\[TNAdS\]) where now \[unlikely\] (r) &=& g\^2(r\^2-[s]{}\^2)\^2-\^2 . It will be useful to note that the four roots of $\Omega(r)$ in (\[unlikely\]) are \[roots\] {
[c]{} r\_4\
r\_3
} &=& ,\
{
[c]{} r\_2\
r\_1
} &=& . Notice that these are all complex if $|s|< 1/(2g)$. The gauge field is given by (after a suitable gauge transformation) \[noninstanton\] A &=& - \_3 . As $r\rightarrow \infty$ this tends to \[globalpart\] A && A\^[(3)]{} -s\_3 , which is (up to analytic continuation) what we had in the previous example (\[A3\]).
Killing spinors {#killing-spinors-1 .unnumbered}
---------------
Taking the same Clifford algebra and spinor conventions as the previous section, and again using the orthonormal frame (\[frame\]), one can verify that the integrability condition for the Killing spinor equation (\[KSE2\]) leads to the algebraic relation \_-&=& (
[cc]{} & 0\
0 &
)\_+ . Here recall that $\epsilon_\pm$ are two-component spinors, and $r_i$, $i=1,2,3,4$, are the four roots of $\Omega$ in (\[roots\]). Substituting into the $\mu=r$ component of (\[KSE2\]) then leads to decoupled first order ODEs, which may be solved to give \[spinors2\] \_+ &=& (
[c]{} \_+\
\_-
) , \_- = (
[c]{} \_+\
\_-
) , where $\chi_\pm$ are independent of $r$. The large $r$ expansion of these is given by \_+ &=& r\^[1/2]{} ,\
\_- &=& r\^[1/2]{} , where the two-component spinor $\chi$ is again given by (\[chispinor\]). Notice this is the same as (\[spinorexpand\]), up to analytic continuation. Again using the metric expansion and three-metric in (\[threemetric\]), we may extract the coefficient of $r^{1/2}$ in (\[KSE2\]). A very similar computation to that in the previous section then leads to the three-dimensional Killing spinor equation \[3dkse2\] \_\^[(3)]{}- \_+ g A\^[(3)]{}\_\_\^[ ]{}& = & 0 . Setting $g=1$ and again identifying the squashing parameter $v=1/(2s)$, notice this is identical to our original equation (\[newks\]), but where we have replaced $A^{(3)}\rightarrow -{\mathrm{i}}A^{(3)}$. Of course, given the relative difference in Wick rotations of the gauge field in two the cases, this was precisely to be expected. In fact, comparing the $A^{(3)}$ (\[globalpart\]) in this section with its counterpart (\[A3\]) in the previous section, we see that equation (\[3dkse2\]) is in fact *identical* to (\[newks\]), due to the factor of ${\mathrm{i}}$ difference in (\[globalpart\]), (\[A3\]).
The solution to (\[3dkse2\]) is therefore given by an appropriate analytic continuation of the solution presented in [@Imamura:2011wg], and reads \[3dspin\] &=& \^[\_3/2]{} \^[-1]{}\_0 , where $\mathtt{g}\in SU(2)$, $\chi_0$ is a constant two-component spinor, and v &=& , where $v=1/(2s)$. In terms of Euler angles $(\psi,\theta,\varphi)$, recall that \[su2\] &=& (
[cc]{}\^[(+)/2]{} & \^[-(-)/2]{}\
-\^[(-)/2]{} & \^[-(+)/2]{}
) .
Regularity of the metric {#regularity-of-the-metric .unnumbered}
------------------------
We must again consider regularity of the metric (\[TNAdS\]). A complete metric will necessarily close off at the largest root $r_0$ of $\Omega(r)$, which must satisfy $r_0\geq s$. From (\[roots\]) we see that either $r_0=r_+$ or $r_0=r_-$, where it is convenient to define \[rpm\] r\_+ & & r\_4 , r\_- r\_2 . the coordinate $\psi$ must have period $2\pi/n$, for some positive integer $n$, so that the surfaces of constant $r$ are Lens spaces $S^3/{\mathbb{Z}}_n$. Assuming that $r_0>s$ is strict, then the metric (\[TNAdS\]) will have the topology of a complex line bundle $M_4=\mathcal{O}(-n)\rightarrow S^2$ over $S^2$, where $r-r_0$ is the radial direction away from the zero section.
Regularity of the metric near to the $S^2$ zero section at $r=r_0$ requires \[regularity\] || &=& . This conditon ensures that near to $r=r_0$ the metric (\[TNAdS\]) takes the form s\^2\_4 && \^2 + \^2\^2 + (r\_0\^2-s\^2)(\^2+\^2\^2) , near to $\rho=0$. Here note that $n\psi/2$ has period $2\pi$. Imposing (\[regularity\]) at $r_0=r_\pm$ gives \[Qvalues\] Q &=& Q\_([s]{}) . In turn, substituting $Q=Q_\pm(s)$ into (\[rpm\]) one then finds \[rootschoice\] r\_(Q\_([s]{})) &=& . Recall that in order to have a smooth metric, we require $r_0>s$. Imposing this for $r_0=r_\pm(Q_\pm({s}))$ gives r\_(Q\_([s]{})) - s &=& f\_n\^(2gs) , where the function f\_n\^(x) && -x is required to be *positive* for a smooth metric with $s=x/(2g)$. Notice here that $s\geq 1/(2g)$ implies $x\geq 1$. It is straightforward to show that $f_n^-(x)$ is monotonic decreasing on $x\in [1,\infty)$. For simplicity here we will restrict our attention to $n\leq 2$.[^4] The analysis then splits into the cases $\{n=1\}$, $\{n=2\}$, which have a qualitatively different behaviour:
### $n=1$ {#n1 .unnumbered}
It is easy to see that $f_1^\pm(x)< 0$ on $x\in [1,\infty)$, and thus the metric (\[TNAdS\]) cannot be made regular in this case. Specifically, $f_1^\pm(1)=-1/2$: since $f_1^-(x)$ is monotonic decreasing, this rules out taking $r_0=r_-(Q_-(s))$ given by (\[rootschoice\]); on the other hand $f_1^+(x)$ monotonically increases to zero from below as $x\rightarrow\infty$, and we thus also rule out $r_0=r_+(Q_+(s))$ in (\[rootschoice\]).
### $n=2$ {#n2 .unnumbered}
It is easy to see that $f_2^-(x)< 0$ for $x\in (1,\infty)$, while $f_2^+(x)>0$ on the same domain, which means we must set \[Qset\] Q && Q\_+(s) = - , and \[r0\] r\_0(s) & = & , may then be shown to be the largest root of $\Omega(r)$, for all $s\geq 1/(2g)$. In particular, this involves showing that $r_0(s)-r_-(Q_+(s))>0$ for all $s\geq 1/(2g)$, which follows since r\_0(s)-r\_-(Q\_+(s)) &=& h(2gs) , where we have defined h(x) & & + 2- . It is a simple exercise to prove that $h(x)>0$ on $x\in (1,\infty)$.
After this slightly involved analysis, for $n=2$ we end up with a smooth complete metric on $M_4=T^*S^2$, given by (\[TNAdS\]), (\[unlikely\]) with $Q=Q_+(s)$ given by (\[Qset\]), for all $s> 1/(2g)$. The $S^2$ zero section is at $r=r_0(s)$ given by (\[r0\]). The metric is thus of Eguchi-Hanson-AdS type, although we stress that it is *not* Einstein for any $s>1/(2g)$. The large $r$ behaviour is again given by (\[approxmetric\]), so that the conformal boundary is a squashed $S^3/{\mathbb{Z}}_2$. The $s=1/2g$ limit gives a round $S^3/{\mathbb{Z}}_2$ at infinity with the bulk metric being the singular AdS$_4/{\mathbb{Z}}_2$, albeit with a non-trivial torsion gauge field, as we shall see momentarily.
It follows that another interesting difference to the Taub-NUT-AdS solution of the previous section is that the gauge field (\[noninstanton\]) no longer has (anti)-self-dual field strength $F={\mathrm{d}}A$; moreover, the latter has a non-trivial flux. Indeed, although the gauge potential in (\[noninstanton\]) is *singular* on the $S^2$ at $r=r_0$, one can easily see that the field strength $F={\mathrm{d}}A$ is a globally defined smooth two-form on our manifold. One computes the period of this through the $S^2$ at $r_0(s)$ to be \[fluxy\] \_[S\^2]{} F &=& -\
&=& 1 , the last line simply being a remarkable identity satisfied by the largest root $r_0(s)$. Setting $g=1$, we thus see that we have precisely one unit of flux through the $S^2$! It follows that the gauge field $A$ is a connection on the non-trivial line bundle $\mathcal{O}(1)\rightarrow T^*S^2$. The corresponding first Chern class $c_1=[F/2\pi]\in H^2(T^*S^2;{\mathbb{Z}})\cong {\mathbb{Z}}$ is the generator of this group. Moreover, the map $H^2(T^*S^2;{\mathbb{Z}})\rightarrow H^2(S^3/{\mathbb{Z}}_2;{\mathbb{Z}})\cong{\mathbb{Z}}_2$ that restricts the gauge field to the conformal boundary is reduction modulo $2$. Hence at infinity the background gauge field is more precisely given by the global one-form (\[globalpart\]) *plus* the flat non-trivial Wilson line that represents the element $1 \in H^2(S^3/{\mathbb{Z}}_2;{\mathbb{Z}})\cong H_1(S^3/{\mathbb{Z}}_2;{\mathbb{Z}})\cong{\mathbb{Z}}_2$. One would be able to see this explicitly by writing the gauge field $A$ as a one-form that is locally well-defined in coordinate patches, and undergoes appropriate gauge transformations between these coordinate patches. It follows that the gauge field at infinity is more precisely a connection on the non-trivial torsion line bundle over $S^3/{\mathbb{Z}}_2$.
The holographic free energy {#the-holographic-free-energy-1 .unnumbered}
---------------------------
Although we will not pursue the holographic interpretation of this solution in the present paper, below we will compute its holographic free energy using standard formulas. Since the gauge field here is real, the relevant action is the Euclidean action with standard signs S\_ &=& -\^4x(R + 6g\^2 - F\^2) . Notice that upon taking the trace of the Einstein equation, we see that *all* solutions (supersymmetric or not) of $d=4$ gauged supergravity are metrics with constant scalar curvature $R = - 12 g^2$. Using this, a straightforward calculation then gives for the total (bulk plus boundary) gravity part a finite result, after sending the cut-off $r=R\to \infty$. Namely, after setting $g=1$ we get I\^\_ = I\^\_ +I\_\^+ I\_\^& =&\
& - & , where we note that the contribution on the second line comes entirely from the boundary terms. Although the gauge field is not (anti-)-self-dual, it is straightforward to compute its on-shell action, which is still finite, namely we get I\^F\_ &=& - . Therefore for the total on-shell action we obtain I & = & + (s\^2 - )\^[3/2]{} . \[predict\] Notice this makes sense for any $s>1/2$. Moreover, in the $s\to 1/2$ limit the second term vanishes and we are left with a result that is the same as that for the round three-sphere $S^3$. This might seem a contradiction, but in fact if we look back at where this result comes from, we see that in this limit \_[s1/2]{} I\^\_ & = & , which is the correct contribution expected from the (singular) AdS$_4/{\mathbb{Z}}_2$ solution with round $S^3/{\mathbb{Z}}_2$ boundary. However, we get an equal non-zero contribution from the gauge field action \_[s1/2]{} I\^F\_ & =& , despite the fact that the gauge field curvature $F\to 0$ in this limit. The calculation captures correctly the contribution from the flat torsion gauge field, which indeed cannot be turned off continuously since in the bulk has one unit of flux through the vanishing $S^2$ at the ${\mathbb{Z}}_2$ singularity. More precisely, the complement of the singular point has toplogy ${\mathbb{R}}_+\times S^3/{\mathbb{Z}}_2$, and the gauge field is a flat connection on the non-trivial torsion line bundle over this.
Discussion {#discussione}
==========
In this letter we have extended the results of [@Martelli:2011fu], discussing a new class of supersymmetric solutions of $d=4$, $\mathcal{N}=2$ gauged supergravity, which in turn uplift to solutions of eleven-dimensional supergravity. The solutions in section \[nuts\] provide the holographic duals to ${\cal N}=2$ supersymmetric gauge theories on the background of a biaxially squashed three-sphere and a $U(1)$ gauge field, whose localized partition function was recently computed in [@Imamura:2011wg]. In particular, as in [@Martelli:2011fu], we have shown that the bulk metric, gauge field, and Killing spinors reduce precisely to their field theory counterparts on the boundary. Moreover, the holographic free energy is identical to the leading large $N$ contribution to the field theoretic free energy computed from the quiver matrix model. The solution is a special case of the general class of supersymmetric Plebanski-Demianski solutions [@AlonsoAlberca:2000cs], but it differs from the solution discussed in [@Martelli:2011fu] in various respects. The graviphoton field is again an instanton, hence the bulk metric is Einstein, but it is not now diffeomorphic to AdS$_4$. The results of [@Martelli:2011fu], and of this letter, suggest that the AdS/CFT correspondence is a useful setting for studying supersymmetric gauge theories on curved backgrounds.
We conclude noting that although the results presented here share a number of similarities with those in [@Chamblin:1998pz; @Hawking:1998ct], there are some crucial differences that are worth summarizing. In contrast to the solutions we have discussed, the AdS-Taub-NUT and AdS-Taub-Bolt solutions in [@Chamblin:1998pz; @Hawking:1998ct] are *not* supersymmetric, and moreover no gauge field was turned on. In addition, while those solutions have the same biaxially squashed $S^3$ boundary, the boundary of our Eguchi-Hanson-AdS solution has the different topology $S^3/{\mathbb{Z}}_2$. Therefore, although we have computed the free energy for both families, it does not make sense to compare them along the lines of [@Chamblin:1998pz; @Hawking:1998ct]. It would be very interesting to understand the precise field theory dual interpretation of the Eguchi-Hanson-AdS solution discussed here.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Jan Gutowski for a useful discussion. D. M. is supported by an EPSRC Advanced Fellowship EP/D07150X/3 and J. F. S. by a Royal Society University Research Fellowship.
[99]{}
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[^1]: This is denoted $N$ in [@AlonsoAlberca:2000cs].
[^2]: Notice that when $1-4s^2 \geq 0$ this term becomes negative. The calculation is however valid for any value of $s>0$.
[^3]: To recover the result for $S^2\times S^1$ boundary, one should first change coordinates back to the form in (\[intermedi\]), and then set ${s}=0$ there. In these coordinates, with $\tau \in [0, 2\pi]$ the gravitational contribution to the free energy is half that of the round sphere.
[^4]: In the first version of this paper it was argued that $n>2$ breaks supersymmetry; however, this is incorrect.
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The cross-section for the process $e^+e^-\rightarrow W^+W^-$ has been measured with the data sample collected by DELPHI at an average centre-of-mass energy of 189 GeV and corresponding to an integrated luminosity of 155 pb$^{-1}$. Based on the 2392 events selected as $WW$ candidates, the cross-section for the doubly resonant process $e^+e^-\rightarrow W^+W^-$ has been measured to be $15.83 \pm 0.38~\mbox{(stat)} \pm 0.20~\mbox{(syst)~pb}$. The branching fractions of the $W$ decay were also measured and found to be in good agreement with the Standard Model expectation. From these a value of the CKM mixing matrix element $ |V_{cs}| = 1.001 \pm 0.040~\mbox{(stat)} \pm 0.020~\mbox{(syst)}$ was derived.
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Introduction
============
The cross-section for the doubly resonant production of $W$ bosons has been measured with the data sample collected by DELPHI at the average centre-of-mass energy of $188.63 \pm 0.04$ GeV. Depending on the decay mode of each $W$ boson, fully hadronic, mixed hadronic-leptonic (“semileptonic”) or fully leptonic final states were obtained, for which the Standard Model branching fractions are 45.6%, 43.9% and 10.5%, respectively. The detector was essentially unchanged compared to previous years and detailed descriptions of the DELPHI apparatus and its performance can be found in [@DET; @PERF]. The luminosity was measured using the Small Angle Tile Calorimeter [@STIC]. The total integrated luminosity corresponds to 155 pb$^{-1}$; its systematic error is estimated to be $\pm 0.6\%$, which is dominated by the experimental uncertainty on the Bhabha measurements of $\pm 0.5\%$. The luminosities used for the different selections correspond to those data for which all elements of the detector essential to each specific analysis were fully functional. The criteria for the selection of $WW$ events are reviewed in section 2. Generally they follow those used for the cross-section measurements at lower centre-of-mass energies [@wwpap161; @wwpap172; @wwpap183], but for the 4-jet final state a more efficient selection has been applied using a neural network, and the selection of leptons has been modified to improve the efficiency for $\tau$ leptons. In section 3 the total cross-section and the branching fractions of the $W$ boson are presented.
The cross-sections determined in this analysis correspond to $W$ pair production through the three doubly resonant tree-level diagrams (“CC03 diagrams” [@CC03]) involving $s$-channel $\gamma$ and $Z$ exchange and $t$-channel $\nu$ exchange. The selection efficiencies were defined with respect to these diagrams only and were determined using the full simulation program DELSIM [@DELSIM] with the PYTHIA 5.7 event generator [@PYTHIA]. In addition to the production via the CC03 diagrams, the four-fermion final states corresponding to some decay modes may be produced via other Standard Model diagrams involving either zero, one, or two massive vector bosons. Corrections which account for the interference between the CC03 diagrams and the additional diagrams are generally expected to be negligible at this energy, except for final states with electrons or positrons. In these cases correction factors were determined from simulation using the 4-fermion generator EXCALIBUR [@EXCALIBUR] and were found to be consistent with unity within an uncertainty of $\pm 2$%.
Event selection and cross-sections {#sec:cs}
==================================
Fully hadronic final state
--------------------------
A feed-forward neural network (see e.g. [@rojas; @bishop]) was used to improve the selection quality of $W^{+} W^{-} \rightarrow q
\overline{q} q \overline{q}$ from 2-fermion (mainly $Z/\gamma
\rightarrow q \overline{q}$) and 4-fermion background (mainly $Z Z
\rightarrow q\bar{q}x\bar{x}$). Compared to an analysis based on sequential cuts [@wwpap183], the selection efficiency for the signal was increased by about 10% for the same purity. The network is based on the JETNET package [@jetnet], uses the standard back-propagation algorithm, and consists of three layers with 13 input nodes, 7 hidden nodes and one output node.
A preselection of the events was performed with the following criteria:
- [a reconstructed effective centre-of-mass energy [@SPRIME] $\sqrt{s'} > 115$ GeV;]{}
- [4 or more reconstructed jets when clustering with LUCLUS [@LUCLUS] at $d_{join} = 4.0~{\mathrm GeV}/c$;]{}
- [total particle multiplicity $\geq 3$ for each jet.]{}
Each event was forced into a 4-jet configuration. The following jet and event observables were chosen as input variables, taking into account previous neural network studies to optimize input variables for the $WW$ and 2-fermion separation [@somap]:
1. [the difference between the maximum and minimum jet energy after a 4C fit, imposing 4-momentum conservation on the event;]{}
2. [the minimum angle between two jets after the 4C fit;]{}
3. [the value of $d_{join}$ from the cluster algorithm LUCLUS for the migration of 4 jets into 3 jets;]{}
4. [the minimum particle multiplicity of all jets;]{}
5. [the reconstructed effective centre-of-mass energy $\sqrt{s'}$;]{}
6. [the maximum probability (for all possible jet pairings) for a 2C fit (two objects with $W$-mass);]{}
7. [thrust;]{}
8. [sphericity;]{}
9. [the mean rapidity of all particles with respect to the thrust axis;]{}
10. [the sum of the cubes of the magnitudes of the momenta of the 7 highest momentum particles $\sum_{i=1}^{7} |\vec{p}_{i}|^{3}$;]{}
11. [the minimum jet broadening $B_{min}$ [@catani];]{}
12. [the Fox-Wolfram-moment H3 [@fox];]{}
13. [the Fox-Wolfram-moment H4.]{}
The training of the neural network was performed with 3500 signal events ($WW{\rightarrow}{\mbox{${q\bar{q}} $}}{\mbox{${q\bar{q}} $}}$) and 3500 $Z/\gamma{\rightarrow}{\mbox{${q\bar{q}} $}}$ background events simulated with the PYTHIA 5.7 event generator. Afterwards the network output was calculated for other independent samples of simulated $WW$, $Z/\gamma$ and $ZZ$ events and for the real data. Figure \[fig1\] shows the distribution of the neural network output for data and simulated events.
For each bin the fractional efficiencies of the selection of $WW$ decays and the background contributions were estimated from simulation. Events were finally retained if the neural network output variable was larger than -0.5. The resulting selection efficiencies for the $WW$ channels are listed in the second column of Table \[qqxx\] together with the estimated backgrounds.
--------------------------------------------------------------------------------------------- ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ -- --
channel ${\mbox{$ j j j j $}}$ ${\mbox{$ j j e \nu $}}$ ${\mbox{$ j j \mu \nu $}}$ ${\mbox{$ j j \tau \nu $}}$
${\mbox{${q\bar{q}} $}}{\mbox{${q\bar{q}} $}}$ [**0.887**]{} 0. 0. 0.018
${\mbox{$ {\mbox{${q\bar{q}} $}}e \nu $}}$ 0.009 [**0.661**]{} 0. 0.115
${\mbox{$ {\mbox{${q\bar{q}} $}}\mu \nu $}}$ 0.004 0. [**0.865**]{} 0.056
${\mbox{$ {\mbox{${q\bar{q}} $}}\tau \nu $}}$ 0.028 0.039 0.034 [**0.491**]{}
background (pb) 1.788 0.159 0.075 0.437
selected events 1298 259 328 324
luminosity (pb$^{-1}$)
--------------------------------------------------------------------------------------------- ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ -- --
: \[qqxx\] Selection efficiencies, background and data for the hadronic and semileptonic final states.
A relative uncertainty on the efficiency of $\pm 0.6\%$ was estimated from the following studies:
- [Comparison of the simulation with real data, which were taken at a centre-of-mass energy of 91.2 GeV with the same detector and trigger configuration and analysed with the same reconstruction software as the 189 GeV data. For this comparison, the technique of mixing Lorentz-boosted $Z$ events was used, which transformed two independent hadronic $Z$ decays into a pseudo $W$ pair event by applying an appropriate boost to the particles of each $Z$ decay. In this study a difference of 0.17% was observed.]{}
- [Variation of the efficiency using different hadronization models (JETSET 7.4 [@PYTHIA] and ARIADNE [@ARIADNE]) giving 0.59%. Within this error variations of the efficiency from different modelling of Bose-Einstein correlations in the generator were found to be negligable; this is also expected for final state interactions between quarks from different W bosons (“Colour Reconnection”).]{}
The cross-section for the expected total background was estimated from the simulations to be $(1.79 \pm 0.09)$ pb. The main contribution (1.42 pb) comes from ${\mbox{${q\bar{q}(\gamma)} $}}$ events with gluon radiation, the rest from non-$WW$ 4-fermion final states. The systematic uncertainty on the background was estimated from the variation of the selection efficiencies for the different backgrounds when different hadronization models were used (JETSET 7.4 and ARIADNE). A total uncertainty of $\pm 5\%$ was estimated from this variation (4.5%) and from differences between data and simulation due to imperfections of the generator models. Furthermore, the influences of the different parameters of the neural network structure, its learning algorithm, and of the preselection have been investigated and found to be negligible.
A total of 1298 events were selected in the data sample. The cross-section for fully hadronic events was obtained from a binned maximum likelihood fit to the distribution of the neural network output variable above -0.5, taking into account the expected background in each bin. The result is
$$\sigma_{WW}^{qqqq}= \sigma_{WW}^{tot} \times
{\mathrm{BR}}(WW\rightarrow {\mbox{${q\bar{q}} $}}{\mbox{${q\bar{q}} $}})
= 7.36 \pm 0.26~\mbox{(stat)} \pm 0.10~\mbox{(syst)} ~~\mbox{pb},$$
where BR$(WW\rightarrow {\mbox{${q\bar{q}} $}}{\mbox{${q\bar{q}} $}})$ is the probability for the $WW$ pair to give a purely hadronic final state. The first error is statistical and the second is systematic. The systematic error includes contributions from the uncertainties on efficiency and background, and on the luminosity.
Semileptonic final state
------------------------
Events in which one of the $W$ bosons decays into $l\nu$ and the other one into quarks are characterized by two hadronic jets, one isolated lepton (coming either directly from the $W$ decay or from the cascade decay $W \rightarrow \tau\nu \rightarrow e\nu\nu\nu \: {\mathrm or} \: \mu\nu\nu\nu$) or a low multiplicity jet due to a $\tau$ decay, and missing momentum resulting from the neutrino. The major background comes from ${\mbox{${q\bar{q}(\gamma)} $}}$ production and from four-fermion final states containing two quarks and two leptons of the same flavour.
Events were required to show hadronic activity (at least 6 charged particles), to have a total visible energy of at least 50 GeV and to be compatible with a 3-jet topology on application of the LUCLUS [@LUCLUS] clustering algorithm with a value of $d_{join}$ between 2 and 20 GeV/$c$. A lepton had to be found in the event according to one of the following criteria:
- [a single charged particle identified as described in [@wwpap172] as an electron or a muon, and with at least 5 GeV of energy;]{}
- [a single charged particle of momentum, $p$, above 5 GeV/$c$, not identified as a lepton, but isolated from other particles by $p\cdot\theta_{iso}>100~{\mathrm GeV}/c\cdot$degrees, where $\theta_{iso}$ is the angle formed with the closest charged particle with a momentum of at least $1~{\mathrm GeV}/c$;]{}
- [a low multiplicity jet (with less than 4 charged particles) with an energy above 5 GeV, polar angle of its axis with respect to the beam axis between $20{^\circ}$ and $160{^\circ}$, and with at least 15% of its energy carried by charged particles.]{}
When several candidate leptons with the same flavour were found in the event, the one with highest $p\cdot\theta_{iso}$ (single-track case) or smallest opening angle (jet case) was chosen. Tests on $WW$ simulation show that the correct lepton was chosen in more than 99% of the ambiguous events.
Events not compatible with a 3-jet topology or with the lepton too close to fragmentation products or inside a hadronic jet were recovered by looking for particles inside jets with energy above 30 GeV and identified as electrons or muons. In this case additional cuts were imposed on the impact parameter of the lepton with respect to the beam spot and on the angles which the missing momentum formed with respect to the beam direction and the lepton itself.
The background contribution arising from the radiative return to the $Z$ peak was highly suppressed by rejecting events with the direction of the missing momentum close to the beam axis or with a detected photon with an energy above 50 GeV. The cut on the polar angle of the missing momentum was tighter for $q\bar{q}\tau\nu$ candidate events. The four-fermion neutral current background ($q\bar{q}l\bar{l}$) was suppressed by rejecting those events in which a second isolated and energetic lepton of the same flavour as the main candidate was found. Non-resonant contributions to the $q\bar{q}l\nu$ final state were reduced by requiring the invariant mass of the hadronic system to be larger than $20~{\mathrm GeV}/c^2$ and of the lepton-missing momentum system to be larger than $10~{\mathrm GeV}/c^2$.
The different leptonic decays were classified in the following way:
- $WW\rightarrow q\bar{q}\mu\nu$
The lepton was identified as a muon. The contamination from $q\bar{q}\tau\nu$ with $\tau \rightarrow \mu\nu\nu$ was suppressed by requiring that, if the muon momentum was below $45~{\mathrm GeV}/c$, either the missing mass in the event was below $55~{\mathrm GeV}/c^2$, or the fitted mass from a 2C kinematic fit with both $W$ bosons constrained to have the same mass was above $75~{\mathrm GeV}/c^2$.
- $WW\rightarrow q\bar{q}e\nu$
The lepton was identified as an electron. A cut on the aplanarity of the event, defined as the angle between the lepton direction and the plane of the two jets, was applied in order to reduce radiative and non-radiative QCD background. The contribution from the process $ee\rightarrow Zee$ was reduced by imposing requirements on the invariant mass of the electron and the missing momentum (assumed to be a single electron in the beam pipe). A procedure identical to that just described for the muon channel was then applied to reduce the contamination of $q\bar{q}\tau\nu$ events.
- $WW\rightarrow q\bar{q}\tau\nu$
The event was not classified as an electron or a muon decay. In order to suppress the background from $e^+e^-$ annihilations into $q\bar{q}(\gamma)$, events containing a 1-prong candidate $\tau$ had to have aplanarity above $20{^\circ}$. For multi-prong $\tau$ events a cut on the effective centre-of-mass energy $\sqrt{s'}$ was added, requiring it to be between 105 and 175 GeV. In both cases the hadronic system was rescaled to the beam energy and cuts were applied to reject very low (less than $15~{\mathrm GeV}/c^2$) and very high (more than $90~{\mathrm GeV}/c^2$) invariant masses in the resulting jet-jet system or lepton-missing momentum system.
Figure \[fig2\] shows the distribution of the momentum of the selected leptons compared with the expectations from the simulation of signal and backgrounds. The numbers of selected events, efficiencies and backgrounds for each lepton flavour are shown in Table \[qqxx\]. The efficiencies include corrections to account for an imperfect description of the misidentification of electrons or muons as taus in the simulation, for which an uncertainty of 1.0% for electrons and 0.6% for muons has been taken into account in the determination of the $W$ branching ratios. The overall efficiency of the selection of $WW\rightarrow q\bar{q}l\nu$ events was estimated to be $(75.4~\pm~0.7)$%, varying significantly with lepton type: 92% for $\mu$, 78% for $e$ and 56% for $\tau$ (see Table \[qqxx\]). The total expected background was estimated to be $(671~\pm~40)$ fb. The errors on efficiency and background include all systematic uncertainties.
A total of 911 events were selected as semileptonic $W$ decays; the number of events observed in the different lepton channels was found to be consistent with lepton universality. With the values given in Table \[qqxx\] for selected events, efficiencies and backgrounds, and assuming lepton universality, a likelihood fit yields a cross-section: $$\sigma_{WW}^{l\nu qq} = \sigma_{WW}^{tot}
\times BR(WW \rightarrow l\nu {\mbox{${q\bar{q}} $}}) =
6.77\pm0.26(\mbox{stat})\pm 0.12(\mbox{syst})~\mbox{pb}.$$ The systematic error includes contributions from efficiency and background, four-fermion interference in the electron channel, imperfect track reconstruction description in the simulation and uncertainty in the luminosity measurement.
Fully leptonic final state
--------------------------
Events in which both $W$ bosons decay into $l\nu$ are characterized by low multiplicity, a clean two-jet topology with two energetic, acollinear and acoplanar leptons of opposite charge, and with large missing momentum and energy. The relevant backgrounds are di-leptons from $e^+e^- \rightarrow Z (\gamma)$, Bhabha scattering, two-photon collisions, $Z e^+e^-$, $ZZ$ and single $W$ events.
The selection was performed in three steps. First a leptonic preselection was made followed by $\mu$/$e$/$\tau$ identification in both jets. Finally different cuts were applied for each channel to reject the remaining background, which was different in each case.
The leptonic preselection aimed to select a sample enriched in leptonic events. All particles in the event were clustered into jets using the LUCLUS algorithm [@LUCLUS] ($d_{join}=6.5~{\mathrm GeV}/c$) and only events with two reconstructed jets, containing at least one charged particle each, were retained. A charged particle multiplicity between 2 and 6 was required and at least one jet had to have only one charged particle. The leading particle (that with the largest momentum) in each jet was required to have polar angle $\mid \cos\theta_l \mid < 0.98$. In order to reduce the background from two-photon collisions and radiative di-lepton events, the event acoplanarity, $\theta_{acop}$, defined as the acollinearity of the two jet directions projected onto the plane perpendicular to the beam axis, had to be above $5{^\circ}$. In addition, the total momentum transverse to the beam direction, $P_t$, had to exceed 4% of the centre-of-mass energy $\sqrt{s}$. The associated electromagnetic energy for both leading particles was required to be less than $0.4\cdot \sqrt{s}$ to reject Bhabha scattering.
For this sample each particle was identified as $\mu$, $e$ or hadron. Slightly different criteria for lepton identification were applied, depending on whether the particle was in the barrel region ($43^\circ<\theta<137^\circ$), in the forward region ($\theta<37^\circ$ or $\theta>143^\circ$), or in between. A particle was identified as a muon if at least one hit in the muon chambers was associated to it, or if it had deposited energy in the outermost layer of the hadron calorimeter; in addition the energy deposited in the other layers had to be compatible with that from a minimum ionizing particle. For the identification of a particle as an electron the energies deposited in the electromagnetic calorimeters, in the different layers of the hadron calorimeter, and in addition the energy loss in the time projection chamber were used. A lepton was identified as a cascade decay from $W \rightarrow \tau \nu_\tau$ if the momentum was lower than $20~{\mathrm GeV}/c$.
After the preselection and the channel identification, different cuts were applied depending on the channel in order to reject the remaining background. For all channels except $WW \rightarrow \mu \nu \mu \nu$, the visible energy of the particles with $\mid \cos\theta \mid<0.9$ had to exceed $0.06 \cdot \sqrt{s}$. For all channels with at least one $W$ decaying into $\tau \nu$, the invariant mass of each jet had to be below $3~{\mathrm GeV}/c^2$, the momentum of the leading particle of a candidate $\tau$ jet below $0.4 \cdot \sqrt{s}$, and $\theta_{acop}$ above $9{^\circ}$. In addition the following criteria were required for the individual channels:
- $WW\rightarrow e \nu e \nu$
The most important background comes from radiative Bhabha scattering. Therefore a cut on the neutral energy was imposed, and the acoplanarity had to be greater than $7^\circ$. In addition a minimum transverse energy was required and the momenta of both leading particles had to be less than 45% of the centre-of-mass energy.
- $WW\rightarrow \mu \nu \mu \nu$
In order to reject radiative di-muon events, the transverse energy was required to satisfy $0.2\cdot \sqrt{s} < E_{t} < 0.8 \cdot \sqrt{s}$, and the neutral energy had to be less than 2 GeV.
- $WW\rightarrow \tau \nu \tau \nu$
In order to reduce remaining background from $Z (\gamma)$-decays and from $\gamma\gamma \rightarrow \ell\ell$ processes, tighter cuts were applied on the acoplanarity, the transverse momentum and the total transverse energy of the jets. Finally the acollinearity was required to be between $10^\circ$ and $150^\circ$.
- $WW\rightarrow e \nu \mu \nu$
The neutral energy was required to be less than 20 GeV.
- $WW\rightarrow \tau \nu e \nu$
A minimum transverse energy was required. If one of the leading particles was not in the barrel region, additional cuts on the momentum of each leading particle and on the acollinearity were applied to reduce the background from two-photon collisions.
- $WW\rightarrow \tau \nu \mu \nu$
The acoplanarity was required to be greater than $11^\circ$ if at least one leading particle was outside the barrel region.
The distribution of the acoplanarity angle, after having applied all cuts except the one on the acoplanarity, is shown in Figure \[fig3\]. The numbers of selected events, efficiencies and backgrounds in each channel are shown in Table \[lnulnu\]. The overall $l\nu l\nu$ efficiency was ($62.9 \pm 1.6$) %. The efficiencies have been corrected to account for an imperfect simulation of the misidentification of electrons and muons as tau leptons in a similar way as for the semileptonic channel, and the uncertainty in the track reconstruction efficiency is taken into account in the total systematic error. Inefficiencies of the trigger are estimated to be $\leq 0.1\%$. The residual background from non-$W$ and single-$W$ events is $0.134 \pm 0.032$ pb, where the error includes all systematic effects introduced by the selection criteria.
------------------------ --------------------- ------------------ -------------------- --------------- ----------------- ------------------- --
channel $\tau \nu \tau \nu$ $\tau \nu e \nu$ $\tau \nu \mu \nu$ $e \nu e \nu$ $e \nu \mu \nu$ $\mu \nu \mu \nu$
$\tau \nu \tau \nu$ [**0.252**]{} 0.069 0.083 0.005 0.008 0.003
$\tau \nu e \nu$ 0.040 [**0.433**]{} 0.012 0.044 0.057 0.
$\tau \nu \mu \nu$ 0.019 0.008 [**0.540**]{} 0.0 0.043 0.047
$ e \nu e \nu$ 0.005 0.114 0. [**0.474**]{} 0. 0.
$ e \nu \mu \nu$ 0.004 0.038 0.090 0.001 [**0.589**]{} 0.
$ \mu \nu \mu \nu$ 0.001 0. 0.058 0. 0.002 [**0.655**]{}
background (pb) 0.020 0.038 0.026 0.030 0.006 0.014
selected events 15 40 43 20 38 27
luminosity (pb$^{-1}$)
------------------------ --------------------- ------------------ -------------------- --------------- ----------------- ------------------- --
: \[lnulnu\] Selection efficiencies, background and data for the fully leptonic final states.
A total of 183 events were selected in the data sample; the number of events observed in the different di-lepton channels was found to be consistent with lepton universality. With the values given in Table \[lnulnu\] for the selected events, efficiencies and backgrounds, and assuming lepton universality, a likelihood fit yields a cross-section
$$\sigma_{WW}^{\ell\nu \ell\nu}= \sigma_{WW}^{tot} \times
{\mathrm{BR}}(WW\rightarrow \ell\nu \ell\nu )
= 1.68 \pm 0.14 ~\mbox{(stat)} \pm 0.07~\mbox{(syst)}
~~\mbox{pb}.$$
The systematic error has contributions from the efficiency and background determination, four-fermion interferences in the final states with electrons and from the measurement of the luminosity.
Determination of total cross-section and branching fractions
============================================================
The total cross-section for $WW$ production and the $W$ leptonic branching fractions were obtained from likelihood fits to the numbers of events observed in each final state. The input numbers are those given in Tables \[qqxx\] and \[lnulnu\], except for the fully hadronic final state where the binned distribution of the neural network output was used.
From all the final states combined, the leptonic branching fractions with their correlation matrix were obtained as shown in Table \[brlept1\]. They are consistent with lepton universality. The fit was repeated assuming lepton universality, and the results for the leptonic and derived hadronic branching fraction are also given in Table \[brlept1\]. The hadronic branching fraction is in agreement with the Standard Model prediction of 0.675.
channel branching fraction stat. error syst. error syst. from QCD bkg
------------------------- -------------------- ------------- ------------- --------------------
$W \rightarrow e\nu$ 0.1019 0.0064 0.0025 0.0005
$W \rightarrow \mu\nu$ 0.1076 0.0056 0.0012 0.0005
$W \rightarrow \tau\nu$ 0.1109 0.0087 0.0031 0.0003
: \[brlept1\] $W$ branching fractions from 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
Correlations $W \rightarrow e\nu$ $W \rightarrow \mu\nu$ $W \rightarrow \tau\nu$
------------------------- ---------------------- ------------------------ -------------------------
$W \rightarrow e\nu$ 1.00 -0.02 -0.39
$W \rightarrow \mu\nu$ -0.02 1.00 -0.29
$W \rightarrow \tau\nu$ -0.39 -0.29 1.00
: \[brlept1\] $W$ branching fractions from 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
----------------------------------- -------------------- ------------- ------------- --------------------
channel branching fraction stat. error syst. error syst. from QCD bkg
$W \rightarrow \ell\nu$ 0.1066 0.0028 0.0013 0.0004
$W \rightarrow {\mathrm hadrons}$ 0.6803 0.0084 0.0040 0.0013
----------------------------------- -------------------- ------------- ------------- --------------------
: \[brlept1\] $W$ branching fractions from 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
Assuming the other parameters of the Standard Model, i.e. elements $|V_{ud}|$, $|V_{us}|$, $|V_{ub}|$, $|V_{cd}|$ and $|V_{cb}|$ of the CKM matrix, lepton couplings to $W$ bosons, and the strong coupling constant $\alpha_S$, to be fixed at the values given in [@pdg], the measured hadronic branching fraction can be converted [@Vcs] into $$|V_{cs}| = 1.001 \pm 0.040~\mbox{(stat)} \pm 0.020~\mbox{(syst)},$$ where the uncertainties of the Standard Model parameters are included in the systematic error.
The total cross-section for $WW$ production, with the assumption of Standard Model values for all branching fractions, was found to be $$\sigma_{WW}^{tot} = 15.83 \pm 0.38~\mbox{(stat)} \pm 0.20~\mbox{(syst)}
~~\mbox{pb}.$$
This result is shown in figure \[fig4\] together with the measurements at lower centre-of-mass energies [@wwpap161; @wwpap172; @wwpap183], and with the Standard Model prediction using GENTLE [@gentle].
The measurement of the branching fractions can be improved by combining the present measurement with those at lower centre-of-mass energies [@wwpap161; @wwpap172; @wwpap183]. These results, obtained conservatively assuming full correlation of systematics between different energies, are summarized in Table \[brlept2\].
channel branching fraction stat. error syst. error syst. from QCD bkg
------------------------- -------------------- ------------- ------------- --------------------
$W \rightarrow e\nu$ 0.1018 0.0054 0.0026 0.0005
$W \rightarrow \mu\nu$ 0.1092 0.0048 0.0012 0.0006
$W \rightarrow \tau\nu$ 0.1105 0.0075 0.0032 0.0004
: \[brlept2\] $W$ branching fractions from the combined 161, 172, 183 and 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
Correlations $W \rightarrow e\nu$ $W \rightarrow \mu\nu$ $W \rightarrow \tau\nu$
------------------------- ---------------------- ------------------------ -------------------------
$W \rightarrow e\nu$ 1.00 -0.02 -0.38
$W \rightarrow \mu\nu$ -0.02 1.00 -0.30
$W \rightarrow \tau\nu$ -0.38 -0.30 1.00
: \[brlept2\] $W$ branching fractions from the combined 161, 172, 183 and 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
----------------------------------- -------------------- ------------- ------------- --------------------
channel branching fraction stat. error syst. error syst. from QCD bkg
$W \rightarrow \ell\nu$ 0.1071 0.0024 0.0014 0.0005
$W \rightarrow {\mathrm hadrons}$ 0.6789 0.0073 0.0043 0.0015
----------------------------------- -------------------- ------------- ------------- --------------------
: \[brlept2\] $W$ branching fractions from the combined 161, 172, 183 and 189 GeV data and correlation matrix for the leptonic branching fractions. The uncertainty from the QCD background (column 5) is included in the systematic error (column 4).
Summary
=======
From a data sample of $155 {\mathrm{~pb^{-1}}}$ integrated luminosity, collected by DELPHI in collisions at a centre-of-mass energy of 188.63 GeV, the individual leptonic branching fractions were found to be in agreement with lepton universality and the $W$ hadronic branching fraction was measured to be $${\mathrm{BR}}(W \rightarrow q\bar{q})
=0.680\pm 0.008({\mathrm stat}) \pm 0.004 ({\mathrm syst}),$$ in agreement with the Standard Model prediction of 0.675 and compatible with measurements at lower energies by other LEP experiments [@aleph; @l3; @opal]. The total cross-section for the doubly resonant $WW$ process was measured to be $$\sigma_{WW}^{tot} = 15.83
\pm 0.38 ({\mathrm stat}) \pm 0.20 ({\mathrm syst})~{\mathrm pb},$$ assuming Standard Model branching fractions.
Acknowledgements {#acknowledgements .unnumbered}
----------------
3 mm We are greatly indebted to our technical collaborators, to the members of the CERN-SL Division for the excellent performance of the LEP collider, and to the funding agencies for their support in building and operating the DELPHI detector.\
We acknowledge in particular the support of\
Austrian Federal Ministry of Science and Traffics, GZ 616.364/2-III/2a/98,\
FNRS–FWO, Belgium,\
FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil,\
Czech Ministry of Industry and Trade, GA CR 202/96/0450 and GA AVCR A1010521,\
Danish Natural Research Council,\
Commission of the European Communities (DG XII),\
Direction des Sciences de la Mati$\grave{\mbox{\rm e}}$re, CEA, France,\
Bundesministerium f$\ddot{\mbox{\rm u}}$r Bildung, Wissenschaft, Forschung und Technologie, Germany,\
General Secretariat for Research and Technology, Greece,\
National Science Foundation (NWO) and Foundation for Research on Matter (FOM), The Netherlands,\
Norwegian Research Council,\
State Committee for Scientific Research, Poland, 2P03B06015, 2P03B1116 and SPUB/P03/178/98,\
JNICT–Junta Nacional de Investigação Científica e Tecnol$\acute{\mbox{\rm o}}$gica, Portugal,\
Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134,\
Ministry of Science and Technology of the Republic of Slovenia,\
CICYT, Spain, AEN96–1661 and AEN96-1681,\
The Swedish Natural Science Research Council,\
Particle Physics and Astronomy Research Council, UK,\
Department of Energy, USA, DE–FG02–94ER40817.\
[99]{}
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F. A. Berends, R. Kleiss and R. Pittau, [*EXCALIBUR*]{}, Physics at LEP2, eds. G. Altarelli, T. Sjöstrand and F. Zwirner, CERN 96-01 (1996) Vol 2, 23. R. Rojas, [*Neural Networks - A Systematic Introduction*]{}, Springer-Verlag Berlin Heidelberg (1996). C. M. Bishop, [*Neural Networks for Pattern Recognition*]{}, Oxford University Press (1995). L. L[ø]{}nnblad, C. Peterson, H.Pi and T. R[ø]{}gnvaldsson, [*[Jetnet]{} 3.1 - A Neural Network program for jet discrimination and other High Energy Physics triggering situations*]{}, Department of Theoretical Physics, University of Lund, Sweden (1994). P. Abreu et al, Nucl. Instr. & Meth. [**A427**]{} (1999) 487. T. Sjöstrand, [*PYTHIA 5.7 / JETSET 7.4*]{}, CERN-TH.7112/93 (1993). K.-H. Becks, J. Drees, U. Flagmeyer and U. Müller, Nucl. Instr. & Meth. [**A426**]{} (1999) 599. [catani]{} S. Catani, G. Turnock and B. R. Webber, Phys. Lett. [**B295**]{} (1992) 269. [fox]{} G. C. Fox and S. Wolfram, Nucl. Phys. [**B149**]{} (1979) 413. [ARIADNE]{} L. Lönnblad, Comp. Phys. Comm. [**71**]{} (1992) 15. Particle Data Group, E. Phys. J. [**C3**]{} (1998) 1. [Vcs]{} DELPHI Collaboration, P. Abreu et al., Phys. Lett. [**B439**]{} (1998) 209. [gentle]{} D. Bardin et al., Comp. Phys. Comm. [**104**]{} (1997) 161. [aleph]{} ALEPH Collaboration, R. Barate et al., Phys. Lett. [**B453**]{} (1999) 107. [l3]{} L3 Collaboration, M. Acciarri et al., Phys. Lett. [**B436**]{} (1998) 437. [opal]{} OPAL Collaboration, G. Abbiendi et al., Eur. Phys. J. [**C8**]{} (1999) 191.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Consider the situation where a data analyst wishes to carry out an analysis on a given dataset. It is widely recognized that most of the analyst’s time will be taken up with *data engineering* tasks such as acquiring, understanding, cleaning and preparing the data. In this paper we provide a description and classification of such tasks into high-levels groups, namely data organization, data quality and feature engineering. We also make available four datasets and example analyses that exhibit a wide variety of these problems, to help encourage the development of tools and techniques to help reduce this burden and push forward research towards the automation or semi-automation of the data engineering process.'
author:
- Alfredo Nazabal
- 'Christopher K.I. Williams'
- Giovanni Colavizza
- Camila Rangel Smith
- Angus Williams
bibliography:
- 'autods\_ckiw.bib'
title: |
Data Engineering for Data Analytics:\
A Classification of the Issues, and Case Studies
---
Introduction {#sec:introduction}
============
Overview of Case Studies \[sec:case-studies\]
=============================================
Classification of Wrangling Challenges \[sec:class\]
====================================================
Related Work \[sec:rel-work\]
=============================
Conclusions \[sec:conc\]
========================
In this paper we have identified three high-level groups of data wrangling problems, those related with obtaining a proper representation of the data (data organization), those related to assessing [and improving]{} the quality of the data (data quality), and feature engineering issues, which heavily depend on the task at hand and the model employed to solve it. Furthermore, we have presented the full analysis of four use cases, where we have provided a systematic pipeline for each of the datasets to clean them while identifying and classifying the main problems the data scientists faced during the wrangling steps. We hope that this work helps to further explore and understand the field of data engineering, and to value a part of every data scientist’s work that most of the time goes unnoticed both in research and industry. Additionally, we would like to encourage practitioners to provide their raw data and the scripts necessary to clean it in order to advance the field. In future work we would like to study data engineering workflows across multiple datasets in order to identify (if possible) common structures concerning the ordering of the various wrangling operations. We note that there can be feedback loops in the process, as described e.g. in [@crisp-dm-00].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in by The Alan Turing Institute under the EPSRC grant EP/N510129/1 and by funding provided by the UK Government’s Defence & Security Programme in support of the Alan Turing Institute.
We would like to thank the AIDA team including Taha Ceritli, Jiaoyan Chen, James Geddes, Zoubin Ghahramani, Ian Horrocks, Ernesto Jimenez-Ruiz, Charles Sutton, Tomas Petrick and Gerrit van den Burg for helpful discussions and comments on the paper.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study the magnetic response of a superconducting double strip, *i.e.*, two parallel coplanar thin strips of width $2w$, thickness $d \ll w$ and of infinite length, separated by a gap of width $2s$ and subject to a perpendicular magnetic field $H$. The magnetic properties of this system are governed by the presence of a geometric energy barrier for vortex penetration which we investigate as a function of applied field $H$ and gap parameter $s$. The new results deal with the case of a narrow gap $s \ll w$, where the field penetration from the inner edges is facilitated by large flux focusing. Upon reducing the gap width $2s$, we observe a considerable rearrangement of the screening currents, leading to a strong reduction of the penetration field and the overall magnetization loop, with a suppression factor reaching $\sim
(d/w)^{1/2}$ as the gap drops below the sample thickness, $2s < d$. We compare our results with similar systems of different shapes (elliptic, rectangular platelet) and include effects of surface barriers as well. Furthermore, we verify that corrections arising from the magnetic response of the Shubnikov phase in the penetrated state are small and can be omitted. Extending the analysis to multiple strips, we determine the specific sequence of flux penetrations into the different strips. Our studies are relevant for the understanding of platelet shaped samples with cracks or the penetration into layered superconductors at oblique magnetic fields.
author:
- 'R. Willa'
- 'V.B. Geshkenbein'
- 'G. Blatter'
title: Suppression of Geometric Barrier in Type II Superconducting Strips
---
Introduction {#sec:introduction}
============
The characteristic properties of a superconductor are its diamagnetic response [@Meissner_33] $M$ to an external magnetic field $H$ and its ability to transport electric current without dissipation[@Onnes_11]. In the Meissner phase the magnetic induction $B = H + 4 \pi M$ vanishes inside the superconductor and the linear response $M = -H/4\pi$ is that of a perfect (bulk) diamagnet. In type II superconductors, a sufficiently large magnetic field $H>{H_{p}}$ penetrates the material via quantized flux lines (with flux $\Phi_{0}=hc/2e$); we denote with ${H_{p}}$ the field of first penetration. Within the mixed (or Shubnikov[@Shubnikov_37]) phase the presence of vortices reduces the bulk diamagnetic signal and the magnetization $M(H)$ decreases in magnitude. The magnetic properties of the material then depend on the behavior of the vortex state. In this paper, we determine the magnetic response of superconducting samples of more complex shape, in particular a double strip, two parallel coplanar thin strips of infinite length and subject to a perpendicular magnetic field $H$, see [Fig. ]{}\[fig:sketch-double-strip\]. The response of such a system is hysteretic and dominated by the so-called geometrical barrier[@Zeldov_94; @Benkraouda_96], i.e., an energy barrier retarding the magnetic field penetration. Our main result is an apparent suppression of the geometrical barrier for the situation where the two strips are closeby, i.e., separated by a narrow gap or crack. Such a suppression of geometrical barriers may be of practical interest in experiments, as has been the case in disentangling the vortex lattice melting- and irreversibility lines in layered BiSCCO superconductors [@Majer_95] or in separating apart the phenomenon of bulk vortex pinning by defects. So far, the geometrical barrier has been deliberately suppressed by polishing the sample into the shape of a prism [@Majer_95]; the suppression of the geometrical barrier observed when tilting the magnetic field applied to the sample [@Segev_11] and attributed to the appearance of Josephson vortex stacks resembles the mechanism reported in the present paper.
![Side-view representation ($xz$-plane) of two flat superconducting strips (parallel to $y$) subject to a perpendicular magnetic field $H$ (directed along $z$). The cross-sections of the strips have a width $2w$ and a thickness $d \ll w$, while the separation $2s$ between their inner edges measures the width of the gap. The outer edges of the strips at $\pm (2w + s)$ are denoted by $\pm W$. Any position in the $xz$-plane is described by the complex coordinate $\xi = x + i z$.[]{data-label="fig:sketch-double-strip"}](plot_09.eps){width="7cm"}
The precise shape of the magnetization curve depends on the specific configuration assumed by the vortices after penetration, which is determined by the sample shape and its surface properties (we assume a sample free of defects). The sample surface is relevant in the determination of the penetration field ${H_{p}}$ as defined in the asymptotic region far away from the sample. A flat surface parallel to the field generates an image vortex which results in a surface barrier hindering vortices from entering the sample [@Bean_64; @Clem_74]. The metastable Meissner state survives until the local field at the surface is increased beyond the critical value ${H_{s}}$ which is of the order of the thermodynamic critical field $H_c$, ${H_{s}}\sim H_c > H_{c1}$, with $H_{c1}$ the lower critical field. For a non-ideal surface the effective surface barrier is reduced and assumes a value ${H_{s}}$ between $H_{c1}$ and $H_c$.
The sample shape is relevant, too, in the determination of the penetration field ${H_{p}}$. This is well known for elliptic-shaped samples, cf. Fig.\[fig:geometries\], where the magnetic field is enhanced near the sample edge: for a cylindrical shaped diamagnetic (i.e., $\mu = 0$) sample with an elliptic cross section of height $d$ and width $2w$, the demagnetization factor[@Osborn_45] $n= 2w / (2w+d)$ generates a field enhancement ${H_{\mathrm{edge}}}= (1-n)^{-1} H$. Correspondingly, the penetration field is given by ${H_{p}}= (1-n){H_{s}}= d/(2w+d) {H_{s}}$. Once the penetration field is reached, vortices enter the sample, reversibly in the absence of a surface barrier (i.e., if ${H_{s}}=
H_{c1}$) and irreversibly else. Without surface barrier, the vortices distribute homogeneously inside the sample, a result that is consistent with the constant induction inside a magnetic ellipsoid[@Landau_60]. On a microscopic level, this corresponds to an exact matching of the energy gain of vortex motion in the field of the screening current and the energy cost $\varepsilon_l$ associated with the increasing vortex length upon penetration, see Fig. \[fig:geometries\].
![Top: sketch of field enhancement near the edges of an elliptic- (left) and a rectangular- (right) shaped sample. Below penetration $H < {H_{p}}$, for both geometries, the field is enhanced by the factor $\sim \sqrt{w/d}$ a distance $d$ away from the edges. The field remains unchanged on approaching the rectangular edge but increases by a further factor $\sim \sqrt{w/d}$ for the elliptic geometry. Upon increasing $H$ beyond ${H_{p}}$, the field penetrates homogeneously into the elliptic shaped sample and concentrates in a central dome for the rectangular sample. This is due to the different potential landscapes $U_{\mathrm{geo}}(x)$ (see bottom sketch) felt by the vortices penetrating the sample at $H \sim {H_{p}}$, flat for the ellipse (dotted line) and attractive for the rectangle (solid line). Note that the penetration fields differ by the factor $\sqrt{d/w}$ for the elliptic and the rectangular sample. The sketch illustrates the situation without additional surface barrier.[]{data-label="fig:geometries"}](plot_00.eps){width="45.00000%"}
For a platelet shaped sample (of width $2w$ and thickness $d$) with a rectangular edge, the field at the boundary is enhanced as well, although (effectively) less than for the elliptic sample. A distance $d$ away from the edge[@footnote:enhancement-ellipse], the applied field $H$ is enhanced by a factor $\sim (w/d)^{1/2}$, resulting in a penetration field ${H_{p}}\sim
(d/w)^{1/2} {H_{s}}$. At this field strength, the barrier for vortex entry into the sample has vanished and vortices move to the center of the sample where they accumulate in a dome-shaped form, cf. Fig. \[fig:geometries\]. Under further increase of the external field $H$, the vortex dome grows both in height and width until the sample is fully penetrated. In this geometry, the cost $\varepsilon_l d$ to create the vortex is payed right upon vortex entry at the sample edge; beyond the edge region the energy gain in the current field $I(x)$ is no longer balanced by the energy cost and the vortex is driven to the sample center. Hence the field penetration into the platelet shaped sample is irreversible even in the absence of a surface barrier, what is due to the presence of a geometric barrier defined through the energy cost for flux entry. It is this type of geometric barrier effects [@Zeldov_94; @Benkraouda_96] which is at the focus of the present paper.
Another situation arises in dirty samples where vortices are pinned onto defects. Once the surface and geometrical barriers are overcome, the vortex arrangement may be dominated by bulk pinning and the magnetic induction (or magnetization) is given by a Bean profile[@Bean_62]. What is common to all three cases, surface-, geometric-, and bulk pinning is the irreversible, hysteretic behavior of the magnetization $M(H)$ with changing external field $H$. In this paper, we concentrate on the defect-free case and thus ignore possible modifications due to bulk pinning.
The motivation to study geometrical barriers in samples of complex shape is manifold: Originally, the understanding of the flux penetration and vortex lattice melting in layered high-$T_c$ superconductors necessitated a proper analysis of the vortex state in platelet-shaped samples[@Zeldov_94]. On the technological side, the structuring of current-carrying strips [@Benkraouda_98; @Mawatari_01] enhances their critical current as the incorporation of slits generates geometrical barriers hindering vortex motion. Recently, Segev *et al.*[@Segev_11] observed a structured vortex dome in layered $\mathrm{Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}}$ samples subject to a tilted magnetic field. This finding can be interpreted as arising from stacks of in-plane (Josephson) vortices reducing the superconducting order parameter[@Koshelev_99] and acting as weak links for the perpendicular field (pancake vortices[@Feigelman_90; @Clem_91]). Our analysis of vortex penetration into a double-strip with a narrow gap, see Fig.\[fig:sketch-double-strip\], may serve as a first step towards the understanding of flux penetration in this geometry.
From a general perspective, the magnetic response associated with superconducting samples can be calculated numerically. Effects of complex sample shapes, inhomogeneous material equations, and time-dependent perturbations can then be studied quantitatively[@Brandt_99]. On the other hand, analytic approaches give more qualitative insights into the system’s behavior. Earlier work on geometrical barriers in samples with more complex shapes considered the case of two coplanar thin strips in the Meissner phase[@Brojeny_02] and the full magnetization curve for a strip-shaped sample with a slit[@Mawatari_03], i.e., two strips shunted at their ends; this ring-type topology with circulating currents exhibits a markedly different magnetization $M(H)$ as compared to our unshunted situation. The situation of an unshunted double stripline in the critical state was investigated in Ref.[\[\]]{}. In our work, we go beyond these results in various ways, including the situation where the sample thickness $d$ plays an important role.
![Sketch of the geometric energy barrier $U_{b}$ for vortex penetration as a function of the applied field $H$ and the gap parameter $s$, see also [Fig. ]{}\[fig:sketch-double-strip\]. In this Figure we neglect an additional surface barrier, i.e., ${H_{s}}= H_{c1}$. The thick black curve marks the geometric barrier height $U_{b}^{\mathrm{eq}}(s)$ at the equilibrium field ${H_{\mathrm{eq}}}$ as defined in [Eq. ]{} and provides a measure for the irreversibility of the sample. Note the rapid decrease of the geometric barrier $U_{b}^{\mathrm{eq}}(s\ll d,H)$ with increasing field $H$ at small separation $s$ between the two strips; the small geometric barrier $U_{b}^{\mathrm{eq}}(s)$ tells that irreversibility is reduced when $s~\ll~d$. Still, a finite irreversibility remains with the geometrical barrier rapidly reinstalled when reversing the applied field.[]{data-label="fig:geometric-barrier"}](plot_21_mod.eps){width="45.00000%"}
The most pertinent new result is the dramatic suppression of the geometrical barrier which we illustrate in [Fig. ]{}\[fig:geometric-barrier\]. This suppression is driven by a large flux-focussing into the gap between the strips, forcing the flux penetration into the sample to start from the inner edges. In tracing the evolution of the penetration field ${H_{p}}$ as a function of separation $s$ between the strips, we find it decay from ${H_{p}}\sim
\sqrt{d/w} \,{H_{s}}$ at large $s$ to ${H_{p}}\sim \sqrt{sd/w^2} \log(w/s)\, {H_{s}}$ at intermediate separation $d < s < w$ to ${H_{p}}\sim (d/w) \, {H_{s}}$ at small $s \ll
d$; the latter coincides with the result for the elliptic sample where the geometrical barrier is absent alltogether. We emphasize, however, that the narrow-gap double-strip still differs from the ellipse as the geometrical barrier remains present but rapidly collapses from $\varepsilon_l d$ to zero with increasing field, hence maintaining the hysteretic magnetization. The latter strongly decreases with the separation $s$ between strips as well: Within the individual strips, the penetrated field assumes a dome-like shape which is increasingly skewed towards the gap when $s$ becomes small. Following the change in shape of the magnetization curve through the various regimes, we find it to shrink by a factor $\propto (s/w)^{1/2} \log(w/s)$ when $s < w$ and by a factor $\propto (d/w)^{1/2}$ for narrow gaps $s \ll d$ when compared to the single platelet sample; this decay of the magnetization with decreasing $s$ ends up in a flat and nearly constant value $M = -({H_{s}}/4\pi)
(4wd)$ at small $s \log(W/s) \ll d $.
In the following, we briefly recall the key features of the magnetic response for elliptically shaped strips in Sec. \[sec:sec:ellipse\] and proceed with the description of coplanar parallel rectangular strips for the case where the thickness $d$ is the smallest geometric length in the problem (Sec. \[sec:sec:formalism\]). We review the appearance and consequences of a geometric barrier in a single strip (Sec. \[sec:sec:single-strip\]) and continue with the analysis of two adjacent strips (Sec.\[sec:sec:double-strip\]) discussing the behavior of the Meissner- and penetrated states. In Sec. \[sec:finite-thickness\], we analyze the double strip for the situation where the separation $2s$ between the strips is smaller than the strip thickness $d$, $s \ll d$. Section \[sec:several-strips\] is devoted to multi-strips and a summary and conclusions are given in Sec. \[sec:conclusions\].
Thin Strips {#sec:thin-strips}
===========
Introduction - Elliptical strip {#sec:sec:ellipse}
-------------------------------
Before considering samples with rectangular geometries, it is instructive to revisit the magnetic properties of a flat superconducting strip with an *elliptic* cross-section. The strip extends infinitely in the $y$-direction and the semi-axes along $x$ and $z$ are $w$ and $d/2$ ($d \ll
w$) respectively, with the upper/lower sample surface parametrized by $z_{\pm}(x) = \pm(d/2w)\sqrt{w^2-x^2}$. The magnetic field $H$ is applied parallel to the $z$-axis; outside the sample, $\boldsymbol{B} =
\boldsymbol{H}$, while $B_{\mathrm{el}} = \mu(B_{\mathrm{el}})
H_{\mathrm{el}}$ is constant and parallel to the $z$-axis inside the elliptic sample[@Landau_60], a consequence of the special elliptic shape. Here, $$\begin{aligned}
\label{eq:mu-from-free-energy}
\mu(B) = \frac{B}{4\pi}\Big(\frac{dF}{dB}\Big)^{-1}\end{aligned}$$ is the magnetic permeability of the material as obtained from the free energy density $F(B)$. The magnetic field at the sample edges $(\pm w,0)$ is continuous[@Landau_60], $H_{\mathrm{el}} = {H_{\mathrm{edge}}}$, where ${H_{\mathrm{edge}}}$ denotes the magnetic field strength at the sample edge. The latter is modified due to demagnetization effects of the sample which are described by the geometric demagnetizing factor[@Osborn_45] $n = 2w / (2w + d) \approx 1 -
d/2w$. Exploiting the fact that the magnetic induction $B_\mathrm{el}$ is constant within the ellipse, we decompose the total field $\boldsymbol{B}(x,z)$ into two components, a constant one $\boldsymbol{B}_\mathrm{el} = (0,0,B_\mathrm{el})$, and the remaining field $\boldsymbol{B}_0(x,z)$ which does not penetrate the sample. Far away from the sample, all fields point along $z$, $\boldsymbol{B}_0 \equiv (0,0,B_0^\infty)$ and we have $B_\mathrm{el} + B_0^\infty = H$. The component $\boldsymbol{B}_0(x,z)$ then describes the field of a perfectly diamagnetic ellipse in the reduced external field $B_0^\infty = H-B_\mathrm{el}$. The magnetic field at the sample edge ($x = \pm w$) points along $z$, involves the two components $B_\mathrm{el}$ and $B_0 = B_0^\infty/(1-n)$, the latter enhanced by demagnetization effects, and reads $$\begin{aligned}
\label{eq:edge-field-general}
{H_{\mathrm{edge}}}&= B_\mathrm{el} + \frac{B_0^\infty}{1-n}.\end{aligned}$$ Using $B_0^\infty = H-B_{\mathrm{el}}$ as well as $B_{\mathrm{el}} =
\mu(B_{\mathrm{el}}) H_{\mathrm{el}} = \mu(B_{\mathrm{el}})
{H_{\mathrm{edge}}}$, we obtain the standard formula for the field strength inside the sample[@Landau_60] $$\begin{aligned}
\label{eq:elliptic-induction}
B_{\mathrm{el}} = \frac{\mu(B_{\mathrm{el}})}{1-n[1-\mu(B_{\mathrm{el}})]}H,\end{aligned}$$ where the value for $B_{\mathrm{el}}$ has to be determined self-consistently. For notational simplicity we denote by $\mu$ the value for $\mu(B_{\mathrm{el}})$ after solving the above equation.
The $\boldsymbol{B}$-field at the surface outside of the ellipse has both a normal ($\perp$) and a tangential ($\|$) component. Their magnitudes can be determined from the boundary conditions[@Landau_60], telling that $B_{\perp}$ and $B_{\|}/\mu$ are continuous across the surface. For the upper surface $z=z_+(x)$ of the ellipse we find $$\begin{aligned}
\label{eq:elliptic-field-parallel-perp}
\boldsymbol{(}B_{\|}(x),B_{\perp}(x) \boldsymbol{)}
&= \frac{H}{1-n(1-\mu)}\boldsymbol{(}\sin(\alpha), \mu \cos(\alpha)\boldsymbol{)},\end{aligned}$$ where $$\begin{aligned}
\alpha(x) &= \arctan\bigg(\frac{d}{2w}\frac{-x}{\sqrt{w^2-x^2}}\bigg)\end{aligned}$$ measures the angle between the external field orientation ($z$-axis) and the direction normal to the elliptic surface at the position $\boldsymbol{(}x,
z_{+}(x)\boldsymbol{)}$. In most of the strip region (when $w-|x| \gg d^2/w$) the surface of the ellipse is almost parallel to the $x$-axis and the above field expression simplifies to $$\begin{aligned}
\label{eq:elliptic-field-parallel-perp-approx}
\boldsymbol{(}B_{\|}(x),B_{\perp}(x) \boldsymbol{)}
&\approx \frac{H}{1-n(1-\mu)}\Big(\frac{-x(1-n)}{\sqrt{w^2-x^2}},\mu \Big).\end{aligned}$$ The discontinuity of the field parallel to the boundary determines the surface current that generates the magnetization of the sample. Using Ampère’s law and defining the sheet current density $I(x) = \int_{z_-}^{z_+}
dz\, j(x,z)$ across the sample, we find $$\begin{aligned}
I(x)&\approx \frac{2 c}{4\pi} \big[ B_{\|}(x) - B_{\mathrm{el}} \sin(\alpha)\big]\\
\label{eq:approx-field-at-the-surface-of-ellipse}
&\approx -\frac{Hc}{2\pi} \frac{(1-n)(1-\mu)}{1-n(1-\mu)}
\frac{x}{\sqrt{w^2-x^2}}\\
\label{eq:current-density-ellipse}
&= -\frac{(H-B_{\mathrm{el}})c}{2\pi} \frac{x}{\sqrt{w^2-x^2}}.\end{aligned}$$ The factor 2 originates from the two current contributions at the upper and lower sample surface. The last expression shows that only the expelled component $B_0^\infty = H-B_{\mathrm{el}}$ contributes to the shielding currents. The magnetization $M$ (per unit length) is obtained from the relation $4\pi M / A = B_{\mathrm{el}} - H_{\mathrm{el}}$, where $A = \pi w
d/2$ is the area of the strip’s cross-section. Using $H_{\mathrm{el}} = {H_{\mathrm{edge}}}$ and [Eq. ]{} gives for the magnetization $$\begin{aligned}
\label{eq:magnetization-ellipse-general}
M&= -\frac{B_{0}^{\infty}}{4} w^2
= -\frac{H}{4} \frac{(1-n)(1-\mu)}{1-n(1-\mu)} w^2.\end{aligned}$$ In the last equality we used $B_0^\infty = H-B_{\mathrm{el}}$ and [Eq. ]{}.
### Meissner state {#sec:sec:sec:meissner-ellipse}
At low fields, the superconducting elliptic strip remains in the Meissner state ($\mu = 0$), resulting in a vanishing induction, i.e., $B_\mathrm{el} =
B=0$. The field strength at the edge, see [Eq. ]{}, is enhanced by the geometric factor $1/(1-n) \approx 2w/d$ as compared to the applied field $H$. At the sample surface, the field is everywhere tangential and its strength is given by $H \sin(\alpha)/(1-n)$ $(\approx -H x/
\sqrt{w^{2}- x^{2}})$ as obtained from [Eqs. ]{} and . The resulting sheet current density inside the sample is obtained from [Eq. ]{}, $$\begin{aligned}
\label{eq:meissner-current-ellipse}
I(x) &\approx -\frac{H c}{2\pi} \frac{x}{\sqrt{w^2-x^2}}.\end{aligned}$$ The perfectly diamagnetic response \[[Eq. ]{} with $\mu = 0$\] $$\begin{aligned}
\label{eq:magnetization-meissner-ellipse}
M &= -\frac{H}{4}w^{2}.$$ lasts until the magnetic flux starts penetrating the superconducting sample in the form of vortices.
To bring a vortex to the position $x$ inside the sample costs an energy $U_{\scriptscriptstyle{L}}(x) = \varepsilon_{l} \ell(x)$, gradually rising with the vortex length $\ell(x) = z_{+}(x) - z_{-}(x)$ from zero at the sample edges to $d$ in the sample center; here, the line-energy $\varepsilon_{l} = \varepsilon_{0}
\log(\lambda/\xi) = \Phi_{0}(dF/dB)|_{B=0}$ is the cost per unit length associated with the nucleation of a single vortex in the bulk superconductor. On the other hand, the work gained from the Lorentz force \[due to the current $I(x)$ in [Eq. ]{}\] drives the vortex entrance. The two energy contributions can be combined to an effective potential landscape[@Likharev_71] for a single vortex $$\begin{aligned}
\label{eq:energy-profile}
U_{\mathrm{geo}}(x)
&= U_{\scriptscriptstyle{L}}(x) - \frac{\Phi_{0}}{c}
\int\limits_{w}^{x} du\,I(u).\end{aligned}$$ In the elliptical geometry, the functional form of the driving energy due to the current [Eq. ]{} coincides with the geometrical thickness $\ell(x) = d\sqrt{1-x^{2}/w^{2}}$ of the sample and the energy profile reduces to $$\begin{aligned}
U_{\mathrm{geo}}(x) &= \varepsilon_{l}\ell(x)
\bigg(1- \frac{H \Phi_{0}}{4\pi \varepsilon_{l}} \frac{2w}{d}\bigg).\end{aligned}$$ The barrier then vanishes throughout the sample at the penetration field $$\begin{aligned}
\label{eq:single-penetration-field-ellipse}
{H_{p}}&= \frac{4\pi \varepsilon_{l}}{\Phi_{0}}\frac{d}{2w} = H_{c1} \frac{d}{2w}\end{aligned}$$ where the *local* field strength at the edge reaches $H_{c1}$ and the magnetization (per unit length) as obtained from [Eq. ]{} amounts to $$\begin{aligned}
\label{eq:single-maximal-magnetization-ellipse}
M_{p} = -\frac{H_{c1}}{8} wd = -\frac{H_{c1}}{4\pi} \frac{\pi w d}{2},\end{aligned}$$ with $\pi w d/2$ the cross-section of the strip.
### Penetrated state {#sec:sec:sec:penetrated-ellipse}
Beyond the field of first penetration ${H_{p}}$, vortices homogeneously flood the sample, and the potential landscape takes the form (we replace $B_\mathrm{el}
\to B$) $$\begin{aligned}
\label{eq:energy-profile-penetrated-state}
U_{\mathrm{geo}}(x) &= \varepsilon_{l}(B)\frac{d}{w}\sqrt{w^{2}-x^{2}}
- \frac{\Phi_{0}}{c} \int\limits_{w}^{x} du\,I(u).\end{aligned}$$ The line energy $\varepsilon_{l}(B)$ describes the energy difference (per unit length) between the vortex state and the homogeneous field configuration, i.e., $$\begin{aligned}
\label{eq:effective-line-energy}
\varepsilon_{l}(B)
&= \Phi_{0}\frac{d}{d B}\bigg[F(B) -\frac{B^{2}}{8\pi}\bigg]
= \frac{\Phi_{0} B}{4\pi} \frac{1 -\mu(B)}{\mu(B)}\end{aligned}$$ with $F(B)$ the free energy density of the superconducting state. The second term on the right-hand side of [Eq. ]{} is modified as well, since only the non-penetrating (diamagnetic) part $H-B$ of the field drives the diamagnetic currents in [Eq. ]{}. The resulting state remains in equilibrium for all $H > {H_{p}}$, i.e., $U_{\mathrm{geo}}(x) \equiv 0$, and the reversible magnetic response follows the form in Eq. $$\begin{aligned}
\label{eq:magnetization-superconducting-ellipse}
M&= -\frac{H-B}{4} w^2\end{aligned}$$ with $B$ determined by the self-consistency equation . A finite surface barrier as discussed further below will retard the vortex penetration and generate a hysteretic response.
In order to illustrate the above results, we consider a superconductor with the Abrikosov (bulk) induction[@Tinkham_96] $$\begin{aligned}
\label{eq:toy-bulk-magn}
B &= C_{1} H_{c1} \Big[\log\Big(\frac{C_{2}H_{c1}}{H-H_{c1}}\Big)\Big]^{-2}\end{aligned}$$ near the penetration field, with $C_{1,2}$ constants of order unity. In this equation, $H = H_\mathrm{edge}$ is the local field strength at the surface of the bulk sample. The magnetic permeability $\mu(B)$, can be extracted from the above expression via the relation $\mu(B) = B/H(B)$ and we find $$\begin{aligned}
\label{eq:toy-permeability}
\mu(B) = \frac{B}{H_{c1}}
\bigg[1 + C_{2}\exp\bigg(\!-\sqrt{\frac{C_{1}H_{c1}}{B}}\ \bigg)\bigg]^{-1}.\end{aligned}$$ The linear slope $1/H_{c1}$ of the permeability near $B=0$ follows from the vertical onset of the induction (see [Eq. ]{}) beyond $H_{c1}$. Dropping the exponential term in [Eq. ]{} close to the penetration ($B \ll H_{c1}$) and substituting $\mu$ to the self-consistency equation we obtain the induction $$\begin{aligned}
\label{eq:ellipcit-induction}
B(H) = (H-{H_{p}})/n,\end{aligned}$$ resulting in a linear decrease of the diamagnetic response, $$\begin{aligned}
\label{eq:elliptic-magnetization}
M(H) = -\frac{{H_{p}}}{4n} w^{2}\,\Big(1-\frac{H}{H_{c1}}\Big).\end{aligned}$$ Note that for small inductions $B \ll H_{c1}$, the diamagnetic response is very different from the usual bulk Abrikosov magnetization (see, e.g., Ref. [\[\]]{}). The linear decrease in [Eq. ]{} extrapolates to $M=0$ at $H=H_{c1}$. The full solution of [Eq. ]{} for the permeability leads to the magnetic response illustrated in [Fig. ]{}\[fig:elliptic-magnetization\].
![Magnetic response of a superconducting elliptic strip with demagnetizing factor $n \approx d/2w$ (here $n=0.9$) as obtained from [Eq. ]{} and with material properties described by (solid line). For comparison, we show the bulk (Abrikosov) magnetization with the same permeability $\mu(B)$ (thin solid line). The vertical onset in the bulk magnetization goes over into the linear reduction of $M$ in the ellipse, extrapolating to $M=0$ at $H = H_{c1}$ (thin dashed line).[]{data-label="fig:elliptic-magnetization"}](plot_15.eps){width=".45\textwidth"}
Rectangular strips - Formalism {#sec:sec:formalism}
------------------------------
Having familiarized ourselves with the results for the elliptic strip, we turn our attention to strips with rectangular shape, i.e., samples with constant thickness $d$ as opposed to the ellipse where the height is changing over the entire sample width. Specifically, we will consider (smooth) sample edges with a typical radius of curvature $\gtrsim d$ in contrast to the much sharper edge of the ellipse where the radius of curvature is $d^{2}/4w \ll d$. We consider a set of coplanar (in the $xy$-plane) and parallel superconducting strips of infinite length (along $y$), each with a rectangular shape of width $2w$ (along $x$) and thickness $d \ll 2w$ (along $z$), subject to a perpendicular magnetic field $H$ along $z$. The strip thickness $d$ is assumed to be the smallest geometric length and is set to zero in the following mathematical analysis; its finite value is properly reinstalled through appropriate boundary conditions. Because the system is effectively two-dimensional, we express the magnetic field $\boldsymbol{B}(x,z)$ in the $xz$-plane through the complex function[@Zeldov_94] $\mathcal{B}(\xi) = B_{z}(x,z) +
iB_{x}(x,z)$, with the two-dimensional coordinate $(x,z)$ replaced by the complex variable $\xi = x + i z$. The magnetostatic problem of solving the Laplace equation ($\Delta \boldsymbol{B} = 0$) for $\boldsymbol{B}$ is translated to a problem in complex analysis, where the holomorphic function $\mathcal{B}(\xi)$ satisfies the Cauchy-Riemann equations (correspnding to the magnetostatic equations $\nabla\cdot \boldsymbol{B}=0$ and $\nabla \wedge
\boldsymbol{B}=0$) in the superconductor-free region; the presence of the superconductor is accounted for through appropriate boundary conditions. The latter derive from two physical conditions: on the one hand, no vortices are present in regions where current is flowing, i.e., $$\begin{aligned}
\label{eq:bcBzI}
B_z(x) &= 0\quad \mathrm{ when } \quad I(x) \neq 0.\intertext{Here and below we simply call `current' the sheet current density $I(x)$ flowing between $z_{\pm}=\pm
d/2$. On the other hand, no currents flow in the
vortex-filled regions,
}
\label{eq:bcIBz}
I(x) &= 0 \quad \mathrm{ when } \quad B_{z}(x) \neq 0.\end{aligned}$$ This last condition neglects the microscopic structure of the vortex state by treating the penetrated region as magnetically inactive, $\mu = 1$; the accuracy of this simplification will be discussed later in this section. Using Ampère’s law $$\begin{aligned}
\label{eq:ampere-law}
I(x) = \frac{c}{2\pi} B_{x}(x,0^{+})
= \frac{c}{2\pi} \mathrm{Im}[\mathcal{B}(x + i 0^{+})],\end{aligned}$$ the boundary conditions and transform to $$\begin{aligned}
\label{eq:bcBzBx}
B_z(x) &= 0 \quad \mathrm{ when }\quad B_{x}(x) \neq 0,\ \mathrm{and}\\
\label{eq:bcBxBz}
B_x(x) &= 0 \quad \mathrm{ when }\quad B_{z}(x) \neq 0. \end{aligned}$$ For a single strip centered a the origin ($\xi = 0$) the holomorphic field $$\begin{aligned}
\label{eq:ant-single-strip-function}
\mathcal{B}(\xi) &= H \sqrt{\frac{\xi^{2}-{b_{0}}^{2}(H)}{\xi^{2}-w^{2}}}\end{aligned}$$ is known to satisfy all the above requirements[@Zeldov_94]; the parameter ${b_{0}}$ then determines the field configuration of the entire system. For the double strip studied below, the corresponding expression reads $$\begin{aligned}
\label{eq:ant-double-strip-function}
\mathcal{B}(\xi) &=
H \sqrt{\frac{[\xi^{2}-{b_{1}}^{2}(H)][\xi^{2}-{b_{2}}^{2}(H)]}
{(\xi^{2}-s^{2})(\xi^{2}-W^{2})}}.\end{aligned}$$ Here, the strips are arranged symmetrically, extending between $\pm s$ and $\pm W$ (with $W=s+2w$) on the $x$-axis.
In order to specify the field and current distributions for these geometries, the parameters ${b_{0}}$, ${b_{1}}$, and ${b_{2}}$ (with $0\leq{b_{0}}<w$ and $s < {b_{1}}\leq
{b_{2}}< W$) describing the boundaries of the field-filled region have to be determined from two physical conditions: First, the net current along each strip vanishes, i.e., $$\begin{aligned}
\label{eq:current-neutrality-condition-general}
\int_{\mathrm{strip}}\!\!\!\!\!dx\,I(x) &= 0.\end{aligned}$$ This (first) condition is independent of the magnetic state of the strips, Meissner or Shubnikov. The second condition regulates the penetration process of vortices into the superconducting sample. In the Meissner phase, no field penetrates the superconductor and the width of the vortex dome vanishes, imposing the (second) condition $$\begin{aligned}
\label{eq:con2}
\left.\begin{aligned}
{b_{0}}&= 0 &&\textrm{for the single, or}\\
{b_{1}}&= {b_{2}}&&\textrm{for the double}
\end{aligned}\right.\end{aligned}$$ strip geometry. The second condition for the penetrated state derives from the analysis of vortex penetration at the sample edge. We consider a smooth edge of shape $z_{\pm}(r) = \pm \ell(r)/2$ with $r$ measured from the sample edge, rising to $\ell = d$ within a distance $r \approx d/2$ (e.g., $\ell(r<d/2) =
\sqrt{2rd}$). The (tangential) field ${H_{\mathrm{edge}}}$ at the surface is assumed constant and generates a current density $j =c {H_{\mathrm{edge}}}/4 \pi\lambda$ at the sample boundary, with $\lambda$ denoting the London penetration depth, $\lambda \ll
d$. A simple geometrical consideration provides us with the sheet current $I(r) = 2
(c{H_{\mathrm{edge}}}/4\pi) \sqrt{1+[\ell'(r)/2]^2}$ and using [Eq. ]{}, we obtain the rise of the vortex energy near the edge $$\begin{aligned}
\label{eq:Ugeo-edge-general}
U_{\mathrm{geo}}(r) &= \varepsilon_l \ell(r)
- \frac{\Phi_{0}{H_{\mathrm{edge}}}}{2\pi} \int\limits_{0}^{r}
du\, \sqrt{1+[\ell'(u)/2]^2}.\end{aligned}$$ For a smooth edge with radius of curvature $\gtrsim d$ we have $\ell' \gg 1$ for $r \ll d$ (consistent with a roughly constant field ${H_{\mathrm{edge}}}$) and we can simplify the above expression to read $$\begin{aligned}
U_{\mathrm{geo}}(r) &= \varepsilon_{l} \ell(r)
\Big[1 - \frac{\Phi_{0}{H_{\mathrm{edge}}}}{4\pi \varepsilon_{l}}\Big].\end{aligned}$$ Hence, we find that the energy barrier for vortex entry is eliminated when the local field strength reaches the first critical field ${H_{\mathrm{edge}}}= H_{c1} = 4\pi
\varepsilon_{l}/\Phi_{0}$. Once the edge region of width $d$ has been overcome, the vortices are driven to the sample center where they arrange within the vortex dome. The vortices deep inside the sample reduce the field at the edge and the penetration of flux is stopped when ${H_{\mathrm{edge}}}$ drops below $H_{c1}$. With a further increase of the external field, vortices continue to penetrate the sample when the condition ${H_{\mathrm{edge}}}= H_{c1}$ is satisfied again. This stop and go criterion for vortex penetration then is the second condition imposed on the fields in [Eqs. ]{} and and determines, together with [Eq. ]{}, the parameters ${b_{0}}$, ${b_{1}}$, and ${b_{2}}$.
The above discussion ignores the possible presence of a surface barrier[@Clem_74] appearing on small length scales below $\lambda$. In the most effective case, this barrier further retards the penetration of vortices until the local field reaches the critical strength ${H_{\mathrm{edge}}}\sim H_c$. In order to deal with the general situation accounting for effects due to a surface barrier we denote the local critical field for vortex penetration by ${H_{s}}$ ($H_{c1} < {H_{s}}< H_{c}$). The second condition determining the fields [Eqs. ]{} and in the penetrated ($H > {H_{p}}$) state then can be cast in the form $$\begin{aligned}
\label{eq:condition-penetration-field}
{H_{\mathrm{edge}}}&= {H_{s}}.\end{aligned}$$ The above equation replaces the condition [Eq. ]{} valid for the Meissner phase. In the regime of very high fields, $H > {H_{s}}$, diamagnetic screening becomes small and the field strength at the sample edge lines up with the applied field, ${H_{\mathrm{edge}}}\approx H$; however, this large-field limit will not be considered below.
Finally, we comment on the precision of this second condition: The field strengths in [Eqs. ]{} and show square-root singularities near the sample edges. The description of the spacial dependence of the field when approaching the edges to distances smaller than $d$ then requires a detailed analysis of the edge region. On the other hand, the typical scale for the field strength needed for overcoming the edge region can be obtained by the considerations presented above, once we have a proper definition for the edge field ${H_{\mathrm{edge}}}$ at our disposal. Below, we identify this field strength with the field evaluated a distance $d/2$ away from the edge, ${H_{\mathrm{edge}}}=
B_{z}(r=-d/2)$.
The surface barrier retarding the penetration of flux appears on the small length scale between $\lambda$ (at low fields of order $H_{c1}$) and $\xi$ (near $H_c$). On the contrary, the *geometric energy barrier* is a macroscopic object appearing on the scale $d$. We define the *geometric barrier* $U_{b}$ of a platelet sample as the maximum of [Eq. ]{} that is reached near $d$. The second term in [Eq. ]{} then reduces the geometric barrier linearly to zero at ${H_{\mathrm{edge}}}= H_{c1}$ and the barrier takes the functional form $$\begin{aligned}
\label{eq:geometric-barrier}
U_{b} = \varepsilon_{l} d \Big(1 - \frac{{H_{\mathrm{edge}}}}{H_{c1}}\Big)
= \varepsilon_{l} d \Big(1 - \frac{H}{{H_{p}}}\frac{{H_{s}}}{H_{c1}}\Big)\end{aligned}$$ where the first (second) equality expresses the barrier in terms of the local (asymptotic) field (note that field penetration only starts when ${H_{\mathrm{edge}}}= {H_{s}}$, where the additional surface barrier has disappeared). While the geometric barrier only vanishes when the local field reaches $H_{c1}$, the vortex state may become thermodynamically stable at a lower *equilibrium* field ${H_{\mathrm{eq}}}$, defined as the applied field where a global minimum of the energy profile [Eq. ]{} develops inside the sample. For the single (double) strip, this minimum appears at $x_{0} = 0$ ($x_{0} = {b}$) and ${H_{\mathrm{eq}}}$ is determined from the condition $$\begin{aligned}
\label{eq:def-heq}
\varepsilon_{l} d - \frac{\Phi_{0}}{c}
\int\limits_{e}^{x_{0}} du\, I(u)\Bigg|_{H = {H_{\mathrm{eq}}}} \!\!\!\!= 0,\end{aligned}$$ where $e$ denotes the position of the sample edge penetrated first, $e = w$ for the single strip and $e=s$ for the double strip, see Sec.\[sec:sec:double-strip\]. The geometrical barrier at the thermodynamic field ${H_{\mathrm{eq}}}$ $$\begin{aligned}
\label{eq:eq-geometric-barrier}
U_{b}^{\mathrm{eq}}
= \varepsilon_{l} d \Big(1 - \frac{{H_{\mathrm{eq}}}}{{H_{p}}}\frac{{H_{s}}}{H_{c1}}\Big)\end{aligned}$$ then provides us with a measure for the irreversibility of the system, see Fig.\[fig:geometric-barrier\].
Having analyzed and determined the conditions determining the parameters ${b_{0}}$, ${b_{1}}$, and ${b_{2}}$ in the expressions and for the magnetic field, we now are in a position to evaluate the magnetic response (magnetization) of the sample. For this purpose, we make use of Ampère’s law and write the holomorphic field in the form (Biot-Savart, see also Ref. [\[\]]{}) $$\begin{aligned}
\label{eq:biot-savart}
\mathcal{B}(\xi) &= H -
\frac{2}{c}\int_{\text{strips}}\!\!\!\! du \, \frac{I(u)}{\xi - u}.\end{aligned}$$ This field assumes the asymptotic form (we expand for $|\xi|\gg w$) $$\begin{aligned}
\label{eq:multipole-expansion}
\mathcal{B}(\xi) &= H -
\frac{2}{c\, \xi^{2}}\int_{\text{strips}}\!\!\!\!\!\!
du\, u\, I(u) + \mathcal{O}(\xi^{-4}),\end{aligned}$$ where we have used that the total current in each strip vanishes. The second term in [Eq. ]{} describes the field of a line of magnetic dipoles distributed along the $y$-axis ($\xi = 0$). We thus identify the magnetization $M$ per unit length (from here on called magnetization) with the expression $$\begin{aligned}
\label{eq:definition-magnetization}
M &= \frac{1}{c} \int_{\mathrm{strips}}\!\!\!\!du\,u\, I(u).\end{aligned}$$ This result differs from the usual textbook formula[@Jackson_62; @Landau_60] $$\begin{aligned}
\label{eq:magnetization-landau-lifshitz}
\mathcal{M} = \frac{1}{2c} \int d^{3}r\;\boldsymbol{r}
\times \boldsymbol{j}(\boldsymbol{r})\end{aligned}$$ relating the total magnetic moment $\mathcal{M}$ to its generating current density $\boldsymbol{j}(\boldsymbol{r})$ flowing in a loop. The translation invariant 2D result can easily be shown to be consistent with the 3D textbook formula for a finite size ($2L$ along $y$) strip taking also into account the currents $j_{x}(y)$ flowing near the ends $y = \pm L$ of the strips and closing the loop.
Formally expanding the left-hand side of [Eq. ]{} in $\xi^{-2}$ and comparing terms, the magnetization can be rewritten as $$\begin{aligned}
\label{eq:magnetization-through-B}
M(H) &= -\frac{1}{2} \left.\frac{\partial
\mathcal{B}(\xi)}{\partial (1/\xi^{2})}\right|_{\xi^{-2} \to 0}\!.\end{aligned}$$ The magnetic responses of the single and double strip geometries \[as obtained from [Eqs. ]{}, , and \] take the particularly simple form $$\begin{aligned}
\label{eq:magnetization-via-B-single}
M(H) &= -\frac{H}{4} (w^{2}-{b_{0}}^{2}),\\
\label{eq:magnetization-via-B-double}
M(H) &= -\frac{H}{4} (W^{2} + s^{2} - {b_{1}}^{2} - {b_{2}}^{2}).\end{aligned}$$
Single strip {#sec:sec:single-strip}
------------
We briefly review the physics of geometrical barriers for a single strip derived by Zeldov and co-workers[@Zeldov_94]. The function $\mathcal{B}(\xi)$, holomorphic in the superconductor-free region and satisfying the required boundary conditions, is given by [Eq. ]{}. On the $x$-axis ($z = 0$), the magnetic field component along $z$ is given by $$\begin{aligned}
\label{eq:single-field-general}
B_{z}(x) &= \left\{\begin{aligned}
&H\sqrt{\frac{{b_{0}}^{2}-x^{2}}{w^{2}-x^{2}}}
&& \mathrm{for\ } |x| \leq {b_{0}},\\
&H\sqrt{\frac{x^{2}-{b_{0}}^{2}}{x^{2}-w^{2}}}
&& \mathrm{for\ }w \leq |x|,\\
&0
&& \mathrm{for\ }{b_{0}}\leq |x| \leq w.
\end{aligned}\right.\end{aligned}$$ The region $|x| \leq {b_{0}}$ describes the field-penetrated part of the sample where $B_z$ is finite. The current $I(x)$ flows in the complementary regions ${b_{0}}\leq |x| \leq w$ inside the strip; making use of [Eq. ]{} and Ampere’s law in the form of [Eq. ]{} we obtain the current $$\begin{aligned}
\label{eq:single-current-general}
I(x) &= -\frac{H c}{2\pi} \frac{x}{|x|}
\sqrt{\frac{x^{2}-{b_{0}}^{2}}{w^{2}-x^{2}}}.\end{aligned}$$ The anti-symmetry of $I(x)$ guarantees the vanishing of the total current as required by [Eq. ]{}. The diamagnetic response resulting from these currents can be obtained with the formula given in [Eq. ]{} or directly via [Eq. ]{}.
### Meissner state {#sec:sec:sec:Meissner-1}
In the Meissner state the field is fully expelled from the strip, ${b_{0}}=0$, and [Eqs. ]{} and simplify to $$\begin{aligned}
\label{eq:single-field-meissner}
B_{z}(x) &= \left\{\begin{aligned}
&H\frac{x}{\sqrt{x^{2}-w^{2}}}
&& \mathrm{for\ }w \leq |x|,\\
&0
&& \mathrm{for\ } |x| \leq w.,
\end{aligned}\right.\end{aligned}$$ and $$\begin{aligned}
\label{eq:single-current-meissner}
I(x) &= -\frac{H c}{2\pi}
\frac{x}{\sqrt{w^{2}-x^{2}}},\end{aligned}$$ respectively. This anti-symmetric current density preserves the Meissner state and is identical to the one for the elliptic strip discussed before, see [Eq. ]{}. The divergencies in [Eq. ]{} at $x =\pm w$ have to be cut at the distance $\sim d$ away from the edges and we choose the specific value $d/2$. The local field strength at the edge (we drop corrections of higher order in $d/w$) $$\begin{aligned}
\label{eq:single-field-enhancement}
{H_{\mathrm{edge}}}&\equiv B_{z}(w + d/2) \simeq H\sqrt{\frac{w}{d}}\end{aligned}$$ then is enhanced by the factor $\sqrt{w/d}$. This enhancement is parametrically smaller as compared to the flat ellipsoid with corresponding dimensions where the enhancement factor is $2w/d$. The response of the superconducting strip in the Meissner state produces the magnetization \[see [Eq. ]{} with ${b_{0}}= 0$\] $$\begin{aligned}
\label{eq:single-magnetization-meissner}
M(H) &= - \frac{H}{4} w^{2},\end{aligned}$$ corresponding to the expulsion of the field $H$ from a region of size $\sim
w^{2}$. Similar to the currents, the diamagnetic response is identical with that of an elliptic sample, see [Eq. ]{}.
The Meissner state becomes unstable at $H = {H_{p}}$ as determined by the condition [Eq. ]{}; with the field enhancement given in [Eq. ]{}, we find $$\begin{aligned}
\label{eq:single-penetration-field}
{H_{p}}& \simeq {H_{s}}\sqrt{\frac{d}{w}}\end{aligned}$$ and the (maximum) magnetization at penetration reads $$\begin{aligned}
\label{eq:single-m-at-penetration-field}
M_{p} &= - \frac{{H_{s}}}{4} w^2 \sqrt{\frac{d}{w}}
= - \frac{{H_{p}}}{4} w^2.\end{aligned}$$ As discussed above, the precise value for ${H_{p}}$ depends on the details of the edge geometry; the latter will modify the result by a numerical factor of order unity and affect all further results in this section in a straightforward way. For an elliptic strip, the larger field enhancement near the edges causes the penetration field [Eq. ]{} to be parametrically ($\sim\!\! \sqrt{d/w}$) smaller than that of the platelet sample.
Although penetration is delayed to ${H_{p}}$, a field-filled state is thermodynamically stable (yet inaccessible due to the geometric barrier) beyond the equilibrium field $$\begin{aligned}
\label{eq:heq-single}
{H_{\mathrm{eq}}}= H_{c1} \frac{d}{2w}\end{aligned}$$ as obtained from evaluating [Eq. ]{}. The geometric barrier height \[from [Eq. ]{} with ${H_{s}}= H_{c1}$\] at that specific field amounts to $$\begin{aligned}
\label{eq:eq-barrier-single}
U_{b}^{\mathrm{eq}} = \varepsilon_{l} d \,
\big(1 - \sqrt{d/4w}\big).\end{aligned}$$
### Penetrated state {#sec:sec:sec:Shubnikov-1}
Increasing the external field $H$ beyond ${H_{p}}$, vortices accumulate inside the strip in a dome-like density distribution of width $2{b_{0}}$. The field (current) profile along the $x$-axis ($z=0$) is given by the general form \[\]. The absence of a net current inside the strip is satisfied by symmetry, $I(-x) =
-I(x)$. The evolution $$\begin{aligned}
\label{eq:single-dome-width}
{b_{0}}^{2}(H) &\simeq w^{2} [1-({H_{p}}/H)^{2}]\end{aligned}$$ of the dome width as a function of the applied field $H$ is determined by imposing a critical field strength at the edges, i.e., by solving [Eq. ]{} for ${H_{\mathrm{edge}}}= B_{z}(w +
d/2)$. The induction in the vortex dome takes the maximal value $({b_{0}}/w)H$ at the gap center. For a largely penetrated strip, $w-{b_{0}}\ll w$, the induction is almost uniform and equal to the external field, $B(x) \approx
H$. The presence of vortices inside the superconductor reduces the diamagnetic response, see [Eq. ]{} $$\begin{aligned}
\label{eq:single-penetrated-magnetization}
M(H) &\simeq - \frac{{H_{p}}^{2}}{4H} w^{2}
= - \frac{{H_{s}}^{2}}{4H} w d .\end{aligned}$$ The applicability of the expressions and is limited to the regime where the screening currents flow in regions much wider than the sample thickness ($w-{b_{0}}\gg d$), a limit reached when the external field $H$ is very large, of order ${H_{s}}$. At this point, the strip is almost uniformly penetrated by the field with $B_{z} \approx {H_{s}}$, while the remaining screening currents flow in a narrow region of width $\sim d$ near the edges, maintaining a diamagnetic response $$\begin{aligned}
\label{eq:magnetization-at-full-penetration}
M({H_{s}}) &\approx - \frac{{H_{p}}}{4} w^{2}
\sqrt{\frac{d}{w}}.\end{aligned}$$ Predictions on the system’s behavior for very large applied fields $H > {H_{s}}$ require a precise knowledge of the field distribution near the sample edge, a topic which is beyond our present analysis.
The penetration process of vortices across a geometric energy barrier in a platelet strip features a hysteretic behavior[@Zeldov_94; @Zeldov_94_2]; upon reduction of the external field from a maximal value $H^{\star}$, the flux $\phi_{d}^{\star}=\phi_{d}(H^{\star})$ of vortices through the sample, where $$\begin{aligned}
\label{eq:definition-flux-through-dome}
\phi_{d} = \int\limits_{-{b_{0}}}^{{b_{0}}} dx\, B_{z}(x),\end{aligned}$$ is trapped unless the vortex dome boundaries reach the sample edges. Evaluating the above flux with the field , we find $$\begin{aligned}
\label{eq:flux-through-dome}
\phi_{d} = 2wH\Big[\operatorname{E}({b_{0}}/w) - \frac{w^{2}-{b_{0}}^{2}}{w^{2}}\operatorname{K}({b_{0}}/w)\Big],\end{aligned}$$ with $\operatorname{K}$ ($\operatorname{E}$) the complete elliptic integral of the first (second) kind defined according to standard textbooks on mathematical functions; e.g., see Eqs. (17.2.18)-(17.3.3) of Ref. [\[\]]{}, $$\begin{aligned}
\label{eq:K}
\operatorname{K}(\kappa) &= \int_{0}^{\pi/2}
\frac{d\theta}{\sqrt{1-\kappa^{2}\sin\!{}^{2}(\theta)}},\\
\label{eq:E}
\operatorname{E}(\kappa) &= \int_{0}^{\pi/2}
d\theta \sqrt{1-\kappa^{2} \sin\!{}^{2} (\theta)}.\end{aligned}$$ For $\kappa \ll 1$, the elliptic functions show the limiting behavior $$\begin{aligned}
\label{eq:K-small}
\operatorname{K}(\kappa) &= \frac{\pi}{2}\Big[ 1 + \frac{\kappa^{2}}{4}
+ \frac{9\kappa^{4}}{64}+ \mathcal{O}(\kappa^{6}) \Big],\\
\label{eq:E-small}
\operatorname{E}(\kappa) &= \frac{\pi}{2}\Big[ 1 - \frac{\kappa^{2}}{4}
- \frac{3\kappa^{4}}{64}+ \mathcal{O}(\kappa^{6}) \Big],\end{aligned}$$ while for the opposite limit, $\kappa = \sqrt{1-\nu}$ with $\nu \ll 1$, we find $$\begin{aligned}
\label{eq:K-large}
\operatorname{K}(\sqrt{1-\nu}) &= \frac{1}{2}\log\Big(\frac{16}{\nu}\Big)
- \frac{\nu}{8}\Big[2 - \log\Big(\frac{16}{\nu}\Big)\Big]
+ \mathcal{O}(\nu^{2}),\\
\label{eq:E-large}
\operatorname{E}(\sqrt{1-\nu}) &= 1
- \frac{\nu}{4}\Big[1
- \log\Big(\frac{16}{\nu}\Big)\Big]
+ \mathcal{O}(\nu^{2}).\end{aligned}$$ The constraint $\phi_{d}(H<H^{\star}) = \phi_{d}^{\star}$ reduces to a condition for the dome width ${b_{0}}(H)$ of the form $$\begin{aligned}
\label{eq:descending-branch-condition-single}
\operatorname{E}({b_{0}}/w) - \frac{w^{2}-{b_{0}}^{2}}{w^{2}}\operatorname{K}({b_{0}}/w)
&= \frac{\phi_{d}^{\star}}{2w\, H},\end{aligned}$$ in agreement with Ref. [\[\]]{}. The left-hand side is limited by unity from above (for ${b_{0}}= w$). Upon decreasing $H$, the vortex dome expands over the sample until reaching the edge. Since for a large dome, $w - {b_{0}}\ll w$, the induction is uniform and equal to $H$, we find that vortices leave the sample at $H = H_{\mathrm{ex}}= \phi_{d}^{\star}/2w$ where formally ${b_{0}}= w$ and $M=0$.
![The magnetization for the descending field branches are shown for different values of the turning field $H^{\star}$ ($H^{\star}/{H_{p}}$ = 1.25, 1.5, 2.5, 3). The numerical solution of [Eq. ]{} (thick solid lines) is compared to the magnetic response obtained from small domes (thin solid lines) featuring a constant Meissner slope, see [Eq. ]{}. For $H^{\star}$ = 1.25${H_{p}}$ and 1.5${H_{p}}$, the dotted curves show the magnetic response as obtained from a next-to-leading order expansion of [Eq. ]{} in ${b_{0}}/w$, see [Eq. ]{}. For $H^{\star}$ = 2.5${H_{p}}$ and 3${H_{p}}$, where the dome is sufficiently large at $H^{\star}$, i.e., $\nu^{\star} = ({H_{p}}/H^{\star}) \ll 1$, the magnetization is well described by the expression shown as dashed lines.[]{data-label="fig:descending-branch-single"}](plot_19.eps){width="48.00000%"}
For a narrow dome ${b_{0}}/w \ll 1$, the above condition can be simplified using the asymptotic expressions and for the elliptic functions; to lowest (quadratic) order in $\kappa = {b_{0}}/w$, we find $$\begin{aligned}
\label{eq:descending-branch-single-small-dome-1}
\frac{{b_{0}}^{2}}{w^{2}} &= \frac{4}{\pi}\frac{H_{\mathrm{ex}}}{H}.\end{aligned}$$ The growth of the dome width ${b_{0}}^{2} = {b_{0}}^{\star}{}^{2} H^{\star}/H$ \[with ${b_{0}}^{\star} = {b_{0}}(H^{\star})$\] results in a magnetic response of the form $$\begin{aligned}
\label{eq:magnetization-descending-single-small-dome-1}
M(H) &= -\frac{H}{4} w^{2} + \frac{H^{\star}}{4} {{b_{0}}^{\star} }^2\end{aligned}$$ with a slope identical to the Meissner state. Higher order corrections (quartic in $\kappa = {b_{0}}/w$) are straight forwardly obtained from [Eqs. ]{} and : the condition then yields $$\begin{aligned}
\label{eq:descending-branch-single-small-dome-4}
\frac{{b_{0}}^{2}}{w^{2}} &= 4 \bigg\{
\sqrt{1 + \frac{H^{\star}}{H}
\Big[
\Big(1 + \frac{1}{4}\frac{{b_{0}}^{\star}{}^{2}}{w^{2}}
\Big)^{2} - 1
\Big]
} -1
\bigg\}.\end{aligned}$$ Inserting this solution into the expression for the magnetization, the Meissner slope is corrected according to $$\begin{aligned}
\nonumber
\frac{dM}{dH} &= - \frac{w^{2}}{4} \bigg\{
1 - \frac{1}{2}\Big(\frac{H^{\star}}{H}\Big)^{2}
\Big[
\Big(1 + \frac{1}{4}\frac{{b_{0}}^{\star}{}^{2}}{w^{2}}
\Big)^{2} - 1
\Big]^{2}
\bigg\}\\
&\approx - \frac{w^{2}}{4} \Big[1- \frac{1}{8}
\Big(
\frac{H^{\star}}{H} \frac{{b_{0}}^{\star}{}^{2}}{w^{2}}
\Big)^{2}
\Big].\end{aligned}$$ The rapid growth of the dome-width on both the field-increasing (filling the dome with additional flux) and decreasing (expanding the dome at fixed flux) branches leads to a fast violation of the condition ${b_{0}}\ll w$ assumed above and hence these results have a rather limited range of validity. Another limit is reached when ${b_{0}}$ is large, $w - {b_{0}}\ll w$. Defining $\nu = 1 -
{b_{0}}^{2}/w^{2}$, the asymptotic expressions and can be used to simplify (up to linear order in $\nu$) the condition to $$\begin{aligned}
\label{eq:simplified-returning-branch}
1 - \frac{\nu}{4} \Big[\log\Big(\frac{16}{\nu}\Big) + 1\Big]
&= \frac{H_{\mathrm{ex}}}{H}.\end{aligned}$$ In most of the $M$-$H$-diagram, the system’s magnetic response on the descending branch then is given by $M(H) = -H w^{2} \nu(H)/4$. Taking the derivative of $M$ with respect to $H$, the slope of the descending branch can be evaluated and, after some reordering, we find that $$\begin{aligned}
\label{eq:slope}
\frac{dM}{dH} &= -\frac{w^{2}}{4} \frac{4 - \nu}{\log(16/\nu)}.\end{aligned}$$ The derivative deviates from the Meissner slope $-w^{2}/4$ by a numerical factor which assumes the value $\approx 1.01$ for $\nu \sim 1/2$, when the previous approach of a narrow dome predicts a perfect Meissner slope \[see [Eq. ]{}\]. In the regime of applicability, where $\nu$ may change by several orders of magnitude, the factor $(4-\nu)/\log(16/\nu)$ changes noticeably but not parametrically. Typically, the slope of the descending branch is numerically close to the Meissner slope within the parameter range under consideration, see Fig. \[fig:descending-branch-single\]. For large reversal fields $H^{\star} \gg {H_{p}}$, we replace the parameter $\nu(H)$ by its value at the field reversal $\nu(H^\star) = \nu^{\star} = 1 - {b_{0}}^{\star}{}^{2}/w^{2} =
({H_{p}}/H^\star)^{2}$, where we have used [Eq. ]{}. The magnetization $$\begin{aligned}
\label{eq:analytic-returning-single-strip}
M(H) = M(H^{\star}) - \frac{H-H^{\star}}{4} w^{2}
\frac{4 - \nu^{\star}}{\log(16/\nu^{\star})} \end{aligned}$$ as obtained from [Eq. ]{} and integration from $H^{\star}$ to $H$ provides a good description of the descending branch in this regime, see [Fig. ]{}\[fig:descending-branch-single\].
As the boundaries of the dome approach the edges of the strip to a distance $\sim d$ (which is the case when $H \approx [1+ \mathcal{O}(d/w)]
H_{\mathrm{ex}}$) the precise geometric shape of the sample edge needs to be taken into account, requiring a more accurate analysis going beyond the present description. An attempt to cope with this situation has been undertaken by Zeldov and co-workers in Refs.[\[\]]{}.
### Magnetization of the vortex dome {#sec:sec:sec:magnetic-medium-1}
The physical properties of quantized flux lines appeared in the above analysis merely as a criterion for vortex entry at the sample edges. The vortex dome in the penetrated state has been described by a smooth field $B_z(x) \neq 0$ residing in a magnetically inactive medium with $\mu = 1$ whose extend $[-b_0,b_0]$ derives from the solution ${\cal B}(\xi)$ of the boundary value problem. In reality, the vortex state in the dome is described by a field $h(x)$ modulated on the scale of the inter-vortex distance due to vortex currents. In the following, we show that the currents associated with the vortex state in the dome generate a magnetization which remains small as compared to the magnetization produced by the screening currents flowing in the field-free regions.
An analogous problem appears in the context of surface barriers as discussed by Clem[@Clem_74] and by Koshelev[@Koshelev_94]: quite similar to our analysis, in Ref. [\[\]]{} the vortex-penetrated bulk, separated from the boundary by a layer of screening (Meissner) currents, has been described by an induction $B_z$ averaged over the inter-vortex spacing. This approximation neglects all field and current modulations due to the vortex state and the resulting magnetization density is given by[@Clem_74] $$\begin{aligned}
\label{eq:magnetization-surf-barrier-Clem}
m(H) &= -\frac{H}{4\pi} \big[1 - \sqrt{1 - ({H_{s}}/H)^{2}}\,\big].\end{aligned}$$ A way to account for the local currents in the vortex state has been proposed by Koshelev[@Koshelev_94], who found that these contribute a paramagnetic correction $\delta m = (\sqrt{3}/48) (\Phi_{0}/4\pi \lambda^{2})$ to the magnetization density $m(H)$ in the limit $B \gg \Phi_{0}/\lambda^{2}$. Following a similar ideology as in Ref. [\[\]]{}, we describe the flux-filled region in terms of a vortex lattice along $z$ with vortex rows aligned along $y$ and separated by ${b_{\scriptscriptstyle{\triangle}}}$ in the $x$-direction with ${b_{\scriptscriptstyle{\triangle}}}^{2} = (3/4)^{1/2}\Phi_{0}/B_{z}$. While in Ref.[\[\]]{} $B_z(x)$ was determined self-consistently, here, we estimate the corrections to the magnetization by adopting the averaged field $B_z(x)$ obtained from the above analytic solution. In our strip geometry, the spacing ${b_{\scriptscriptstyle{\triangle}}}$ between vortex-rows slowly varies along $x$, as the induction $B_{z}$ changes on macroscopic length scales. The connection between the local field $h(x)$ and the induction $B_{z}(x)$ is given by the average $$\begin{aligned}
\label{eq:average-induction}
B_{z}(x) = \frac{1}{{b_{\scriptscriptstyle{\triangle}}}}\int\limits_{x-{b_{\scriptscriptstyle{\triangle}}}/2}^{x+{b_{\scriptscriptstyle{\triangle}}}/2} dx' h(x').\end{aligned}$$ The local field $h(x)$ satisfies the one-dimensional London equation $\lambda^{2} h''(x) +
h(x) = 0$ between the vortex rows with the boundary conditions replaced by the constraint . For a slowly varying dome profile, i.e., ${b_{\scriptscriptstyle{\triangle}}}\partial_{x} B_{z}(x) \ll B_{z}(x)$, we obtain the field modulation between vortex rows $$\begin{aligned}
h(x) &\approx B_{z}(x_{c}) \frac{{b_{\scriptscriptstyle{\triangle}}}}{2\lambda}
\frac{\cosh[(x-x_{c})/\lambda]}{\sinh({b_{\scriptscriptstyle{\triangle}}}/2\lambda)},\end{aligned}$$ with $x_{c}$ the center between the two adjacent rows and $|x-x_{c}| < {b_{\scriptscriptstyle{\triangle}}}/2$. Ampère’s law then provides us with the current profile $$\begin{aligned}
j(x) &\approx - \frac{B_{z}(x_{c})c}{4\pi} \frac{{b_{\scriptscriptstyle{\triangle}}}}{2\lambda^{2}}
\frac{\sinh[(x-x_{c})/\lambda]}{\sinh({b_{\scriptscriptstyle{\triangle}}}/2\lambda)}\end{aligned}$$ and we can evaluate the associated average magnetization density at the vortex location $x_v$ $$\begin{aligned}
m(x_{v}) &\approx \frac{1}{{b_{\scriptscriptstyle{\triangle}}}c}
\int\limits_{x_{v}-{b_{\scriptscriptstyle{\triangle}}}/2}^{x_{v}+{b_{\scriptscriptstyle{\triangle}}}/2} dx' x' j(x')\\
\label{eq:material-magnetization}
&\approx \frac{B_{z}(x_{v})}{4\pi}
\Big[1 - \frac{{b_{\scriptscriptstyle{\triangle}}}}{2\lambda} \frac{1}{\sinh({b_{\scriptscriptstyle{\triangle}}}/2\lambda)}\Big].\end{aligned}$$ For small fields $B_{z} \ll H_{c1}$, we find that $m(x) \approx
B_{z}(x)/4\pi$, while the magnetization density saturates at $(\Phi_{0}/4\pi\lambda^{2}) \sqrt{3}/48$ for large fields $B_{z} \gg H_{c1}$, consistent with the results presented in Ref. [\[\]]{}. In order to estimate the correction to the strips’ magnetic response, we introduce the upper bound $$\begin{aligned}
\label{eq:ub-magnetization}
m(x) \leq \frac{B_{z}(x)}{4\pi} \frac{H_{c1}}{H_{c1}+B_{z}(x)}\end{aligned}$$ with the correct asymptotic behavior for $B \ll H_{c1}$ and logarithmically \[$\propto \log(\lambda/\xi)$\] overestimating the magnetization when $B \gg
H_{c1}$. Integrating $m(x)$ over the dome and replacing the dome profile $B_z(x)$ by its maximum $H {b_{0}}/w$ at the center, see [Eq. ]{}, we obtain the bound $$\begin{aligned}
\label{eq:magnetic-correction-estimated}
\delta M < \frac{H}{4\pi} \frac{H_{c1}}{H_{c1}
+ H {b_{0}}/w} \frac{{b_{0}}^{2}}{w^{2}}2w d.\end{aligned}$$ For a small dome, ${b_{0}}\ll w$, this expression simplifies to $$\begin{aligned}
\label{eq:magnetic-correction-estimated-low-field}
\delta M < \frac{H}{4\pi} \Big(1 - \frac{{H_{p}}^{2}}{H^{2}} \Big)2w d,\end{aligned}$$ whereas for a large part of the penetrated region $d \ll w-{b_{0}}(H) \ll w$, we find that $$\begin{aligned}
\label{eq:magnetic-correction-estimated-high-field}
\delta M < \frac{H}{4\pi} \frac{H_{c1}}{H_{c1} + H} 2w d.\end{aligned}$$ As a result, the correction $\delta M$ due to the reversible magnetization measured on the magnetization $M$ of the screening currents [Eq. ]{} is bounded from above by $$\begin{aligned}
\label{eq:relative-corrections-single}
\frac{\delta M}{M}
&< \frac{2}{\pi} \frac{H^{2}}{{H_{s}}^{2}} \frac{H_{c1}}{H_{c1}+H}.\end{aligned}$$ In the absence of a surface barrier (${H_{s}}= H_{c1}$) these corrections are small and become of order unity at the largest fields $H \sim H_{c1}$ where our analysis applies. In the presence of a large surface barrier where ${H_{s}}\gg H_{c1}$, the corrections are even smaller and reach a maximum $\sim
H_{c1}/{H_{s}}\ll 1$ when $H \sim {H_{s}}$. We conclude that the corrections arising from the vortex currents can be omitted in the single strip geometry.
Double strip {#sec:sec:double-strip}
------------
We now investigate the double-strip configuration defined in [Fig. ]{}\[fig:sketch-double-strip\], a system of two coplanar, parallel strips of width $2w$ each and separated by a gap $2s$. Assuming a gap that is large as compared to the strip thickness, $s \gg d$, the system can be treated within the framework introduced in [Sec. ]{}\[sec:sec:formalism\]. The holomorphic function has been presented in [Eq. ]{}, from which the \[symmetric, $B_z(-x) = B_z(x)$\] field and \[anti-symmetric, $I(-x) =
-I(x)$\] current distribution on the $x$-axis can be readily deduced $$\begin{aligned}
\label{eq:double-field-general}
\frac{B_{z}(x)}{H} &= \left\{ \begin{aligned}
&\sqrt{\frac{({b_{1}}^{2}-x^{2})({b_{2}}^{2}-x^{2})}
{(s^{2}-x^{2})(W^{2}-x^{2})}}
&& \mathrm{for\ } 0 \leq x \leq s,\\
&\sqrt{\frac{(x^{2}-{b_{1}}^{2})({b_{2}}^{2}-x^{2})}
{(x^{2}-s^{2})(W^{2}-x^{2})}}
&& \mathrm{for\ } {b_{1}}\leq x \leq {b_{2}},\\
&\sqrt{\frac{(x^{2}-{b_{1}}^{2})(x^{2}-{b_{2}}^{2})}
{(x^{2}-s^{2})(x^{2}-W^{2})}}
&& \mathrm{for\ } W \leq x,\\
&0
&& \mathrm{otherwise},
\end{aligned}\right.\end{aligned}$$ and $$\begin{aligned}
\label{eq:double-current-general}
\frac{2\pi I(x)}{cH} &= \left\{ \begin{aligned}
&\sqrt{\frac{({b_{1}}^{2}-x^{2})({b_{2}}^{2}-x^{2})}
{(x^{2}-s^{2})(W^{2}-x^{2})}}
&& \mathrm{for\ } s \leq x \leq {b_{1}},\\
&-\sqrt{\frac{(x^{2}-{b_{1}}^{2})(x^{2}-{b_{2}}^{2})}
{(x^{2}-s^{2})(W^{2}-x^{2})}}
&& \mathrm{for\ } {b_{2}}\leq x \leq W,\\
&0
&& \mathrm{otherwise}.
\end{aligned}\right.\end{aligned}$$ The resulting magnetization is given by [Eq. ]{}.
### Meissner state {#sec:sec:sec:Meissner-2}
In the (low-field) Meissner state the parameters ${b_{1}}$, ${b_{2}}$ in [Eq. ]{} coincide, ${b_{1}}= {b_{2}}= {b}$, with $\pm {b}$ marking the the positions inside the strips where the current density changes sign (see [Fig. ]{}\[fig:double-strip-meissner-d-ll-s\]). The magnetic field component $B_{z}$ \[from [Eq. ]{}\] is non-vanishing whenever $|x|\leq s$ or $W \leq |x|$ and reads $$\begin{aligned}
\label{eq:double-field-meissner}
B_{z}(x) &=
H \frac{|x^{2}-{b}^{2}|}
{\sqrt{(x^{2}-s^{2})(x^{2}-W^{2})}}.\end{aligned}$$ In the complementary region $s \leq |x| \leq W$, the screening current $$\begin{aligned}
\label{eq:double-current-meissner}
I(x) &= - \frac{H c}{2\pi} \frac{x}{|x|} \frac{x^{2}-{b}^{2}}
{\sqrt{(x^{2}-s^{2})(W^{2}-x^{2})}}\end{aligned}$$ guarantees a perfect diamagnetic (Meissner) response $$\begin{aligned}
\label{eq:double-strip-magnetization-meissner}
M(H) &= -\frac{H}{4} (W^{2} + s^{2} - 2{b}^{2}),\end{aligned}$$ with ${b}$ independent of $H$. The condition that no net current flows along each strip requires that $$\begin{aligned}
\label{eq:double-zero-current-cond}
\int\limits_{s}^{W} \frac{dx\,x^{2}}{\sqrt{(x^{2}-s^{2})(W^{2}-x^{2})}} &=
\int\limits_{s}^{W} \frac{dx\,{b}^{2}}{\sqrt{(x^{2}-s^{2})(W^{2}-x^{2})}},\end{aligned}$$ from which we find the value of ${b}$, $$\begin{aligned}
\label{eq:double-zero-current-result}
{b}^{2} &= W^{2} \frac{\operatorname{E}(\kappa')}{\operatorname{K}(\kappa')},\end{aligned}$$ in agreement with Ref. [\[\]]{}. Here, $\operatorname{K}$ ($\operatorname{E}$) is the complete elliptic integral of the first (second) kind, as defined in [Eq. ]{} \[\], and $\kappa' = \sqrt{1-\kappa^{2}}$ is the complementary modulus of $\kappa = s/W$. For large gaps, the double strip behaves as two independent strips: indeed, for $s/w \to \infty$, the parameter ${b}$ approaches the sample center $w+s$ and the magnetization assumes the asymptotic value $M(H) \to -H w^{2}/2$, twice that of an isolated strip, see [Eq. ]{}.
![Normalized current density $2 \pi I(x)/H c$ (solid line) flowing along the $y$-direction and dimensionless magnetic field $B_{z}(x)/H$ (dashed line) of a double-strip in the Meissner state. The two strips with width $2w$ and thickness $d$ ($d/w \to 0$) are separated by a gap $2s$ (here $w/s=100$). According to [Eqs. ]{} and , the local current reverts its sign at $\pm {b}$, with ${b}\approx 0.38 \, W$. The magnetic field inside the gap between the strips (see inset) is far above the range of this graph, $B_{z}(|x|<s)/H
\geq {b}^{2}/Ws \approx 30$.[]{data-label="fig:double-strip-meissner-d-ll-s"}](plot_01_mod.eps){width="48.00000%"}
Let us then focus on the opposite limit $s \ll W=2w+s$, where the right hand side of [Eq. ]{} shows a logarithmic divergence $\propto \log(W/s)$, while the left hand side is regular; in this limit, the parameter ${b}$ takes the asymptotic form $$\begin{aligned}
\label{eq:a-over-W-ratio}
{b}^{2} &= \frac{W^{2}}{\log{(4W/s)}},\end{aligned}$$ and the position ${b}$ where the current $I(x)$ changes sign is no longer at the sample center but has shifted towards the inner edge, see Figs.\[fig:double-strip-meissner-d-ll-s\], \[fig:b1-n-b2\] and \[fig:a-of-s\]. The magnetization (per unit length) to leading order in $s/W$ reads, $$\begin{aligned}
\label{eq:meiss-mag-double-strip}
M(H) &= -\frac{H}{4}W^{2}\Big[1 - \frac{2}{\log(4W/s)}\Big].\end{aligned}$$ In the limit $s/W \to 0$, the Meissner slope approaches that of a single strip with double width, see [Eq. ]{}. We conclude that over the full range of gap widths $s$ (from $s \gg W$ down to $s/W \to 0$) the slope in the magnetization of the Meissner state increases only by a factor 2.
For the double strip geometry, the flux (per unit length) $\phi_{g}$ passing through the gap $|x| < s$ is defined as the $z$-component of the magnetic field integrated over the gap width, $$\begin{aligned}
\label{eq:flux}
\phi_{g} &=\!\!\int\limits_{-s}^{s}\! dx\,B_{z}(x)
= 2 W \Big[\operatorname{E}(\kappa)
- \Big(1-\frac{{b}^{2}}{W^{2}}\Big)\operatorname{K}(\kappa)\Big] H,\end{aligned}$$ where the elliptic functions are evaluated at $\kappa = s/W$. In the regime of almost independent strips, $s \gg W$, the flux $2s H$ of the homogeneous field in the empty gap region is enhanced by half of the flux $\phi_{b} = 4w
H$ blocked by the two strips, thus adding up to $\phi_{g} \approx (2s + 2w)H$. In the opposite limit $s \ll W$, the expression for the flux in the gap simplifies to $$\begin{aligned}
\label{eq:flux-t-s-l}
\phi_{g} &\simeq \frac{\pi {b}^{2}}{W} H \simeq \frac{\pi W}{\log(4W/s)} H.\end{aligned}$$ An essential part (up to a logarithmic factor) of the blocked flux $\phi_{b} = 2W H$ is pushed through the gap. This slow reduction of $\phi_{g}$ upon reducing $s$ goes hand in hand with an enhancement of the field strength at the gap center $$\begin{aligned}
\label{eq:double-field-enhancement-gap-center}
B_{z}(0) &= H \frac{{b}^{2}}{s W} = \frac{2}{\pi}\frac{\phi_{g}}{2s} = H \frac{W/s}{\log(4W/s)}\end{aligned}$$ and near the inner edges $$\begin{aligned}
\label{eq:double-field-enhancement}
B_{z}(s - d/2) &\simeq H\frac{{b}^{2}}{\sqrt{sd}\,W}
= H\frac{W/\sqrt{sd}}{\log(4W/s)}.\end{aligned}$$ This last expression is parametrically larger than the enhancement observed at the edge of an isolated strip, see [Eq. ]{}. Note that the field inside the gap is far from constant, but increases by a factor $\sqrt{s/d}$ from the gap center to one strip edge, see inset in [Fig. ]{}\[fig:double-strip-meissner-d-ll-s\]). On the other hand, the field strength near the outer edges $$\begin{aligned}
\label{eq:double-field-enhancement-outer}
B_{z}(W + d/2) &\simeq H \frac{W^{2}-{b}^{2}}{W\sqrt{Wd}}
= H\sqrt{\frac{W}{d}}\Big[1 - \frac{1}{\log(4W/s)}\Big] \end{aligned}$$ is comparable to that of an isolated strip, see [Eq. ]{}. From this analysis we conclude that the local critical field ${H_{s}}$ is first reached near the inner edges, such that the penetration of vortices occurs from *inside*. The field of first penetration ${H_{p}}$ then is determined by the condition $$\begin{aligned}
\label{eq:criticality-condition-double-strip}
{H_{\mathrm{edge}}}&= B_{z}(s - d/2) = {H_{s}}\end{aligned}$$ and making use of [Eq. ]{} we find that the penetration field for small gaps $s \ll W$ $$\begin{aligned}
\label{eq:double-penetration-field}
{H_{p}}&\simeq {H_{s}}\sqrt{\frac{s d}
{W^{2}}} \frac{W^{2}}{{b}^{2}}
= {H_{s}}\sqrt{\frac{s d}{W^{2}}} \log{(4W/s)}\end{aligned}$$ is substantially reduced as compared to the one for isolated strips ${H_{p}}\simeq {H_{s}}\sqrt{d/w}$. As discussed for the single strip, see [Eq. ]{} and thereafter, the precise edge geometry will alter the above expression for ${H_{p}}$ by a numerical factor of order unity (the same factor as for the single strip), a correction that will be neglected in the following. At penetration $H={H_{p}}$, the Meissner state reaches the maximal diamagnetic response \[see [Eq. ]{}\] $$\begin{aligned}
\label{eq:max-magnetization-double-thin-strip}
M_{p} &= -\frac{{H_{s}}}{4}W \sqrt{s d}\, \big[\log{(4W/s)} - 2\big].\end{aligned}$$ Upon reducing the gap width $s$, the penetration field diminishes and the geometrical barrier is more strongly suppressed, see [Eq. ]{}. Vortices become energetically favorable (deep) inside the sample beyond the equilibrium field (we use [Eq. ]{} in the regime $s \ll W$) $$\begin{aligned}
\label{eq:heq-double-thin}
{H_{\mathrm{eq}}}= H_{c1} \frac{d}{2W} \Big\{1
- \frac{\log[4\log(4W/s)]+1}{2 \log(4W/s)}\Big\}^{-1},\end{aligned}$$ resulting in a geometric barrier at ${H_{\mathrm{eq}}}$ which decreases with $s$, $$\begin{aligned}
\label{eq:eq-barrier-double-thin}
\frac{U_{b}^{\mathrm{eq}}(s)}{\varepsilon_{l} d}= 1 - \frac{\sqrt{d/s}}
{2 \log(4W/s) - \log[4\log(4W/s)]-1}.\end{aligned}$$
### Penetrated state {#sec:sec:sec:Shubnikov-2-tsl}
Increasing the external field beyond its critical value, ${H_{p}}$, vortices penetrate the superconductor from the inner edges at $x = \pm s$ and accumulate near the position ${b}$ inside the strips where the potential $U_{\mathrm{geo}}(x)$ is minimal. The field and currents take the general form given in [Eqs. ]{} and , with the non-trivial vortex state determined by the two boundaries of the vortex dome ${b_{1}}$ and ${b_{2}}$.
![Dimensionless field $B_{z}(x)/H$ (dashed line) and current $2\pi
I(x)/H c$ (solid line) as a function of $x$ (for $z=0$) for the same geometry ($w/s = 100$ and $s/d = 100$) as in [Fig. ]{}\[fig:double-strip-meissner-d-ll-s\] and an external field $H = 1.2 {H_{p}}$ above first penetration. In the penetrated state above ${H_{p}}$, vortices accumulate in a finite region inside each strip (the vortex dome), with boundaries given by $\pm{b_{1}}$ and $\pm{b_{2}}$.[]{data-label="fig:double-strip-penetrated-d-ll-s"}](plot_05.eps){width="48.00000%"}
These two parameters satisfy the constraint of vanishing net current in each strip $$\begin{aligned}
\label{eq:double-no-net-current}
\int_{s}^{{b_{1}}} dx\, I(x) + \int_{{b_{2}}}^{W} dx\, I(x) &= 0,\end{aligned}$$ together with the condition \[from [Eq. ]{}\] $$\begin{aligned}
\label{eq:critical-condition-penetrated-double}
{H_{\mathrm{edge}}}= B_{z}(s-d/2) &= {H_{s}}.\end{aligned}$$ While this constraint locks the field strength at the inner edge to ${H_{s}}$, the field strength near the outer edge continuously grows, but remains below ${H_{s}}$. In [Fig. ]{}\[fig:double-strip-penetrated-d-ll-s\] we show the field and current profiles in the penetrated state for $H = 1.2{H_{p}}$ as obtained from solving [Eqs. ]{} and numerically. The evolution of the dome’s boundaries and its width ${b_{2}}- {b_{1}}$ with increasing field is shown in [Fig. ]{}\[fig:b1-n-b2\]. The maximal field value in the dome can be estimated with the interpolation formula $B_\mathrm{dome}
\sim H (b_2-b_1)/W$.
In order to find analytic results describing the penetrated state, we have to simplify the problem of determining the parameters ${b_{1}}$, ${b_{2}}$. Evaluating the condition for the field and expressing the result through the penetration field ${H_{p}}$, the dome boundaries ${b_{1}}(H)$ and ${b_{2}}(H)$ are related via $$\begin{aligned}
\label{eq:critical-current-condition}
\frac{\sqrt{{b_{1}}^{2}-s^{2}}\ {b_{2}}}{{b}^{2}} = \frac{{H_{p}}}{H},\end{aligned}$$ where ${b}$ is the (field-independent) zero-current location in the Meissner state, [Eq. ]{}.
It turns out that a perturbative calculation around the penetration field with the small parameter $h = (H-{H_{p}})/{H_{p}}\ll 1$ produces results with a very limited range of validity. This is due to the rapid growth of the dome width ${b_{2}}- {b_{1}}$ with increasing $h$, leading to a fast break-down of the approximation. Approaching the problem from the high field limit $H \gg {H_{p}}$ is more successful: starting from the regime where the dome extends over a large fraction of the strip ${b_{1}}\ll {b_{2}}$, we can adopt another perturbative approach which provides accurate results all the way down to ${H_{p}}$. We use the Ansatz ${b_{2}}= W (1-\nu)^{1/2}$ with $\nu(H) < 1$. For $s\ll {b_{1}}$, where [Eq. ]{} simplifies to $$\begin{aligned}
\label{eq:appr-critical-current-condition}
{b_{1}}= W \frac{{b}^{2}}{W^{2}} \frac{{H_{p}}}{H} \frac{1}{\sqrt{1-\nu}},\end{aligned}$$ the constraint of vanishing net current in the strips can be written as $$\begin{aligned}
\label{eq:appr-neutral-current-condition}
\frac{{H_{p}}}{H}\frac{{b}^{2}}{W^{2}}
\left[\log{\Big(\frac{4W}{s}
\frac{{b}^{2}}{W^{2}}
\frac{{H_{p}}}{H}
\frac{1}{\sqrt{1-\nu}}\Big)}-1\right]&\\[.5em]
= \operatorname{E}(\sqrt{\nu})-&(1-\nu)\operatorname{K}(\sqrt{\nu}).\nonumber\end{aligned}$$ Solving this equation to leading order in ${H_{p}}/H$ where $\nu \ll 1$, we find that $$\begin{aligned}
\label{eq:mu-t-s-l}
\nu(H) &= \frac{4}{\pi}\frac{{H_{p}}}{H}\frac{{b}^{2}}{W^{2}}
\left[\log{\Big(\frac{4W}{s}
\frac{{b}^{2}}{W^{2}}
\frac{{H_{p}}}{H}\Big)}-1\right].\end{aligned}$$
![Evolution of the dome edges ${b_{1}}(H)$ and ${b_{2}}(H)$ with increasing field $H$ for parameters $w/s = 100$ and $s/d = 100$. For small fields $H
< {H_{p}}$, the double strip is in the Meissner phase and ${b_{1}}= {b_{2}}= {b}$, where ${b}$ is shifted away from the sample center $w+s$. In the penetrated state $H>{H_{p}}$, vortices accumulate inside the strip and the dome width ${b_{2}}- {b_{1}}$ widens. The field and current profiles for the field $H = 1.2 {H_{p}}$ are shown in [Fig. ]{}\[fig:double-strip-penetrated-d-ll-s\].[]{data-label="fig:b1-n-b2"}](plot_13_mod.eps){width=".48\textwidth"}
To leading order in $\nu(H)$, the magnetic response in [Eq. ]{} takes the form $M \simeq -H \nu(H)
W^{2}/4$, resulting in a logarithmic field-dependence $$\begin{aligned}
\label{eq:appr-mag-t-s-l-original-parameters}
M(H) &\simeq -\frac{{H_{s}}}{4\pi} (2W d)
\bigg\{2\sqrt{\frac{s}{d}}
\Big[\log{\Big(\frac{4 {H_{s}}}{H}\sqrt{\frac{d}{s}\,}\Big)}-1\Big]
\bigg\}.\end{aligned}$$
![Magnetization of a double strip system obtained from numerical evaluation (solid line) and from the analytic solutions (dashed lines) for parameters $w/s = 100$ and $s/d = 100$. The expression in [Eq. ]{} is applicable in the field range ${H_{p}}\ll H \ll H|_{{b_{1}}\sim2s}$. It turns out, that the analytic approximation is accurate almost down to ${H_{p}}$, where the magnetization is $M_{p} = M({H_{p}})$, see [Eq. ]{}. For very large fields, $H > H|_{{b_{1}}\sim2s}$, where the distance between the dome boundary ${b_{1}}$ and the sample edge $s$ falls below $s$, the magnetic response is well described by the asymptotic result in [Eq. ]{}. The dome reaches the edges at a distance $d$ only when $H \sim {H_{s}}\gg H|_{{b_{1}}\sim 2s}$. Both approximations and are shown in their domain of applicability.[]{data-label="fig:numeric-vs-analytic"}](plot_07.eps){width=".43\textwidth"}
Because of the simplification in [Eq. ]{}, the validity of the result is limited to fields $H \ll
H|_{{b_{1}}\sim2s}\sim ({b}^{2}/sW){H_{p}}\approx \sqrt{d/s\,} {H_{s}}$, where the restriction $s \ll {b_{1}}$ is satisfied. As shown in [Fig. ]{}\[fig:numeric-vs-analytic\], the expression is in good agreement with the numerical solution and describes the evolution of the magnetic response over a large range of fields ${H_{p}}\lesssim H \ll H|_{{b_{1}}\sim2s}$.
For ${b_{1}}- s \ll s$, the same Ansatz ${b_{2}}= W (1-\nu)^{1/2}$ allows to simplify the constraint [Eq. ]{} to $$\begin{aligned}
\label{eq:no-net-current-for-b1minuss-ll-s}
{b_{1}}&= s + W\nu/2,\end{aligned}$$ while [Eq. ]{} takes the form $$\begin{aligned}
\label{eq:appr-critical-current-condition-2}
s + W\nu/2 &= \sqrt{s^{2} +
W^{2}\Big(\frac{{H_{p}}}{H}\frac{{b}^{2}}{W^{2}}\Big)^{2}}.\end{aligned}$$ To leading order in ${H_{p}}/H$ we find $$\begin{aligned}
\nu(H) &= \frac{W}{s}\Big(\frac{{H_{p}}}{H} \frac{{b}^{2}}{W^{2}}\Big)^{2}\end{aligned}$$ and the magnetization reads $$\begin{aligned}
\label{eq:magnetization-in-large-field-assymptotic}
M(H) = -\frac{{H_{s}}^{2}}{4H} Wd.\end{aligned}$$ For the strongly penetrated double strip, when the current-carrying regions are smaller than $s$ but still wider than $d$, the mutual influence of the two strips becomes negligible. The magnetization thus approaches that of two independent single strips of width $W$ each \[see [Eq. ]{}\].
Pushing the above ‘thin strip’ solution obtained for $s \gg d$ to the limit $s
= d$, we find for the penetration field in [Eq. ]{} $$\begin{aligned}
\label{eq:penetration-field-s-is-d-thin-strips}
{H_{p}}&\approx {H_{s}}\frac{d}{W} \log{(4W/d)},\end{aligned}$$ which is substantially smaller than that of an isolated strip as given in [Eq. ]{}. Similarly, in this limit the magnetization as approximated by [Eq. ]{} becomes $$\begin{aligned}
\label{eq:appr-mag-s-eq-d}
M(H) &= -\frac{{H_{s}}}{4\pi} 4Wd \,\big[\log{(4 {H_{s}}/H)}-1\big].\end{aligned}$$ This expression is valid for fields up to $H|_{{b_{1}}\sim 2d} \sim {H_{s}}$ where the dome reaches the edges and is consistent with the limit $s
{\scriptstyle{\ \nearrow\ }} d$ approaching the thickness $d$ from below as discussed in Sec. \[sec:finite-thickness\] below.
In order to understand the penetration mechanism in the double strip for the full range of strip separations $2s$, below we extend our analysis to a system where the gap width $2s$ is much smaller than the thickness $d$ of the strips, $s \ll d$, see [Sec. ]{}\[sec:finite-thickness\]. Before doing that, we briefly elaborate on the corrections due to the vortex structure in the dome.
![The magnetic response of the double strip is shown for different separation $s$ (solid lines) and a fixed ratio $w/d=10^{3}$. For $s$ larger or equal to $d$, we adopt the thin-strip approach of [Sec. ]{}\[sec:sec:double-strip\], while for $s < d$ (see top expansion), the determination of the magnetization curves has to account for the finite thickness of the strips as described in [Sec. ]{}\[sec:finite-thickness\]. The magnetization $M_{p}$ at the penetration field ${H_{p}}$ is largest for isolated strips ($s/w \to \infty$) and reduces upon decreasing $s$. The parametric curve $({H_{p}}, M_{p})$ as a function of $s$ is indicated by the dotted line. In the limit $s/w \to 0$ the slope of the magnetization curve in the Meissner state doubles as compared to that for isolated strips ($s/w \to \infty$). The dashed line indicates the magnetization of a single strip of width $2W$, corresponding to $s = 0$.[]{data-label="fig:magnetization-s=d-compared-to-single-and-isolated-strip"}](plot_04_mod_test.eps){width=".48\textwidth"}
### Magnetization of the vortex dome {#sec:sec:sec:magnetic-medium-2}
To estimate the quantitative effects arising from the currents around the flux lines in the vortex dome, we give an upper bound to the corrections of the magnetic response in [Eqs. ]{} and . Following the analysis presented in [Sec. ]{}\[sec:sec:sec:magnetic-medium-1\], we find an upper bound $$\begin{aligned}
\label{eq:magn-correction-estimated-2}
\delta M < \frac{H}{4\pi} \frac{H_{c1}}{H_{c1}+H} 2W d\end{aligned}$$ for the magnetization corrections. In the regime $s \ll W$ the relative correction to the magnetic response is bounded by $$\begin{aligned}
\label{eq:relative-corrections-double-1}
\frac{\delta M}{M} &< \frac{H}{{H_{s}}}
\frac{H_{c1}}{H_{c1}+H}
\sqrt{\frac{d}{4s}}\frac{1}{\log(4{H_{s}}/H\sqrt{d/s})-1}\end{aligned}$$ in the low field range $H < {H_{s}}\sqrt{d/s}$ \[see [Eq. ]{}\] and by $$\begin{aligned}
\label{eq:relative-corrections-double-2}
\frac{\delta M}{M} &< \frac{H^{2}}{{H_{s}}^{2}}
\frac{H_{c1}}{H_{c1}+H} \frac{2}{\pi}\end{aligned}$$ for higher fields field, $H > {H_{s}}\sqrt{d/s}$ \[see [Eq. ]{}\]. The first expression is always small by the order $d/s$, while the second expression predicts small corrections $\propto (H/{H_{s}})^{2}$ in the field range $H \ll {H_{s}}$, spanning the range of validity for the results presented in this section. We conclude, that the corrections arising due to the vortex state inside the superconducting strips are small, justifying the simplified model for the penetrated state ($\mu = 1$) used in our analysis.
Strips with Finite thickness $\boldsymbol{d}$ {#sec:finite-thickness}
=============================================
Introduction {#sec:f-t-formalism}
------------
We now explore the double strip geometry for narrow gaps $2s \ll d$. In order to simplify our discussion, the penetration depth $\lambda$ is assumed to be negligible[@footnote:finite-lambda], $\lambda \ll s$. With the gap-width $s$ the smallest geometric length and using $d \ll w$, the results are presented to leading order in $s/d$ and $d/w$, respectively; in particular, the half-width $W$ of the system is approximated by the width $2w$ of one strip.
The solutions for infinitely thin strips derived in the previous sections have been regularized near the sample edges with a cut-off $\delta$ of the order of the thickness, $\delta~\sim~d$. This approach is not appropriate anymore when the spacial solution near (inside) the gap is determined by the length scale $s$ rather than $d$. The appropriate boundary conditions then have to be taken into account on the entire rectangular cross-section and the strips cannot be treated as infinitely thin anymore.
![Left panel: Field lines for the estuary problem (solid lines in the ${\tilde{\xi}}$-plane) as calculated numerically from [Eq. ]{}. Right panel: the field lines of a point source in the upper half $\zeta$-plane from which the estuary flow is derived via the inverse Schwarz-Christoffel transformation $\zeta({\tilde{\xi}})$. The field lines of the estuary problem approach that of a point source (dashed lines in the left panel) within a distance $s$ away from the opening.[]{data-label="fig:estuary-flow"}](plot_02_mod.eps){width="45.00000%"}
The detailed derivation of the field distribution in the vicinity of the narrow ($2s$) and elongated ($d$) gap (see [Fig. ]{}\[fig:estuary-flow\]) presented in [Sec. ]{}\[sec:sec:sec:near-field\] below will provide us with a uniform field inside the gap of strength $$\begin{aligned}
\label{eq:anticipate-field-in-the-gap}
B_{g} &= \frac{\phi_{g}}{2s},\end{aligned}$$ where the flux $\phi_{g}$ through the gap has to be determined consistently with the field distribution far away from the gap. For distances $s <
|\boldsymbol{r}| \ll w$ away from the upper ($+$) and lower ($-$) gap opening, the field assumes the form of a monopole with radial decay $$\begin{aligned}
\label{eq:anticipate-field-away-from-the-opening}
\boldsymbol{B}(\boldsymbol{r}) &= \pm \frac{\phi_{g}}{\pi}
\frac{\boldsymbol{r}}{|\boldsymbol{r}|^{2}}.\end{aligned}$$ The corresponding result expressed through the holomorphic field reads $$\begin{aligned}
\label{eq:ant-field-away-from-the-opening}
\mathcal{B}(\xi) &= i \,\frac{\phi_{g}}{\pi\xi}.\end{aligned}$$ In [Sec. ]{}\[sec:sec:sec:far-field\] we find the field distribution far away from the gap, match the far-field solution with the solution in the gap, and thereby find the flux $\phi_{g}$ through the gap. Along with this derivation, we will discuss the consequences on the double strip solution originating from the current and field distribution in and around the gap.
Estuary Problem {#sec:sec:sec:near-field}
---------------
The field distribution inside the gap and near the opening at $\xi_{{\mathrm{out}}} = 0 + i d/2$ is described by a so-called estuary flow, i.e., the flow into open space of an incompressible fluid leaving a canal of width $2s$ and large (infinite) length $d$, see Fig.\[fig:estuary-flow\]. We define the shifted coordinate system ${\tilde{\xi}}= \xi -
\xi_{{\mathrm{out}}}$ centered at the gap opening two-dimensional estuary geometry and determine the holomorphic function $\mathcal{B}({\tilde{\xi}})$. For a diamagnetic superconductor, the field component perpendicular to the surface vanishes everywhere such that $\mathcal{B}({\tilde{\xi}})$ is purely real ($B_{x} = 0$ and $B_{z} \neq 0$) at the surfaces inside the gap ($\operatorname{Re}{[{\tilde{\xi}}]}
= \pm s$, $\operatorname{Im}{[{\tilde{\xi}}]} <0$) and imaginary ($B_{x} \neq 0$ and $B_{z} = 0$) on the surfaces along $x$, i.e., for $\operatorname{Im}{[{\tilde{\xi}}]} = 0$ and $|\!\operatorname{Re}{[{\tilde{\xi}}]}|
\geq s$.
![Illustration of the stereographic projection of a (conventional) triangle from the Euclidean plane (top left) to the Riemann sphere (top right). Replacing one edge of this triangle by its complement (passing through infinity) generates an unbound triangle. This situation is illustrated both in the Euclidean plane (bottom left) and on the Riemann sphere (bottom right) after a stereographic projection. []{data-label="fig:stereographic-projection-example"}](plot_17_4in1.eps){width="48.00000%"}
This boundary value problem can be solved with the help of a Schwarz-Christoffel transformation[@Henrici_74] describing a biholomorphic mapping of the upper complex half-plane $\zeta$, $\operatorname{Im}{[\zeta]} \geq 0$, onto the inner of a polygon. Indeed, the field-allowed region in the estuary geometry is a special case of an unbounded triangle (visualized in Figs.\[fig:stereographic-projection-example\] and \[fig:stereographic-projection\]), with vertices ${\tilde{\xi}}_{v}$ at $-s$, $-i\infty$, and $s$ and internal angles $3\pi/2$, $0$, and $3\pi/2$. The corresponding Schwarz-Christoffel transformation takes the form $$\begin{aligned}
\label{eq:sc-transformation}
{\tilde{\xi}}(\zeta) &= s + \frac{2}{\pi}
\bigg[\sqrt{\zeta^{2}-s^{2}}
- 2s \arctan{\sqrt{\frac{\zeta - s}{\zeta + s}}}\bigg]\end{aligned}$$ and maps the upper half-plane $\zeta$ ([Fig. ]{}\[fig:estuary-flow\], right) to the estuary plane ${\tilde{\xi}}$ ([Fig. ]{}\[fig:estuary-flow\], left). The flux $\phi_{g}$ emanating from the vertex at ${\tilde{\xi}}_{v} = -i \infty$ in the estuary is conserved in the transformation [Eq. ]{} and maps to a point source of strength $\phi_{g}$ at $\zeta = 0$, with field lines dispersing into the upper half plane $\operatorname{Im}[\zeta] \geq 0$. The complex potential[@Thomson_60] $$\begin{aligned}
\label{eq:complex-potenital}
\bar\Omega(\zeta) = \frac{i\phi_g}{\pi} \log\zeta\end{aligned}$$ is generating the field $\mathcal{\bar B}(\zeta)= d\bar\Omega/d\zeta = i\phi_{g}/\pi \zeta$ of this point source in the upper half-plane. Transforming back to the estuary geometry, the potential $\Omega({\tilde{\xi}}) = \bar
\Omega[\zeta({\tilde{\xi}})]$ generates the field $$\begin{aligned}
\label{eq:estuary-solution}
\mathcal{B}({\tilde{\xi}})= \frac{d\Omega}{d{\tilde{\xi}}} = \frac{i\phi_g}{2} \frac{1}{\sqrt{\zeta({\tilde{\xi}})\big.^2-s^2}}.\end{aligned}$$ The last equality was obtained by using the Schwarz-Christoffel transformation . Alternatively, the analysis on the level of fields involves the solution $\mathcal{\bar B}(\zeta) = i \phi_{g}/\pi \zeta$ for a point source and the transformation back involves an additional derivative, $\mathcal{B}({\tilde{\xi}}) = (d\zeta/d{\tilde{\xi}}) \mathcal{\bar
B}[\zeta({\tilde{\xi}})]$. In [Fig. ]{}\[fig:estuary-flow\], we show the resulting field lines of [Eq. ]{} as obtained from inverting [Eq. ]{} numerically.
In our further discussion it is sufficient to determine the field distribution in the asymptotic regimes where analytic results are available. Deep inside the gap ($-\operatorname{Im}{[{\tilde{\xi}}]} \gg s$) the inverse of [Eq. ]{} takes the form $$\begin{aligned}
\label{eq:gap-inversion}
\zeta({\tilde{\xi}}) &= 2s \, e^{- [i \pi ({\tilde{\xi}}- 1)/ 2s] -1}\end{aligned}$$ and using [Eq. ]{}, we find \[up to corrections $\propto
\exp({\pi \tilde{z}/2s})$\] a uniform field directed along $z$ of strength $$\begin{aligned}
\label{eq:field-in-the-gap}
B_{g} &= \frac{\phi_{g}}{2s}.\end{aligned}$$ Near the corner of the estuary, $|{\tilde{\xi}}- s| \ll s$, the transformation [Eq. ]{} reads $$\begin{aligned}
\label{eq:sc-near-corner-s}
\frac{{\tilde{\xi}}-s}{2s} &\sim \frac{2}{3\pi} \left(\frac{\zeta-s}{2s}\right)^{3/2}\end{aligned}$$ and a similar expression is found near ${\tilde{\xi}}= -s$. For both corners, the holomorphic field shows a power law singularity $\propto |{\tilde{\xi}}\pm s|^{-1/3}$, which will be regularized in a real sample by the partial penetration (at a depth $\sim s$) of vortices into the sample corners.
![Visualization of the field-allowed region (light gray) of the estuary geometry on the Riemann sphere (filled region) via a stereographic projection. The triangular shape of the boundary of the estuary, with vertices ${\tilde{\xi}}_{v}$ at $\pm s$ and $-i\infty$ is clearly visible on the Riemann sphere representation, see also [Fig. ]{}\[fig:stereographic-projection-example\]. Here, the north pole corresponds to the origin of the complex plane ${\tilde{\xi}}$, while the complex infinity is projected onto the south pole. A line in the original plane ${\tilde{\xi}}$ is mapped to a circle (passing through the south pole) on the Riemann sphere.[]{data-label="fig:stereographic-projection"}](plot_10.eps){width="48.00000%"}
Far away from the opening, $|{\tilde{\xi}}| \gg s$ and $\operatorname{Im}{[{\tilde{\xi}}]} \geq 0$, the inverse transform becomes $\zeta({\tilde{\xi}}) = \pi {\tilde{\xi}}/ 2$ and the holomorphic function assumes the limiting form $$\begin{aligned}
\label{eq:field-away-from-the-opening}
\mathcal{B}({\tilde{\xi}}) &= i \,\frac{\phi_{g}}{\pi{\tilde{\xi}}}\end{aligned}$$ describing a point source of strength $\phi_{g}$ located at ${\tilde{\xi}}= 0$.
Narrow gap double strip {#sec:sec:ngds}
-----------------------
\[sec:sec:sec:far-field\] Away from the gap and from the outer strip edges, a thin-strip description similar to the one discussed in [Sec. ]{}\[sec:thin-strips\] is applicable, with the holomorphic field taking the form $$\begin{aligned}
\label{eq:double-field-general-narrow-gap}
\mathcal{B}(\xi) &= H \sqrt{\frac{(\xi^{2}-{b_{1}}^{2})(\xi^{2}-{b_{2}}^{2})}
{\xi^{2}(\xi^{2}-W^{2})}}.\end{aligned}$$ The factor $\xi^{2}$ in the denominator \[replacing $(\xi^{2}-s^{2})$ in [Eq. ]{}\] captures the flux emanating from the point-like source as derived in [Eq. ]{}. From the above expression, the field distribution along the $x$-axis is given by $$\begin{aligned}
\label{eq:double-field-general-narrow-gap_xz}
\frac{B_{z}(x)}{H} &= \left\{ \begin{aligned}
&\sqrt{\frac{(x^{2}-{b_{1}}^{2})({b_{2}}^{2}-x^{2})}
{x^{2}(W^{2}-x^{2})}},
&& \mathrm{for\ } {b_{1}}\leq x \leq {b_{2}},\\
&\sqrt{\frac{(x^{2}-{b_{1}}^{2})(x^{2}-{b_{2}}^{2})}
{x^{2}(x^{2}-W^{2})}},
&& \mathrm{for\ } W \leq x.
\end{aligned}\right.\end{aligned}$$ Comparing [Eqs. ]{} and in the regime $|\xi| \ll {b_{1}}$, we find the flux $$\begin{aligned}
\label{eq:flux-general-narrow-gap}
\phi_{g}= H W \pi {b_{1}}{b_{2}}/W^{2}\end{aligned}$$ and the uniform field strength inside the gap $|x|
< s$ takes the form $$\begin{aligned}
\label{eq:field-general-in-the-gap}
B_{g} = H \frac{\pi W}{2 s} \frac{{b_{1}}{b_{2}}}{W^{2}}.\end{aligned}$$ Note, that for $s \ll d \ll w$, the difference between the shifted coordinate ${\tilde{\xi}}$ and $\xi$ is beyond our resolution, such that ${\tilde{\xi}}= \xi$.
The current contribution from the region away from the gap is obtained from the holomorphic field in [Eq. ]{} via Ampères law and reads $$\begin{aligned}
\label{eq:double-current-general-narrow-gap}
I(x) &= \left\{\begin{aligned}
&\frac{H c}{2\pi}
\sqrt{\frac{({b_{1}}^{2}-x^{2})({b_{2}}^{2}-x^{2})}
{x^{2}(W^{2}-x^{2})}},
&& s \leq |x| \leq {b_{1}},\\
&\frac{-H c}{2\pi}
\sqrt{\frac{(x^{2}-{b_{1}}^{2})(x^{2}-{b_{2}}^{2})}
{x^{2}(W^{2}-x^{2})}},
&& {b_{2}}\leq |x| \leq W,\\
&0,
&& \mathrm{otherwise}.
\end{aligned}\right.\end{aligned}$$ The $1/x$ dependence of the current is applicable only for $|x| \gg s$. However, it turns out that the deviation of $I(x)$ (as obtained from solving [Eq. ]{} numerically) from $1/x$ is not relevant for the further analysis, and the expression given above for the sheet current density $I(x)$ can be used down to $|x| = s$.
The homogeneous field inside the gap is generated by a screening current density $$\begin{aligned}
\label{eq:current-density-in-the-gap}
j_{g}(x,|z|<d/2) &= \frac{x}{|x|} \frac{B_{g}\, c}{4\pi} \,\delta(|x|-s)\end{aligned}$$ flowing along $y$ at the gap surfaces ($x=\pm s$, $|z| \leq d/2$). Here $\delta$ is the Dirac delta function, which accounts for the assumption $\lambda \to 0$. The two current channels at $x = \pm s$ provide a significant contribution to the total current in the strips. Note that these channels exist for $s \gg d$ as well; for large gaps their contribution to the total current is negligible, though. To treat these currents on equal footing with the sheet current flowing in the strips ($s \leq x \leq W$) we define the sheet current density for the gap currents $$\begin{aligned}
\label{eq:current-in-the-gap}
I_{g}(x) &= \frac{x}{|x|}
\frac{B_{g} c}{4\pi} \,d\,\delta(|x|-s)\end{aligned}$$ by integrating [Eq. ]{} over the strip thickness $d$.
The currents flowing along the vertical surfaces at the outer edges ($|x| =
W$) are parametrically smaller as compared to the contributions near the gap ($|x| = s$) and are neglected here. The two dominant current contributions then add up to the total current, $I_{\mathrm{tot}}(x) = I(x) +
I_{g}(x)$. This current distribution, when compared to the thin strip case, corresponds to a rearrangement of the current densities towards the inner edges of the strips, see [Fig. ]{}\[fig:double-strip-meissner-s-ll-d\].
![Dimensionless current $2\pi I(x)/H c$ (thick solid line) and reduced magnetic field $B_{z}(x)/H$ (dashed line) as a function of $x$ and for $z=0$ in a double strip in the Meissner state. The profiles are calculated for the case $s \ll d \ll 2w$ with the parameters $w/d = d/s = 100$. The current profile inside the strips changes sign at $\pm {b}$, with $|{b}| \approx 0.21
w$. The additional current $I_{g}(x)$ from [Eq. ]{} flowing near the inner edges of the strips changes the condition of zero net current dramatically. The thin lines show the current and field profiles of the double strip in the thin strip limit, $d/s \to 0$, for fixed $s = 10^{-4}w$.[]{data-label="fig:double-strip-meissner-s-ll-d"}](plot_03_mod.eps){width="45.00000%"}
The diamagnetic contribution from the current $I(x)$ in the strip, $$\begin{aligned}
\label{eq:double-magnetization-general-narrow-gap}
M(H) &= -\frac{H}{4}(W^{2}-{b_{1}}^{2}-{b_{2}}^{2}),\end{aligned}$$ as obtained from evaluating [Eq. ]{} with the field , is parametrically larger ($\propto W/d$) than the paramagnetic contribution $$\begin{aligned}
\label{eq:magnetization-in-the-gap}
M_{g}(H) &= \frac{H}{4} W^{2} \frac{{b_{1}}{b_{2}}d}{W^{3}}\end{aligned}$$ from the current $I_{g}(x)$ along the gap surface and we neglect the latter in the following. The problem is then, once again, reduced to finding the parameters ${b_{1}}$ and ${b_{2}}$ within the Meissner- and penetrated states.
### Meissner state {#sec:sec:sec:Meissner-3}
In the Meissner state were ${b_{1}}= {b_{2}}= {b}$, the field $B_{z}$ \[[Eq. ]{}\] along the $x$-axis simplifies to $$\begin{aligned}
\label{eq:double-field-meissner-narrow-gap}
B_{z}(x) &=
H \frac{x^{2}-{b}^{2}}
{|x| \sqrt{x^{2}-W^{2}}}.\end{aligned}$$ for $|x|>W$ and is constant \[[Eq. ]{}\], $$\begin{aligned}
\label{eq:field-meissner-in-the-gap}
B_{g} = H \pi \frac{{b}^{2}}{2 s W},\end{aligned}$$ inside the gap ($|x|<s$). The total sheet current density reads $$\begin{aligned}
\label{eq:double-current-meissner-narrow-gap}
I_{\mathrm{tot}}(x) &= - \frac{H c}{2\pi}\bigg[
\frac{x^{2}-{b}^{2}}{x \sqrt{W^{2}-x^{2}}}
- \frac{x}{|x|}
\frac{\pi {b}^{2}}{4sW} \,d\,\delta(|x|-s) \bigg]\end{aligned}$$ and the general expression for the magnetization takes the form $$\begin{aligned}
\label{eq:double-magnetization-meissner-narrow-gap}
M(H) &= -\frac{H}{4}(W^{2}-2{b}^{2}).\end{aligned}$$ The value of the parameter ${b}$ is fixed by the constraint of vanishing net current given as $$\begin{aligned}
\label{eq:no-net-current-explicit}
\int\limits_{s}^{W} \frac{dx \, x}{\sqrt{W^{2}-x^{2}}}
= {b}^{2} \bigg[ \frac{\pi d}{4sW} + \int\limits_{s}^{W}
\frac{dx}{x \sqrt{W^{2}-x^{2}}}\bigg].\end{aligned}$$ The above integrals simplify in the limit $s \ll W$ and the parameter ${b}$ takes the asymptotic form $$\begin{aligned}
\label{eq:a-over-W-ratio-2}
{b}^{2} &= \frac{W^{2}}{\pi d/4 s + \log\big(2W/s\big)}.\end{aligned}$$ In contrast to the result for thin strips \[see [Eq. ]{}\], where ${b}^{2}$ changes logarithmically with $s$, in the present case the dependence on $s$ is dominated by the linear term $d/s$ in the denominator. As a result, the parameter ${b}$ is substantially reduced when $s \ll d$, see [Fig. ]{}\[fig:a-of-s\], which is due to the additional currents $I_g$ flowing at the (vertical) gap surface and producing a substantial rearrangement of the overall current density as shown in [Fig. ]{}\[fig:double-strip-meissner-s-ll-d\].
We note that the numerical factor $\pi/4$ of the term $d/s$ in the above expression is precisely known since it derives from the current $I_{g}(x)$ originating from screening the uniform field inside the gap, [Eq. ]{}. The prefactor under the logarithm, however, will be modified if the field distribution at the opening of the estuary is accurately taken into account. Indeed, approaching the corner $(s,d/2)$ from both surfaces $(x,d/2)$ and $(s,z)$ the field deviates from the assumed behavior $B_{x}(x)\propto 1/x$ and $B_{z}(z) = \mathrm{const}$ \[following from [Eqs. ]{} and respectively\]. The precise field distribution (and its related current profile) can be derived by solving [Eq. ]{} numerically and inserting the result into [Eq. ]{}. Neglecting partial penetration of the edge corners, [Eq. ]{} will be modified to $$\begin{aligned}
{b}^{2} &= \frac{W^{2}}{\pi d/4 s + \log\big(2.38 W/s\big)}.\end{aligned}$$ Since the precision of this expression also suffers from corrections (e.g. from partial penetration of the edge corners), we will use the relation in the following.
The diamagnetic response in the Meissner phase follows from and reduces to $$\begin{aligned}
\label{eq:mag-meissner-double-strip-s-ll-d}
M(H) &\approx -\frac{H}{4}W^{2} \bigg[1-\frac{8s/\pi d}{1 + (4s/\pi
d)\log\big(2W/s\big)}\bigg].\end{aligned}$$ This result approaches that of a single strip of double width \[[*cf. *]{}[Eq. ]{} with $w \to W$\] upon reducing $s$ far below $d$.
![The parameter ${b}$ characterizing the Meissner phase of the double strip is plotted against the half-width $s$ of the gap between the strips. All lengths are normalized to the half width $w$ of the strips. The fixed strip thickness $d = 10^{-3}w$ separates two regimes; in the thin strip regime $d \ll s$, ${b}$ depends on $s$ via [Eq. ]{}. For $s \ll d$, ${b}(s)$ is given through [Eq. ]{}. In between, i.e., for $s \sim d$, a smooth cross-over (dashed line) connects the two limits. In both far asymptotic limits $s \lll d$ and $s \ggg
d$, the position ${b}(s)$ follows a simple behavior ${b}(s) = 2w\sqrt{4s/\pi d}$ and ${b}(s) = w + s$, respectively (thin lines).[]{data-label="fig:a-of-s"}](plot_08.eps){width="45.00000%"}
Using [Eq. ]{} in the expression for the flux through the gap, we find that $$\begin{aligned}
\label{eq:flux-meissner-finite-thickness}
\phi_{g} &= H 2W \frac{2s/d}{1 + (4s/\pi d)\log\big(2W/s\big)}\end{aligned}$$ shrinks (up to logarithmic corrections) linearly with decreasing $s$ (note that ${b_{1}}{b_{2}}= {b}^{2}$). Compared to the blocked flux $\phi_{b} \approx
H\, 2W$, only a small fraction $\sim 2s/d$ passes through the narrow gap of width $2s$ and length $d$. Consequently, the field strength inside the gap, $$\begin{aligned}
\label{eq:double-field-meissner-in-the-gap}
B_{g} = H \frac{2W}{d} \frac{1}{1 + (4s/\pi d)\log\big(2W/s\big)},\end{aligned}$$ does not diverge for $s/d \to 0$ but saturates at $H\, 2W/d$.
As the field profile inside the gap from where vortices start penetrating the sample is precisely known, the present penetration process is more accurately described than the one for the thin-strip limit where the strips are separated by a distance larger than $d$. Corrections originating from the precise field distribution near the opening of the estuary affect the results only to the next-to-leading order.
As before, the penetration starts when the field inside the gap reaches the strength ${H_{s}}$, i.e., at the penetration field $$\begin{aligned}
\label{eq:double-penetration-field-narrow-gap}
{H_{p}}&= {H_{s}}\frac{2s}{\pi W} \frac{W^{2}}{{b}^{2}}
= {H_{s}}\frac{d}{2W}
\Big[1 + \frac{4s}{\pi d} \log \Big(\frac{2W}{s}\Big)\Big].\end{aligned}$$ In the limit $s/d \to 0$, the penetration field asymptotically reaches the value ${H_{s}}d/2W$, that is the penetration field of the elliptic strip, [*cf. *]{}[Eq. ]{}. The retardation of field penetration originating from the geometrical barrier has completely disappeared in this limit. At penetration, $H = {H_{p}}$, the diamagnetic response $$\begin{aligned}
\label{eq:magnetization-at-penetration-point}
M_{p} &= -\frac{{H_{s}}}{16}(2Wd)
\Big\{1 + \frac{2s}{\pi d} \Big[2\log \Big(\frac{2W}{s}\Big)-1\Big]\Big\}\\
&= -\frac{{H_{p}}}{4}W^{2} \Big[1-\frac{8s/\pi d}{1 + (4s/\pi
d)\log\big(2W/s\big)}\Big],\end{aligned}$$ has collapsed by a factor $\sim (d/W)^{1/2}$ as compared to that of a single strip, [Eq. ]{}, for which the geometrical barrier is fully active. This so-called ‘suppression of the geometrical barrier’, the collapse of ${H_{p}}$ and of $M(H)$, is a central result of this work. Although in the limiting case $s/d \to 0$, the Meissner response and the field of first penetration coincide with that of an elliptically shaped strip, beyond ${H_{p}}$, the magnetic signatures of the double strip still differs substantially from those of the elliptic sample, see Figs.\[fig:numeric-vs-analytic-f-t-l\] and \[fig:descending-double-strip\] as well as the discussion below.
Upon decreasing the gap width $s$, the penetration field ${H_{p}}$ is reduced, what leads to a stronger suppression of the geometrical barrier as follows from [Eq. ]{}. The calculation of the equilibrium field defined through [Eq. ]{} provides the result $$\begin{aligned}
\label{eq:double-eq-field-narrow-gap}
{H_{\mathrm{eq}}}= H_{c1} \frac{d}{2W} \Big[1 - \frac{\log(W^{2}/b^{2})-1}
{\pi d / 2s + \log(4W^{2}/s^{2})}\Big]^{-1},\end{aligned}$$ approaching $H_{c1} d/2W$ and the corresponding geometrical barrier height vanishes as $s \log(s)$, $$\begin{aligned}
U_{b}^{\mathrm{eq}} &= \varepsilon_{l} d \Big(1 -
\Big\{1 + \frac{2s}{\pi d}
\big[\log(4{b}^{2}/s^{2})-1\big]\Big\}^{-1}\Big)\\
&\approx \varepsilon_{l} d\ \frac{2s}{\pi d}
\big[\log(16W^{2}/\pi s d)-1\big],\end{aligned}$$ where we have assumed that $s \log(W/s) \ll d$ for the last equality.
### Penetrated state {#sec:sec:sec:Shubnikov-2-ft}
The field and current distributions [Eqs. ]{}, and [Eqs. ]{}, describe the penetrated state once the parameters ${b_{1}}$ and ${b_{2}}$ have been found; the latter have to respect the limits ${b_{1}}-s \gg s$ and $W-{b_{2}}\gg d$ and are determined by the usual conditions governing the evolution of the vortex dome, the vanishing of the total currents in the strips, $$\begin{aligned}
\label{eq:no-net-current-2}
\int\limits_{s_{-}}^{{b_{1}}} dx\, I_{\mathrm{tot}}(x)
+ \int\limits_{{b_{2}}}^{W} dx\, I_{\mathrm{tot}}(x) &= 0\end{aligned}$$ and the condition of criticality at the edge regulating the vortex entrance, here $B_{g} = {H_{s}}$. The latter condition is equivalent to the requirement that the flux $\phi_{g}$ in saturates at ${H_{s}}2s$, or $$\begin{aligned}
\label{eq:critical-field-condition}
\frac{{b_{1}}{b_{2}}}{{b}^{2}} &= \frac{{H_{p}}}{H},\end{aligned}$$ as expressed through the penetration field ${H_{p}}$ and the zero-current position $b$ of the Meissner state.
As before, the perturbative calculation around the penetration field ${H_{p}}$ is very limited due to the rapid growth of the dome width ${b_{2}}-{b_{1}}$ beyond ${H_{p}}$ and we concentrate on the high-field expansion where the vortex domes occupy a large fraction of the strips ${b_{1}}\ll {b_{2}}$, providing results over a large field-range. The two conditions regulating the dome evolution then can be simplified and an analytic solution can be given. With the Ansatz ${b_{2}}= W (1-\nu)^{1/2}$ with $\nu < 1$, the inner dome edge $$\begin{aligned}
\label{eq:bone}
{b_{1}}&= \frac{{b}^{2}}{W} \frac{{H_{p}}}{H} \frac{1}{\sqrt{1-\nu}}\end{aligned}$$ is expressed through $\nu$ with the help of [Eq. ]{}. Assuming $s \ll {b_{1}}$ and ${b_{1}}\ll
{b_{2}}$, the requirement of vanishing net current in [Eq. ]{} simplifies to $$\begin{aligned}
\label{eq:appr-neutral-current-condition-f-t-l}
\frac{{H_{p}}}{H}\frac{{b}^{2}}{W^{2}}
\left[\frac{W^{2}}{{b}^{2}} +
\log{\Big(\frac{{b}^{2}}{W^{2}}
\frac{{H_{p}}}{H}
\frac{1}{\sqrt{1-\nu}}
\Big)}-1\right] &\\[.5em]
= \operatorname{E}(\sqrt{\nu})-(1&-\nu)\operatorname{K}(\sqrt{\nu}).\nonumber\end{aligned}$$ For large fields $H \gg {H_{p}}$, where $\nu$ is small, the above equation can be expanded in $\nu$. Solving for $\nu(H)$ to second order in ${H_{p}}/H$, we obtain $$\begin{aligned}
\label{eq:mu-second-order-f-t-l-0}
\nu(H) &\approx \frac{4}{\pi}\frac{{H_{p}}}{H}
\bigg\{ 1 + \frac{{b}^2}{W^2}
\bigg[\log{\left(\frac{{b}^2}{W^2}\frac{{H_{p}}}{H}\right)}-1\bigg]
\bigg\}\\
&\ - \frac{2}{\pi^2}\frac{{H_{p}}^2}{H^2}
\bigg\{1+\frac{{b}^2}{W^2}
\bigg[\log{\bigg(\frac{{b}^2}{W^2}\frac{{H_{p}}}{H}\bigg)}-1\bigg]
\bigg\}^2\nonumber\\
&\ + \frac{8}{\pi^2}\frac{{H_{p}}^2}{H^2}
\frac{{b}^2}{W^2}
\bigg\{1+\frac{{b}^2}{W^2}
\bigg[\log{\bigg(\frac{{b}^2}{W^2}\frac{{H_{p}}}{H}\bigg)}-1\bigg]
\bigg\}.\nonumber
$$ The magnetic response given in [Eq. ]{} simplifies to $M(H) = - H
\nu(H)W^{2}/4$ and [Fig. ]{}\[fig:numeric-vs-analytic-f-t-l\] shows the result of combining this expression with $\nu(H)$ from [Eq. ]{}. Although the range of applicability ${H_{p}}\ll H$ of the above expression does not a-priori cover the regime near penetration, the results are still in good agreement with the numerical solution down to $H \approx {H_{p}}$.
![The diamagnetic response $M(H)$ (solid line) for the double strip in the limit $s \ll d \ll w$ (here $w/d = d/s = 100$) as obtained from solving [Eqs. ]{} and numerically. In addition the dotted (dashed) line shows the analytic result for the magnetization $M(H) = -H \nu(H) W^{2}/4$, where $\nu(H)$ is obtained from solving [Eq. ]{} to linear (quadratic) order in ${H_{p}}/H$, see [Eq. ]{}. It is necessary to express the solution to second order in ${H_{p}}/H$ as the first order solution gives only poor results close to ${H_{p}}$. The magnetization of a single elliptic strip of width $2W$ and thickness $d$ (long thin dashes) is reversible and with linear slope beyond ${H_{p}}$ (as also shown in [Fig. ]{}\[fig:elliptic-magnetization\]).[]{data-label="fig:numeric-vs-analytic-f-t-l"}](plot_11_merged.eps){width="45.00000%"}
Neglecting irrelevant terms of order $({b}/W)^{2}({H_{p}}/H)^{2}$ and higher in [Eq. ]{}, the magnetization reads $$\begin{aligned}
\label{eq:mag-second-order-f-t-l}
M(H) &\approx \bar{M}\bigg\{
1 + \frac{{b}^2}{W^2} \bigg[
\log{\bigg(\frac{{b}^2}{W^2}\frac{{H_{p}}}{H} \bigg)}-1
\bigg]
- \frac{1}{2\pi}\frac{{H_{p}}}{H}
\bigg\},\end{aligned}$$ with $\bar{M} = -{H_{p}}W^{2}/\pi$. In contrast to the thin strip case \[see [Eq. ]{}\], where the magnetization depends logarithmically $\propto \log{({H_{s}}/H)}$ on the applied field, the magnetic response in the present limit is dominated by a field-independent contribution, $$\begin{aligned}
\label{eq:appr-mag-f-t-l}
\bar{M} &\approx -\frac{{H_{p}}}{\pi} W^{2} \approx -\frac{{H_{s}}}{4\pi} (2Wd),\end{aligned}$$ producing an almost flat magnetization. This flatness is the result of the particular current distribution inside the strips: The current flowing close to the inner edge is dominated by the contribution $I_{g}(x)$ from the gap, i.e., $$\begin{aligned}
\label{eq:current-estimation-near-the-gap}
\int\limits_{s_{-}}^{{b_{1}}} dx\, I_{\mathrm{tot}}(x)
&\approx \int\limits_{s_{-}}^{s_{+}} dx\, I_{g}(x)
= \frac{{H_{s}}c}{4\pi} \,d.\end{aligned}$$ To satisfy the condition of vansihing net current, the current density $I(x)$ between ${b_{2}}$ and $W$ has to compensate the gap contribution, leading to $$\begin{aligned}
\label{eq:current-estimation-outer-edge}
\int\limits_{{b_{2}}}^{W} dx\, I_{\mathrm{tot}}(x)
&\approx -\frac{{H_{s}}c}{4\pi} \,d.\end{aligned}$$ Once the dome occupies a large fraction of the sample, these currents flow at the outer edge, i.e., a distance $\sim W$ away from the origin and produce the dominant (field-independent) contribution $-{H_{s}}(2Wd)/4\pi$ \[[*cf. *]{}[Eq. ]{}\] to the magnetization at large fields (the factor 2 originates from the integration over both strips). Note that in the limit $s/d
\to 0$, the leveling out of the magnetization at the value given in [Eq. ]{} is by a factor $4/\pi$ larger than its value at penetration ${H_{p}}$ \[see [Eq. ]{}\].
Although almost constant, the magnetization assumes a maximal diamagnetic signal $$\begin{aligned}
\label{eq:max-magnetization}
M(H_{m}) &= \bar{M} \bigg\{1 + \frac{4s}{\pi d}
\bigg[\log\bigg(\frac{32s^{2}}{\pi d^{2}}\bigg) -2\bigg]\bigg\}\end{aligned}$$ at the applied field $$\begin{aligned}
\label{eq:H-max}
H_{m} &\approx {H_{p}}\frac{d}{8s}
\approx \frac{{H_{s}}}{16}\frac{d^{2}}{s W}.\end{aligned}$$ For $s/d \lesssim d/16W$, the diamagnetic response monotonically increases up to $H \sim {H_{s}}$. On the other hand, we may extrapolate the expression to $s \lesssim d$ and predict a value of the gap parameter $s
\sim d/8$ where $H_{m}$ merges with the penetration field ${H_{p}}$ upon increasing $s$. The “flatness” of the magnetization curve in the penetrated state is quantified by relating the slope $M'(H)$ \[as obtained from [Eq. ]{}\] to the Meissner slope $-
W^{2}/4$, yielding $$\begin{aligned}
\label{eq:relative-magnetization-slope}
-\frac{4 M'(H)}{W^{2}} = \frac{2}{\pi^{2}}
\bigg(\frac{{H_{p}}^{2}}{H^{2}} - \frac{8s}{d}\frac{{H_{p}}}{H}\bigg) \ll 1.\end{aligned}$$ As for thin strips and wide gaps (see [Sec. ]{}\[sec:sec:sec:Shubnikov-2-tsl\]), we can push the results obtained in the limit $s \ll d$ to the extreme case $s \to d$. The penetration field $$\begin{aligned}
\label{eq:penetration-field-s-is-d-thick-strips}
{H_{p}}&\approx {H_{s}}\frac{d}{2W}
\Big[1 + \frac{4}{\pi} \log \Big(\frac{2W}{d}\Big)\Big]\end{aligned}$$ as obtained from [Eq. ]{} with $s = d$, agrees up to numbers of order unity with the result obtained from the opposite limit $s \gg d$, see [Eq. ]{}. Taking the limit $s \to d$ from the regime of small gaps $s \ll d$, the magnetization is dominated by the first term in [Eq. ]{} and simplifies to $$\begin{aligned}
\label{eq:magnetization-s-is-d-thick-strips}
M(H) &\approx -\frac{{H_{s}}}{4\pi} (4Wd) \Big[\frac{2}{\pi}
\log{\Big(\frac{4{H_{s}}}{\pi H}\Big)} + \frac{4 - \pi}{2\pi} \Big].\end{aligned}$$ This expression agrees well with the corresponding expression obtained in the limit $s {\scriptstyle{\ \searrow\
}}d$ approaching the thickness $d$ from above.
Although the penetration field asymptotically ($s/d \to 0$) approaches the equilibrium field , a finite irreversibility persists and the geometric barrier rapidly reappears upon reducing the applied field. Upon decreasing the magnetic field from a maximal value $H^{\star}$, the vortex dome expands while keeping the trapped flux constant, $$\begin{aligned}
\label{eq:double-flux-through-dome}
\phi_{d}(H) &\equiv \int\limits_{{b_{1}}}^{{b_{2}}} dx\,
H \sqrt{\frac{(x^{2}-{b_{1}}^{2})({b_{2}}^{2}-x^{2})}
{x^{2}(W^{2}-x^{2})}}
= \phi_{d}^{\star},\end{aligned}$$ where $\phi_{d}^{\star} = \phi_{d}(H^{\star})$. This constraint for the decreasing field replaces the constraint $B_{g}={H_{s}}$ for the increasing field. Excluding a narrow field range $H^{\star} \simeq {H_{p}}$, the dome extends over a large fraction of the sample and the constraint of conserved trapped flux can be simplified under the assumptions ${b_{1}}\ll {b_{2}}$ and $W-{b_{2}}\ll W$ to read $$\begin{aligned}
H W \Big\{1 - \frac{\nu}{4}
\Big[\log\Big(\frac{16}{\nu}\Big) + 1\Big] \Big\}
&= \phi_{d}^{\star},\end{aligned}$$ where we used $\nu(H) = 1-[{b_{2}}(H)/W]^{2} \ll 1$ as before. Similar to the single strip calculations \[see [Eq. ]{}\], the slope of the magnetic response $M(H) = - H \nu(H)W^{2}/4$ is given by $$\begin{aligned}
\label{eq:slope-2}
\frac{dM}{dH} &= -\frac{W^{2}}{4} \frac{4 - \nu}{\log(16/\nu)},\end{aligned}$$ which is numerically close to that of the Meissner phase ($-W^{2}/4$). At the onset of the descending branch, i.e., $H^{\star}-H \ll
H^{\star}$, we find an analytic expression for the magnetization of the form $$\begin{aligned}
\label{eq:analytic-descending-branch-2}
M(H) = M(H^{\star}) - \frac{H-H^{\star}}{4} W^{2}
\frac{4 - \nu^{\star}}{\log(16/\nu^{\star})},\end{aligned}$$ where $\nu^{\star} = \nu(H^{\star})$ is obtained from [Eq. ]{}. The above expression and the result of an exact numerical calculation of the magnetization are shown in [Fig. ]{}\[fig:descending-double-strip\].
![Numerical solution for the magnetization of the descending branch for narrow gaps $s \ll d$. The two conditions of vanishing net current and conserved trapped flux is solved for $H^{\star} = n {H_{p}}$ (with integer $2\leq
n \leq 5$). Parameters are $w/d = d/s = 100$. The analytic result (dashed lines) as obtained from an expansion close to $H = H^{\star}$ gives a reasonable description of the numerical solution over a wide field range.[]{data-label="fig:descending-double-strip"}](plot_18.eps){width=".45\textwidth"}
### Magnetization of the vortex dome {#sec:sec:sec:magnetic-medium-3}
In the limit $s \ll d$, the diamagnetic response of the double strip is flat and small by the factor $\sim \sqrt{d/W}$ as compared to the single strip at $H_p$; hence, we should verify that the magnetic response of the vortex state in the flux-filled region does not substantially alter the above results. Following again the analysis discussed earlier in [Sec. ]{}\[sec:sec:sec:magnetic-medium-1\], the corrections to the magnetic response are bounded from above by the function $$\begin{aligned}
\label{eq:magn-correction-estimated-3}
\delta M < \frac{H}{4\pi} \frac{H_{c1}}{H_{c1}+H} 2W d\end{aligned}$$ leading to relative corrections $$\begin{aligned}
\label{eq:relative-corrections-double-3}
\frac{\delta M}{M} &< \frac{H}{{H_{s}}} \frac{H_{c1}}{H_{c1}+H} \frac{4}{\pi}\end{aligned}$$ that are small as long as $H \ll {H_{s}}$. Without surface barrier, ${H_{s}}=
H_{c1}$, the corrections become of order unity only when $H \sim H_{c1}$. On the other hand, for a large surface barrier ${H_{s}}\gg H_{c1}$, the corrections remain small when $H \sim {H_{s}}$. We conclude, that the contribution of the equilibrium magnetization of the Shubnikov state to the overall magnetization of the double strip geometry (with $s \ll d \ll w$) is small and can, in most cases, be neglected in the entire field range $H < H_{c1}$.
Several strips {#sec:several-strips}
==============
So far, we have given a detailed description of the single and double strip geometries. A discussion of three coplanar strips will reveal additional features as compared to the previous systems, and allows for a qualitative understanding of the response of a system of a *finite* number $n \geq 3$ of coplanar strips in a parallel arrangement.
The general holomorphic field for $n$ ($n \geq 1$) parallel strips arranged symmetrically around the origin $\xi=0$ assumes the form $$\begin{aligned}
\label{eq:ant-n-strip-function}
\mathcal{B}(\xi) &=
H \sqrt{\prod\limits_{i}\frac{\xi^{2}-b_{i}^{2}(H)}{\xi^{2}-e_{i}^{2}}},\end{aligned}$$ where $\pm e_{i}$ denote the strip edges and the parameters $b_{i}(H)$ define the boundaries of the vortex states. For an even number $n = 2m$ of strips, the $k$th strip ($0 < k \leq m$) as counted along the positive $x$-axis ranges from $e_{2k-1}$ to $e_{2k}$ and vortices fill the region $b_{2k-1}$ to $b_{2k}$. Every strip has a symmetric counterpart on the negative $x$-axis. For an odd number $n=2m+1$ of strips, the above remains unchanged except for an additional innermost strip ranging from $-e_{0}$ to $e_{0}$ with a dome between $-{b_{0}}$ and ${b_{0}}$. The product in [Eq. ]{} runs from 1 to $n$ (from 0 to $n-1$) for the even (odd) numbered configurations. The expressions and are special cases for the single and double strip geometries. The magnetization of the $n$-strip system as obtained from [Eq. ]{} reads $$\begin{aligned}
\label{eq:n-strip-magnetization}
M(H) &= -\frac{H}{4} \sum\limits_{i}(e_{i}^{2}-b_{i}^{2}).\end{aligned}$$ In this section, we consider strips of equal width $2w$ and separated by a gap $2s$. We also limit the analysis to the thin strip case, i.e., the thickness $d$ of the strips is the smallest of all geometric lengths.
Three strips {#sec:sec:three-strips}
------------
The holomorphic field for three parallel strips reads $$\begin{aligned}
\label{eq:3-strip-function}
\mathcal{B}(\xi) &= H \sqrt{
\frac{[\xi^{2}-{b_{0}}^{2}(H)]
[\xi^{2}-{b_{1}}^{2}(H)]
[\xi^{2}-{b_{2}}^{2}(H)]}
{(\xi^{2}-e_{0}^{2})
(\xi^{2}-e_{1}^{2})
(\xi^{2}-e_{2}^{2})}},\end{aligned}$$ with ${e_{0}}= w$, ${e_{1}}= w + 2s$, and ${e_{2}}= 3w + 2s$.
### Meissner state {#sec:sec:sec:3-strip-Meissner}
For small fields, where the entire system is in the Meissner state, i.e., no vortices have penetrated in either of the strips, the holomorphic field reduces to $$\begin{aligned}
\label{eq:field-function-three-strips}
\mathcal{B}(\xi) &= H\sqrt{\frac{\xi^{2} (\xi^{2}-{b}^{2})^{2}}
{(\xi^{2}-{e_{0}}^{2})(\xi^{2}-{e_{1}}^{2})(\xi^{2}-{e_{2}}^{2})}},\end{aligned}$$ where the remaining parameter ${b}$ ($={b_{1}}= {b_{2}}$, note that ${b_{0}}= 0$) is determined from requiring a vanishing total current in the outer strip pair, $$\begin{aligned}
\label{eq:neutral-current-3-strips}
\int\limits_{{e_{1}}}^{{e_{2}}} &\frac{dx\, x\, b^{2}}
{\sqrt{(x^{2}-{e_{0}}^{2})(x^{2}-{e_{1}}^{2})({e_{2}}^{2}-x^{2})}} \\
&\quad\qquad\qquad\qquad= \int\limits_{{e_{1}}}^{{e_{2}}}
\frac{dx\, x^{3}}{\sqrt{(x^{2}-{e_{0}}^{2})(x^{2}-{e_{1}}^{2})({e_{2}}^{2}-x^{2})}}.
\nonumber\end{aligned}$$ With the substitution $x^{2} \to {e_{1}}^{2} + ({e_{2}}^{2}-{e_{1}}^{2}) t^{2}$ the solution can formally be expressed through $$\begin{aligned}
\label{eq:a-for-3-strips}
{b}^{2} &= {e_{0}}^{2} + ({e_{1}}^{2} - {e_{0}}^{2})\frac{\operatorname{E}(\kappa)}{\operatorname{K}(\kappa)},\end{aligned}$$ where the elliptic integrals, defined in [Eqs. ]{} and , are evaluated at the imaginary argument $\kappa = \sqrt{({e_{2}}^{2}-{e_{1}}^{2})
/({e_{0}}^{2}-{e_{1}}^{2})}$, $\kappa^{2}<0$. In two asymptotic regimes the above result simplifies to $$\begin{aligned}
\label{eq:a-for-3-strips-limits}
{b}^{2} &\simeq \left\{\begin{aligned}
&4(w+s)^{2} &\mathrm{for\ } s \gg w,\\
&w^{2} \Big[1 + \frac{16}{\log{(32w/s)}}\Big] &\mathrm{for\ } s \ll w.\\
\end{aligned}\right.\end{aligned}$$ The first limit ($s \gg w$) describes three almost isolated strips, while in the latter case of nearby strips with $s \ll w$ a logarithmic dependence of ${b}$ on $s$ shows up, analogous to the expression for two strips. Focusing on the regime of nearby strips $s \ll w$, we find that the flux $$\begin{aligned}
\label{eq:flux-3-strips}
\phi_{g} &=\int\limits_{e_{0}}^{e_{1}}\! dx\,B_{z}(x)
\approx H 6w \frac{\pi \sqrt{2}}{3} \frac{1}{\log{(32w/s)}}\end{aligned}$$ passing through each of the two gaps carries a substantial fraction of the flux $\phi_{\mathrm{b}} = H 6w$ that is blocked by the strips. The field enhancement $\sim H\sqrt{w^{2}/sd}$ at the strip edges ${e_{0}}$ and ${e_{1}}$ \[see [Eq. ]{}\] is found to be parametrically $\sim\sqrt{w/s}$ larger than at the outermost edge ${e_{2}}$. A more detailed calculation reveals, that the field strength is largest near ${e_{0}}$, followed by a slightly lower field near ${e_{1}}$, $$\begin{aligned}
\frac{B_{z}({e_{1}}-d/2)}{B_{z}({e_{0}}+d/2)}
= 1 - \frac{21 w^{2} - 5 {b}^{2}}{{b}^{2}-w^{2}}\frac{s}{4w}.\end{aligned}$$ We conclude that the critical field ${H_{s}}$ \[as discussed in [Eq. ]{} \] is first reached at the edges $\pm
{e_{0}}$ (strip index $k=0$) where the field enhancement is most pronounced. Thus, the geometrical barrier is first suppressed in the central strip and vortices start to penetrate the innermost strip beyond $$\begin{aligned}
\label{eq:penetration-field-3-strips}
{H_{p}}^{k=0} &\approx {H_{s}}\sqrt{\frac{s d}{8 w^{2}}} \log(32w/s).\end{aligned}$$ This critical field is parametrically similar to the field of first penetration of the double strip geometry, see [Eq. ]{}.
### Penetrated state(s) {#sec:sec:sec:3-strip-Penetrated}
In general, for a multiple strip geometry, the strips are not equivalent and the penetration of vortices starts at a different field value for each strip. The penetration sequence may depend on the geometrical setup as well as on the boundary condition at $y \to \pm \infty$ (shunted vs. unshunted ends). In particular, we shall compare our results to the findings by Mawatari *et al.*, who considered a system of three *shunted* strips in Ref. [\[\]]{}. As the external field $H$ increases beyond ${H_{p}}^{k=0}$, vortices populate the innermost strip (${b_{0}}\neq 0$), while the two other strips remain free of flux (${b_{1}}= {b_{2}}= {b}$). The field distribution then is given by $$\begin{aligned}
\label{eq:field-function-three-strips-penetrated}
\mathcal{B}(\xi) &= H\sqrt{\frac{(\xi^{2}-{b_{0}}^{2}) (\xi^{2}-{b}^{2})^{2}}
{(\xi^{2}-{e_{0}}^{2})(\xi^{2}-{e_{1}}^{2})(\xi^{2}-{e_{2}}^{2})}},\end{aligned}$$ where the two parameters ${b_{0}}$ and ${b}$ (now both depending on $H$) are fixed by the constraints of critical field strength ${H_{s}}$ near the edge ${e_{0}}$ and vanishing net current $$\begin{aligned}
\label{eq:vanishing-current-three-strips}
\int\limits_{{e_{1}}}^{{e_{2}}} dx\,I(x) = 0\end{aligned}$$ in the outer strips. The outer strip pair will first be penetrated by vortices only at a higher field ${H_{p}}^{k=1}$, where a critical field strength ${H_{s}}$ is reached at the edge ${e_{1}}$. At this particular field, the requirement that the field strength is critical at both edges ${e_{0}}$ and ${e_{1}}$ while the outer dome has not yet developed (${b_{1}}= {b_{2}}= {b}$), gives a relation between ${b_{0}}$ and ${b}$ of the form $$\begin{aligned}
\label{eq:condition-for-hp2}
{b_{0}}^{2} &= w^{2}\Big[1 - \frac{8({b}^{2}-w^{2})}
{16w^{2} + 5({b}^{2}-w^{2})}\Big].\end{aligned}$$ Inserting this relation ${b_{0}}({b})$ into the constraint of vanishing net current in the outer strip fixes the last degree of freedom ${b}$ and permits to express the second penetration field through $$\begin{aligned}
\label{eq:second-penetration-field}
{H_{p}}^{k=1} &= {H_{s}}\sqrt{\frac{32 w^{4}sd}
{(w^{2}-{b_{0}}({b})^{2})({b}^{2}-w^{2})^{2}}}.\end{aligned}$$
![Penetration fields ${H_{p}}^{k=0}$ (dotted) and ${H_{p}}^{k=1}$ (dashed) as a function of $s/w$ for a system of three coplanar superconducting strips. The vertical scale is fixed by the specific choice of the ratio $d/w$. The position of the parameters ${b_{0}}$ and ${b}={b_{1}}={b_{2}}$ corresponding to these two penetration fields is shown in figure \[fig:dome-evolution-at-hp2\].[]{data-label="fig:hp2"}](plot_16_bis.eps){width=".482\textwidth"}
Solving [Eqs. ]{} and numerically, we show the results for the two penetration fields ${H_{p}}^{k=0}$ and ${H_{p}}^{k=1}$ in [Fig. ]{}\[fig:hp2\]. Using the same numerical solution, we visualize in [Fig. ]{}\[fig:dome-evolution-at-hp2\] the dome boundaries ${b_{0}}$ and ${b}$ within the strips at the first (second) penetration field ${H_{p}}^{k=0}$ (${H_{p}}^{k=1}$) for different values of the gap width $2s$. We observe that ${b}(H)$ changes only little between ${H_{p}}^{k=0}$ and ${H_{p}}^{k=1}$. This finding allows us to give an estimate for ${H_{p}}^{k=1}$; indeed, inserting ${b}({H_{p}}^{k=1}) \approx
{b}({H_{p}}^{k=0})$ in [Eq. ]{} where ${b}({H_{p}}^{k=0})$ is taken from [Eq. ]{}, we find $$\begin{aligned}
\frac{{H_{p}}^{k=1}}{{H_{p}}^{k=0}} &
\approx \sqrt{\frac{w^{2}}{w^{2}-{b_{0}}({b})^{2}}}
\approx \sqrt{\frac{5+\log(32w/s)}{8}}.\end{aligned}$$
![Position of the vortex dome boundaries ${b_{0}}$ and ${b}= {b_{1}}= {b_{2}}$ of the three-strip system at the penetration fields ${H_{p}}^{k=0}$ (dotted lines) and ${H_{p}}^{k=1}$ (dashed lines) (see also [Fig. ]{}\[fig:hp2\]) as a function of $s/w$. The strip areas along the positive $x$-axis are indicated in gray. For $H \leq {H_{p}}^{k=0}$, the system is described by one non-vanishing parameter ${b}$ (${b_{0}}= 0$); its dependence on $s$ is shown as a dotted line. Increasing $H$ beyond ${H_{p}}^{k=0}$, a vortex dome forms in the innermost strip ($0< {b_{0}}<
w$) reaching a finite width at the second penetration field ${H_{p}}^{k=1}$ (dashed line). For that field, the position ${b}$ has shifted towards the center of the outer strip $2w+2s$ (dashed line). Beyond ${H_{p}}^{k=1}$, a pair of domes forms in the two outer strips (${b_{1}}\neq {b_{2}}$, not shown here).[]{data-label="fig:dome-evolution-at-hp2"}](plot_16.eps){width=".45\textwidth"}
Beyond ${H_{p}}^{k=1}$, all three strips are penetrated, and the dome widths are determined by the restriction of no net current in the outer strips and the two critical field conditions at the edges ${e_{0}}$ and ${e_{1}}$. Note, that the central strip is penetrated from both edges while the strips of the pair $k=1$ are penetrated from the inner edges $\pm{e_{1}}$ only.
The order in which the strips are populated with vortices depends on the specification of the problem. Indeed, if the same geometrical configuration was *shunted* at both ends ($y \to \pm \infty$), as considered in Ref. [\[\]]{}, the outer strip pair would be populated by vortices from $\pm e_2$, while the innermost strip remains free of flux until much higher fields.
Many strips - General picture {#sec:sec:many-strips}
-----------------------------
For any finite number of *unshunted* strips, the field enhancement in the Meissner state is strongest at the innermost edges, i.e., the inner edges at $\pm{e_{1}}$ for an even number of strips and the two edges at $\pm{e_{0}}$ of the central strip for an odd number of strips (the only case where a strip is penetrated from both sides). Subsequently, the strips are always penetrated asymmetrically from the inner edges. Specifically, when the applied field is raised beyond the first penetration field, vortices start to populate the innermost strip(s) through the respective edges, while all other strips are still free of flux. Under further increase of $H$, the critical field strength ${H_{s}}$ is successively reached at the inner strip edges $e_{2k-1}$ and vortices penetrate the strip pair $k$, when $H > {H_{p}}^{k}$, with ${H_{p}}^{k} > {H_{p}}^{k-1}$, with $k$ starting from 1 (2) in the case of an odd (even) number of strips. In the limit of a large number of strips, the field strengths in the different gaps are almost the same, such that vortex penetration starts within a narrow field range in all the strips.
Summary and Conclusions {#sec:conclusions}
=======================
In summary, we have investigated the magnetic response of (two) aligned superconducting strips subject to a perpendicular magnetic field $H$. The penetration of vortices in these systems is dominated by a macroscopic energy barrier, the so-called geometrical barrier. We have found that a narrow slit between the strips completely suppresses this geometrical barrier as manifested in an early field penetration and the collapse of the hysteretic magnetization loop. We have compared the results for a pair of rectangular (platelet shaped) strips to those of various other shapes and geometries. Of particular interest is the comparison with a single elliptic strip, i.e., the generic shape defining demagnetization effects, and a single platelet strip, the simplest system exhibiting a geometric barrier. In the elliptic case, vortices penetrate the sample above a field ${H_{p}}= {H_{s}}d/2w$ ($H_{c1} \leq {H_{s}}< H_{c}$), distribute uniformly inside the sample, and produce a reversible response (in the absence of a surface barrier). The penetration of vortices in a platelet sample is impeded by a geometrical energy barrier (and potentially an additional surface barrier) at the sample edges, ${H_{p}}= {H_{s}}\sqrt{d/w}$. Once this barrier is overcome, vortices occupy a finite region inside the sample (vortex dome), while the rest carries the diamagnetic shielding currents. Upon decreasing the applied field, the strip shows a irreversible response, where the penetrated flux is trapped inside the sample until the vortex dome expands to the sample edges. These qualitative features remain valid for an array of rectangular strips.
The attention of the present work has mainly focused on the double strip. In the regime of a small gap parameter $s \ll w$, where the strip system is equivalent to a single strip of width $2W = 4w + 2s$ cut in half by a narrow gap, the geometrical barrier is overcome at much lower applied fields $H$. When $d$ is the smallest geometric length, the situation still resembles the one of the single strip; modifications concern the field of first penetration, ${H_{p}}= {H_{s}}\sqrt{d/W} \boldsymbol{[}\sqrt{s/W} \log(4W/s)\boldsymbol{]}$, the exclusive penetration from the inner edges, and the asymmetric shape of the vortex domes leaning towards the gap. When the gap width $2s$ drops below the thickness $d$, the currents rearrange strongly, piling up at the inner surfaces and channeling a larger field through the central opening. In the limit $s/d \to 0$, the geometrical barrier is maximally suppressed and the penetration field ${H_{p}}= {H_{s}}d/2W$ of the double strip coincides with that of a single ellipse with aspect ratio $d/2W$. In contrast to the previously discussed cases where the magnetization decreases beyond penetration, here the magnetic response levels off at the magnitude $M = {H_{s}}W d/2\pi$, a factor $4/\pi$ above the magnetization at the penetration field.
In order to study the irreversibility due to the geometric barrier, we have examined the descending branch in the magnetization upon reduction of the field. Remarkably, our analytic results show, that the initial slope of the descending branch is close to the Meissner response, with a correction factor approaching unity when the reversing field approaches ${H_{p}}$ from above; reversing the field at larger $H$, the Meissner slope is changed by a factor $(4-\nu)/\log(16/\nu)$, where $\nu$ depends only on the point $(H,M)$ where the slope is evaluated, $\nu = -4M/Hw^2$. Surprisingly, the correction factor remains close to unity over a wide range of $\nu < 1$. The latter result is equally valid for both the single and narrow-gap ($s \ll d$) double strip.
In addition, we have examined the influence of the vortex currents in the Shubnikov phase; while our complex-analysis approach describes the vortex-phase in terms of a non-magnetic medium, it intrinsically exhibits a finite magnetic response. We have shown that the magnetization due to the structure of the vortex state in the dome remains small within the region $H <
{H_{s}}$ where our analysis is valid. Finally, we have extended our analysis to $n$ ($\geq 3$) strips and have given a qualitative discussion of the field penetration for this more complex geometry.
The suppression of the geometrical barrier can be beneficial in many circumstances, as the hysteretic behavior due to geometrical effects often obscures other interesting physical phenomena. E.g., this has been the case in the identification of the vortex-lattice melting-transition in platelet-shaped layered BiSCCO samples, where the irreversibility line potentially interferes with the first-order melting line: polishing the sample into a prism shape, the geometrical barrier could be suppressed, what allowed to demonstrate experimentally that melting and irreversibility are uncorrelated phenomena [@Majer_95]. Another example is the competition between bulk pinning of vortices and pinning due to surface- and shape effects as analyzed in the present work: again, the suppression of the geometrical barrier provides access to an unambiguous study of bulk pinning phenomena. In evaluating different means to suppress the geometrical barrier, the generation of a simple gap or crack in the sample appears as a rather simple alternative. Recently, the suppression of geometrical barriers in platelet BiSCCO samples has also been observed when tilting the magnetic field [@Segev_11]. This finding has been related to the appearance of Josephson vortex stacks due to the parallel field component weakening the superconductor and channeling the perpendicular component of the magnetic field into the sample. Relating our present study to this experiment, we have modeled a stack of Josephson vortices by a sample crack (of width $2s$) and observe a similar suppression of the geometrical barrier.
Another topic where the suppression of geometrical barriers is advantageous is the generation of low-density vortex states, which are difficult to realize in bulk samples due to the rapid accumulation of vortices when increasing $H$ beyond ${H_{s}}$. In elliptic samples, low vortex densities of the order of ${H_{p}}/\Phi_{0} \sim {H_{s}}(d/w)/\Phi_0$ could be achieved; however, it appears difficult to fabricate samples with this shape. In a realistic platelet-shape sample, typical vortex densities are larger, of order ${H_{s}}\sqrt{d/w}
/\Phi_{0}$. Introducing a narrow gap in the sample suppresses the geometrical barrier and low vortex densities ${H_{s}}(d/w)\Phi_{0}$ can be reached.
Further possible applications of the narrow-gap double strip include the lensing of magnetic fields near the gap, what may be useful for focusing weak magnetic signals. Finally, the analysis and results discussed in this paper may be of relevance in the design of superconducting atom chips for the manipulation of ultra-cold atoms [@Bernon_13].
We acknowledge illuminating discussions with Alexei Koshelev and Eli Zeldov and the financial support of the Swiss National Fonds under the program NCCR MaNEP.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $\,T^{j,k}_{N}:L^{p}(B)\,
\rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B}
\,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx,
\quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n}\,$ and $\,N\,>>1.$ We compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of the operator norms $\,\parallel T^{j,k}_{N} \parallel_
{L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,\,q\in [1,+\infty].\,$ We prove that, except for the dimension $n=1,\,$ this asymptotic behaviour depends on the linearity or quadraticity of the phase in $s$ only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schrödinger equation.
author:
- 'Ahmed A. Abdelhakim'
bibliography:
- 'mybibfile.bib'
title: '$L^p$-$L^q$ boundedness of integral operators with oscillatory kernels: Linear versus quadratic phases'
---
Strichartz estimates for the Schrödinger equation ,Oscillatory integrals,$L^{p}-L^{q}$ boundedness 35B45, 35Q55, 42B20
### 1. A remark on a counterexample to inhomogeneous Strichartz estimates for the Schrödinger equation and motivation {#a-remark-on-a-counterexample-to-inhomogeneous-strichartz-estimates-for-the-schrödinger-equation-and-motivation .unnumbered}
Consider the Cauchy problem for the inhomogeneous free Schrödinger equation with zero initial data $$\begin{aligned}
\label{shreq}
\imath \partial_{t}u+\Delta u\,=\, F(t,x),\qquad (t,x)\in (0,\infty)\times{\mathbb{R}}^{n},\qquad u(0,x)\,=\,0.\end{aligned}$$ Space time estimates of the form $$\begin{aligned}
\label{est1}
||u||_{L^{q}_{t}\left(
\mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)}\;\lesssim\;
||F||_{L^{{\widetilde{q}}^{\prime}}_{t}
\left(\mathbb{R};L^{{\widetilde{r}}
^{\prime}}_{x}({\mathbb{R}}^{n})\right)},\end{aligned}$$ have been known as inhomogeneous Strichartz estimates. The results obtained so far (see [@damianoinhom; @Kato; @keeltao; @vilela; @YoungwooKoh]) are not conclusive when it comes to determining the optimal values of the Lebesue exponents $\,q$, $r$, $\tilde{q}\,$ and $\,\tilde{r}\,$ for which the estimate (\[est1\]) holds. Trying to further understand this problem, we [@ahmed1] found new necessary conditions on these exponents values. The counterexample in [@ahmed1], like Example 6.10 in [@damianoinhom], contains an oscillatory factor with high frequency. More precisely, we used a forcing term given by $$\begin{aligned}
\label{myphase}
F(t,x)= e^{-\imath\, N^2\,t } \,\chi_{[0,\frac{\eta}{N}]}(t)\,
\chi_{B\left(\frac{\eta}{N}\right)}{(x)}\end{aligned}$$ where $\,\eta>0\,$ is a fixed small number, $\,N>>1\,$ and $ B\left(\frac{\eta}{N}\right)$ is the ball with radius $\,\eta/N\,$ about the origin. While in [@damianoinhom] the stationary phase method is applied to the inhomogeneity $$\begin{aligned}
\label{fphase}
F (t,x)= e^{-2\imath\, N^2\,t^{2} }\,\chi_{[0,1]}(t)\,\chi_{B\left(\frac{\eta}{N}\right)}{(x)}.\end{aligned}$$ When $\,t\in [2,3],\,$ both data in (\[myphase\]) and (\[fphase\]) force the corresponding solution $u(t,x)$ to concentrate in a spherical shell centered at the origin with radius about $N.$ This agrees with the dispersive nature of the Schrödinger operator. The shell thickness is different in both cases though. It is about $1$ in the case of the data (\[myphase\]) but about $N$ in the case of (\[fphase\]). The necessary conditions obtained are respectively $$\begin{aligned}
\frac{1}{q}\geq\frac{n-1}{\widetilde{r}}-\frac{n}{r},
\qquad \quad\frac{1}{\widetilde{q}}\geq\frac{n-1}{r}-
\frac{n}{\widetilde{r}}\end{aligned}$$ and $$\begin{aligned}
\label{necess1}
|\frac{1}{r}-\frac{1}{\widetilde{r}}|\leq
\frac{1}{n}.\end{aligned}$$ Observe that the oscillatory function in (\[myphase\]) has a linear phase and is applied for the short time period of length $\:1/\sqrt{\text{frequency}}.\,$ The oscillatory function in (\[fphase\]) on the other hand has a quadratic phase and the oscillation is put to work for a whole time unit. We noticed that the phase in [@damianoinhom] need not be quadratic and we can get the necessary condition (\[necess1\]) using the data $$\begin{aligned}
\label{fphase1}
F_{l} (t,x)= e^{-\imath\, N^2\,t}\,
\chi_{[0,1]}(t)\,\chi_{B\left(\frac{\eta}{N}\right)}{(x)}\end{aligned}$$ where the phase in the oscillatory function is linear. Before we show this, we recall the following approximation of oscillatory integrals according to the principle of stationary phase.
\[stationary\] (see [@stein], Proposition 2 Chapter VIII and Lemma 5.6 in [@damianorem]) Consider the oscillatory integral $\;I(\lambda)=\displaystyle \int_{a}^{b}e^{\imath \lambda \phi(s)}\psi(s)d s.\;$ Let the phase $\,\phi \in C^5([a,b])\,$ and the amplitude $\psi\in C^3([a,b])$ such that
(i)
: $\;\phi^{\prime} (z)=0\,$ for a point $\;z\,\in\, ]a+c, b-c[\;$ with $\,c\,$ a positive constant,
(ii)
: $\;|\phi^{\prime} (s)|\,\gtrsim\, 1,\;$ for all $\;s\,\in\, [a, a+c]\,\cup\, [b-c, b],$
(iii)
: $\;|\phi^{\prime\prime} (s)|\,\gtrsim\, 1,$
(iv)
: $\;\psi^{(j)}\,$ and $\,\phi^{(j+3)}\;$ are uniformly bounded on $[a,b]$ for all $j=0,1,2$.
$$\begin{aligned}
\hspace*{-1 cm}\mbox{Then}\qquad \qquad I(\lambda)\,=\, {\,\sqrt{\frac{2 \pi}{\lambda|\phi^{\prime\prime} (z)|}} \,\psi(z)\,e^{\imath \,\lambda\, \phi(z)+\imath\,\mbox{\small sgn}\left(
\phi^{\prime\prime} (z)\right)\, \frac{\pi}{4}}}+\mathcal{O}\left(\frac{1}{\lambda}\right),
\vspace*{-0.22 cm}\end{aligned}$$
where the implicit constant in the $\mathcal{O}-$symbol is absolute.
The norm of the inhomogeneous term $F_{l} $ in (\[fphase1\]) has the estimate $$\begin{aligned}
\label{normf}
\parallel F_{l} \parallel_{L^{\tilde{q}^{\prime}}
([0,1];L^{\tilde{r}^{\prime}}(\mathbb{R}^{n}))}\,\approx\,
{\eta}^{n-\frac{n}{\tilde{r}}}\,
{N}^{-n}\,{N}^{\frac{n}{\tilde{r}}}.\end{aligned}$$ For the solution of (\[shreq\]), we have the explicit formula $$\begin{aligned}
\label{solshro}
u(t,x) \,=\, (4 \pi)^{-\frac{n}{2}}\int_{0}^{t}
(t-s)^{-\frac{n}{2}}\int_{{\mathbb{R}}^{n}}
e^{\imath\frac{|x-y|^2}{4(t-s)}}\,F(s,y)\,dy\, ds.\end{aligned}$$ Let us estimate the solution $u_{l}(t,x)$ that corresponds to $F_{l}.$ We shall restrict our attention to the region $$\begin{aligned}
\Omega_{\eta,N}=\left\{
(t,x)\in [2,3] \times \mathbb{R}^{n}\!\!:\,
2(t-{3}/{4})N+\eta N^{-1}\,<|x|<\,2(t-{1}/{4})N-\eta N^{-1}\right\}.\end{aligned}$$ It will be momentarily seen that this is the region where we can exploit Lemma \[stationary\] to approximate $\, u_{l}(t,x).$ Substituting from (\[fphase1\]) into (\[solshro\]) then applying Fubini’s theorem we get $$\begin{aligned}
\label{corsol}
u_{l}(t,x)\:=\:
(4\pi)^{-\frac{n}{2}} \,\int_{B({\eta}/{N})}
\,I_{N}(t,x,y)\, d y\end{aligned}$$ where $\,I_{N}(t,x,y)\,$ is the oscillatory integral $$\begin{aligned}
\label{inoscints}
I_{N}(t,x,y)\;=\;\int_{0}^{1}\,
e^{\imath N^2\, \phi_{N}{(s,t,x,y)}}\,\psi{(s,t)}\, d s,\end{aligned}$$ with the phase $\;
\displaystyle \phi_{N}{(s,t,x,y)} =\frac{ |x-y|^2}{4\,N^2}\frac{1}{t-s}-s\,$ and amplitude $\, \psi{(s,t)}=(t-s)^{-\frac{n}{2}}.$ For simplicity, we write $\phi(.)$ and $\psi(.)$ in place of $\,\phi_{N}{(.,t,x,y)}\,$ and $\,\psi{(.,t)}\,$ respectively. Next, we verify the conditions (**i**) - (**iv**) for $\,\phi\,$ and $\,\psi.$ Let $\,(t,x)\in \Omega_{\eta,N}\,$ and $\,y\in B(\eta/N).$ Observe then that $\;\displaystyle t-{3}/{4}<{|x-y|}/{2N}<t-{1}/{4}\;$ and $\; t-s \in [1,3]. $ Therefore
(i)
: If $\,z\,$ is such that $\,\phi^{\prime}(z)=0\,$ then $\,\displaystyle
z =t-{|x-y|}/{2N}.\,$ Moreover, $\displaystyle \,z\in\:]{1}/{4},{3}/{4} [.$
(ii)
: $\phi^{\prime}$ is monotonically increaing so $\;\displaystyle \min_{s\in[0,1]}{\phi^{\prime}(s)}
= \phi^{\prime}(0)=\frac{|x-y|^2}{4 N^2\, t^2}
> \left(1-\frac{3}{4t}\right)^{2}
\gtrsim 1.$
(iii)
: $\,\displaystyle
\phi^{\prime\prime}(s)\,=\,
\frac{|x-y|^2}{2 N^2}\frac{1 }{(t- s)^3}\,\approx\,1.$
(iv)
: $\;\displaystyle \phi^{(j)}(s)\,=\,
\frac{|x-y|^2}{4N^2}\frac{ j!}{(t- s)^{(j+1)}} \,\approx\,1,\; j=3,4,5$, $\;\;\;\psi(s)\,=\,
(t- s)^{-\frac{n}{2}}
\,\approx\,1$,\
$\psi^{\prime}(s)\,=\,
\frac{n}{2}(t- s)^{-\frac{n}{2}-1}
\,\approx\,1$, $\qquad \psi^{\prime\prime}(s)\,=\,
\frac{n}{2}(\frac{n}{2}+1)(t- s)^{-\frac{n}{2}-2}\,\approx\,1.$
Now, applying Lemma \[stationary\] to the oscillatory integral $\,I_{N}(t,x,y)\,$ in (\[inoscints\]) yields $$\begin{aligned}
\label{noosciny0}
I_{N}(t,x,y)\,=\,
{\,\sqrt{\frac{2 \pi}{\phi_N^{\prime\prime}(z,t,x,y)}} \,\psi(z,t)\,
\frac{e^{ \frac{\pi}{4} \imath}}{N}
\,e^{\imath N^2 \phi_N(z,t,x,y)}}+
\mathcal{O}\left(\frac{1}{N^2}\right).\end{aligned}$$ Since $\,\phi_N(z,t,x,y)+t=|x-y|/N\, $ and since $\, N\left(|x-y|-|x|\right)= \mathcal{O}\left(\eta\right)\,$ whenever\
$\,(t,x)\in \Omega_{\eta,N},\;y\in B(\eta/N).$ Then $\,\,N^2\,\phi_N(z,t,x,y)+N^2\,t
=N\,|x|+\mathcal{O}\left(\eta\right).$ Hence $$\begin{aligned}
\label{noosciny}
e^{\imath N^2\,\phi_N(z,t,x,y)}=
e^{\imath\left(N\,|x|-N^2\,t\right)}
\,e^{\mathcal{O}\left(\eta\right)}
=e^{\imath\left(N\,|x|-N^2\,t\right)}\,
\left(1+\mathcal{O}\left(\eta\right)\right).\end{aligned}$$ Inserting (\[noosciny\]) into (\[noosciny0\]) then returning to (\[corsol\]), we discover $$\begin{aligned}
u_{l}(t,x)\:=\:&
\frac{(4\pi)^{\frac{1-n}{2}} }{\sqrt{2}}\frac{e^{ \frac{\pi}{4} \imath}}{N} \,e^{\imath\left(N\,|x|-N^2\,t\right)}\,
\int_{B({\eta}/{N})}\,{
\,\frac{\psi(z,t)}{\sqrt{\phi_N^{\prime\prime}(z,
t,x,y)}} \,
\,\left(1+\mathcal{O}\left(\eta\right)\right)}
\,d y\\&\;+
\mathcal{O}\left(\frac{1}{N^2}\right)\,
\int_{B({\eta}/{N})}\,\,
d y.\end{aligned}$$ Recalling that $\,\psi,\:
\phi^{\prime\prime}\approx 1,\,$ we immediately deduce the estimate $$\begin{aligned}
&| u_{l}(t,x)|\,\gtrsim\,\frac{|B(\eta/N)|}{N}
\,\approx\,\eta^{n}\,N^{-(1+n)},\quad
(t,x)\in \Omega_{\eta,N}.\quad \text{Thus, for all}\;\;
t\in [2,3],\\
&\hspace*{-1 cm}
||u_{l}(t,x)||_{L^{r}_{x}\left({\mathbb{R}}^{n}\right)}
\,\geq\,
\left( \int_{2(t-{3}/{4})N+\eta N^{-1}\,<\,|x|\,<\,2(t-{1}/{4})N-\eta N^{-1}}\,
| u_{l}(t,x)|^{r}\,dx\,\right)^{\frac{1}{r}}
\,\gtrsim\,\eta^{n}\,N^{-(1+n)+\frac{n}{r}}.\end{aligned}$$ Consequently $$\begin{aligned}
\label{normul}
||u_{l}||_{L^{q}_{t}\left(
\mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)}
\,\geq\,||u_{l}||_{L^{q}_{t}\left(
[2,3];L^{r}_{x}({\mathbb{R}}^{n})\right)}
\,\gtrsim\,\eta^{n}\,N^{-(1+n)+\frac{n}{r}}.\end{aligned}$$ Lastly, it follows from (\[normf\]) and (\[normul\]) that $$\begin{aligned}
||u_{l}||_{L^{q}_{t}\left(
\mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)}/
\parallel F_{l} \parallel_{L^{\tilde{q}^{\prime}}
([0,1];L^{\tilde{r}^{\prime}}(\mathbb{R}^{n}))}\;
\gtrsim\;
\eta^{\frac{n}{\tilde{r}}}\,
N^{\frac{n}{r}-\frac{n}{\tilde{r}}-1}\end{aligned}$$ which, for a fixed $\,\eta,\,$ blows up as $\,N\rightarrow +\infty\,$ if $\,\displaystyle \frac{n}{r}-\frac{n}{\tilde{r}}>1.\,$ In the light of duality this implies the necessary condition (\[necess1\]).
These examples made us wonder how exactly different are linear oscillations from quadratic ones if we capture the cancellations in Lebesgue spaces. One way to see this is to consider the operators $\,T^{j,k}_{N}:L^{p}(B)\,
\rightarrow\,L^{q}([0,1])\,$ defined by $$\begin{aligned}
\label{intop}
T^{j,k}_{N}f(s):=\int_{B}
\,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx,
\qquad (j,k)\in\{1,2\}^{2},\end{aligned}$$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n},\,$ and compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of their operator norms for all $\,p,\,q\in [1,+\infty].\,$ Let $\,C_{j,k,n}:[0,1]^{2}\rightarrow \mathbb{R}\,$ be the functions defined by $$\begin{aligned}
C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)\,:=\,
\alpha \quad \text{if}\qquad
\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}
\;\approx\; N^{ - \alpha}.\end{aligned}$$ We discover that $\,C_{j,k,n}\,$ is a continuous function with range $\,[0,{1}/{4}]\,$ when $n=1,$ $j=2$ and $\,[0,{1}/{2}{k}]\,$ otherwise (see the figure below). We actually prove that
\[mainthm\] $$C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)\;=\;
\left\{
\begin{array}{ll}
\frac{1}{4}\, \sigma\left(\frac{1}{p},\frac{1}{q}\right), & \hbox{$n=1,\;$ $j=2$;} \\\\
\frac{1}{2\,k}\, \sigma\left(\frac{1}{p},\frac{1}{q}\right), & \hbox{
$n\geq j$.}
\end{array}
\right.$$ where $$\label{sgmab}
\sigma(a,b):=\left\{
\begin{array}{ll}
2b , & \hbox{$\; 0\leq a \leq 1-b,\;\;
0\leq b \leq \frac{1}{2}$;} \\
2(1-a) , & \hbox{$\; \frac{1}{2}\leq a \leq 1,\;\;a+ b \geq 1$;} \\
1 , & \hbox{$\;0\leq a \leq \frac{1}{2},\;\;\frac{1}{2}\leq b \leq 1$.}
\end{array}
\right.$$
$$\begin{aligned}
\begin{tikzpicture} [scale=9]
\draw[->] (0.0, 0) -- (0.6, 0) node[below] {$\frac{1}{p}$};
\draw[->] (0,0.0) -- (0, 0.6) node[left] {$\frac{1}{q}$};
\draw (0.5, 0) node[below] {${1}$};
\draw (0, 0.5) node[left] {${1}$};
\draw (0.25, 0) node[below] {$\frac{1}{2}$};
\draw (0, 0.25) node[left] {$\frac{1}{2}$};
\draw
(0, 0) -- (0.5, 0.0) -- (0.5, 0.5) -- (0.0, 0.5) -- cycle;
\draw
(0.5, 0.0) -- (0.25, 0.25)-- (0.25, 0.5);
\draw (0.25, 0.25) -- (0.0, 0.25);
\draw [loosely dotted] (0.25, 0.0) -- (0.25, 0.25);
\draw (0.165,0.125) node{$\;\frac{1}{2}\frac{1}{q}$};
\draw (0.125,0.375) node{$\;\frac{1}{4}$};
\draw (0.375,0.3) node{$\;
\frac{1}{2}(1-\frac{1}{p})$};
\draw (0.3, -0.1) node[below] {$C_{2,k,1}$};
\end{tikzpicture}\qquad\qquad\quad
\begin{tikzpicture} [scale=9]
\draw[->] (0.0, 0) -- (0.6, 0) node[below] {$\frac{1}{p}$};
\draw[->] (0,0.0) -- (0, 0.6) node[left] {$\frac{1}{q}$};
\draw (0.5, 0) node[below] {${1}$};
\draw (0, 0.5) node[left] {${1}$};
\draw (0.25, 0) node[below] {$\frac{1}{2}$};
\draw (0, 0.25) node[left] {$\frac{1}{2}$};
\draw
(0, 0) -- (0.5, 0.0) -- (0.5, 0.5) -- (0.0, 0.5) -- cycle;
\draw
(0.5, 0.0) -- (0.25, 0.25)-- (0.25, 0.5);
\draw (0.25, 0.25) -- (0.0, 0.25);
\draw [loosely dotted] (0.25, 0.0) -- (0.25, 0.25);
\draw (0.165,0.125) node{$\;\frac{1}{k}\frac{1}{q}$};
\draw (0.125,0.375) node{$\;\frac{1}{2}\frac{1}{k}$};
\draw (0.375,0.3) node{$\;
\frac{1}{k}(1-\frac{1}{p})$};
\draw (0.3, -0.1) node[below] {$C_{j,k,n}$};
\end{tikzpicture}\end{aligned}$$
For each $\,p,q\in [1,\infty],$ and all dimension $\,n>1,\,$ the asymptotic behaviour of $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}\,$ as $\,n\rightarrow +\infty\,$ is determined only by the linearity or quadraticity of the phase in $s$. The role of the power $j$ of $x$ appears exclusively in the dimension $n=1.$
There is nothing special about neither the unit interval nor the unit ball in defining the operators $T^{j,k}_{N}$. Actually we shall make use of Hölder inclusions of $L^{p}$ spaces on measurable sets of finite measure (see Lemma \[holder\] below). So we may take any suitable two such sets provided their finite measures are asymptotically equivalent to a constant independent of $ N $ as $ N\rightarrow +\infty.$
Foschi [@damianorem] studied a discrete version of an operator a little simpler than the integral operator $\,T^{1,1}_{N}.\,$ He considered the operator $\,D_{N}:\ell^{p}(\mathbb{C}^{N})
\rightarrow L^{q}(-\pi,\pi)\,$ that assigns to each vector $\,a=(a_{0},a_{1},...a_{N-1})\in {\mathbb{C}}^{N}\,$ the $\,2\pi$-periodic trigonometric polynomial $\,D_{N}a(t)=\sum_{m=0}^{N-1}a_{m}\,e^{\imath\,m\,t}\,$ and described the asymptotic behaviour of $\, \displaystyle\sup_{a\in \mathbb{C}^{N}-\{0\}}
{{\parallel D_{N}a\parallel_{L^{q}([-\pi,\pi])} }/
{\parallel a\parallel_{\ell^{p}\left(\mathbb{C}^{N} \right)}}}$ as $N\rightarrow+\infty,$ for all $ 1\leq p,\,q\leq+\infty.$ The norms there are defined by $$\begin{aligned}
&\parallel a \parallel_{\ell^{p}}=
\left( \sum_{m=0}^{N-1}|a_{m}|^{p}\right)^{\frac{1}{p}},
\quad
1\leq p <\infty, \qquad
\parallel a \parallel_{\ell^{\infty}}=
\max_{0\leq m\leq N-1}|a_{m}|,\\
&
\parallel f \parallel_{L^{q}}=
\left(\frac{1}{2\pi} \int_{-\pi}^{\pi}|f(t)|^{q}dt\right)^{\frac{1}{q}},
\quad
1\leq q <\infty, \qquad
\parallel f \parallel_{L^{\infty}}=
\max_{|t|\leq \pi}|f(t)|.\end{aligned}$$ This was followed by a similar investigation (see Section 5 in [@damianorem]) of a linear integral operator with an oscillatory kernel $\, L_{N}: L^{p}([0,1])\rightarrow L^{q}([0,1])\,$ defined by $$\begin{aligned}
L_{N}f(t)\,:=\,\int_{0}^{1}\,
e^{\imath N/(1+t+s)}\,\frac{f(s)}{(1+t+s)^{\gamma}}\,ds,
\quad\text{\small for some fixed}\;\; \gamma \geq 0.\end{aligned}$$
### 2. Proof of Theorem \[mainthm\] {#proof-of-theorem-mainthm .unnumbered}
In order to show Theorem \[mainthm\], we shall go through the following steps.\
******. Find lower bounds for $\,\parallel T^{j,k}_{N} \parallel_
{L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,q\in [1,+\infty]\,$:\
Test the ratio $ \parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}/
\parallel f \parallel_{L^{p}(B)} $ for functions $f \in {L^{p}(B)}$ that kill or at least slow down the oscillations in the integrals $T^{j,k}_{N}f.\,$ Of course this ratio is majorized by $\displaystyle \parallel T^{j,k}_{N} \parallel_
{L^{p}(B)\rightarrow L^{q}([0,1])}=\sup_{f\in L^{p}(B)-\{0\}}
{{\parallel T^{j,k}_{N}f\parallel_{L^{q}([0,1])} }/{\parallel f\parallel_{L^{p}\left(B\right)}}}. $ But what is really interesting is the fact that such functions likely maximize the ratio as well.\
******. We find upper bounds for $\,\parallel T^{j,k}_{N} \parallel_
{L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,q\in [1,+\infty].$ Thanks to interpolation and Hölder’s inequality, we merely need an upper bound for $\parallel T^{j,k}_{N} \parallel_
{L^{2}(B)\rightarrow L^{2}([0,1]).}$
\[holder\] Let $\,T^{j,k}_{N}:L^{p}(B)\,
\rightarrow\,L^{q}([0,1])\,$ be as in (\[intop\]). Assume that $$\begin{aligned}
\label{en11}
\parallel T^{j,k}_{N}f \parallel_{L^{2}([0,1])}
\,\leq\, c_{j,k,N}
\parallel f \parallel_{L^{2}(B)}.\end{aligned}$$ Then $$\begin{aligned}
\label{consigma}
\parallel T^{j,k}_{N} \parallel_{L^{p}(B)
\rightarrow L^{q}([0,1])}
\;\lesssim_{p,q,n}\; c^{\sigma\left(\frac{1}{p},\frac{1}{q}
\right)}_{j,k,N}\end{aligned}$$ where $\,\sigma:[0,1]^{2}\rightarrow [0,1]\,$ is the continuous function in (\[sgmab\]).
If we take absolute values of both sides of (\[intop\]) we get the trivial estimate\
$\;\parallel T^{j,k}_{N}f\parallel_{L^{\infty}([0,1])}
\,\leq\,\parallel f\parallel_{L^{1}\left(B\right)}.$ Interpolating this with (\[en11\]) using Riesz-Thorin theorem ([@loukas]) implies $$\begin{aligned}
\label{int1}
\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}
\,\leq\, c^{2\left(1-\frac{1}{p}\right)}_{j,k,N} \parallel f \parallel_{L^{p}(B)},
\qquad \frac{1}{2}\leq\frac{1}{p}\leq 1,\;\;
\frac{1}{q}=1-\frac{1}{p}.\end{aligned}$$ Since, by Hölder’s inequality, $\; \parallel T^{j,k}_{N}f \parallel_{L^{\bar{q}}([0,1])}
\,\leq\,\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}\;$ whenever\
$\,1\leq \bar{q}\leq q\leq \infty,$ then $$\begin{aligned}
\label{int2}
\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}
\,\leq\, c^{2\left(1-\frac{1}{p}\right)}_{j,k,N} \parallel f \parallel_{L^{p}(B)},
\qquad \frac{1}{2}\leq\frac{1}{p}\leq 1,\;\;
1-\frac{1}{p}\leq\frac{1}{q}\leq 1.\end{aligned}$$ Applying Hölder’s inequality once more we find that if $\;1\leq p\leq\bar{p}\leq \infty,\,$ then $$\begin{aligned}
\nonumber&\hspace{-1 cm}\parallel f \parallel_{L^{p}(B)}
\,\leq\,|B|^{\frac{1}{p}-\frac{1}{\bar{p}}}\,
\parallel f \parallel_{L^{\bar{p}}(B)}.
\;\; \text{Therefore by}\;(\ref{int1})\;
\text{we have}\\
\label{int3}
&\hspace{-0.6 cm}\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}
\,\leq\, |B|^{1-\frac{1}{p}-\frac{1}{q}}\,
c^{2/q}_{j,k,N} \parallel f \parallel_{L^{p}(B)},
\quad 0\leq\frac{1}{q}\leq \frac{1}{2},\;\;
0\leq\frac{1}{p}\leq 1-\frac{1}{q}.\end{aligned}$$ Moreover, since we know from (\[int2\]) that $$\begin{aligned}
\nonumber &\hspace*{-1 cm}\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}
\,\leq\, c_{j,k,N} \parallel f \parallel_{L^{2}(B)},
\quad \frac{1}{2}\leq\frac{1}{q}\leq 1,\quad
\text{then}\\
&\label{int4}
\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}
\,\leq\, |B|^{\frac{1}{2}-\frac{1}{p}}\,
c_{j,k,N} \parallel f \parallel_{L^{p}(B)},
\quad 0\leq\frac{1}{p}\leq \frac{1}{2},\;\;
\frac{1}{2}\leq\frac{1}{q}\leq 1.\end{aligned}$$
If the constants in inequalities (\[int1\]) - (\[int4\]) were sharp, they would be precisely the values of the corresponding norms $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}.$ Unfortunately, we are not able to compute the optimal constant $\,c_{j,k,N}\,$ in the energy estimate (\[en11\]). Nevertheless, the constants $\,c^{\sigma\left(\frac{1}{p},\frac{1}{q}
\right)}_{j,k,N}\,$ in (\[consigma\]) would be good enough for our purpose if, for each $p,q\in [1,+\infty],$ they were asymptotically equivalent, as $ N\rightarrow +\infty$, to the corresponding lower bounds of $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}\,$ that we compute in *Step 1*.\
****.**\
(i) **Focusing data**\
When $\,x\in B(\eta /N^{\frac{1}{j}})\,$ we have $\;\displaystyle e^{\imath N{|x|}^{j}s^{k}}=
e^{\mathcal{O}\left(\eta\right)}=
1+\mathcal{O}\left(\eta\right), \;\;
\text{\small for all}\;s\in [0,1].$ Thus, if we take $f_{j}$ to be the focusing functions $\,f_{j}=\displaystyle \chi_{B(\eta /N^{\frac{1}{j}})}\,$ then $\; \displaystyle \parallel f \parallel_{L^{p}(B)}\,=\,
|B(\eta /N^{\frac{1}{j}})|^{\frac{1}{p}}\;$ and $$\begin{aligned}
T^{j,k}_{N}f_{j}(s)\,=\,\int_{B(\eta /N^{\frac{1}{j}})}
\,e^{\imath N{|x|}^{j}s^{k}}\,dx=
\int_{B(\eta /N^{\frac{1}{j}})}
\,\left(1+\mathcal{O}\left(\eta\right)\right)\,dx
\,\gtrsim \,|B(\eta /N^{\frac{1}{j}})|\end{aligned}$$ for all $ \;0\leq s\leq 1.\;$ Consequently, since $\eta$ is fixed, $$\begin{aligned}
\label{lb1}
\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}
\,\geq\,\frac{\parallel T^{j,k}_{N}f_{j} \parallel_{L^{q}([0,1])}}{
\parallel f_{j} \parallel_{L^{p}(B)}}
\;\gtrsim\;
N^{-\frac{n}{j}\left(1-\frac{1}{p}\right)}.\end{aligned}$$ The figure below illustrates the one dimensional case. $$\begin{aligned}
&\hspace{-1 cm} \begin{tikzpicture} [scale=10]
\fill[fill = black!10] (0,0.35)--(0.048,0.35)--(0.048,0.0)--
(0,0)--cycle;
\draw[->] (0, 0) -- (0.38, 0) node[below] {\small$x$};
\draw[->] (0,0.0) -- (0, 0.4) node[left] {\small$ f_{j}(x)$};
\draw (0.35, 0) node[below] {\small ${1}$};
\draw (0,0.35) node[left] {\small ${1}$};
\draw[thick] (0,0.35)--(0.05,0.35);
\draw[thick] (0.05,0.0001)--(0.35,0.0001);
\draw (0.07,0.0)node[below] { \footnotesize {$N^{-\frac{1}{j}}$}};
\draw[dotted] (0.35, 0) -- (0.35,0.35)--(0,0.35);
\end{tikzpicture}\quad
\begin{tikzpicture} [scale=3.5]
\draw[->] (0.0, 0) -- (1.1, 0) node[below] {\small $s$};
\draw[->] (0.0,0.0) -- (0.0,1.1);
\draw (0.9,1.2) node[left] { \footnotesize {$N^{1/j}$\text{Re}$\left\{T^{j,k}_{N}f_{j}(s)\right\}$}};
\draw (1,0) node[below] {\small${1}$};
\draw (1,-0.001) --(1,0.003);
\draw (0.0,0.95) node[left] {\small${1}$};
\draw [blue,samples=500,domain=0.0:1] plot
(\x, {cos(0.65*\x r)});
\draw [red,samples=500,domain=0.0:1] plot
(\x, {cos(0.65*\x*\x r)});
\draw(0.7,1) node[right,thick] {\tiny {$k=2$}};
\draw(0.6,0.8) node[right,thick] {\tiny {$k=1$}};
\draw(0.0,-0.15) node {};
\end{tikzpicture}\qquad
\begin{tikzpicture} [scale=3.5]
\draw[->] (0.0, 0) -- (1.1, 0) node[below] {\small $s$};
\draw[->] (0.0,0.0) -- (0.0,1.1);
\draw (0.9,1.2) node[left] { \footnotesize {$N^{1/j}$\text{Im}$\left\{T^{j,k}_{N}f_{j}(s)\right\}$}};
\draw (1,0) node[below] {\small${1}$};
\draw (1,-0.001) --(1,0.003);
\draw (-0.001,1) --(0.003,1);
\draw (0.0,0.95) node[left] {\small${1}$};
\draw [blue,samples=500,domain=0.0:1] plot
(\x, {sin(0.65*\x r)});
\draw [red,samples=500,domain=0.0:1] plot
(\x, {sin(0.65*\x*\x r)});
\draw(0.7,0.3) node[right,thick] {\tiny {$k=2$}};
\draw(0.6,0.6) node[right,thick] {\tiny {$k=1$}};
\draw(0.0,-0.15) node {};
\end{tikzpicture}\\
&\text{\small \emph{Both real and imaginary parts of the functions}}
\;T^{1,k}_{N}f_{1}\; \text{\small\emph{and}}\;
T^{2,k}_{N}f_{2}\; \text{\small \emph{have the same profile}}.\end{aligned}$$ (ii) **Constant data**\
Let $\,g(x)=1.\,$ Whenever $\,\displaystyle s \in [0,\eta/N^{\frac{1}{k}}]\,$ we have $\;\imath N{|x|}^{j}s^{k}\,=\,\mathcal{O}\left(\eta\right)\;$ for all $\;x\in B\;$ and it follows that $\;\displaystyle e^{\imath N{|x|}^{j}s^{k}}=
1+\mathcal{O}\left(\eta\right).\;$ Hence, when $\,\displaystyle s \in [0,\eta/N^{\frac{1}{k}}],\,$ $$\begin{aligned}
T^{j,k}_{N}g(s)\,=\,
\int_{B}\,e^{\imath N{|x|}^{j}s^{k}}\,dx\,=\,
\int_{B}\,\left(1+\mathcal{O}\left(\eta\right)\right)\,dx
\,\gtrsim 1.\end{aligned}$$ Therefore, recalling that $\eta$ is fixed, $$\label{lb20}
\int_{0}^{1}|T^{j,k}_{N}g(s)|^{q}\,ds
\,\geq\,
\int_{0}^{\eta/N^{\frac{1}{k}}}|T^{j,k}_{N}g(s)|^{q}\,ds
\,\gtrsim\,
\int_{0}^{\eta/N^{\frac{1}{k}}}\,ds
\,\approx\,N^{-\frac{1}{k}}.$$ In view of (\[lb20\]), we deduce that $$\begin{aligned}
\label{lb2}
\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}
\,\geq\,\frac{\parallel T^{j,k}_{N} g \parallel_{L^{q}([0,1])}}{
\parallel g \parallel_{L^{p}(B)}}
\;\gtrsim\;
N^{-\frac{1}{k}\frac{1}{q}}.\end{aligned}$$ By rescaling, it is easy to verify that the estimate (\[lb2\]) follows for any complex-valued constant function $g$. The figure below shows the behaviour of $ T_{N}^{j,k}g $ on $[0,1]$ in the dimension $n=1.$ $$\begin{aligned}
\hspace*{-0.5 cm}
\begin{tikzpicture} [scale=10]
\draw[->] (0.0, 0.0) -- (0.48,0.0) node[below right] {$s$};
\draw[->] (0.0,0.0) -- (0.0,0.52);
\draw(0.25,0.44) node[above] {\scriptsize $
\text{Re}\left\{T^{1,k}_{N}g(s)\right\}=2$};
\draw(0.45,0.44) node[above] {\large $ \frac{\sin{(Ns^{k})}}{N\,s^{k}}$};
\draw (0.0,0.49) node[left] {\scriptsize ${2}$};
\draw (0.1,0.2) node[left] {\scriptsize $k=1$};
\draw (0.175,0.3) node[left] {\scriptsize $k=2$};
\draw [very thin,blue,samples=500,domain=0.001:0.45] plot
(\x, {2*sin((250*\x) r)/((1000*\x) )});
\draw [very thin,red,samples=500,domain=0.0012:0.45] plot
(\x, {2*sin((250*\x*\x) r)/((1000*\x*\x) )});
\end{tikzpicture}\qquad\qquad\qquad
\begin{tikzpicture} [scale=10]
\draw(0.23,0.44) node[above] {\scriptsize $
\text{Im}\left\{T^{1,k}_{N}g(s)\right\}=4$};
\draw(0.46,0.44) node[above] {\large $ \frac{\sin^{2}{(Ns^{k}/2)}}{N\,s^{k}}$};
\draw (0.0,0.49) node[left] {\scriptsize ${2}$};
\draw (0.0,0.49) --(0.001,0.49);
\draw (0.09,0.38) node[left] {\scriptsize $k=1$};
\draw (0.2,0.38) node[left] {\scriptsize $k=2$};
\draw [very thin,red,samples=500,domain=0.001:0.45] plot
(\x, {4*sin((100*\x*\x) r)*sin((100*\x*\x) r)/((800*\x*\x) )});
\draw [very thin,blue,samples=500,domain=0.0001:0.45] plot
(\x, {4*sin((50*\x) r)*sin((50*\x) r)/((400*\x) )});
\draw (-0.084, -0.084) node[below] {};
\draw[->] (0.0, 0.0) -- (0.48,0.0) node[below right] {$s$};
\draw[->] (0.0,0.0) -- (0.0,0.52);
\end{tikzpicture}\end{aligned}$$ $$\begin{aligned}
&\hspace*{-1.58 cm}\begin{tabular}{c c}
\hspace{-4.75 cm}\vspace{0.25 cm} \scriptsize $2$& \\
\hspace{-1.7 cm} \scriptsize $ k=2$&\\
\hspace{-3.3 cm} \scriptsize $ k=1$&\\
&\hspace{-1 cm} \scriptsize $ k=1\quad k=2$
\vspace{-2.5 cm}\\
\hspace{0.5 cm}\scriptsize{ \text{Re}$\left\{T^{2,k}_{N}g(s)\right\}$}
&\hspace{2 cm}\scriptsize
{\text{Im}$\left\{T^{2,k}_{N}g(s)
\right\}$} \vspace{-1.5 cm}\\
\includegraphics[scale=0.32]{t1001.pdf}
&\qquad\quad\;
\includegraphics[scale=0.32]{t2001.pdf}\vspace{-1 cm}\\
\hspace{5.5 cm} \small$ s$& \hspace{7 cm}
\small $ s$
\end{tabular}\\
&\hspace{-2.3 cm} \text{\small\emph{Functions}}\; \text{\small \emph{Re}}\small \{ T^{1,k}_{N}g(s)\}\;
\text{\small\emph{vanish and}}\;\text{\small \emph{Re}}\small \{T^{2,k}_{N}g(s)\}\;
\text{\small \emph{change monotonicity, for the first time, when }}\;s=\sqrt[k]{\pi/N}\end{aligned}$$ (iii) **Oscillatory data**\
Consider the oscillatory function $\,h(x)=e^{2\imath N \left(|x|^2-|x|\right)}.\,$ Using polar coordinates we can write $$\begin{aligned}
T^{j,k}_{N}h(s)\,=\,
\int_{S^{n-1}}
\int_{0}^{1}\,e^{\imath N
\,\left({\rho}^{j}s^{k}+2\rho^2-2\rho\right)}\,
\rho^{n-1}\,d\rho \,d\omega\,=\,
\omega_{n-1}
\:I^{j,k}_{N}(s)\end{aligned}$$ where $I^{j,k}_{N}(s) $ is the oscillatory integral given by $$\begin{aligned}
\label{inoscint}
I^{j,k}_{N}(s) =
\int_{0}^{1}\,e^{\imath N
\,\phi_{j,k}(\rho;s)}\,
\rho^{n-1}\,d\rho\end{aligned}$$ with the phase $\displaystyle \phi_{j,k}(\rho;s) ={\rho}^{j}s^{k}+2\rho^2-2\rho. $\
The quadratic function $\displaystyle \rho\rightarrow\phi_{j,k}(\rho;s)$, after a suitable translation along the vertical axis, has a single nondegenerate stationary point that happens to lie well inside $]\frac{1}{5},\frac{4}{5}[.$ Indeed, one can simply write $$\begin{aligned}
\phi_{j,k}(\rho;s)=\left\{
\begin{array}{ll}
2\left(\rho-\frac{2-s^{k}}{4}\right)^{2}-
\frac{\left(2-s^{k}\right)^{2}}{8}, & \hbox{$j=1$;} \\
\left(2+s^{k}\right)\left(\rho-\frac{1}{2+s^{k}}\right)^{2}-
\frac{1}{\left(2+s^{k}\right)^{2}}, & \hbox{$j=2$.}
\end{array}
\right.\end{aligned}$$ Notice also that $\, \left(2-s^{k}\right)/4\in[\frac{1}{4},\frac{1}{2}]\,$ and $\,\left(2+s^{k}\right)^{-1}\in[\frac{1}{3},
\frac{1}{2}]\,$ when $\,s\in [0,1].$ In fact, this is what we were after when we used the oscillatory function $h$ with its particular quadratic phase. Let us see how we benefit from this. We shall work on the integral $\,I^{1,k}_{N}(s)\,$ and the applicability of the same procedure to the integral $\,I^{2,k}_{N}(s)\,$ will be obvious. For simplicity, let $z$ denote $\,\left(2-s^{k}\right)/4.\,$ Then $$\begin{aligned}
\nonumber
e^{2\imath N\,z^2}\,I^{1,k}_{N}(s) =&
\,\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,
\rho^{n-1}\,d\rho\\
\label{feq}=&\,z^{n-1}\,\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho+
\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,
\left(\rho^{n-1}-z^{n-1}\right)\,d\rho.\end{aligned}$$ We compute $$\begin{aligned}
\label{t1}
\hspace{-1 cm}
\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho=
\int_{-\infty}^{+\infty}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho-
\int_{-\infty}^{0}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho-
\int_{1}^{+\infty}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho.\end{aligned}$$ Using the identity (See Exercise 2.26 in [@taobook]) $$\begin{aligned}
\int_{-\infty}^{+\infty}
\,e^{-ax^2}\,e^{bx}\,dx=\sqrt{\frac{\pi}{a}}
\,e^{b^2/4a},\quad a,b \in \mathbb{C},\;
\textrm{Re}(a) >0 \qquad \text{we get}\end{aligned}$$ $$\begin{aligned}
\label{11}
\hspace{-0.5 cm}\int_{-\infty}^{+\infty}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho
=\sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath}.\end{aligned}$$ And since $$\begin{aligned}
\hspace*{-1 cm}
\left|\;\int_{-\infty}^{0}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,\partial_{\rho}
\left(\rho-z\right)^{-1}\,d\rho \right|\,\leq\,
\frac{1}{z},\quad \left|\;
\int_{1}^{+\infty}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,\partial_{\rho}
\left(\rho-z\right)^{-1}\,d\rho \right|\,\leq\,
\frac{1}{1-z},\end{aligned}$$ then integration by parts implies $$\begin{aligned}
\label{22}&\int_{-\infty}^{0}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho=
\frac{\imath\, e^{2\imath N z^{2}}}{4 N z}
+\mathcal{O}\left(\frac{1}{Nz}\right),\\
\label{33}&\int_{1}^{+\infty}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho=
\frac{\imath\, e^{2\imath N\left(1-z\right)^{2}}}{4 N\left(1-z\right)}
+\mathcal{O}\left(\frac{1}{N\left(1-z\right)}\right).\end{aligned}$$ Recalling that $\,\frac{1}{4}\leq z\leq \frac{1}{2}\;$ and using (\[11\]), (\[22\]), (\[33\]) in (\[t1\]) we obtain $$\begin{aligned}
\label{44}
\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho\,=\,
\sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath}
+\mathcal{O}\left(\frac{1}{N}\right).\end{aligned}$$ This gives us an estimate for the first integral on the right hand side of (\[feq\]). The second integral is $\;\mathcal{O}\left({1}/{N}\right).\;$ This follows from integration by parts and the smoothness of the polynomial $\;P(\rho;z):={\left(\rho^{n-1}-z^{n-1}\right)}/{\left(\rho-z\right)}=
\sum_{\ell=0}^{n-2}\,\rho^{n-2-\ell}\,z^{\ell}\;$ as we can write $$\begin{aligned}
\int_{0}^{1}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,
\left(\rho^{n-1}-z^{n-1}\right)\,d\rho\,=\,
\frac{1}{4\imath N}
\int_{0}^{1}\,
P(\rho;z)\,
\partial_{\rho}\,e^{2\imath N
\,\left(\rho-z\right)^{2}}\,d\rho.\end{aligned}$$ Plugging (\[44\]) together with the latter estimate into (\[feq\]) we get that $$\begin{aligned}
\label{sm}
e^{2\imath N\,z^2}\,I^{1,k}_{N}(s)
\,=\,z^{n-1}\,
\sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath}
+\mathcal{O}\left(\frac{1}{N}\right).\end{aligned}$$ From (\[sm\]) follows the estimate $$\begin{aligned}
\left|I^{1,k}_{N}(s)\right|
\,\gtrsim\,N^{-1/2}.\end{aligned}$$ An explanation for the estimate above comes from the fact that the function $\,\lambda_{N}(\rho;z)=
\cos{\left(2N\,\left(\rho-z\right)^{2}\right)}\,$ remains positive for $\;|\rho-z|<\sqrt{\left(\pi/4N\right)}\;$ and the further we move from the stationary point $\rho=z$ it, unlike the slowly varying factor $\rho^{n-1}$, oscillates rapidly for large $N$ so that, when summing over $\rho$, integrals over neighbouring halfwaves where $\lambda_{N}$ changes sign almost cancel. See the figure below. An identical estimate for $\,I^{2,k}_{N}(s)\,$ follows applying the same argument above. The approach adopted here is standard. It represents the key idea of the proof of the stationary phase method illustrated by Lemma \[stationary\]. $$\begin{aligned}
\begin{tikzpicture}[yscale=1.5]
\fill[fill = black!50] (3*pi/8,0) -- plot [domain=3*pi/8:11*pi/13] (\x,{cos(64*\x*\x )}) -- (11*pi/13,0) -- cycle;
\fill[fill = black!50] (-3*pi/8,0) -- plot [domain=-3*pi/8:-11*pi/13] (\x,{cos(64*\x*\x )}) -- (-11*pi/13,0) -- cycle;
\fill[fill = black!25] (11*pi/13,0) -- plot [domain=11*pi/13:235*pi/208] (\x,{cos(64*\x*\x )}) -- (235*pi/208,0) -- cycle;
\fill[fill = black!25] (-11*pi/13,0) -- plot [domain=-11*pi/13:-235*pi/208] (\x,{cos(64*\x*\x )}) -- (-235*pi/208,0) -- cycle;
\fill[fill = black!5] (235*pi/208,0) -- plot [domain=235*pi/208:141*pi/104] (\x,{cos(64*\x*\x )}) -- (141*pi/104,0) -- cycle;
\fill[fill = black!5] (-235*pi/208,0) -- plot [domain=-235*pi/208:-141*pi/104] (\x,{cos(64*\x*\x )}) -- (-141*pi/104,0) -- cycle;
\draw [ <->] (-6.5,0) -- (6.5,0);
\draw [help lines,dashed,<-] (0,1.3) -- (0,0);
\draw (0,0) node[below] {$\rho=z$};
\draw (-5,1.3) node[above]{$ \cos{\left(N\,\left(\rho-z\right)^{2}\right)}$}; ;
\draw [thick,samples=500,domain=-2*pi:2*pi] plot
(\x, {cos(64*\x*\x )});
\draw [ <-](-3*pi/8+0.01,-0.7)
--(-2*pi/8+0.18,-0.7);
\draw [ ->](2*pi/8-0.1,-0.7)
--(3*pi/8-0.01,-0.7);
\draw (6.5,0) node[right] {$\rho$};
\draw (0,-0.7) node {$\sqrt{{\pi}/{2N}}$};
\draw [help lines,dashed] (-3*pi/8,1) -- (-3*pi/8,-1);
\draw [help lines,dashed] (3*pi/8,1) -- (3*pi/8,0-1);
\end{tikzpicture}\end{aligned}$$ Finally, since $\;\displaystyle \parallel h \parallel_{L^{p}(B)}\,=\, |B|^{{1}/{p}}\,\approx\,1,\;$ then $$\begin{aligned}
\label{lb3}
\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right)
\rightarrow L^{q}([0,1])}
\,\geq\,\frac{\parallel T^{j,k}_{N} h \parallel_{L^{q}([0,1])}}{
\parallel h \parallel_{L^{p}(B)}}
\;\gtrsim\;
N^{-\frac{1}{2}}.\end{aligned}$$ Putting (\[lb1\]), (\[lb2\]) and (\[lb3\]) together we deduce $$\begin{aligned}
\parallel T^{j,k}_{N} \parallel_{L^{p}(B)
\rightarrow L^{q}([0,1])}
\;\;\gtrsim\;
N^{-\min\left\{\frac{n}{j}\left(1-\frac{1}{p}\right),
\,\frac{1}{k}\frac{1}{q},\,\frac{1}{2}\right\}}
\,=\,N^{- C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)}.\end{aligned}$$ ****.** The $\,L^{2} - L^{2}\,$ estimate takes the form: $$\begin{aligned}
\label{energy}
\left.
\begin{array}{ll}
\vspace{0.3 cm}
\parallel T^{j,k}_{N}f\parallel_{L^{2}([0,1])}
\;\lesssim \; N^{-1/2k}\,\parallel f \parallel_{L^{2}\left(B\right)}, & \hbox{$n\geq j$,} \\
\parallel T^{2,k}_{N}f\parallel_{L^{2}([0,1])}
\;\lesssim \; N^{-n/2j}\,\parallel f \parallel_{L^{2}\left(B\right)}, & \hbox{$n=1$.}
\end{array}
\right \}\end{aligned}$$ Besides (\[lb2\]), the estimate (\[energy\]) demonstrates the difference between linear ($k=1$) and quadratic ($k=2$) oscillations. Let $\,x\in {\mathbb{R}}^{n}-\{0\}.\,$ The phase $\;s \longrightarrow {|x|}^{j}\,s^{k}\;$ of the oscillatory factor in (\[intop\]) is non-stationary when $\,k=1.\,$ While in the case $\,k=2,\,$ it is stationary with the nondegenerate critical point $s=0.\,$ This is where non-stationary and stationary phase methods (see lemmas \[nonstationary\] and \[stationary0\] below) for estimating oscillatory integrals come into play. As expected from (\[lb1\]), the role of $j$ appears only in the dimension $n=1.$ Using the estimate (\[energy\]) in Lemma \[holder\] we infer $$\begin{aligned}
\parallel T^{j,k}_{N} \parallel_{L^{p}(B)
\rightarrow L^{q}([0,1])}
\;\;\lesssim\; N^{- C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)}.\end{aligned}$$
### 3. Proof of the energy estimate (\[energy\]) {#proof-of-the-energy-estimate-energy .unnumbered}
To prove the estimate (\[energy\]) we need lemmas \[kernelsk\], \[kernelsq\] and \[even\] that we give below. Lemma \[kernelsk\] is based on the assertions of lemmas \[nonstationary\] and \[stationary0\].
\[nonstationary\] ([@stein], Proposition 1 Chapter VIII) Let $\,\psi \in C^{\infty}_{c}\left(\mathbb{R}
\right)\,$ and let $\displaystyle\; I(\lambda)=
\int_{\mathbb{R}}\,\psi(s)\,e^{\imath \,\lambda\,s}\,ds. \,$ Then $\;\displaystyle |I(\lambda)|\;\lesssim\;
\min{\left\{
\frac{1}{1+|\lambda|},
\frac{1}{1+\lambda^{2}}\right\}}.$
Observing that $\;\displaystyle \int_{0}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\,=\,
\frac{1}{2}\int_{-1}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\;$ and arguing as in (\[t1\])-(\[44\]) implies the estimate in Lemma \[stationary0\].
\[stationary0\] $$\begin{aligned}
\left|\int_{0}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\right|\;\lesssim \; \max{\left\{\frac{1}{1+\sqrt{|\lambda|}},
\frac{1}{1+|\lambda|}\right\}}.\end{aligned}$$
\[kernelsk\] Let $\,\psi \in C^{\infty}_{c}\left(\mathbb{R}
\right)\,$ and let $\;K_{N}^{j,k}:{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}
\longrightarrow {\mathbb{C}}\;$ be defined by $$\begin{aligned}
K_{N}^{j,k}(x,y):=
\left\{
\begin{array}{ll}
\displaystyle \int_{\mathbb{R}}\,\psi(s)\,e^{\imath N \left({|x|}^{j}-{|y|}^{j}\right)s}\,ds, & \hbox{$k=1$;} \\\\
\displaystyle \int_{0}^{1}\,\,e^{\imath N \left({|x|}^{j}-{|y|}^{j}\right)s^{2}}\,ds , & \hbox{$k=2$.}
\end{array}
\right.\end{aligned}$$ Then $$\begin{aligned}
\label{kernelsk1}
&\hspace*{-1 cm}
|K_{N}^{j,1}(x,y)|\;\lesssim\;
\min{\left\{
\left(1+N\,\left|{|x|}^{j}-{|y|}^{j}\right|\right)^{-1},
\left(1+N^{2} \, \left({|x|}^{j}-{|y|}^{j}\right)^{2}\right)^{-1}
\right\}},
\\ \label{kernelsk2}
&\hspace*{-1 cm} |K_{N}^{j,2}(x,y)|\;\lesssim\;
\max{\left\{\left( 1+\sqrt{N}\, \sqrt{\left|{|x|}^{j}-{|y|}^{j}\right|}\right)^{-1},
\left(1+N\, \left|{|x|}^{j}-{|y|}^{j}\right|\right)^{-1}\right\}}.\end{aligned}$$
The next lemma is mainly a consequence of Young’s inequality.
\[kernelsq\] Let $\,p,q,r\geq 1\,$ and $\,1/p+1/q+1/r =2.\,$ Let $\,f\in L^{p}(B),\,$ $\,g\in L^{q}(B)\,$ and $\,h\in L^{r}([0,1]).\,$ Then $$\begin{aligned}
\left|\,\int_{{B}}\,\int_{{B}}\,
f(x)\,f(y)\,h(|x|^{m}-|y|^{m})\,dx\,dy\,\right|\;\lesssim
\;\parallel f \parallel_{L^{p}(B)}\,
\parallel g \parallel_{L^{q}(B)}\,
\parallel h \parallel_{L^{r}([0,1])}\end{aligned}$$ provided $\,m\leq n$.
Switching to polar coordinates by setting $\,x=r_{1}\theta_{1}\,$ and $\,y=r_{2}\theta_{2}\,$ then applying Fubini’s theorem gives $$\begin{aligned}
\label{newlemma1}
\left|\,\int_{{B}}\,\int_{{B}}\,
f(x)\,f(y)\,h(|x|^{m}-|y|^{m})\,dx\,dy\,\right|
\,\leq\,\int_{S^{n-1}}\,\int_{S^{n-1}}\,
|Q(\theta_{1},\theta_{2})|
\,d\theta_{1}\,d\theta_{2}\end{aligned}$$ where $$\begin{aligned}
Q(\theta_{1},\theta_{2})\,=\,
\int_{0}^{1}\,\int_{0}^{1}\,
f(r_{1}\theta_{1})\,g(r_{2}\theta_{2})
\,h\left({r_{1}}^{m}-{r_{2}}^{m}\right)
\,r_{1}^{n-1}\,r_{2}^{n-1}\,dr_{1}\,dr_{2}.\end{aligned}$$ Changing variables $\:r_{i}^{m}\,\longrightarrow\, \rho_{i}\:$ then using Young’s inequality we get $$\begin{aligned}
\hspace*{-1 cm}
|Q(\theta_{1},\theta_{2})|\,\lesssim\,
\left(\int_{0}^{1}\left|f(\sqrt[m]{\rho_{1}}\,\theta_{1})
\right|^{p}\,\rho_{1}^{p\frac{n-m}{m}}\,d\rho_{1}\right)
^{\frac{1}{p}}
\left(\int_{0}^{1}\left|g(\sqrt[m]{\rho_{2}}\,\theta_{2})
\right|^{q}\,\rho_{2}^{q\frac{n-m}{m}}\,d\rho_{2}\right)
^{\frac{1}{q}}
\parallel h \parallel_{L^{r}([0,1])}.\end{aligned}$$ Reversing the variables change in the first two integrals on the right-hand side of the latter estimate we obtain $$\begin{aligned}
\label{newlemma2}
\hspace*{-1 cm}
\nonumber |Q(\theta_{1},\theta_{2})|\,\lesssim&\,
\left(\int_{0}^{1}\left|f({r_{1}}\,\theta_{1})
\right|^{p}\,r_{1}^{(p-1)(n-m)}\,
r_{1}^{n-1}\,dr_{1}\right)
^{\frac{1}{p}}\\&\;\nonumber
\left(\int_{0}^{1}\left|g({r_{2}}\,\theta_{2})
\right|^{q}\,r_{2}^{(p-1)(n-m)}\,
r_{2}^{n-1}\,dr_{2}\right)
^{\frac{1}{q}}\,
\parallel h \parallel_{L^{r}([0,1])}\\
\leq&
\left(\int_{0}^{1}\left|f({r_{1}}\,\theta_{1})
\right|^{p}\,r_{1}^{n-1}\,dr_{1}\right)
^{\frac{1}{p}}
\left(\int_{0}^{1}\left|g({r_{2}}\,\theta_{2})
\right|^{q}\,r_{2}^{n-1}\,dr_{2}\right)
^{\frac{1}{q}}
\,\parallel h \parallel_{L^{r}([0,1])}\end{aligned}$$ as long as $\,m\leq n.$ Invoking Hölder’s inequality it follows that $$\begin{aligned}
\nonumber &\int_{S^{n-1}}\,
\left(\int_{0}^{1}\left|f({r_{1}}\,\theta_{1})
\right|^{p}\,
r_{1}^{n-1}\,dr_{1}\right)^{\frac{1}{p}}\,d\theta_{1}\\
\label{newlemma3} &\hspace{0.8 cm}\leq\;\omega_{n-1}^{1-\frac{1}{p}}\;
\left( \int_{S^{n-1}}\,\int_{0}^{1}
\left|f({r_{1}}\,\theta_{1})
\right|^{p}\,r_{1}^{n-1}\,dr_{1}\,d
\theta_{1}\right)^{\frac{1}{p}}\;=
\;\omega_{n-1}^{1-\frac{1}{p}}\;
\parallel f \parallel_{L^{p}(B)},\\
\nonumber &
\int_{S^{n-1}}\,
\left(\int_{0}^{1}\left|g({r_{2}}\,\theta_{2})
\right|^{q}\,
r_{2}^{n-1}\,dr_{2}\right)^{\frac{1}{q}}\,d\theta_{2}\\
\label{newlemma4} &
\hspace{0.8 cm}\leq\;\omega_{n-1}^{1-\frac{1}{q}}\;
\left( \int_{S^{n-1}}\,\int_{0}^{1}
\left|g({r_{2}}\,\theta_{2})
\right|^{q}\,r_{2}^{n-1}\,dr_{2}\,d
\theta_{2}\right)^{\frac{1}{q}}\;=
\;\omega_{n-1}^{1-\frac{1}{q}}\;
\parallel g \parallel_{L^{q}(B)}.\end{aligned}$$ Returning to (\[newlemma1\]) with the estimates (\[newlemma2\]), (\[newlemma3\]) and (\[newlemma4\]) concludes the proof.
Remark \[even0\] together with Lemma \[homogeneous\] are needed to show Lemma \[even\].
\[even0\] Suppose that the integral $$\begin{aligned}
J\,=\,
\int_{-b_{1}}^{b_{1}}...\int_{-b_{m}}^{b_{m}}
\,K(t_{1},...,t_{m})\,
f_{1}(t_{1})...f_{m}(t_{m})\,dt_{1}...dt_{m}\end{aligned}$$ exists. If $\,K\,$ is even in all its variables then $$\begin{aligned}
J\,=\,
\int_{0}^{b_{1}}...
\int_{0}^{b_{m}}
\,K(t_{1},...,t_{m})\,
\prod_{i=1}^{m}\left(f_{i}(t_{i})+f_{i}(-t_{i})\right)
\,dt_{1}...dt_{m}.\end{aligned}$$ This follows easily from the fact that the integrand in the second expression for $\,J\,$ is even in all variables.
Lemma \[homogeneous\] discusses the boundedness of a bilinear form with a homogeneous kernel.
\[homogeneous\] Let $\,f\in L^{p}([0,1])\,$ and $\,g\in L^{q}([0,1])\,$ with $\,1\leq p \leq +\infty\,$ and $\,1/p\, +\, 1/q=1.\,$ Assume that $\,K:{[0,1]}\times
{[0,1]}\longrightarrow {\mathbb{R}}\,$ is homogeneous of degree $-1,\,$ that is, $\,K(\lambda x, \lambda y)=
\lambda^{-1} K(x,y),\,$ for $\,\lambda>0.\,$ Assume also that $$\begin{aligned}
\int_{0}^{+\infty}
\,\left|K(x,1)\right|\,{x}^{-\frac{1}{p}}\,dx
\,\lesssim\,1 \qquad \text{or } \qquad
\int_{0}^{+\infty}
\,\left|K(1,y)\right|\,{y}^{-\frac{1}{q}}\,dy
\,\lesssim\,1.\end{aligned}$$ Then $$\begin{aligned}
\left|\int_{0}^{1}\int_{0}^{1}\,
K(x,y)\,f(x)\,g(y)\,dx\,dy\right|\;\lesssim\;
\parallel f \parallel_{L^{p}([0,1])}\,
\parallel g \parallel_{L^{q}([0,1])}.\end{aligned}$$
In [@hardy], one can find a proof for the case when the integrals that define the bilinear form are taken over $\,[0,+\infty[.\,$ We treat this slightly trickier case of finite range without using the result in [@hardy].
Let $\,\displaystyle Q(f,g)\,=\,
\int_{0}^{1}\int_{0}^{1}\,
K(x,y)\,f(x)\,g(y)\,dx\,dy.\,$ Using a change of variables, $\,x\rightarrow y.u,\,$ and exploiting the homogeneity of the kernel we have $$\begin{aligned}
\hspace*{-0.8 cm} Q(f,g)\,=\,
\int_{0}^{1}y\,g(y)\int_{0}^{\frac{1}{y}}
K(y.u,y)\,f(y.u)\,du\,dy\,=\,
\int_{0}^{1}g(y)\int_{0}^{\frac{1}{y}}
K(u,1)\,f(y.u)\,du\,dy.\end{aligned}$$ By Fubini’s theorem we may write $$\begin{aligned}
\label{qfg}
\hspace*{-1 cm}
Q(f,g)=\int_{0}^{1} K(u,1) \int_{0}^{1} f(y.u)\,g(y)\,dy\,du+
\int_{1}^{+\infty} K(u,1) \int_{0}^{\frac{1}{u}} f(y.u)\,g(y)\,dy\,du.\end{aligned}$$ But by Hölder’s inequality we have $$\begin{aligned}
\hspace*{-0.8 cm}
\left|\int_{0}^{1}\,f(y.u)\,g(y)\,dy\right|
\;\leq&\;\left(\int_{0}^{1}\,|f(y.u)|^{p}\,dy\right)
^{\frac{1}{p}}
\left(\int_{0}^{1}\,|g(y)|^{q}\,dy\right)^{\frac{1}{q}}\\
\,&\hspace*{-2 cm}=u^{-\frac{1}{p}}\,
\left(\int_{0}^{u}\,|f(x)|^{p}\,dx\right)
^{\frac{1}{p}}\,
\parallel g \parallel_{L^{q}([0,1])}
\;\leq\;u^{-\frac{1}{p}}\,\parallel f \parallel_{L^{q}([0,1])}\,
\parallel g \parallel_{L^{q}([0,1])}\end{aligned}$$ for all $\,0 < u < 1.\,$ Similarly $$\begin{aligned}
\hspace*{-0.8 cm}
\left|\int_{0}^{\frac{1}{u}}\,f(y.u)\,g(y)\,dy\right|
\;\leq&\;\left(\int_{0}^{\frac{1}{u}}
\,|f(y.u)|^{p}\,dy\right)
^{\frac{1}{p}}\,
\left(\int_{0}^{\frac{1}{u}}\,|g(y)|^{q}
\,dy\right)^{\frac{1}{q}}\\
\,&\hspace*{-3.4 cm}=u^{-\frac{1}{p}}\,
\left(\int_{0}^{1}\,|f(x)|^{p}\,dx\right)
^{\frac{1}{p}}\,
\left(\int_{0}^{\frac{1}{u}}\,|g(y)|^{q}
\,dy\right)^{\frac{1}{q}}
\;\leq\;u^{-\frac{1}{p}}\,\parallel f \parallel_{L^{q}([0,1])}\,
\parallel g \parallel_{L^{q}([0,1])}\end{aligned}$$ for all $\,1< u < +\infty.\,$ Using the last two inequalities together with the triangle inequality in (\[qfg\]) we get $$\begin{aligned}
\hspace*{-1 cm}
|Q(f,g)|\leq& \,
\,\parallel f \parallel_{L^{q}([0,1])}\,
\parallel g \parallel_{L^{q}([0,1])}\,
\left(\int_{0}^{1} |K(u,1)| \,u^{-\frac{1}{p}}\,du+
\int_{1}^{+\infty} |K(u,1)|\,u^{-\frac{1}{p}}\,du
\right)\\
\lesssim&\;
\parallel f \parallel_{L^{q}([0,1])}\,
\parallel g \parallel_{L^{q}([0,1])},
\qquad \text{when} \quad
\int_{0}^{+\infty} |K(x,1)|\,x^{-\frac{1}{p}}\,dx
\,\lesssim\,1.\end{aligned}$$ When $\;\displaystyle \int_{0}^{+\infty}\left|K(1,y)\right|
{y}^{-\frac{1}{q}}dy\lesssim 1\;$ the assertion follows analogously.
If $\,K(x,y)=\left( x + y \right)^{-1}\,$ in Lemma \[homogeneous\] we get Hilbert’s inequality.
\[even\] Let $\,f,g \in L^2([-1,1]).$ Then $$\begin{aligned}
\label{even1}&\int_{-1}^{1}\,\int_{-1}^{1}\,
\frac{|f(x)||g(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy \;\lesssim\; \frac{1}{\sqrt{N}}\,
\parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}, \\
\label{even2} &\int_{-1}^{1}\,\int_{-1}^{1}\,
\frac{|f(x)|\,|g(y)|}{
\sqrt{\left|{x}^{2}-{y}^{2}\right|}}
\,dx\,dy \;\lesssim\; \parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}.\end{aligned}$$
Beginning with the estimate (\[even1\]), Remark \[even0\] suggests estimating\
\
$\; \displaystyle \int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(\pm x)||g(\pm y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy.\;$ Let $\;\displaystyle W_{N}(f,g):=
\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(x)||g(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy.$\
\
If $\,x,y \geq 0\,$ and $\,|x-y|>>1/\sqrt{N}\,$ then we also have $\,x+y>>1/\sqrt{N}\,$ and consequently $\,N\left|x^2-y^2\right|>>1.\,$ Therefore $$\begin{aligned}
\hspace{-0.8 cm}
\nonumber W_{N}(f,g)&\approx
\int\,\int_{
\substack{0\leq x,y\leq1,\\
|x-y|\lesssim\; 1/\sqrt{N}}}
\frac{|f(x)||g(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy+
\int\,\int_{
\substack{0\leq x,y\leq1,\\
|x-y|>> 1/\sqrt{N}}}
\frac{|f(x)||g(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy\\
\nonumber &\lesssim
\int\,\int_{
\substack{0\leq x,y\leq1,\\
|x-y|\lesssim\; 1/\sqrt{N}}}
{|f(x)||g(y)|}\,dx\,dy+
\frac{1}{N}\,\int\,\int_{
\substack{0\leq x,y\leq1,\\
|x-y|>> 1/\sqrt{N}}}
\frac{|f(x)||g(y)|}{\left|x^2-y^2\right|}
\,dx\,dy\\
\label{h1} &\lesssim
\int_{0}^{1}\,\int_{0}^{1}\,
\chi_{N}{\left(|x-y|\right)}{|f(x)||g(y)|}\,dx\,dy+
\frac{1}{\sqrt{N}}\,\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(x)||g(y)|}{x+y}
\,dx\,dy\end{aligned}$$ where $\,\chi_{N}\,$ is the characteristic function of the interval $\,[0,1/\sqrt{N}\,].$ By Young’s inequality we have $$\begin{aligned}
\label{h2}
\int_{0}^{1}\,\int_{0}^{1}\,
\chi_{N}{\left(|x-y|\right)}{|f(x)||g(y)|}
\,dx\,dy\,\leq\,
\frac{1}{\sqrt{N}}\,
\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ And by Hilbert’s inequality $$\begin{aligned}
\label{h3}
\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(x)||g(y)|}{x+y}
\,dx\,dy\,\lesssim \;\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ Using (\[h2\]) together with (\[h3\]) in (\[h1\]) we obtain $$\begin{aligned}
\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(x)||g(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy\,\lesssim\,\frac{1}{\sqrt{N}}\,
\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ In obtaining (\[h1\]), we worked only on the kernel of $W_{N}.$ It is therefore easy to see that replacing the function $\,x\rightarrow f(x)\,$ by the function $\,x\rightarrow f(-x)\,$ or $\,y\rightarrow g(y)\,$ by $\,y\rightarrow g(-y)\,$ then repeating the routine above eventually leads to the estimate $$\begin{aligned}
\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(\pm x)||g(\pm y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy\,\lesssim\,\frac{1}{\sqrt{N}}\,
\parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}.\end{aligned}$$ This proves (\[even1\]). Taking advantage of Remark \[even0\] again and arguing like before, it suffices to\
\
estimate $\displaystyle
V(f,g)=\int_{0}^{1}\,\int_{0}^{1}\,
\frac{|f(x)|\,|g(y)|}{\sqrt{\left|{x}^{2}-{y}^{2}\right|}}
\,dx\,dy.\;$ Since $\; \displaystyle \int_{0}^{+\infty}\frac{dz}{\sqrt{z}\,\sqrt{|1-z^2|}}
\,\approx\,1,$\
\
a direct application of Lemma \[homogeneous\] then gives $\,V(f,g)\,\lesssim\,\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.$
We are now ready to prove (\[energy\]). We do this for each of the cases $k=1$ and $k=2$ separately.\
**The phase is linear in $\textbf{s}\,$ $\,(k=1)$**:\
Let $\psi$ be a nonnegative smooth cutoff function such that $\,{supp}\:\psi \subset\;]-1,2[\,$ and $\,\psi(s)=1\,$ on $\,[0,1]$. Since $\,|T^{j,1}_{N} f |^2 \,=\, T^{j,1}_{N} f\;\;\overline{T^{j,1}_{N} f}.\,$ Then $$\begin{aligned}
&\hspace{-1 cm}\parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])}
\,=
\int_{0}^{1}\,|T^{j,1}_{N} f(s)|^{2}\,ds
\,\leq\,\int_{\mathbb{R}}\psi(s)\,|T^{j,1}_{N} f(s)|^{2}\,ds\\
&\hspace{-1 cm}=\;\int_{\mathbb{R}}\psi(s)\, T^{j,1}_{N} f(s)\;\overline{T^{j,1}_{N} f(s)}\,ds\,
=\;
\int_{\mathbb{R}}\psi(s)\, \int_{{B}}\,\int_{{B}}\,e^{\imath N \left({|x|}^{j}-{|y|}^{j}\right)s}\,
f(x)\,\overline{f(y)}\,dx\,dy\,ds.\end{aligned}$$ Let $f\in L^{2}(B)$. Applying Fubini’s theorem we get $$\begin{aligned}
\label{energy01}
\parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\;\leq\;
\int_{{B}}\,\int_{{B}}\,K_{N}^{j,1}(x,y)\,
f(x)\,\overline{f(y)}\,dx\,dy.\end{aligned}$$ In the light of the estimate (\[kernelsk1\]) of Lemma \[kernelsk\], it follows that $$\begin{aligned}
\label{energy11}
\parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\;\lesssim\;
\int_{{B}}\,\int_{{B}}\,
\frac{|f(x)|\,|f(y)|}{1+N^{2} \, \left({|x|}^{j}-{|y|}^{j}\right)^{2}}
\,dx\,dy.\end{aligned}$$ Since $\displaystyle
\int_{0}^{1}\frac{dz}{1+N^2 z^2}\approx \frac{1}{N},\,$ then, applying Lemma \[kernelsq\] with $\,h(z)=\left(1+N^2 z^2\right)^{-1}\,$ to the\
\
estimate (\[energy11\]), we obtain $$\begin{aligned}
\label{e1}
\parallel T^{j,1}_{N} f \parallel_{L^{2}([0,1])}\;\lesssim\;
\frac{1}{\sqrt{N}}\,\parallel f \parallel_{L^{2}(B)},
\qquad\text{for all dimensions}\;\;n\geq j.\end{aligned}$$ To finish this case, it remains to estimate $\,T^{2,1}f\,$ in the dimension $\,n=1.$ In view of (\[kernelsk1\]) and (\[energy01\]), we have $$\begin{aligned}
\hspace{-1 cm}
\parallel T^{2,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\:\lesssim\,
\int_{-1}^{1}\,\int_{-1}^{1}\,
\frac{|f(x)|\,|f(y)|}{1+N\,
\left|x^2-y^2\right|}
\,dx\,dy.\end{aligned}$$ Hence, by (\[even1\]) of Lemma \[even\], $$\begin{aligned}
\label{e2}
\parallel T^{2,1}_{N} f \parallel_{L^{2}([0,1])}\:
\lesssim\,\frac{1}{N^{1/4}}\,
\parallel f \parallel_{L^{2}([-1,1])}.\end{aligned}$$ **The phase is quadratic in $\textbf{s}\,$ $\,(k=2)$**:\
For $f\in L^{2}(B)$, using Fubini’s theorem then employing the estimate (\[kernelsk2\]) implies $$\begin{aligned}
\label{energy21}
\hspace{-0.6 cm}
\parallel T^{j,2}_{N} f \parallel^{2}_{L^{2}([0,1])}\:=\,
\int_{{B}}\,\int_{{B}}\,K_{N}^{j,2}(x,y)\,
f(x)\,\overline{f(y)}\,dx\,dy
\,\lesssim\, G^{j}_{N}(f)+H^{j}_{N}(f)\end{aligned}$$ where $$\begin{aligned}
G^{j}_{N}(f)\,=&\, \int_{{B}}\,\int_{{B}}\,
\frac{|f(x)|\,|f(y)|}{1+\sqrt{N}\, \sqrt{\left|{|x|}^{j}-{|y|}^{j}\right|}}
\,dx\,dy,\\
H^{j}_{N}(f)\,=&\, \int_{{B}}\,\int_{{B}}\,
\frac{|f(x)|\,|f(y)|}{1+N\, \left|{|x|}^{j}-{|y|}^{j}\right|}
\,dx\,dy.\end{aligned}$$ Since $\;\displaystyle \int_{0}^{1}\,\frac{dz}{1+\sqrt{N}\,\sqrt{z}}
\,\approx\, \frac{1}{\sqrt{N}},\quad
\int_{0}^{1}\,\frac{dz}{1+N\,z}
\,=\,
\text{\large o}\left(\frac{1}{\sqrt{N}}\right),
\quad \text{as}\;\;\; N\longrightarrow+\infty,
$\
\
then applying Lemma \[kernelsq\] to both $\,G^{j}_{N}(f)\,$ and $\,H^{j}_{N}(f)\,$ gives the estimate $$\begin{aligned}
\label{energy22}
G^{j}_{N}(f)+
H^{j}_{N}(f)\;\lesssim\; \frac{1}{\sqrt{N}}
\parallel f \parallel^{2}_{L^{2}(B)},
\qquad n\geq j.\end{aligned}$$ It remains to control $\:G^{2}_{N}(f)\,$ and $\,
H^{2}_{N}(f)\:$ in the dimension $\,n=1.\,$ But when $\,n=1,$ $$\begin{aligned}
\hspace*{-0.4 cm}
G^{2}_{N}(f)\,=&\, \int_{-1}^{1}\,\int_{-1}^{1}\,
\frac{|f(x)|\,|f(y)|}{1+\sqrt{N}\, \sqrt{\left|{x}^{2}-{y}^{2}\right|}}
\,dx\,dy\\ \leq&\,\frac{1}{\sqrt{N}}\,
\int_{-1}^{1}\,\int_{-1}^{1}\,
\frac{|f(x)|\,|f(y)|}{
\sqrt{\left|{x}^{2}-{y}^{2}\right|}}
\,dx\,dy
\,\lesssim\,
\frac{1}{\sqrt{N}}\,\parallel f \parallel^{2}_{L^{2}([-1,1])}\quad
\text{by}\;\; (\ref{even2})\; \text{of}\;
\text{Lemma} \;\ref{even}.\end{aligned}$$ An identical estimate holds for $H^{2}_{N}(f)$ in the dimension $n=1$ because of (\[even1\]). Combining this with (\[energy22\]) and using them in (\[energy21\]) yields $$\begin{aligned}
\label{e3}
\parallel T^{j,2}_{N} f \parallel_{L^{2}([0,1])}
\;\lesssim\;\frac{1}{{N}^{1/4}}
\parallel f \parallel_{L^{2}(B)}.\end{aligned}$$ Finally, bringing the estimates (\[e1\]), (\[e2\]) and (\[e3\]) together results in (\[energy\]).
References {#references .unnumbered}
==========
[10]{} Ahmed A. Abdelhakim, A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation, Journal of Mathematical Analysis and Applications, 414 (2014), 767-772. Damiano Foschi, Some remarks on the $L^{p}-L^{q}$ boundedness of trigonometric sums and oscillatory integrals, Communications on pure and applied analysis, 4 (2005), 569-588. Damiano Foschi, Inhomogeneous Strichartz estimates, Journal of Hyperbolic Differential Equations, 2 (2005), 1–24.
Loukas Grafakos, Classical Fourier Analysis, 2nd ed., Springer, 2008. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, UK, 1952. T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223–238. M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998), 955–980. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 2006. M. C. Vilela, Strichartz estimates for the nonhomogeneous Schrödinger equation, Transactions of the American Mathematical Society, 359 (2007), 2123–2136. Youngwoo Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, Journal of Mathematical Analysis and Applications, 373 (2011), 147–160.
Mathematics Department, Faculty of Science\
Assiut University, Assiut,71516, Egypt\
[email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the important problem of estimating parameter sensitivities for stochastic models of reaction networks that describe the dynamics as a continuous-time Markov process over a discrete lattice. These sensitivity values are useful for understanding network properties, validating their design and identifying the pivotal model parameters. Many methods for sensitivity estimation have been developed, but their computational feasibility suffers from the critical bottleneck of requiring time-consuming Monte Carlo simulations of the exact reaction dynamics. To circumvent this problem one needs to devise methods that speed up the computations while suffering acceptable and quantifiable loss of accuracy. We develop such a method by first deriving a novel integral representation of parameter sensitivity and then demonstrating that this integral may be approximated by any convergent tau-leap method. Our method is easy to implement, works with any tau-leap simulation scheme and its accuracy is proved to be similar to that of the underlying tau-leap scheme. We demonstrate the efficiency of our methods through numerical examples. We also compare our method with the tau-leap versions of certain finite-difference schemes that are commonly used for sensitivity estimations.\
author:
- 'Ankit Gupta[^1]'
- 'Muruhan Rathinam[^2]'
- 'Mustafa Khammash[^3]'
title: ' [**Estimation of parameter sensitivities for stochastic reaction networks using tau-leap simulations** ]{}'
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Condition]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Definition]{} \[theorem\][Hypothesis]{} \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Description]{} \[theorem\][Assumption]{}
¶
|
[**Keywords:**]{} parameter sensitivity; reaction networks; Markov process; tau-leap simulations\
[**Mathematical Subject Classification (2010):**]{} 60J22; 60J27; 60H35; 65C05.
Introduction {#sec:intro}
============
The study of chemical reaction networks is an essential component of the emerging fields of Systems and Synthetic Biology [@Alon; @Vilar; @Gardner]. Traditionally chemical reaction networks were modeled in the deterministic setting, where the dynamics is represented by a set of ordinary differential equations (ODEs) or partial differential equations (PDEs). In the study of intracellular chemical reactions, some chemical species are present in low copy numbers. Since the behavior of individual molecules is best described by a stochastic process, in the low molecular copy number regime, the copy numbers of the molecular species itself is better modeled by a stochastic process than by ODEs [@GP]. Only in the limit of large molecular copy numbers, one expects the deterministic models to be accurate [@DASurvey]. While our work in this paper is focused on biochemical reaction networks as primary examples, we emphasize that the mathematical framework of reaction networks can also be used to describe a wide range of other phenomena in fields such as Epidemiology [@Hethcote] and Ecology [@Bascompte].
Suppose $\theta$ is a parameter (like ambient temperature, cell-volume, ATP concentration etc.) that influences the rate of firing of reactions. Let $( X_\theta (t) )_{t \geq 0 }$ be the $\theta$-dependent Markov process representing the reaction dynamics, and suppose that for some real-valued function $f$ and observation time $T$, our output of interest is $f ( X_\theta(T) ) $. This output is a random variable and we are interested in determining the sensitivity of its expectation $\E( f( X_\theta(T) ) )$ w.r.t. infinitesimal changes in the parameter $\theta$. We define this sensitivity value, denoted by $S_\theta(f,T)$, as the partial derivative $$\begin{aligned}
\label{defn_paramsens}
S_\theta(f,T) := \frac{\partial}{\partial \theta} \E( f( X_\theta(T) ) ). \end{aligned}$$ Determining these parametric-sensitivity values are useful in many applications, such as, understanding network design and its robustness properties [@Stelling], identifying critical reaction components, inferring model parameters [@Fink2009] and fine-tuning a system’s behavior [@Feng].
Generally the sensitivities of the form cannot be directly evaluated, but instead, they need to be estimated with Monte Carlo simulations of the dynamics $( X_\theta (t) )_{t \geq 0 }$. Many methods have been developed for this task [@IRN; @Gir; @KSR1; @KSR2; @DA; @Our; @Gupta2], but they all rely on exact simulations of $( X_\theta (t) )_{t \geq 0 }$ that can be performed using schemes such as Gillespie’s *stochastic simulation algorithm* (SSA) [@GP]. This severely constrains the computational feasibility of these sensitivity estimation methods because these exact simulations become highly impractical if the rate of occurrence of reactions is high [@GillespieRev], which is typically the case. The main difficulty is that that exact simulation schemes keep track of each reaction event which is very time-consuming. To avoid this problem, tau-leaping methods have been developed that proceed by combining many reaction-firings over small time intervals [@tleap1]. Tau-leap methods have been shown to produce good approximations of the reaction dynamics, at a small fraction of the computational cost of exact simulations [@tleap1; @tleap2; @Rathinam2003; @Burrage2004; @AndersonPost; @Rathinam-ElSamad; @Yang-Rathinam; @Yang-Rathinam-Shen; @Tempone2011; @Tempone2014]. Their accuracy and stability has also been investigated theoretically in many papers [@Rathinam1; @Li; @Gang; @Rathinam3; @Cao-Petzold+Stability].
Our goal in this paper is to develop a method that takes advantage of the computational efficiency of tau-leap methods for the purpose of estimating sensitivity values of the form . Since tau-leap methods introduce a bias in the estimation, it is highly desirable to start with an unbiased method for computing sensitivities (instead of biased methods such as the Finite Difference (FD)) and then replace exact SSA simulations by a suitable tau-leap method. Having only one form of bias, modulated by the tau-leap step size, allows one to control the bias more effectively and also facilitates the design of *multilevel* strategies that eliminate or reduce the estimator bias and enhance its computational efficiency [@Anderson2012; @Lester2015; @Tempone2014]. Among the existing methods in the literature, only the [*Girsanov Transformation*]{} (GT) method [@Glynn1; @Gir], the *Auxiliary Path Algorithm*(APA)[@Our] and the *Poisson Path Algorithm* (PPA) [@Gupta2] are unbiased. Since the GT method in general suffers from large variance [@Gupta2; @DA; @Our; @KSR1; @KSR2; @Rathinam2] and the APA/PPA methods are not directly amenable to tau-leap approximation, we develop a variant of the PPA method in which exact SSA simulations are replaced by tau-leap simulations. Our method, called *Tau Integral Path Algorithm* ($\tau$IPA), works with any underlying tau-leap simulation scheme and it is based on a novel integral representation of parameter sensitivity $S_\theta(f,T)$ that we derive in this paper. We provide computational examples that show that using $\tau$IPA we can often *trade-off* a small amount of bias for large savings in the overall computational costs for sensitivity estimation. We prove that the bias incurred by $\tau$IPA depends on the step-size in the same way as the bias of the tau-leap scheme chosen for simulations. Moreover if we substitute the tau-leap simulations in $\tau$IPA with the exact SSA generated simulations, then we obtain a new unbiased method for sensitivity estimation which we call the ‘exact’ IPA or eIPA that is similar to the PPA method in [@Gupta2]. Two main reasons for the high variance of the GT method that have been identified in the existing literature are: 1) low magnitude of the sensitivity parameter $\theta$ (see [@Gupta2; @Our]) and 2) large system-size or volume under the classical volume scaling of the reaction network [@Rathinam2]. The second issue is somewhat resolved by the *centered Girsanov Transformation* (CGT) method [@Rathinam2; @warren2012steady] and our numerical results indicate that the volume scaling behavior of eIPA is similar to CGT (see Section \[ex:bd\]). However eIPA does not suffer from high variance when the sensitivity parameter $\theta$ is small. In addition, when $\theta=0$, GT or CGT methods are not even applicable while eIPA does not suffer from this restriction. These observations make eIPA more appealing than CGT for unbiased estimation of parameter sensitivity.
For the sake of comparison, we use *tau-leap versions* of certain commonly used finite-difference estimators (see [@DA; @KSR1; @morshed2017efficient]) that approximate the infinitesimal derivative in by a finite-difference (see ). Such estimators are computationally faster than $\tau$IPA (in simulation time per trajectory) but they suffer from two sources of bias (finite-differencing and tau-leap approximations) unlike $\tau$IPA which only incurs bias from the latter source. We note that while in some examples the biases nearly cancel each other fortuitously, as a general principle one has no logical reason to expect such cancellation.
This paper is organized as follows. In Section \[sec:prelim\] we describe the stochastic model for reaction dynamics and the sensitivity estimation problem. We also discuss the existing sensitivity estimation methods, the tau-leap simulation schemes and explain the rationale for using such simulations in sensitivity estimation. Section \[sec:mainres\] contains the main results of this paper which include a novel integral representation of the exact sensitivity in Section \[sec:integralrepresentation\], a result on error bounds for the sensitivity estimates of $\tau$IPA in Section \[sec:tauest1\] and the novel tau-leap sensitivity estimation method $\tau$IPA in Section \[sec:tauest2\]. In Section \[sec:num\_Examples\] we provide computational examples to compare our method with other methods and finally in Section \[sec:conc\] we conclude and provide directions for future research.
Preliminaries {#sec:prelim}
=============
Consider a reaction network with $d$ species and $K$ reactions. We describe its kinetics by a continuous time Markov process whose state at any time is a vector in the non-negative integer orthant $\N^d_0$ comprising of the molecular counts of all the $d$ species. The state evolves due to transitions caused by the firing of reactions. We suppose that when the state is $x$, the rate of firing of the $k$-th reaction is given by the *propensity* function $\lambda_k(x)$ and the corresponding state-displacement is denoted by the stoichiometric vector $\zeta_k \in \Z^d$. There are several ways to represent the Markov process $(X(t))_{t \geq 0 }$ that describes the reaction kinetics under these assumptions. We can specify the generator (see Chapter 4 in [@EK]) of this process by the operator $$\begin{aligned}
\label{defn_gen}
\mathbb{A} h (x) = \sum_{k=1}^K \lambda_k(x ) \left( h(x+\zeta_k) - h(x) \right),\end{aligned}$$ where $h$ is any bounded real-valued function on $\N^d_0$. Alternatively we can express the Markov process directly by its random time-change representation (see Chapter 7 in [@EK]) $$\begin{aligned}
\label{rtc_rep1}
X(t) = X(0) + \sum_{k=1}^K Y_k \left( \int_{0}^{t} \lambda_k( X (s) ) ds \right) \zeta_k,\end{aligned}$$ where $\{Y_k : k=1,\dots,K\}$ is a family of independent unit rate Poisson processes. Since the process $(X(t))_{t \geq 0 }$ is Markovian, it can be equivalently specified by writing the *Kolmogorov forward equation* for the evolution of its probability distribution $p_t(x) := \P( X(t) = x )$ at each state $x$: $$\begin{aligned}
\label{defn_cme}
\frac{ d p_{t}(x) }{dt} = & \sum_{k=1}^K p_{t}( x - \zeta_k) \lambda_k(y - \zeta_k) -p_t(x )\sum_{k=1}^K \lambda_k(x).\end{aligned}$$ This set of *coupled* ordinary differential equations (ODEs) is termed as the *Chemical Master Equation* (CME) in the biological literature [@DASurvey]. As the number of ODEs in this set is typically infinite, the CME is nearly impossible to solve directly, except in very restrictive cases. A common strategy is to estimate its solution with *pathwise* simulations of the process $(X(t))_{t \geq 0}$ using Monte Carlo schemes such as Gillespie’s SSA [@GP], the *next reaction method* [@NR], the *modified next reaction method* [@AndMod], and so on. While these schemes are easy to implement, they become computationally infeasible for even moderately large networks, because they account for each and every reaction event. To resolve this issue, tau-leaping methods have been developed which will be described in greater detail in Section \[tau\_leap\_methods\].
We now assume that each propensity function $\lambda_k$ depends on a real-valued system parameter $\theta$. To emphasize this dependence we write the rate of firing of the $k$-th reaction at state $x$ as $\lambda_k(x,\theta)$ instead of $\lambda_k(x)$. Let $( X_\theta(t) )_{t \geq 0 }$ be the Markov process representing the reaction dynamics with these parameter-dependent propensity functions. As stated in the introduction, for a function $f : \N^d_0 \to \R$ and an observation time $T \geq 0$, our goal is to determine the sensitivity value $S_\theta(f,T) $ defined by . This value cannot be computed directly for most examples of interest and so we need to find ways of estimating it using simulations of the process $( X_\theta(t) )_{t \geq 0 }$. Such simulation-based sensitivity estimation methods work by specifying the construction of a random variable $s_\theta(f,T)$ whose expected value is “close" to the true sensitivity value $S_\theta(f,T)$, i.e. $$\begin{aligned}
\label{gen_sens_form}
S_\theta(f,T) \approx \E( s_\theta(f,T) ). \end{aligned}$$ Once such a construction is available, a large number (say $N$) of independent realizations $s_1,\dots,s_N$ of this random variable $s_\theta(f,T)$ are obtained and the sensitivity is estimated by computing their empirical mean $\hat{\mu}_N$ as $$\begin{aligned}
\label{defn_empirical_mean}
\hat{\mu}_N = \frac{1}{N} \sum_{i=1}^N s_i.\end{aligned}$$ This estimator $\hat{\mu}_N$ is a random variable with mean and variance $$\begin{aligned}
\label{defn_mean_var_estimator}
\mu = \E\left( \hat{\mu}_N \right) = \E( s_\theta(f,T) ) \qquad \textnormal{and} \qquad \sigma^2_N = \textnormal{Var}( \hat{\mu}_N) = \frac{ \sigma^2 }{N} \end{aligned}$$ respectively, where $\sigma^2 = \textnormal{Var}(s_\theta(f,T) )$. For a large sample size $N$, the distribution of $ \hat{\mu}_N $ is approximately Gaussian with mean $\mu$ and variance $\sigma^2_N$, due to the Central Limit Theorem. The *standard deviation* $\sigma_N$ measures the *statistical spread* of the estimator $\hat{\mu}_N $, that is inversely proportional to its *statistical precision*. The sample size $N$ must be large enough to ensure that $\sigma_N$ is small relative to $\mu$, i.e. for some small parameter $\epsilon > 0$, we should have $$\begin{aligned}
\label{rel_estimatorcond}
\frac{ \textnormal{RSD} }{ \sqrt{N} } \leq \epsilon,\end{aligned}$$ where $\textnormal{RSD}:= \sigma/| \mu |$ is the *relative standard deviation* of the random variable $s_\theta(f,T)$. If such a condition holds, then $\hat{\mu}_N$ is a reliable estimator for the true sensitivity value $S_\theta(f,T)$ because it is very likely to assume a value close to its mean $\mu= \E( s_\theta(f,T) ) $ which in turn is close to $S_\theta(f,T)$ (see ). In practice both $\mu$ and $\sigma$ are unknown, but we can estimate them as $\mu \approx \hat{\mu}_N$ and $\sigma \approx \sqrt{N} \hat{\sigma}_N$ where $$\begin{aligned}
\label{est_std_dev}
\hat{\sigma}_N = \frac{1}{ \sqrt{N (N-1) } } \sqrt{ \sum_{i=1}^N (s_i - \hat{ \mu}_N)^2 }.\end{aligned}$$ is the estimated standard deviation $\sigma_N$ of the estimator.
The performance of any sensitivity estimation method (say $\mathcal{X}$) depends on the following three key metrics that are based on the properties of random variable $s_\theta(f,T)$:
1. The *bias* $\mathcal{B}(\mathcal{X}) = \E( s_\theta(f,T) ) - S_\theta(f,T) $, which is the error incurred by the approximation .
2. The *variance* $\mathcal{V}(\mathcal{X}) = \textnormal{Var}(s_\theta(f,T) )$ of random variable $s_\theta(f,T)$.
3. The *computational cost* $\mathcal{C}(\mathcal{X})$ of generating one sample of $s_\theta(f,T)$.
The bias $\mathcal{B}(\mathcal{X})$ can be positive or negative, and its absolute value $|\mathcal{B}(\mathcal{X})|$ can be seen as the upper-bound on the statistical accuracy that can be achieved with method $\mathcal{X}$ by increasing the sample size $N$ [@ReviewRahbek]. As mentioned before, the standard deviation $\sigma( \mathcal{X} ) = \sqrt{ \mathcal{V}(\mathcal{X}) }$ measures the statistical precision of the method $\mathcal{X}$ and its magnitude relative to the mean $\mu(\mathcal{X}) = \E( s_\theta(f,T) )$ determines the number of samples $N$ that are needed to produce a reliable estimate. In particular, to satisfy condition for the relative standard deviation $\textnormal{RSD}( \mathcal{X} ) = \sigma( \mathcal{X} ) /|\mu(\mathcal{X})|$, the number of samples $N_\epsilon$ needed would be around $N_\epsilon := (\textnormal{RSD}( \mathcal{X} ) )^2 \epsilon^{-2}$. Hence the total cost of the estimation procedure is $$\begin{aligned}
\label{overallcomputationcost}
N_\epsilon \mathcal{C}(\mathcal{X}) \approx (\textnormal{RSD}( \mathcal{X} ) )^2 \mathcal{C}(\mathcal{X}) \epsilon^{-2} = \frac{ \mathcal{V}( \mathcal{X} ) }{ (\mu(\mathcal{X} ) )^2 } \mathcal{C}(\mathcal{X}) \epsilon^{-2},\end{aligned}$$ where $\mathcal{C}(\mathcal{X})$ is the CPU time required for constructing one realization of $s_\theta(f,T)$. The goal of a good estimation method is to simultaneously minimize the three quantities $|\mathcal{B}(\mathcal{X})|$, $\mathcal{V}(\mathcal{X})$ and $\mathcal{C}(\mathcal{X})$. This creates various conflicts and trade-offs among the existing sensitivity estimation methods as we now discuss.
Biased methods {#sec:biasedmethods}
--------------
A sensitivity estimation method $\mathcal{X}$ is called *biased* if $\mathcal{B}(\mathcal{X}) \neq 0$. The most commonly used biased methods are the *finite-difference schemes* which approximate the infinitesimal derivative in the definition of parameter sensitivity (see ) by a finite-difference of the form $$\begin{aligned}
\label{fd:form1}
S_{\theta,h}(f,T) = \frac{\E\left( f( X_{\theta +h } (T) ) - f( X_\theta(T) ) \right)}{h},\end{aligned}$$ for a small perturbation $h$. The processes $X_\theta$ and $X_{\theta +h}$ represent the Markovian reaction dynamics with values of the sensitive parameter set to $\theta$ and $\theta +h$ respectively. These two processes can be simulated independently [@IRN] but it is generally better to couple them in order to reduce the variance of the associated estimator. The two commonly used coupling strategies are called *Common Reaction Paths* (CRP) [@KSR1] and *Coupled Finite Differences* (CFD) [@DA] and they are based on the random time-change representation .
The finite-difference approximation for the true sensitivity value can be expressed as the expectation $\E( s_{\theta,h}(f,T) )$ of the following random variable $$\begin{aligned}
s_{\theta,h}(f,T) = \frac{f( X_{\theta +h } (T) ) - f( X_\theta(T) ) }{h}.\end{aligned}$$ The three metrics (bias, variance and computational cost) based on this random variable define the performance of CRP and CFD. Since both these methods estimate the same quantity $S_{\theta,h}(f,T)$, they have the same bias (i.e. $\mathcal{B}( \textnormal{CRP}) = \mathcal{B}( \textnormal{CFD})$). However in many cases it is found that the CFD coupling is *tighter* than the CRP coupling, resulting in a lower variance of $s_{\theta,h}(f,T)$ (i.e. $\mathcal{V}( \textnormal{CFD}) < \mathcal{V}( \textnormal{CRP})$) (see [@DA]). For each realization of $s_{\theta,h}(f,T)$, both CRP and CFD require simulation of a coupled trajectory $(X_\theta,X_{\theta +h} )$ in the time interval $[0,T]$. The computational costs of such a simulation is roughly $2 \mathcal{C}_0$, where $\mathcal{C}_0$ is the cost of *exactly* simulating the process $X_\theta$ using Gillespie’s SSA [@GP] or a similar method.[^4]
Finite-difference schemes introduce a bias in the estimate whose size is proportional to the perturbation value $h$ (i.e. $\mathcal{B}(
\textnormal{CRP}) = \mathcal{B}( \textnormal{CFD}) \propto h$), but the constant of proportionality can be quite large in many cases, leading to significant errors even for small values of $h$ [@Gupta2]. Unfortunately we cannot circumvent this problem by choosing a very small $h$ because the variance is proportional to $1/h$ (i.e. $\mathcal{V}(
\mathcal{\textnormal{CRP}} ) , \mathcal{V}( \mathcal{\textnormal{CFD}} )
\propto 1/h$). Therefore if a very small $h$ is selected, the variance will be enormous and the sample-size required to produce a statistically precise estimate will be very large, imposing a heavy computational burden on the estimation procedure [@Gupta2]. This trade-off between bias and variance is the main drawback of finite-difference schemes and there does not exist a strategy for selecting $h$ that optimally balances these two quantities. Note that unlike bias and variance, the computational cost of generating a sample (i.e. $\mathcal{C}( \textnormal{CRP} )$ or $\mathcal{C}( \textnormal{CFD} )$) does not change significantly with $h$, thereby ensuring that regardless of $h$, the total computational burden varies linearly with the required number of samples $N$. Apart from finite-difference schemes, there exists another biased method, called the *regularized pathwise-derivative* method [@KSR2] for estimating the sensitivity value , but we do not discuss this approach in this paper.
Unbiased methods {#subsec:unbiasedmethods}
----------------
A sensitivity estimation method $\mathcal{X}$ is called *unbiased* if $\mathcal{B}(\mathcal{X}) =0$. The main advantage of unbiased methods is that the estimation can in principle be made as accurate as possible by increasing the sample size $N$. The first unbiased method for sensitivity estimation is called the *Girsanov Transformation* (GT) method [@Glynn1; @Gir], which works by estimating the $\theta$-derivative of the probability distribution of $X_\theta$. The GT method is easy to implement and the computation cost of generating each sample is roughly $\mathcal{C}_0$ – the cost of *exact* simulation of the process $X_\theta$. The main issue with the GT method is that generally the variance of its associated random variable $s_\theta(f,T)$ is very large and so the number of samples needed to obtain a statistically precise estimate is very high [@DA; @KSR1]. So far two reasons have been identified for this behavior. Firstly, it has been shown that for mass-action models (see [@DASurvey]) this variance can become unbounded when the magnitude of the sensitive reaction rate-constant $\theta$ approaches zero [@Gupta2; @Our]. This is a serious issue because biological networks often consist of *slow* reactions which are characterized by low values of the associated rate-constants. Furthermore the GT method does not allow one to estimate the sensitivity w.r.t. a rate-constant set to zero. Such sensitivity values are useful for understanding network design as it allows one to probe the effect of presence or absence of reactions. Another reason for the high variance of GT estimator was provided in [@Rathinam2] where it was theoretically established that this variance can grow boundlessly as the system expands in size, i.e. the system volume $V$ tends to infinity. This issue is somewhat ameliorated by the *centered Girsanov Transformation* (CGT) method [@warren2012steady] but the problem with small reaction rate-constants persists.
We now discuss a couple of unbiased methods that have been recently proposed. These methods are called the *Auxiliary Path Algorithm*(APA )[@Our] and the *Poisson Path Algorithm* (PPA) [@Gupta2], and they are based on *exact* representations of the form for the parameter sensitivity . For both the methods, sampling the random variable $s_\theta (f,T)$ requires simulation of a fixed number $M_0$ of *additional* paths of the process $X_\theta$. It was shown in [@Our] that in comparison to the GT method, the computational cost of generating each sample for APA is much higher (i.e. $\mathcal{C}( \textnormal{APA}) \gg \mathcal{C}( \textnormal{GT})$) but this is often compensated by the fact that its variance is much lower (i.e. $\mathcal{V}( \textnormal{APA}) \ll \mathcal{V}( \textnormal{GT})$), resulting in a smaller overall cost of estimation . The reason for the higher sampling cost for APA is that it needs estimates of certain unknown quantities at each jump-time of the process $X_\theta$ in the time interval $[0,T]$, which can be very large in number even for small networks. In PPA, this problem is resolved by *randomly selecting* a small number of these unknown quantities for estimation in such a way that the estimator remains unbiased. Due to this extra randomness, the sample variance for PPA is generally greater than APA (i.e. $\mathcal{V}( \textnormal{PPA}) > \mathcal{V}( \textnormal{APA})$) but the computational cost for realizing each sample is much lower (i.e. $\mathcal{C}( \textnormal{PPA}) \ll \mathcal{C}( \textnormal{APA})$). Moreover in comparison to APA, PPA is far easier to implement and has lower memory requirements, making it an attractive unbiased method for sensitivity estimation. In [@Gupta2] it is shown using many examples that for a given level of statistical accuracy, PPA can be more efficient than GT and also the finite-difference schemes CFD and CRP. The computational cost of generating each sample in PPA is roughly $(2 M_0+1) \mathcal{C}_0$, where $M_0$ is a small number that upper-bounds the expected number of unknown quantities that will be estimated using additional paths. For both APA and PPA, the parameter $M_0$ serves as a *trade-off* factor between the computational cost and the variance - as $M_0$ increases, the cost also increases but the variance decreases. However both these methods remain unbiased for any choice of $M_0$.
The foregoing trade-off relationships for the existing sensitivity estimation methods are summarized in Table \[tab1:tradeoff\].
------ --------------- ------------ ----------- -----------
Type Method Trade-off Trade-off Preserved
$\mathcal{X}$ quantities parameter quantity
CRP
CFD
APA
PPA
------ --------------- ------------ ----------- -----------
: Trade-off relationships among the bias $\mathcal{B}( \mathcal{X} )$, variance $\mathcal{V}( \mathcal{X} )$ and the computational cost $\mathcal{C}( \mathcal{X} )$ for existing sensitivity estimation methods. Here $h$ is the perturbation size for finite-difference schemes [@DA; @KSR1] and $M_0$ quantifies the number of auxiliary paths for APA [@Our] and PPA [@Gupta2]. The cost of *exactly* simulating the underlying process is $\mathcal{C}_0$.[]{data-label="tab1:tradeoff"}
Rationale for using tau-leap schemes for sensitivity estimation {#using_tau_leap_methods}
---------------------------------------------------------------
All the existing sensitivity estimation methods suffer from a critical bottleneck – they are all based on exact simulations of the process $X_\theta$. The computational cost $\mathcal{C}_0$ of generating each trajectory of $X_\theta$ can be exorbitant even for moderately large networks when those networks have some molecular species in moderately large copy numbers and/or reactions firing at multiple timescales ([*stiff systems*]{}). One way to counter this problem is to develop methods that can accurately estimate parameter sensitivities with *approximate* computationally inexpensive simulations of the process $X_\theta$ obtained with tau-leap methods. The use of tau-leap simulations provides a natural way to trade-off a small amount of error with a *potentially* large reduction in the computational costs. The [*explicit*]{} tau-leap method with Poisson random numbers proposed by Gillespie [@tleap1] generally works well in [*non-stiff*]{} situations and when molecular copy numbers are modestly large. The major drawback is that it becomes inefficient for stiff systems where vastly different time scales are present. The [*implicit*]{} tau-leap was proposed to remedy this weakness [@Rathinam2003]. Many other tau-leap methods and step size selection strategies have been proposed to address stiffness and other issues [@tleap2; @Burrage2004; @AndersonPost; @Rathinam-ElSamad; @Yang-Rathinam-Shen; @Yang-Rathinam; @Tempone2014].
In the context of stiff systems, tau-leap methods have not been as successful in maintaining accuracy while reducing computational cost in comparison with the success of stiff solvers for deterministic differential equations. This is because stiffness manifests in a more complex manner in stochastic systems where stability is not the only issue, but accurately capturing the asymptotic distribution of the fast variables is also important [@Rathinam2003; @Cao-Petzold+Stability; @Li-Abdulle-E; @Cipcigan-Rathinam]. We shall limit our attention to non-stiff or modestly stiff systems in this paper.
Our goal in this paper is to develop a method that can estimate parameter sensitivity $S_\theta(f,T)$ of the form using only tau-leap simulations of the process $X_\theta$. This can be done by specifying a random variable $s^{(\tau)}_\theta(f,T)$ which can be constructed with these tau-leap simulations and whose expected value is “close" to the true sensitivity value $S_\theta(f,T)$, i.e. $$\begin{aligned}
\label{gen_sens_form_tau}
S_\theta(f,T) \approx \E( s^{(\tau)}_\theta(f,T) ). \end{aligned}$$ We propose such a random variable $ s^{(\tau)}_\theta(f,T)$ in this paper and provide a simple algorithm for generating the realizations of $s^{(\tau)}_\theta(f,T) $. We theoretically show that under certain reasonable conditions, the associated estimator is *tau-convergent*, which means that the *bias* incurred due to the approximation in converges to $0$, as the maximum step-size $\tau_{
\textnormal{max} }$ or the *coarseness* of the time-discretization mesh goes to $0$. Hence by making this mesh finer and finer, we can make the estimator as accurate as we desire, provided that we are willing to bear the increasing computational costs. In the context of estimating expected values $\E( f( X_\theta(T) ) )$, the property of tau-convergence along with the rate of convergence, has already been established for many tau-leap schemes [@Rathinam1; @Li; @Gang; @Rathinam3]. We use these pre-existing results and obtain a similar tau-convergence result for our sensitivity estimation method. An important feature of our approach is that it is completely flexible, as far as the choice of the tau-leap simulation method is concerned. Furthermore the order of accuracy of our sensitivity estimation method is the same as the order of accuracy of the underlying tau-leap method.
We end this section with observing that incorporating tau-leap schemes in sensitivity estimation opens up a new dimension in attacking this challenging problem. In the trade-off relationships for existing sensitivity estimation methods (see Table \[tab1:tradeoff\]) parameters like $h$ and $M_0$ only allow us to explore one trade-off curve between the variance $\mathcal{V}( \mathcal{X} )$ and some other metric like the bias $\mathcal{B}(
\mathcal{X} )$ (for $ \mathcal{X}$ = CRP, CFD) or the computational cost $\mathcal{C}( \mathcal{X} )$ (for $\mathcal{X}$ = APA, PPA). The main advantage of employing tau-leap schemes is that they provide a mechanism for exploring another trade-off curve between the bias $\mathcal{B}( \mathcal{X}
)$ and the computational cost $\mathcal{C}( \mathcal{X} )$, for the purpose of optimizing the performance of a sensitivity estimation method. In Section \[sec:num\_Examples\], we provide numerical examples to show that with tau-leap simulations we can *indeed* trade-off a small amount of bias with large savings in the computational effort required for estimating parameter sensitivity. Moreover this trade-off relationship appears to be independent of existing trade-off relationships mentioned in Table \[tab1:tradeoff\] because replacing exact simulations in a sensitivity estimation method, with approximate tau-leap simulations, usually does not alter the variance $\mathcal{V}( \mathcal{X} )$ significantly at least when the tau step size is sufficiently small (see Section \[sec:num\_Examples\]). Of course, the computational advantage of tau-leap schemes can only be appropriated if we can incorporate them into existing sensitivity estimation methods. The main contribution of this paper is to develop a method, similar to PPA, that works well with tau-leap schemes (see Section \[sec:mainres\]). For the sake of comparison, we also provide tau-leap versions of the finite-difference schemes (CRP and CFD) in Section \[sec:num\_Examples\].
Sensitivity estimation with tau-leap simulations {#sec:mainres}
================================================
In this section we present our approach for accurately estimating parameter sensitivities of the form with *only* approximate tau-leap simulations of the dynamics. This approach is based on an exact *integral representation* for parameter sensitivity given in Section \[sec:integralrepresentation\]. With this representation at hand, we construct a tau-leap estimator for parameter sensitivity and examine its convergence properties as the time-discretization mesh gets finer and finer (see Sections \[sec:tauest1\] and \[sec:tauest2\]). Thereafter in Section \[sec:tbpa\] we present an algorithm that computes the tau-leap estimator for sensitivity estimation. We start with the description of a generic tau-leap method that approximately simulates the stochastic reaction paths defined by the Markov process $( X(x_0,t))_{t \geq 0 }$ with generator $\mathbb{A}$ (see ) and initial state $x_0$.
A generic tau-leap method {#tau_leap_methods}
-------------------------
For each reaction $k=1,\dots,K$, let $R_k(t)$ be the number of firings of reaction $k$ until time $t$. Due to we can express each $R_k(t)$ as $$\begin{aligned}
R_k(t) = Y_k \left( \int_{0}^{t} \lambda_k( X (x_0, s) ) ds \right) \zeta_k,\end{aligned}$$ where $\{Y_k : k=1,\dots,K\}$ is a family of independent unit rate Poisson processes. From now on we refer to $R(t) = ( R_1(t),\dots,R_K(t))$ as the reaction count vector. For any two time values $s,t \geq 0$ (with $s<t$), the states at these times satisfy $X (x_0, t) = X (x_0, s) + \sum_{k=1}^K ( R_k(t) - R_k(s) ) \zeta_k.$ At any given time $t$ and the computed (approximate) state $x$ at time $t$, a tau-leap method entails taking either a predetermined step of size $\tau > 0$ or choosing step-size $\tau$ as a function of the current state and time, i.e. step-size selection is adapted to the information sigma-algebra generated by the tau-leap process. Next an approximating distribution for the state at time $(t+\tau)$ is generated. This distribution is generally found by approximating the difference $(R(t+\tau)-R(t))$ in the reaction count vector by a random variable $\tilde{R}=(\tilde{R}_1,\dots,\tilde{R}_K)$ whose probability distribution is easy to sample from. The most straightforward choice is given by the simple (explicit) Euler method [@tleap1], which assumes that the propensities are approximately constant in the time interval $[t,t +\tau)$ and conditioned on the information at time $t$, each $\tilde{R}_k$ is an independent Poisson random variable with rate $\lambda_k(x)\tau$. Other distributions for $\tilde{R}=(\tilde{R}_1,\dots,\tilde{R}_K)$ have also been used in the literature to obtain better approximations and particularly to prevent the state-components from becoming negative [@Burrage2004]. The selection method for step size $\tau$ also varies, with the simplest being steps based on a deterministic mesh $0=t_0 < t_1
\dots < t_n =T$ over the observation time interval $[0,T]$. To obtain better accuracy several strategies have been proposed that randomly select $\tau$ based on some criteria such as avoidance of negative state-components or constancy of conditional propensities [@tleap2; @AndersonPost; @Tempone2014].
To represent a generic tau-leap method we shall use a pair of abstract labels $\alpha$ and $\beta$, where $\alpha$ denotes a method, i.e. a choice of distribution for $\tilde{R}$, and $\beta$ denotes a step size selection strategy. We will use $|\beta|$ as a (deterministic) parameter which quantifies the *coarseness* of the time-discretization scheme $\beta$. For instance $\alpha$ may stand for the explicit Euler tau-leap method [@tleap1] and $\beta$ may stand for a deterministic mesh $0 = t_0 < t_1 < \dots < t_n = T$, and in this case the coarseness parameter is $|\beta|=\max (t_j-t_{j-1})$. Typically, tau-leap methods produce approximations of the underlying process at certain leap times that are separated by the step-size $\tau$ and one can interpolate these approximate state values at other time points. The most obvious interpolation is the “sample and hold” method, where the tau-leap process is held constant between the consecutive leap times. In circumstances, such as the explicit Euler tau-leap method with Poisson updates, it is more natural to use interpolation strategies based on the random time-change representation – for example see the “Poisson bridge" approach in [@Tempone2011]. In the following discussion, we suppose that the interpolation strategy is also determined by the label $\alpha$. We shall use $( Z_{\alpha,\beta}(x_0,t) )_{t \geq 0 }$ to denote the tau-leap process, that approximates the exact dynamics $( X(x_0,t))_{t \geq 0 }$, and that results from the application of a tau-leap method $\alpha$ with step size selection strategy $\beta$. This process is defined by the prescription $Z_{\alpha,\beta}(x_0,t_{0}) = x_0$ and $$\label{eq-tau-leap}
Z_{\alpha,\beta}(x_0,t_{i+1}) = Z_{\alpha,\beta}(x_0,t_i) + \sum_{k=1}^K \zeta_k \tilde{R}_{k,i,\alpha,\beta} \quad \textnormal{for} \quad i=1,\dots,\mu,$$ where $\mu$ is the (possibly) random number of time points, $0=t_0 < t_1 < \dots < t_\mu=T$ are the (possibly) random leap times, and $\tilde{R}_{k,i,\alpha,\beta}$ for $i=1,\dots,\mu$ and $k=1,\dots,K$ are random variables whose distribution when conditioned on $Z_{\alpha,\beta}(x,t_i)$ is determined by the method $\alpha$ and step size strategy $\beta$.
\[redtoSSA\] Note that this generic tau-leap method reduces to Gillespie’s SSA [@GP], if at state $Z_{\alpha,\beta}(x_0,t_i) = z$, the next step size $\tau$ is an exponentially distributed random variable with rate $\lambda_0(z) := \sum_{k=1}^K \lambda_k(z)$ and each $ \tilde{R}_{k,i,\alpha,\beta}$ is chosen as $1$ if $k = \eta$ and $0$ otherwise, where $\eta$ is a discrete random variable which assumes the value $i \in \{1,\dots,K\}$ with probability $(\lambda_i(z)/\lambda_0(z))$.
Later we shall establish tau-convergence of our sensitivity estimator by showing that for a fixed tau-leap method $\alpha$, the bias incurred by our estimator converges to $0$ as the coarseness $|\beta|$ of the time-discretization scheme goes to $0$. For this we shall require (weak) convergence of all moments of the tau-leap process to those of the exact process. We now state this requirement more precisely and present a simple lemma that will be needed later. For $p \geq 0$, we say that a function $f : \N_0^d \to \R$ is of class $\sC_p$ if there exists a positive constant $C$ such that $$\begin{aligned}
\label{polynomialgrowth}
| f(x)| \leq C( 1 + \|x\|^p ) \qquad \textnormal{for all } x \in \N^d_0.\end{aligned}$$ We shall require that a tau-leap method $\alpha$ satisfies an order $\gamma>0$ convergent error bound. This is stated formally by Assumption 1 and it can be verified using the results in [@Rathinam3].
[**Assumption 1**]{} Given a tau-leap method $\alpha$, there exist $\gamma>0$, $\delta>0$ and a mapping $\xi:\R_+ \to \R_+$ such that, for every $p \geq 0$ and every final time $T>0$, there exists a constant $C_1(p,T,\alpha)$ satisfying $$\begin{aligned}
&\sup_{t \in [0,T]}|\mathbb{E}(\|Z_{\alpha,\beta}(x_0,t)\|^p) -
\mathbb{E}(\|X(x_0,t)\|^p)|\\
\leq &\sup_{t \in [0,T]} \sum_{y \in \N_0^d} (1 + \|y\|^p)
|\P( Z_{\alpha,\beta}(x_0,t)=y) - \P(X(x_0,t)=y)|\\
\leq &C_1(p,T,\alpha) (1 + \|x_0\|^{\xi(p)}) |\beta|^\gamma,
\end{aligned}$$ for any initial state $x_0$ provided that $|\beta| \leq \delta$. Note that here the second inequality [*is our assumption*]{} while the first inequality always holds. In above, we have assumed that there is a common probability space $(\Omega,\P)$ carrying the exact process $X$ and the tau-leap process $Z_{\alpha,\beta}$.
\[rem:assumption1\] We observe that Assumption 1 essentially assumes order $O(|\beta|^\gamma)$ convergence in the so-called [*$p$-th moment variation norm*]{} (see [@Rathinam3]) of the probability law of $Z_{\alpha,\beta}(x_0,t)$ (on $\Z^d$) to the probability law of $X(x_0,t)$ (on $\Z^d$) and it is not as restrictive as it might seem at first glance. The $p$-th moment variation norm of a signed finite measure $\mu$ on $\Z^d$ which possesses a finite $p$-th moment is defined by $$\|\mu\|_p = \sum_{x \in \Z} \frac{1}{2} (1 + \|x\|^p) |\mu(x)|,$$ and the space $\mathcal{M}_p$ defined by $$\mathcal{M}_p = \{\mu:\Z \to \R \, | \, \|\mu\|_p < \infty\},$$ is isometrically isormorphic to $\ell^1$, the space of absolutely summable sequences and moreover, $\sC_p$ is the dual space of $\mathcal{M}_p$ (see [@Rathinam3]). We note that by the [*Schur property*]{}, weak convergence implies norm convergence in $\ell^1$. In Assumption 1, if we merely assumed weak convergence of order $O(|\beta|^\gamma)$ in $\mathcal{M}_p$, due to the Schur property, we obtain convergence in $p$-th moment variation norm of order $O(|\beta|^{\gamma'})$ for any $\gamma'\in
(0,\gamma)$. Moreover, we note that convergence of tau-leap methods in the moment variation norms have been derived in [@Rathinam3] and apply to a large class of situations including (but not limited to) systems that remain in a bounded subset of the integer state space. We also remark that to our best knowledge, all convergence results on tau-leaping have been limited to considering determinstic time steps. However, in the applied literature, adaptive time step selection methods have been explored numerically, and it is reasonable to expect convergence results to be established in the future for a reasonable class of adaptive step size selection schemes. In this paper, our numerical simulations are restricted to deterministic time steps.
Additionally we will require Assumptions 2 and 3 on moment growth bounds of the exact process as well as the tau-leap process. These assumptions can be verified using the results in [@Rathinam4; @GuptaPLOS; @Rathinam3].\
[**Assumptions 2 and 3**]{} Given a tau-leap method $\alpha$, there exists $\delta>0$ such that for each $T>0$ and $p \geq 0$ there exist constants $C_2(p,T)$ and $C_3(p,T,\alpha)$ satisfying $$\label{eq-Ass23}
\begin{aligned}
\sup_{t \in [0,T]} (1 + \E(\|X(x_0,t)\|^p)) &\leq C_2(p,T) (1 + \|x_0\|^p)\\
\textnormal{and} \qquad \sup_{t \in [0,T]} (1 + \E(\|Z_{\alpha,\beta}(x_0,t)\|^p)) &\leq C_3(p,T,\alpha) (1 + \|x_0\|^p),\\
\end{aligned}$$ for all $t \in [0,T]$, provided $|\beta| \leq \delta$.
We emphasize that constants $C_1$ and $C_3$ in Assumptions 1 and 3, do not depend on the step-size selection strategy $\beta$, and all the three constants in these assumptions may be assumed to be monotonic in $T$ without any loss of generality. The following lemma follows readily from the above assumptions.
\[lem-phi-Ass123\] Consider a function $\phi:\N_0^d \times [0,T] \rightarrow \R$ and suppose that there exists a constant $C>0$ such that $\sup_{t \in [0,T]} |\phi(x,t)| \leq C (1 + \|x\|^p)$ for all $x \in \N_0^d$. Then under Assumptions 1,2 and 3, we have $$\begin{aligned}
\sup_{t \in [0,T]} |\mathbb{E}(\phi(Z_{\alpha,\beta}(x_0,t),t)) -
\mathbb{E}(\phi(X(x_0,t),t))| &\leq C C_1(p,T,\alpha) (1 + \|x_0\|^{\xi(p)}) |\beta|^\gamma,\\
\sup_{t \in [0,T]} |\E(\phi(X(x_0,t),t))| &\leq C C_2(p,T) (1 + \|x_0\|^p)\\
\textnormal{and} \qquad \sup_{t \in [0,T]} |\E(\phi(Z_{\alpha,\beta}(x_0,t),t))| &\leq C C_3(p,T,\alpha) (1 +
\|x_0\|^p),
\end{aligned}$$ provided $|\beta| \leq \delta$.
An integral formula for parameter sensitivity {#sec:integralrepresentation}
---------------------------------------------
Let $(X_\theta(t))_{t \geq 0 }$ be the Markov process representing reaction dynamics with initial state $x_0$ and let $\Psi_{\theta}(x,f,t) $ be defined by $$\begin{aligned}
\label{defdtheta}
\Psi_{\theta}(x,f,t) = \mathbb{E} \left( f(X_\theta(t)) \middle\vert X_{\theta}(0) = x\right),\end{aligned}$$ for any state $x \in \N^d_0$ and time $t \geq 0$. For any $k =1,\dots,K$ and any function $ h : \N^{d}_0 \to \R$, let $\Delta_{\zeta_k}$ denote the difference operator given by $$\begin{aligned}
\Delta_{\zeta_k} h(x) = h(x+\zeta_k) - h(x).\end{aligned}$$ The following theorem expresses the sensitivity value $S_\theta(f,T)$ as the expectation of a random variable which can be computed from the paths of the process $(X_\theta(t))_{t \geq 0 }$ in the time interval $[0,T]$. The proof of this theorem is provided in the Appendix \[sec:appA\].
\[thm:main\] Suppose $(X_\theta(t))_{t \geq 0}$ is the Markov process with generator $\mathbb{A}_\theta$ and initial state $x_0$. Then the sensitivity value $S_{\theta}(f,T) $ is given by $$\begin{aligned}
S_{\theta}(f,T) = \frac{ \partial }{ \partial \theta } \Psi_{\theta}(x_0,f,T) = \sum_{k=1}^K \E\left( \int_{0}^T \frac{ \partial \lambda_k ( X_\theta(t),\theta ) }{ \partial \theta}
\Delta_{\zeta_k} \Psi_{\theta}( X_\theta(t),f,T-t) dt \right).\end{aligned}$$
This formula has the following simple interpretation. Due to an infinitesimal perturbation of parameter $\theta$, the probability that the process $(X_\theta(t))_{t \geq 0}$ has an “extra" jump at time $t$ in the direction $\zeta_k$ is proportional to $$\frac{ \partial \lambda_k ( X_\theta(t),\theta ) }{ \partial \theta}.$$ Moreover the change in the expectation of $f(X_\theta(T))$ at time $T$ due to this “extra" jump at time $t$ is just $$\Delta_{\zeta_k} \Psi_{\theta}( X_\theta(t) +\zeta_k ,f,T-t).$$ The above result shows that the overall sensitivity of the expectation of $f(X_\theta(x,T))$ is just the product of these two terms, integrated over the whole time interval $[0,T]$.
The rest of this section is devoted to the development of a tau-leap estimator for parameter sensitivity using this formula. To simplify our notations, we suppress the dependence on parameter $\theta$, and hence denote $ \lambda_k (\cdot,\theta)$ by $\lambda_k(\cdot)$, $\partial \lambda_k / \partial \theta$ by $\partial \lambda_k$, $S_\theta(f,T)$ by $S(f,T)$, $\Psi_\theta(x,f,t)$ by $\Psi(x,f,t)$ and the process $( X_\theta(t) )_{t \geq 0 }$ by $( X(t) )_{t \geq 0 }$. Due to Theorem \[thm:main\] the sensitivity value $S(f,T)$ can be expressed as $$\begin{aligned}
\label{integral_sensitivity_formula}
S(f,T) = \sum_{k=1}^K \E\left( \int_{0}^T \partial \lambda_k ( X(t) ) \Delta_{\zeta_k} \Psi( X(t),f,T-t) dt \right).\end{aligned}$$
Sensitivity approximation with tau-leap simulations {#sec:tauest1}
---------------------------------------------------
In order to construct a tau-leap estimator for parameter sensitivity using formula , we need to replace both $\partial \lambda_k ( X(t) )$ and $\Delta_{\zeta_k} \Psi( X(t),f,T-t)$ with approximations derived with tau-leap simulations. Recall from Section \[tau\_leap\_methods\] that a generic tau-leap scheme can be described by a pair of abstract labels $\alpha$ and $\beta$, specifying the method and the step-size selection strategy respectively. Assuming such a tau-leap scheme is chosen, let the corresponding tau-leap process $( Z_{ \alpha,\beta }(x,t) )_{ t \geq 0}$ (see ) be an approximation for the exact dynamics starting at state $x$.
Suppose that we use the tau-leap method $\alpha_0$ with the step-size selection strategy $\beta_0$ to approximate $X(t)$ and possibly a different tau-leap method $\alpha_1$ with a time-dependent step-size selection strategy $\beta_1(t)$ to compute an approximation of $\Delta_{\zeta_k} \Psi( X(t),f,T-t)$. This time-dependence in step-size selection is needed because the latter quantity requires simulation of *auxiliary* tau-leap paths in the interval $[0,T-t]$ which varies with $t$. We discuss this in greater detail in the next section. In the following discussion, we will assume that both the tau-leap schemes $(\alpha_0,\beta_0)$ and $(\alpha_1,\beta_1(t))$ satisfy Assumptions 1,2 and 3, with common $\gamma>0, \delta>0$ and with $|\beta|$ replaced by the supremum step-size $$\begin{aligned}
\label{defn_tmax}
\tau_{ \textnormal{max} } = \sup_{t \in [0,T]} \{|\beta_0|,|\beta_1(t)|\} \end{aligned}$$ which is less than $\delta$. We define the tau-leap approximation of $\Psi(x,f,t)$ (see ) by $$\label{eq-tilde-Psi}
\tilde{\Psi}_{\alpha,\beta}(x,f,t) = \mathbb{E}(f(Z_{\alpha,\beta}(x,t))),$$ and make the assumption that the step size selection strategy $\beta_1(t)$ depends on $t$ in such a way that $t \mapsto \tilde{\Psi}_{\alpha_1,\beta_1(t)}(x,f,T-t)$ is a measurable function of $t$. Motivated by formula , we shall approximate the true sensitivity value $S(f,t)$ by $$\label{eq-tildeS}
\tilde{S}(f,T) = \sum_{k=1}^K \E\left( \int_{0}^T \partial \lambda_k( Z_{\alpha_0,\beta_0}(x_0,t))
\Delta_{\zeta_k} \tilde{\Psi}_{\alpha_1,\beta_1(t)}( Z_{\alpha_0,\beta_0}(x_0,t),f,T-t) dt \right),$$ where $x_0$ is the starting state of the process $( X (t))_{t \geq 0}$. The next theorem, proved in the Appendix \[sec:appA\], shows that the bias of this sensitivity approximation is similar to the bias of the underlying tau-leap scheme. In particular if the tau-leap method satisfies order $\gamma$ convergent error bound, then the same is true for the error incurred by the sensitivity approximation. Before we state the theorem, recall that for any $p \geq 0$, a function $f : \N^d_0 \to \R$ is in class $\mathcal{C}_p$ if it satisfies for some constant $C \geq 0$.
\[thm-tau-conv\] Let $f:\N_0^d \to \R$ as well as $\partial \lambda_k$ for each $k=1,\dots,K$ be of class $\sC_p$ for some $p \geq 0$. Suppose that a tau-leap approximation $\tilde{S}(f,T)$ of the exact sensitivity $S(f,T)$ is computed by , where a tau-leap method $\alpha_0$ with step size strategy $\beta_0$ is used to approximate the underlying process $( X(t) )_{t \geq 0}$ and possibly a different tau-leap method $\alpha_1$ with time-dependent step size strategy $\beta_1(t)$ is used to compute approximations $\tilde{\Psi}_{\alpha_1,\beta_1(t)}(x,f,T-t)$ of $\Psi(x,f,T-t)$ at each $t \in [0,T]$. If both the tau-leap methods satisfy Assumptions 1,2 and 3, with common $\gamma>0$ and $\delta>0$, then there exists a constant $\tilde{C}(f,T)$ such that $$|\tilde{S}(f,T)-S(f,T)| \leq \tilde{C}(f,T) \tau_{ \textnormal{max} }^\gamma,$$ where $\tau_{ \textnormal{max} }$ is given by and it is less than $\delta$.
We remark that there are two forms of error analyses in the literature for tau-leap methods. The first type is more conventional where the analysis is carried out for a given system in an interval $[0,T]$ as $\tau_{\max} \to 0$. See [@Rathinam1; @Li; @Rathinam3]. An alternative analysis considers a family of systems parametrized by “system size” $V$, where step size $\tau$ is chosen in relation to $V$ as $\tau = V^{-\beta}$ (where $\beta>0$), and the limit considered as $V \to
\infty$ [@Gang]. As pointed out in [@Rathinam3] both analyses are useful. The first type of analysis with fixed system size is important in that if convergence or more importantly zero-stability (see [@Rathinam3]) does not hold in this conventional sense, then the computed solution can be very erroneous not only when the step size $\tau$ is too large, but also when it is too small! On the other hand, the system size scaling analysis helps explains why tau-leap remains efficient while leaping over several reaction events. In the interest of space, we limit ourselves to the first type in this paper.
A tau-leap estimator for parameter sensitivity {#sec:tauest2}
----------------------------------------------
We now come to the problem of estimating the sensitivity approximation $\tilde{S}(f,T)$ using tau-leap simulations. Expression shows that $\tilde{S}(f,T)$ is the expectation of the random variable $\bar{s}(f,T)$ defined by $$\begin{aligned}
\label{ineffrv}
\bar{s}(f,T) = \sum_{k=1}^K \int_{0}^T \partial \lambda_k(
Z_{\alpha_0,\beta_0}(x_0,t)) \Delta_{\zeta_k} \tilde{\Psi}_{\alpha_1,\beta_1(t)}(Z_{\alpha_0,\beta_0}(x_0,t),f,T-t) dt.\end{aligned}$$ If we can generate samples of this random variable, then the estimation of $\tilde{S}(f,T)$ would be quite straightforward using . However this is not the case as the random variable $\bar{s}(f,T)$ is nearly impossible to generate. This is mainly because it requires computing quantities of the form $$\begin{aligned}
\label{formofunknownquantities}
\Delta_{\zeta_k} \tilde{\Psi}_{\alpha_1,\beta_1(t)}(Z_{\alpha_0,\beta_0}(x_0,t),f,T-t)\end{aligned}$$ at infinitely many time points $t$. These quantities generally do not have an explicit formula and hence they need to be estimated via auxiliary Monte Carlo simulations, which severely restricts the number of such quantities that can be feasibly estimated. We tackle these problems by constructing another random variable $\tilde{s}(f,T)$ whose expected value equals $\tilde{S}(f,T)$, and whose samples can be easily generated using a simple procedure called $\tau$IPA (Tau Integral Path Algorithm) that is described in Section \[sec:tbpa\]. This random variable is constructed by *adding randomness* to the random variable $\bar{s}(f,T)$ in such a way that only a small finite number of unknown quantities of the form require estimation. We now present this construction.\
\
[**Construction of the random variable $\tilde{s}(f,T)$:**]{} Recall from Section \[tau\_leap\_methods\] the description of the tau-leap process $(Z_{\alpha_0,\beta_0}(x_0,t) )_{t \geq 0 }$ which approximates the exact dyamics $( X(t) )_{t \geq 0}$. Let $0=t_0 < t_1 < \dots < t_\mu=T$ be the (possibly random) mesh corresponding to step size selection strategy $\beta_0$. We denote the $\sigma$-algebra generated by the process $( Z_{\alpha_0,\beta_0}(x_0,t) )_{t \geq 0}$ and the random mesh $\beta_0$ over the interval $[0,T]$ by $\sF_T$. Let $\tau_i = t_{i+1}-t_i$ and let $\eta_i$ be the positive integer given by $$\begin{aligned}
\label{defn_etai}
\eta_i = \max\left\{ \left \lceil \frac{ \sum_{k=1}^K | \partial \lambda_k ( Z_{\alpha_0,\beta_0}(x_0, t_i)) | \tau_i }{ C } \right \rceil , 1 \right\}, \end{aligned}$$ where $C$ is a positive constant and $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. The choice of $C$ and its role will be explained later in the section. Define $\sigma_{ij} := t_i + u_{ij} \tau_i $ for each $j=1,\dots,\eta_i$, where each $u_{ij}$ is an independent random variable with distribution $\textnormal{Uniform}[0,1]$. Thus given $t_i$ and $t_{i+1}$, the distribution of each $\sigma_{ij} $ is $\textnormal{Uniform}[t_i, t_{i+1} ]$. Moreover taking expectation over the distribution of $u_{ij}$-s we get $$\begin{aligned}
&\E\left( \frac{\tau_i }{ \eta_i } \sum_{j=1}^{ \eta_i } \partial \lambda_k ( Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )) \,
\Delta_{\zeta_k} \tilde{\Psi}_{\alpha_1,\beta_1(\sigma_{ij})}(Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} ),f, T - \sigma_{ij}) \middle\vert \mathcal{F}_T \right) \\
&= \int_{ t_i }^{ t_{i+1} } \partial \lambda_k (
Z_{\alpha_0,\beta_0}(x_0,t)) \, \Delta_{\zeta_k}\tilde{\Psi}_{\alpha_1,\beta_1(t)}(Z_{\alpha_0,\beta_0}(x_0,t),f,T-t) dt. \notag\end{aligned}$$ In deriving the last equality we have used the substitution $t = t_i + u \tau_i$. This relation along with yields $$\begin{aligned}
\label{defn_sthetanft2}
& \tilde{S}(f,T) = \sum_{k=1}^K \E\left( \sum_{i=0}^{\mu-1} \int_{ t_i }^{ t_{i+1} }
\partial \lambda_k ( Z_{\alpha_0,\beta_0}(x_0,t)) \, \Delta_{\zeta_k}\tilde{\Psi}_{\alpha_1,\beta_1(t)}(Z_{\alpha_0,\beta_0}(x_0,t),f,T-t) dt \right) \\
& = \sum_{k=1}^K \E\left( \sum_{i=0}^{\mu-1} \sum_{j=1}^{ \eta_i } \frac{ \tau_i }{ \eta_i }\partial \lambda_k (
Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij})) \, \Delta_{\zeta_k}\tilde{\Psi}_{\alpha_1,\beta_1(\sigma_{ij} )}(Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} ),f, T - \sigma_{ij} ) \right) \notag\end{aligned}$$ using linearity of the expectation operator. To obtain the states $Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )$ for all the $\sigma_{ij}$-s, we need to interpolate the tau-leap dynamics between the times $t_i$ and $t_{i+1}$.
To proceed further we define a “conditional estimator” $\hat{D}_{kij}$ of the quantity at $t = \sigma_{ij}$ by $$\label{eq-Dki}
\hat{D}_{kij} =
f(Z^{1kij}_{\alpha_1,\beta_1(\sigma_{ij} )}(z+\zeta_k,T-\sigma_{ij})) -
f(Z^{2kij}_{\alpha_1,\beta_1(\sigma_{ij} )}(z,T-\sigma_{ij}))$$ where $z = Z_{\alpha_0,\beta_0}(x_0,\sigma_{ij})$, and $Z^{1kij}$ and $Z^{2kij}$ are instances of tau-leap approximations of the exact dynamics starting at initial states $(z+\zeta_k)$ and $z$ respectively. Both these tau-leap processes use the same method $\alpha_1$ and the same step-size selection strategy $\beta_1(\sigma_{ij})$. Moreover conditioned on $Z_{\alpha_0,\beta_0}(x_0,\sigma_{ij})$ and $\sigma_{ij}$, the processes $Z^{1kij},Z^{2kij}$ and the step-size selection strategy $\beta_1(\sigma_{ij})$ are independent of the process $Z_{\alpha_0,\beta_0}$ and the step-size selection strategy $\beta_0$. Therefore it is immediate that $$\label{diff_estimation}
\E(\hat{D}_{kij} \vert Z_{\alpha_0,\beta_0}(x_0,\sigma_{ij}), \sigma_{ij}) =
\Delta_{\zeta_k}\tilde{\Psi}_{\alpha_1,\beta_1(\sigma_{ij} )}(Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} ),f, T - \sigma_{ij} ),$$ and hence from we obtain the following representation for $\tilde{S}(f,T)$ $$\begin{aligned}
\label{defn_sthetanft3}
\tilde{S}(f,T) = \sum_{k=1}^K \E\left( \sum_{i=0}^{\mu-1} \sum_{j=1}^{ \eta_i } \frac{ \tau_i }{ \eta_i} \partial \lambda_k (Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )) \hat{D}_{kij} \right).\end{aligned}$$ An estimator for $\tilde{S}(f,T) $ based on this formula can require several computations of $\hat{D}_{kij}$. Since each evaluation of $\hat{D}_{kij}$ is computationally expensive, we would like to control the total number of these evaluations by randomizing the decision of whether $\hat{D}_{kij}$ should be evaluated at time $\sigma_{ij}$ or not. Moreover this randomization must be performed without introducing a bias in the estimator. We now describe this process.
Define $R_{kij}$ and $P_{kij}$ by $$\begin{aligned}
\label{defn_rki_rhoki}
R_{kij} = \partial \lambda_k ( Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )) \tau_i \qquad \textnormal{and} \qquad
P_{kij} =\left( \frac{ \left| R_{kij} \right| } { C \eta_i } \right) \wedge 1, \end{aligned}$$ and let $\rho_{kij}$ be an independent $\{0,1\}$-valued random variable whose distribution is Bernoulli with parameter $ P_{kij}$. Since $ \E\left( \rho_{kij} \middle\vert Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} ), \sF_T \right) = P_{kij}$ we have that $$\begin{aligned}
\label{defn_sthetanft4}
\tilde{S}(f,T) = \sum_{k=1}^K \E\left( \sum_{i=0}^{\mu-1} \sum_{j=1}^{ \eta_i } \left( \frac{ R_{kij } }{ P_{kij } \eta_i} \right) \rho_{kij} \hat{D}_{kij} \right),\end{aligned}$$ where we define $R_{kij }/ P_{kij }$ to be $0$ when $R_{kij } = 0$. This formula suggests that $\tilde{S}(f,T) $ can be estimated, without any bias, using realizations of the random variable $$\begin{aligned}
\label{expr:sthetahat}
\tilde{s}(f,T)& = \sum_{k=1}^K \sum_{i = 0}^{ \mu - 1 } \sum_{j=1}^{ \eta_i } \left( \frac{ R_{kij } }{ P_{kij } \eta_i} \right) \rho_{kij} \hat{D}_{kij} .\end{aligned}$$
In generating each realization of $\tilde{s}(f,T)$, the computation of $\hat{D}_{kij}$ is only needed if the Bernoulli random variable $\rho_{kij}$ is $1$. Therefore, if we can effectively control the number of such $\rho_{kij}$-s then we can efficiently generate realizations of $\tilde{s}(f,T)$. This can be achieved using the positive parameter $C$ (see and ) as we soon explain. Based on the construction outlined above, we provide a method in Section \[sec:tbpa\] for obtaining realizations of the random variable $\tilde{s}(f,T)$. We call this method, the *Tau Integral Path Algorithm* ($\tau$IPA), to emphasize the fact that $\tilde{s}(f,T)$ is essentially an approximation of the integral . Using $\tau$IPA we can efficiently generate realizations $s_1,s_2,\dots,s_N$ of $\tilde{s}(f,T)$ and approximately estimate the parameter sensitivity $\tilde{S}(f,T)$ with the estimator .\
\
[**Minimizing the variance of $\tilde{s}(f,T)$:**]{} To improve the efficiency of $\tau$IPA, we must minimize the additional variance due to the extra randomness that has been added to the random variable $\bar{s}(f,T)$ to obtain $\tilde{s}(f,T)$. Since $\E(\tilde{s}(f,T) \vert \mathcal{F}_T) = \bar{s}(f,T)$, this additional variance is equal to $\textnormal{Var}(\tilde{s}(f,T) \vert \mathcal{F}_T)$, and in order to reduce this quantity we focus on reducing the conditional variance $\text{Var}(\hat{D}_{kij} \vert \mathcal{F}_T)$. Recall that $\hat{D}_{kij}$ is given by and for convenience we abbreviate $Z^{lkij}_{\alpha_1,\beta_1(\sigma_{ij})}$ by $Z^l$ for $l=1,2$. The reduction in this conditional variance can be accomplished by tightly coupling the pair of processes $(Z^{1},Z^{2})$. For this purpose we use the split-coupling (see [@DA]) specified by [ $$\begin{aligned}
\label{splitcoupling2_1}
Z^1(t) &=( Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )+ \zeta_k) + \sum_{k = 1}^K Y_k\left( \int_{0}^{t} \lambda_k( Z^1( \alpha(s) ) ,\theta) \wedge \lambda_k( Z^2(\alpha(s) ) ,\theta) ds \right)\zeta_k \\
&+ \sum_{k = 1}^K Y^{(1)}_k\left( \int_{0}^{t} \left(\lambda_k ( Z^1(\alpha((s) ) ,\theta) - \lambda_k ( Z^1( \alpha(s) ) ,\theta) \wedge \lambda_k( Z^2( \alpha(s) ) ,\theta) \right) ds \right) \zeta_k \notag \\ \label{splitcoupling2_2}
Z^2(t) &=Z_{\alpha_0,\beta_0}(x_0, \sigma_{ij} )+ \sum_{k = 1}^K Y_k\left( \int_{0}^{t} \lambda_k( Z^1( \alpha(s) ) ,\theta) \wedge \lambda_k(Z^2( \alpha(s) ) ,\theta) ds \right)\zeta_k \\
&+ \sum_{k = 1}^K Y^{(2)}_k\left( \int_{0}^{t} \left( \lambda_k( Z^2( \alpha((s) ) ,\theta) - \lambda_k( Z^1( \alpha(s) ) ,\theta) \wedge \lambda_k( Z^2( \alpha(s) ) ,\theta) \right) ds \right) \zeta_k, \notag\end{aligned}$$ ]{} where $\{Y_k, Y^{(1)}_k,Y^{(2)}_k : k =1,\dots,K\}$ is an independent family of unit rate Poisson processes. Here $\alpha(s) = t_i$ for $t_i \leq s <
t_{i+1}$, and $\{ t_0,t_1,t_2,\dots\}$ is the sequence of leap-times of the pair of processes $(Z^{1},Z^{2})$ jointly simulated with the tau-leap scheme $( \alpha_1,\beta_1(t) )$. Note that process $\alpha$ is adapted to the filtration generated by processes $(Z^{1},Z^{2})$. Hence a solution to - can be found by explicit construction. The uniqueness of the solution $(Z^{1},Z^{2})$, until the first time $\tau_M$ its norm exceeds some constant $M > 0$, is guaranteed by the local boundedness of the associated generator (see Theorem 4.1 in Chapter 4 of [@EK]). Using Assumption 3 one can show that as $M \to \infty$ we have $\tau_M \to \infty$ a.s. and from this, the uniqueness of the solution $(Z^{1},Z^{2})$ in the whole time-interval $[0,\infty)$ can be established. See Lemma A.1 in [@gupta2014sensitivity] for more details on this argument.\
\
[**Controlling the number of nonzero $ \rho_{kij}$-s:**]{} We now discuss how the positive parameter $C$ can be selected to control the total number of $\rho_{kij}$-s that assume the value $1$ in , which is $\rho_{ \textnormal{tot} } = \sum_{k=1}^K \sum_{i=1}^{\mu -1 } \sum_{j=1}^{ \eta_i } \rho_{kij}$. This is the number of $\hat{D}_{kij}$-s that are required to obtain a realization of $\tilde{s}(f,T)$. It is immediate that given the sigma field $\mathcal{F}_{T}$, $\rho_{ \textnormal{tot} }$ is a $\N_0$-valued random variable whose expectation is given by: $$\begin{aligned}
\E(\rho_{ \textnormal{tot} } \vert \mathcal{F}_{T} ) = \sum_{k=1}^K \sum_{i=1}^{\mu -1 }\sum_{j=1}^{ \eta_i } \E( P_{kij} \vert \mathcal{F}_{T} ) = \sum_{k=1}^K \sum_{i=1}^{\mu -1 } \sum_{j=1}^{ \eta_i } \E\left[ \left( \frac{ | R_{kij} | } { C \eta_i} \right) \wedge 1 \middle\vert \mathcal{F}_{T} \right].\end{aligned}$$ Using $a \wedge b \leq a$ and $$\begin{aligned}
\E\left( | R_{kij} | \vert \mathcal{F}_{T} \right) = \int_{t_i}^{t_{i+1}} \left| \partial \lambda_k (Z_{\alpha_0,\beta_0}(x_0,t)) \right| dt\end{aligned}$$ we obtain $$\begin{aligned}
\label{almost_exact_ineq}
\E\left( \rho_{ \textnormal{tot} } \right) = \E\left(\E(\rho_{ \textnormal{tot} } \vert \mathcal{F}_{T} ) \right) \leq \frac{1}{C} \sum_{k=1}^K \E\left( \int_{0}^T \left| \partial \lambda_k (Z_{\alpha_0,\beta_0}(x_0,t)) \right| dt \right).\end{aligned}$$ We choose a positive integer $M_0$ and set $$\begin{aligned}
\label{normconstant}
C = \frac{1}{M_0} \sum_{k=1}^K \E\left( \int_{0}^T \left| \partial \lambda_k (Z_{\alpha_0,\beta_0}(x_0,t)) \right| dt \right),\end{aligned}$$ where the expectation can be approximately estimated using $N_0$ tau-leap simulations of the dynamics in the time interval $[0,T]$. Such a choice ensures that $\rho_{ \textnormal{tot} } $ is bounded above by $M_0$ on average. In most cases we can expect that $R_{kij}$ to be close to $ \partial \lambda_k ( Z_{\alpha_0,\beta_0}(x_0, t_i )) \tau_i $ and so the choice of $\eta_i$ automatically ensures that $ | R_{kij} | \leq C \eta_i$. Hence inequality is almost exact and with $C$ chosen as we have $\E\left( \rho_{ \textnormal{tot} } \right) \approx M_0$. Therefore $M_0$ can be interpreted as the expected number of coupled auxiliary paths - needed to obtain a realization of $\tilde{s}(f,T)$. This parameter is in the hands of the user and it plays the same role as in PPA (see Section \[subsec:unbiasedmethods\]), namely, it allows one to select the trade-off between the computational cost $\mathcal{C}( \tau \textnormal{IPA} )$ and the variance $\mathcal{V}(\tau \textnormal{IPA}) $. A higher value of $M_0$ reduces the variance while simultaneously increasing the computational cost. Hence it is difficult to ascertain the effect of $M_0$ on the overall estimation cost which depends on the product $\mathcal{C}( \tau \textnormal{IPA} ) \mathcal{V}(\tau \textnormal{IPA})$ (see ). Numerical examples suggest that for low values of $M_0$, the overall estimation cost decreases gradually with increase in $M_0$, but this trend reverses for higher values of $M_0$ (see Section \[sec:num\_Examples\]). More work is needed to examine if this pattern persists more generally and how one can select the optimal value of $M_0$. Note however that $\tau$IPA will provide an unbiased estimator for $\tilde{S}(f,T)$ regardless of the choice of $M_0$. Hence the accuracy of $\tau$IPA does not vary much with $M_0$, which is also seen in the numerical examples.
The Tau Integral Path Algorithm ($\tau$IPA) {#sec:tbpa}
-------------------------------------------
We now provide a detailed description of the method $\tau$IPA which produces realizations of the random variable $\tilde{s}(f,T)$ defined by . Computing the empirical mean of these realizations estimates the approximate parameter sensitivity $\tilde{S}(f,T)$. Throughout this section we assume that the function $rand()$ returns independent samples from the distribution $\textnormal{Uniform}[0,1]$.
The method $\tau$IPA can be adapted to work with any tau-leap scheme, but for concreteness, we assume that an *explicit* tau-leap scheme is used for all the simulations. This means that the current state $z$ and time $t$, are sufficient to determine the distributions of the next time-step $\tau$ and the vector of reaction firings $\tilde{R} = ( \tilde{R}_1,\dots, \tilde{R}_K)$ in the time interval $[t, t+\tau)$. We suppose that a sample from these two distributions can be obtained using the methods $\Call{GetTau}{z,t,T}$[^5] and $\Call{GetReactionFirings}{z,\tau}$ respectively. If we use the simplest tau-leap scheme given in [@tleap1], then reaction firings can be generated as $$\begin{aligned}
\label{poissreactionfirings}
\tilde{R}_k = \Call{Poisson}{ \lambda_k(z) \tau},\end{aligned}$$ for $k=1,\dots,K$, where the function $\Call{Poisson}{r}$ generates an independent Poisson random variable with mean $r$. Once we have the reaction firings $\tilde{R} = ( \tilde{R}_1,\dots, \tilde{R}_K)$, the state at time $(t+ \tau)$ is given by $z' = (z + \sum_{k=1}^K \tilde{R}_k \zeta_k)$ and for any intermediate time-point $\sigma \in (t, t+\tau)$ the state $\hat{z}$ can be obtained using the “Poisson bridge" interpolation (see [@Tempone2011]). However this interpolation approach is equivalent to setting $\hat{z} = (z + \sum_{k=1}^K \tilde{R}^{(1)}_k \zeta_k ) $ and $z'= ( \hat{z} + \sum_{k=1}^K \tilde{R}^{(2)}_k \zeta_k)$, where $ \tilde{R}^{(1)}= ( \tilde{R}^{(1)}_1,\dots, \tilde{R}^{(1)}_K)$ and $ \tilde{R}^{(2)} = ( \tilde{R}^{(2)}_1,\dots, \tilde{R}^{(2)}_K)$ are reaction firing vectors generated according to with $\tau$ replaced by $(\sigma -t)$ and $(t+ \tau - \sigma)$ respectively. This idea can be easily generalized to obtain the interpolated states $\hat{z}_1,\dots,\hat{z}_\eta$ at $\eta$ intermediate times $\sigma_1,\dots,\sigma_\eta \in (t, t+\tau)$ sorted in ascending order, i.e. $\sigma_1 <\dots < \sigma_\eta$.
Let $Z$ denote the tau-leap process approximating the reaction dynamics with initial state $x_0$. Our first task is to select the normalization parameter $C$ according to , by estimating the expectation in the formula using $N_0$ simulations of the process $Z$. This is done using the function\
$\Call{Select-Normalizing-Constant}{x_0,M_0, T}$ (see Algorithm \[estimatenormalization\] in Appendix \[sec:appB\]) where $M_0$ is the expected number of auxiliary paths - that need to be simulated (see Section \[sec:tauest2\]). Once $C$ is chosen, a single realization of $\tilde{s}(f,T)$ can be computed using $\Call{GenerateSample}{x_0,T,C}$ (Algorithm \[gensensvalue\]). This method simulates the tau-leap process $Z$ and at each leap-time $t_i$, the following happens:
1. The next leap size $\tau_i$ ($=\tau$) is chosen and the positive integer $\eta_i$ ($=\eta$) is computed.
2. The intermediate time-points $\sigma_j$-s are generated for $j=1,\dots,\eta$ and sorted in ascending order.
3. For each $j$, the vector of reaction firings $\tilde{R} = ( \tilde{R}_1,\dots, \tilde{R}_K)$ for the time-interval $(\sigma_{j-1},\sigma_j)$ is computed and the interpolated state $\hat{z}_j$ at time $\sigma_j$ is evaluated. Then for each reaction $k$ the following happens:
- The variables $R_{kij}$ ($=R$), $P_{kij}$ ($=P$) and $\rho_{kij}$ ($=\rho$) are generated. The function $\Call{Bernoulli}{P}$ generates an independent Bernoulli random variable with expectation $P$.
- If $\rho_{kij}= 1$ then $\hat{D}_{kij}$ (see ) is evaluated using\
$\Call{EvaluateCoupledDifference}{\hat{z}_j,\hat{z}_j+\zeta_k,\sigma,T}$ (see Algorithm \[gendiffsample\] in\
Appendix \[sec:appB\]) and the sample value is updated according to . This method independently simulates the pair of processes $(Z^{1},Z^{2})$ specified by the split-coupling - in order to compute $\hat{D}_{kij}$. For simplicity we assume that these simulations are carried out by the same tau-leap scheme which generates reaction firings according to .
4. Finally, time $t$ is updated to $(t +\tau)$, reaction firings for the time-interval $[ \sigma_\eta, t)$ are computed and the state is updated accordingly.
Note that in the computation of reaction firings the propensities are evaluated at $z$ rather than any of the interpolated states $\hat{z}_j$.
Set $z = x_0$, $t = 0$ and $s = 0$ Calculate $\tau= \Call{GetTau}{z,t,T}$ and set $$\begin{aligned}
\eta = \max\left\{ \left \lceil \frac{ \sum_{k=1}^K | \partial \lambda_k (z) | \tau }{ C } \right \rceil , 1 \right\}. \end{aligned}$$ For each $j=1,\dots,\eta$ let $ \sigma_j \gets ( t + rand() \times \tau)$. Relabel $\sigma_j$-s to arrange them in ascending order as $\sigma_1 < \sigma_2 <\dots \sigma_\eta$. Also set $\sigma_0 = t$ and $\hat{z}_0 = z$. Set $( \tilde{R}_1,\dots, \tilde{R}_K) = \Call{GetReactionFirings}{z, \sigma_j - \sigma_{j-1} }$ and compute the interpolated state $\hat{z}_j = \hat{z}_{j-1} + \sum_{k=1}^K \tilde{R}_k \zeta_k$. Set $R =\partial \lambda_k(\hat{z}_j) \tau$ and $ \rho =\Call{Bernoulli}{P}$ with $$\begin{aligned}
P = \left( \frac{ \left| R \right| } { C \eta } \right) \wedge 1.\end{aligned}$$ Update $s \gets s +\left( \frac{ R}{P \eta } \right) \Call{EvaluateCoupledDifference}{\hat{z}_j,\hat{z}_j+ \zeta_k,\sigma_j, T} $ Update $t \gets t +\tau$ Set $( \tilde{R}_1,\dots, \tilde{R}_K) = \Call{GetReactionFirings}{z, t- \sigma_\eta }$ Update $z \gets \hat{z}_\eta + \sum_{k=1}^K \tilde{R}_k \zeta_k$ $s$
Numerical Examples {#sec:num_Examples}
==================
In this section we computationally compare six sensitivity estimation methods on many examples. The methods we consider are the following:
1. [**Tau Integral Path Algorithm**]{} or [**$\tau$IPA**]{}: This is the method described in Section \[sec:tbpa\]. The tau-leap scheme we use is the simple Euler method [@tleap1] with Poisson reaction firings and uniform step-size $\tau =\tau_{ \textnormal{max} }$. To avoid the possibility of *leaping-over* the final time $T$ at which the sensitivity is to be estimated, we set $$\begin{aligned}
\Call{GetTau}{z,t,T} = \min\{\tau_{ \textnormal{max} }, T- t\}.\end{aligned}$$ The value of $\tau_{ \textnormal{max} }$ will depend on the example being considered and the default value of parameter $M_0$ is $10$.
2. [**Exact Integral Path Algorithm**]{} or [**eIPA**]{}: This is the method we obtain by replacing the tau-leap simulations in $\tau$IPA with the exact simulations performed with Gillespie’s SSA [@GP]. This replacement can be easily made by choosing the step-size and the reaction firings according to Remark \[redtoSSA\]. Moreover we need to change the method $\Call{EvaluateCoupledDifference}{}$ to the version given in [@Gupta2]. Note that eIPA is a new unbiased method for estimating parameter sensitivity, like the methods in Section \[subsec:unbiasedmethods\]. This method is conceptually similar to PPA [@Gupta2], but unlike PPA, the formula underlying $\tau$IPA does not involve summation over the jumps of the process, which makes it more amenable for incorporating tau-leap schemes.
3. [**Exact Coupled Finite Difference**]{} or [**eCFD**]{}: This is same as the CFD method in [@DA].
4. [**Exact Common Reaction Paths**]{} or [**eCRP**]{}: This is same as the CRP method in [@KSR1].
5. [**Tau Coupled Finite Difference**]{} or [**$\tau$CFD**]{}: This method is the tau-leap version of CFD which has been proposed in [@morshed2017efficient]. Let $(Z_\theta, Z_{\theta +h})$ be the pair of tau-leap processes that approximate the processes $(X_\theta, X_{\theta+h})$, and suppose that at leap time $t_i$ their state is $( Z_\theta(t_i) ,Z_{\theta+h}(t_i) ) = (z_1,z_2)$. If the next step-size is $\tau$, then for every reaction $k=1,\dots,K$, we set the number of firings $( \tilde{R}_{\theta,k} , \tilde{R}_{\theta+h,k})$ for this pair of processes as $\tilde{R}_{\theta,k} = A_k + \Call{Poisson}{ (\lambda_k(z_1) - \lambda_k(z_1) \wedge \lambda_k(z_2) ) \tau}$ and $ \tilde{R}_{\theta+h,k} = A_k + \Call{Poisson}{ (\lambda_k(z_2) - \lambda_k(z_1) \wedge \lambda_k(z_2) ) \tau} $, where $A_k = \Call{Poisson}{ ( \lambda_k(z_1) \wedge \lambda_k(z_2) ) \tau }$. Such a selection of reaction firings emulates the CFD coupling. To facilitate comparison, we choose the tau-leap simulation method to be the same as for $\tau$IPA.
6. [**Tau Common Reaction Paths**]{} or [**$\tau$CRP**]{}: This method can be viewed as the tau-leap version of CRP where the CRP coupling is emulated by coupling the Poisson random variables that generate the reaction firings. Using the same notation as before, if $( Z_\theta(t_i) ,Z_{\theta+h}(t_i) ) = (z_1,z_2)$ and the next step-size is $\tau$, then we set the number of firings $( \tilde{R}_{\theta,k} , \tilde{R}_{\theta+h,k})$ as $\tilde{R}_{\theta,k} = \Call{Poisson}{ \lambda_k(z_1) \tau , k}$ and $ \tilde{R}_{\theta+h,k} = \Call{Poisson}{ \lambda_k(z_2) \tau, k }$ for every reaction $k=1,\dots,K$. Here we assume that there are $K$ parallel streams of independent $\textnormal{Uniform}[0,1]$ random variables (see [@KSR1]), and the method $\Call{Poisson}{ r, k}$ uses the uniform random variable from the $k$-th stream for generating the Poisson random variable with mean $r$. As for $\tau$CFD, the tau-leap simulation method is the same as for $\tau$IPA.
In all the finite-difference schemes, we use perturbation-size $h=0.1$ and we *center* the parameter perturbations to obtain better accuracy. This centering can be easily achieved by substituting $\theta$ with $( \theta - h/2)$ and $(\theta + h)$ with $( \theta + h/2 )$ in the expression and also in the definition of the coupled processes. Since we use Poisson random variables to generate the reaction firings for tau-leap simulations, it is possible that some state-components become negative during the simulation run. In this paper we deal with this problem rather crudely by setting the negative state-components to $0$. We have checked that this does not cause a significant loss of accuracy because the state-components become negative *very rarely*.
Note that among the methods considered here, eIPA is the only unbiased sensitivity estimation method. All the other methods are biased either due to a finite-difference approximation of the derivative (eCFD and eCRP) or due to tau-leap approximation of the sample paths ($\tau$IPA) or due to both these reasons ($\tau$CFD and $\tau$CRP). In the examples, we apply each sensitivity estimation method $\mathcal{X}$ with a sample-size of $N = 10^5$, and compute the estimator mean $\hat{\mu}_N$ , the standard deviation $\hat{\sigma}_N$ , the relative standard deviation $\textnormal{RSD}(\mathcal{X})$ and the computational cost per sample $\mathcal{C}(\mathcal{X})$ (see Section \[sec:prelim\]). Assume that the exact sensitivity value is $s_0$ which is known. We compare the different estimation methods using the following two quantities - the percentage *relative error* (**RE**) defined by $$\begin{aligned}
\label{eqn_reerror}
\textnormal{RE} = \left| \frac{ \hat{\mu}_N -s_0 }{s_0} \right| \times 100,\end{aligned}$$ and the RSD *adjusted computational cost* (**RSDCC**) defined by $$\begin{aligned}
\label{vac_defn}
\textnormal{RSDCC}= ( \textnormal{RSD}(\mathcal{X}) )^2 \mathcal{C}(\mathcal{X}).\end{aligned}$$ The first quantity $\textnormal{RE}$ measures the accuracy of a method, while the second quantity $\textnormal{RSDCC}$ determines the overall computational time that will be required by the method to yield an estimate with the desired statistical precision (see ).
Our numerical results will show that the exact schemes (eIPA, eCFD and eCRP) usually have a higher RSDCC than their tau-leap counterparts ($\tau$IPA, $\tau$CFD and $\tau$CRP), but expectedly their RE is lower. Generally the RE for eIPA is smaller than both eCFD and eCRP because of its unbiasedness and this advantage in accuracy often persists when we compare $\tau$IPA with $\tau$CFD and $\tau$CRP. It can be seen that in most of the cases, the sample variance $\mathcal{V}(\mathcal{X})$ or the estimator standard deviation , remain of similar magnitude, when we switch from an exact scheme to its tau-leap version (see Appendix \[sec:appB\]). This supports our claim in Section \[using\_tau\_leap\_methods\], that substituting exact paths with tau-leap trajectories allows one to trade-off bias with computational costs, and this trade-off relationship is somewhat “orthogonal" to other trade-off relationships shown in Table \[tab1:tradeoff\].
In all the examples below, the propensity functions $\lambda_k$-s for all the reactions have the mass-action form [@DASurvey] unless stated otherwise. Also $\partial$ always denotes the partial-derivative w.r.t. the designated sensitive parameter $\theta$.
Single-species birth-death model {#ex:bd}
--------------------------------
Our first example is a simple birth-death model in which a single species $\mathcal{S}$ is created and destroyed according to the following two reactions: $$\begin{aligned}
\emptyset \stackrel{\theta_1 }{\longrightarrow} \mathcal{S} \stackrel{\theta_2 }{\longrightarrow} \emptyset.\end{aligned}$$ Let $\theta_1=10$, $\theta_2 = 0.1$ and assume that the sensitive parameter is $\theta = \theta_2$. Let $(X(t))_{t \geq 0}$ be the Markov process representing the reaction dynamics. Assume that $X(0)=0$. For $f(x)=x$ we wish to estimate $$\begin{aligned}
S_\theta(f,T) = \partial \E\left( f( X(T) ) \right) = \partial \E\left( X(T) \right)\end{aligned}$$ for $T= 5$ and $T = 10$. For this example, we set $\tau_{ \textnormal{max} }=0.5$. For each $T$ we estimate the sensitivity using all the six methods and the results are displayed in Table \[bdexample\_table\] in Appendix \[sec:appB\]. For this network we can compute the sensitivity $S_\theta(f,T)$ exactly as the propensity functions are affine. These exact values are stated in the *caption* of Table \[bdexample\_table\], and they allow us to compute the RE of a method according to . We also compute the RSDCC[^6] for each method using , and we compare these RE and RSDCC values for all the methods in Figure \[fig:birthdeath\][**A**]{}. From these comparisons we can make the following observations: 1) The exact methods are typically more accurate than the tau-leap methods but they are usually more computationally demanding. 2) For $T = 5$, eCFD/eCRP are far more accurate than $\tau$CFD/$\tau$CRP suggesting that the two sources of bias (finite-difference and tau-leap approximations) are additive in nature. However the same is not true for $T = 10$. 3) For both the cases $T =5$ and $T =10$, $\tau$IPA outperforms $\tau$CFD/$\tau$CRP in terms of accuracy even though it is slightly more computationally expensive. Same is true when we compare eIPA with eCFD/eCRP.
In Figure \[fig:birthdeath\][**B**]{} we numerically analyze the performance of $\tau$IPA w.r.t. its two key parameters - the expected number of auxiliary paths $M_0$ and the maximum tau-leap step-size $\tau_{ \textnormal{max} }$. We see that RE is fairly insensitive to variations in $M_0$ while RSDCC first decreases with $M_0$ up to a certain point, and then it starts increasing with $M_0$. As we are using a first-order explicit tau-leap scheme, it is unsurprising that RE increases *almost linearly* with $\tau_{ \textnormal{max} }$. However, importantly, RSDCC decreases *exponentially* with $\tau_{ \textnormal{max} }$, which makes it possible to use tau-leap simulations to trade-off a small amount of accuracy for a large gain in computational efficiency with $\tau$IPA.
Observe that if we scale the production rate $\theta_1$ by the system-size or volume parameter $V$, then the *concentration process*, derived by dividing the copy-number counts $X(t)$ by $V$, converges to a deterministic ODE limit as $V \to \infty$ (see Chapter 11 in [@EK]). Often it is of interest to determine how the performance of various sensitivity estimation methods scales with the volume parameter $V$. We investigate this issue for the exact schemes (eIPA, eCFD and eCRP) in Figure \[fig:birthdeath\_volume\], by numerically examining the dependence of their RSD, RSDCC and RE on $V$. Here we set the expected number of auxiliary paths $M_0$ for eIPA to be equal to $V$. Note that RSD for finite-difference schemes (eCFD/eCRP) scales like $1/\sqrt{V}$ as was proved in [@Rathinam2] and consequently their RSDCC is of order $1$, because the computational time per sample, which is proportional to the number of reaction events per unit time-interval, is of order $V$. Similar to these finite-difference schemes the RSD for eIPA also scales like $1/\sqrt{V}$, but its RSDCC is of order $V$ as its computational time per sample is of order $V^2$ because to generate each sample for eIPA, $M_0 = V$ auxiliary paths need to be simulated in addition to the main sample path. This computational disadvantage of eIPA is compensated by the fact that accuracy of eIPA improves with volume (i.e. RE decreases with volume), while for the finite-difference schemes it is almost a constant. These numerical results suggest that the computational efficiency of eIPA scales with volume $V$ in the same way as it does for the CGT method (see Section \[subsec:unbiasedmethods\]) whose RSD has been shown to be of order $1$ w.r.t. volume $V$ (see [@Rathinam2]). Despite this similarity in volume scaling, eIPA is still a preferable unbiased method when compared to the CGT method, as its estimator variance does not become unbounded as the magnitude of the sensitive parameter approaches zero (see Section \[subsec:unbiasedmethods\]). The volume-scaling analysis presented here can also be performed for the tau-leap schemes by parameterizing the step-size $\tau_{ \textnormal{max} }$ by volume $V$ as discussed in Section \[sec:tauest1\]. We expect the results to be qualitatively similar to the exact schemes, because, as mentioned previously, it is observed that the sample variance remains similar when we switch from an exact scheme to its tau-leap version (see Appendix \[sec:appB\]). However this needs to be investigated in detail in a future work.
Repressilator Network {#ex:ge}
---------------------
Our second example considers the *Repressilator* network given in [@elowitz2000synthetic], which consists of three mutually repressing gene-expression modules (say 1,2 and 3). Repression occurs at the level of transcription, i.e. production of the three mRNAs $M_1$, $M_2$ and $M_3$, and it is carried out by the corresponding protein molecules $P_1$, $P_2$ and $P_3$ in a cyclic pattern. In other words, protein $P_i$ represses the transcription of mRNA $M_{i-1}$, where we identify $M_0$ with $M_3$. The repression mechanism is modeled with a nonlinear Hill function. The repressilator network consists of $6$ biomolecular species and $12$ reactions described in Table \[tab:repress1\].
No. Reaction Propensity
----- ------------------------------------- -------------------------------------------------
1 $ \emptyset \longrightarrow M_1$ $\lambda_{1}(x) = 1 + 100/(1 + x^{\alpha_1}_5)$
2 $ \emptyset \longrightarrow M_2$ $\lambda_{2}(x) = 1 + 100/(1 + x^{\alpha_2}_6)$
3 $ \emptyset \longrightarrow M_3$ $\lambda_{3}(x) = 1 + 100/(1 + x^{\alpha_3}_4)$
4 $M_1 \longrightarrow \emptyset $ $\lambda_{4}(x) = x_1$
5 $M_2 \longrightarrow \emptyset $ $\lambda_{5}(x) = x_2$
6 $M_3 \longrightarrow \emptyset $ $\lambda_{6}(x) = x_3$
7 $M_1 \longrightarrow M_1 + P_1 $ $\lambda_{7}(x) = 50 x_1$
8 $M_2 \longrightarrow M_2 + P_2 $ $\lambda_{8}(x) = 50 x_2$
9 $M_3 \longrightarrow M_3 + P_3 $ $\lambda_{9}(x) = 50 x_3$
10 $P_1 \longrightarrow \emptyset $ $\lambda_{10}(x) = \gamma_1 x_4$
11 $P_2 \longrightarrow \emptyset $ $\lambda_{11}(x) = \gamma_2 x_5$
12 $P_3 \longrightarrow \emptyset $ $\lambda_{12}(x) = \gamma_3 x_6$
: Reactions for the *Repressilator* network [@elowitz2000synthetic]. Here $x = (x_1,\dots,x_6 )$ denotes the copy-numbers of the 6 network species ordered as $M_1$, $M_2 $, $M_3$, $P_4$, $P_5$ and $P_6$. []{data-label="tab:repress1"}
We set the Hill coefficient $\alpha_i$ for the transcription of each mRNA to be $1$ (see reactions 1-3 in Table \[tab:repress1\]) and the degradation rate constant $\gamma_i$ for each protein to be $0.1$ (see reactions 10-12 in Table \[tab:repress1\]). Let $(X(t))_{t \geq 0}$ be the $\N^6_0$-valued Markov process representing the reaction dynamics, under the species ordering described in the caption of Table \[tab:repress1\]. We assume that $X(0)= (0,0,0,0,0,0)$ and define $f : \N^6_0 \to \R$ by $f(x_1,\dots,x_6) = x_4$. At $T = 10$, our goal is to estimate $$\begin{aligned}
\label{sens_example2}
S_\theta(f,T) = \partial \E \left( f(X(T)) \right) = \partial \E ( X_{4}(T) ),\end{aligned}$$ for $\theta = \alpha_1,\alpha_2, \alpha_3, \gamma_1, \gamma_2, \gamma_3$. These values measure the sensitivity of the mean of protein $P_1$ population at time $T=10$ with respect to the Hill coefficients $\alpha_i$-s and the protein degradation rates $\gamma_j$-s. For this example, we set $\tau_{ \textnormal{max} }=0.01$.
For each $\theta$ we estimate the sensitivity using all the six methods and the results are displayed in Table \[repressexample\_table\] in Appendix \[sec:appB\]. Unlike the previous example, we cannot compute the sensitivity values exactly because of nonlinearity of some of the propensity functions. So we obtain accurate approximations of these values using the unbiased estimator (eIPA) with a large sample size ($N = 10^6$) and they are provided in the *caption* of Table \[repressexample\_table\]. With these values we can compute the REs , which are then compared along with RSDCCs for all the methods in Figure \[fig:repress\]. The results vary with the choice of the sensitive parameter $\theta$, but one can clearly see that $\tau$IPA can be several times more accurate than $\tau$CFD /$\tau$CRP even though its RSDCC is of a similar magnitude. This is especially observable for cases $\theta = \alpha_1, \alpha_3$ and $\gamma_2$. Most notably for the case $\theta = \alpha_1$, the RE for finite-difference schemes is around $800\%$, while it is $1.3\%$ for eIPA and $5\%$ for $\tau$IPA.
Genetic toggle switch {#ex:gts}
---------------------
As our last example we look at a simple network with nonlinear propensity functions. Consider the network of a genetic toggle switch proposed by Gardner et. al. [@Gardner]. This network has two species $\mathcal{U}$ and $\mathcal{V}$ that interact through the following four reactions $$\begin{aligned}
\emptyset \stackrel{\lambda_1 }{\longrightarrow} \mathcal{U} , \ \ \mathcal{U} \stackrel{ \lambda_2 }{\longrightarrow} \emptyset, \ \ \emptyset \stackrel{\lambda_3 }{\longrightarrow} \mathcal{V} \ \textrm{ and } \mathcal{V} \stackrel{ \lambda_4}{\longrightarrow} \emptyset,\end{aligned}$$ where the propensity functions $\lambda_i$-s are given by $$\begin{aligned}
\lambda_1(x_1,x_2) = \frac{\alpha_1}{ 1 +x_2^{\beta} }, \ \ \lambda_2(x_1,x_2) = x_1 , \ \ \lambda_3(x_1,x_2) = \frac{\alpha_2}{ 1 +x_1^{\gamma} } \ \textrm{ and } \ \ \lambda_4(x_1,x_2) = x_2. \end{aligned}$$ In the above expressions, $x_1$ and $x_2$ denote the number of molecules of $\mathcal{U}$ and $\mathcal{V}$ respectively. We set $\alpha_1 =50$, $\alpha_2 = 16$, $\beta = 2.5$ and $\gamma = 1$. Let $(X(t))_{t \geq 0}$ be the $\N^2_0$-valued Markov process representing the reaction dynamics with initial state $(X_{1}(0) , X_{2}(0)) = (0,0)$. For $T = 10$ and $f(x) = x_1$, our goal is to estimate $$\begin{aligned}
S_\theta(f,T) = \partial \E \left( f(X(T)) \right) = \partial \E ( X_1(T) ),\end{aligned}$$ for $\theta = \alpha_1,\alpha_2,\beta$ and $\gamma$. In other words, we would like to measure the sensitivity of the mean of the number of $\mathcal{U}$ molecules at time $T =10$, with respect to all the model parameters. For this example, we set $\tau_{ \textnormal{max} }=0.1$. We estimate these sensitivities with all the six methods and the results are presented in Table \[gtsexexample\_table\] in Appendix \[sec:appB\], and in Figure \[fig:gts\][**A**]{}.
As in the previous example, we estimate the true sensitivity values using the unbiased estimator (eIPA) with a large sample size ($N = 10^6$). These approximate values are given in the *caption* of Table \[gtsexexample\_table\] and they were used in computing the relative errors for Figure \[fig:gts\]. Here we find that eIPA outperforms eCFD/eCRP both in terms of accuracy and computational efficiency for all the parameters. Similarly $\tau$IPA is computationally more efficient than $\tau$CFD/$\tau$CRP for all the parameters, but except for the case $\theta = \alpha_1$, its accuracy is similar to $\tau$CFD/$\tau$CRP. In Figure \[fig:gts\][**B**]{} we numerically examine how the performance of $\tau$IPA is affected by the parameter $M_0$, for a couple of cases. As in Section \[ex:bd\], we find this effect to be quite small for RE but RSDCC first decreases with $M_0$ and then increases.
Conclusions and future work {#sec:conc}
===========================
Estimation of parameter sensitivities for stochastic reaction networks in an important and difficult problem. The main source of difficulty is that all the estimation methods rely on exact simulations of the reaction dynamics performed using Gillespie’s SSA [@GP] or its variants [@NR; @AndMod]. It is well-known that these simulation algorithms are computationally very demanding as they track each and every reaction event which can be very cumbersome. This issue represents the main bottleneck in the use of sensitivity analysis for systems modeled as stochastic reaction networks. The aim of this paper is to develop a method, called *Tau Integral Path Algorithm* ($\tau$IPA), that feasibly deals with this issue by requiring only approximate tau-leap simulations of the reaction dynamics, and still providing provably accurate estimates for the sensitivity values. This method is based on an explicit integral representation for parameter sensitivity that was derived from the formula given in [@Our]. Furthermore, by replacing the tau-leap simulation scheme in $\tau$IPA with an exact simulation scheme like SSA, we obtain a new unbiased method (called eIPA) for sensitivity estimation, that can serve as the natural limit of $\tau$IPA when the step-size $\tau$ gets smaller and smaller.
Using computational examples we compare $\tau$IPA with tau-leap versions of the finite-difference schemes [@DA; @KSR1; @morshed2017efficient] that are commonly employed for sensitivity estimation. We find that in many cases, $\tau$IPA outperforms these tau-leap finite-difference schemes in terms of both accuracy and computational efficiency. This makes $\tau$IPA an appealing method for sensitivity analysis of stochastic reaction networks, where the exact dynamical simulations are computationally infeasible and tau-leap approximations become necessary.
As we argue in Section \[using\_tau\_leap\_methods\], tau-leap simulations provide a natural way to *trade-off* estimator bias with gains in computational speed. Therefore it would be of fundamental importance to extend the ideas in this paper and try to *maximize* the computational gains from tau-leap simulations while sacrificing the *minimum* amount of accuracy. In this context, we now mention two possible directions for future research. The method we proposed here, $\tau$IPA, can work with any underlying tau-leap simulation scheme, but for simplicity we examined it with the most basic tau-leap scheme i.e. an explicit Euler method with a constant (deterministic) step-size and Poissonian reaction firings [@tleap1]. As this tau-leap scheme has several drawbacks (see [@GillespieRev]), it is very likely that $\tau$IPA can yield much better results if a more sophisticated tau-leap scheme is employed, possibly with random step-sizes [@tleap2; @AndersonPost; @Tempone2014], or with Binomial leaps [@Burrage2004] or using implicit step-size selection [@Rathinam2003]. We shall explore these issues in a future paper. Note that $\tau$IPA essentially converts the problem of estimating parameter sensitivities to the problem of estimating a collection of expected values of the process with tau-leap simulations. The latter problem can be efficiently handled using *multilevel* strategies, where estimators are constructed for a range of $\tau$-values, and are suitably coupled to simultaneously reduce the estimator’s bias and variance [@Anderson2012; @Lester2015; @Tempone2014]. A promising approach would be to integrate these multilevel estimators with $\tau$IPA to improve its accuracy and computational efficiency.
Appendix\[sec:APPENDIX\] {#appendixsecappendix .unnumbered}
========================
Proofs of the main results {#sec:appA}
--------------------------
Let $\{\mathcal{F}_t\}$ be the filtration generated by the process\
$(X_\theta(t))_{t \geq 0}$ and let $\sigma_i$ be its $i$-th jump time for $i=1,2,\dots$. We define $\sigma_0 = 0$ for convenience. Since the process $(X_\theta(t))_{t \geq 0}$ is constant between consecutive jump times we can write $$\begin{aligned}
\label{mainthmproof0}
& \E\left( \int_{0}^T \frac{ \partial \lambda_k ( X_\theta (t) , \theta ) }{ \partial \theta } \Delta_{\zeta_k} f(X_\theta(t)) dt \right) \notag \\
& = \sum_{i=0}^\infty \E\left(\frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta } \Delta_{\zeta_k} f(X_\theta( \sigma_i) ) (\sigma_{i+1} \wedge T - \sigma_{i} \wedge T) \right) \notag \\
& = \sum_{i=0}^\infty \E\left( \E\left(\frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta }\Delta_{\zeta_k} f(X_\theta(\sigma_i) ) (\sigma_{i+1} \wedge T - \sigma_{i} \wedge T) \middle\vert \mathcal{F}_{\sigma_i} \right) \right) \notag \\
& = \E\left( \sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta } \left( f(X_\theta(\sigma_i) +\zeta_k) - f(X_\theta(\sigma_i) ) \right) \E\left( \delta_i \middle\vert \mathcal{F}_{\sigma_i}, \sigma_i <T \right) \right),\end{aligned}$$ where $\delta_i = \sigma_{i+1} \wedge T -\sigma_i \wedge T$ and the last equality holds due to linearity of the expectation operator and the fact that $\delta_i = 0$ if $\sigma_i \geq T$. Given $X_{\theta}(\sigma_i) = y $ and $ \sigma_i = u < T$, the distribution of the random variable $\delta_i$ has the *cumulative density function* given by $$\begin{aligned}
\P( \delta_i < s \vert X_{\theta}(\sigma_i) = y ,\sigma_i = u) = \left\{
\begin{array}{cl}
0 & \textnormal{ if } s < 0 \\
1 - e^{ -\lambda_0(y,\theta)s } & \textnormal{ if } 0 \leq s < (T - u) \\
1 & \textnormal{ if } s \geq (T - u).
\end{array}
\right.\end{aligned}$$ This shows that for any continuous function $g: [0,\infty) \to [0,\infty)$ we have $$\begin{aligned}
\label{ibpforg}
& \E\left( \int_{0}^{\delta_i} g(s)ds \middle \vert X_{\theta}(\sigma_i) = y ,\sigma_i = u \right) = e^{-\lambda_0(y,\theta)(T-u) } \int_{0}^{T -u} g(s)ds \\ + & \int_{0}^{ T - u} \lambda_0(y,\theta) e^{ -\lambda_0(y,\theta)s } \left( \int_{0}^{s} g(t) dt \right) ds = \int_{0}^{T-u} e^{ -\lambda_0(y,\theta)s } g(s)ds, \notag\end{aligned}$$ where the last relation holds because by applying integration by parts we get $$\begin{aligned}
& \int_{0}^{ T - u} \lambda_0(y,\theta) e^{ -\lambda_0(y,\theta)s } \left( \int_{0}^{s} g(t) dt \right) ds \\
& = - e^{-\lambda_0(y,\theta)(T-u) } \int_{0}^{T -u} g(s)ds+ \int_{0}^{T-u} e^{ -\lambda_0(y,\theta)s } g(s)ds .\end{aligned}$$ Taking $g \equiv 1$ gives us $\E\left( \delta_i \middle \vert X_{\theta}( \sigma_i) = y ,\sigma_i = u \right) = \int_{0}^{T-u} e^{ -\lambda_0(y,\theta)s } ds$ and therefore $$\begin{aligned}
\E\left( \delta_i \middle\vert \mathcal{F}_{\sigma_i}, \sigma_i <T \right) = \int_{0}^{T-\sigma_i}
e^{ -\lambda_0( X_\theta(\sigma_i),\theta)s } ds= \int_{0}^{T-\sigma_i}
e^{ -\lambda_0( X_\theta(\sigma_i),\theta)(T - \sigma_i - s) } ds.\end{aligned}$$ Substituting this in we obtain $$\begin{aligned}
\label{expectationintegraloperator}
& \E\left( \int_{0}^T \frac{ \partial \lambda_k ( X_\theta (t) , \theta ) }{ \partial \theta } \Delta_{\zeta_k} f(X_\theta(t) ) dt \right) \notag \\
& = \E\left( \sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta }
\Delta_{\zeta_k} f(X_\theta(\sigma_i) ) \int_{0}^{T-\sigma_i}
e^{ -\lambda_0( X_\theta(\sigma_i),\theta) (T - \sigma_i -s) } ds \right).\end{aligned}$$
Theorem 2.3 in [@Our] shows that the sensitivity value $S_\theta (f,T)$ can be expressed as $$\begin{aligned}
\E\left[ \sum_{k = 1}^K \left( \int_{0}^T \frac{ \partial \lambda_k ( X_\theta (t) , \theta ) }{ \partial \theta } \Delta_{\zeta_k} f(X_\theta(t) ) dt + \sum_{ i = 0 : \sigma_i < T }^{\infty} R_{\theta}( X_\theta( \sigma_i) ,f, T -\sigma_i ,k) \right) \right]\end{aligned}$$ where $$\begin{aligned}
R_{\theta}(x,f,t,k) = \frac{ \partial \lambda_k ( x ,\theta ) }{ \partial \theta} \int_{0}^{t} \left( \Delta_{\zeta_k} \Psi_{\theta}(x,f,s) - \Delta_{\zeta_k} f(x) \right) e^{ - \lambda_0(x,\theta) (t-s) } ds.\end{aligned}$$ Using this fact along with we obtain $$\begin{aligned}
&S_{\theta}(f,T) \\
& = \sum_{k = 1}^K \E\left( \sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta } \Delta_{\zeta_k} f(X_\theta( \sigma_i ) ) \int_{0}^{T-\sigma_i}
e^{ -\lambda_0( X_\theta(\sigma_i),\theta)(T-\sigma_i -s) } ds \right) \\ & + \E\left(
\sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta(\sigma_{i}) ,\theta ) }{ \partial \theta} R_{\theta}( X_\theta( \sigma_i) ,f, T -\sigma_i ,k) \right) \\
& = \sum_{k = 1}^K \E\left( \sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta } \Bigg( R_{\theta}( X_\theta(\sigma_i) ,f, T -\sigma_i ,k) \right.\\&\left. + \Delta_{\zeta_k} f(X_\theta( \sigma_i ) ) \int_{0}^{T-\sigma_i}
e^{ -\lambda_0( X_\theta(\sigma_i),\theta)(T-\sigma_i -s) } ds \right) \Bigg) \\
& = \sum_{k = 1}^K \E\left( \sum_{ i = 0 : \sigma_i < T }^{\infty} \frac{ \partial \lambda_k ( X_\theta (\sigma_i) , \theta ) }{ \partial \theta } G_\theta(X_\theta(\sigma_i),f,T - \sigma_i,k) \right), \end{aligned}$$ where $$\begin{aligned}
G_\theta(y,f,t,k) = \int_{0}^{t } \Delta_{\zeta_k}\Psi_{\theta}(y,f,s) e^{ -\lambda_0( y,\theta)(t -s) } ds = \int_{0}^{t } \Delta_{\zeta_k}\Psi_{\theta}(y,f,t-s) e^{ -\lambda_0( y,\theta)s } ds . \end{aligned}$$ However relation with $g(s) = \Delta_{\zeta_k}\Psi_{\theta}(X_\theta(\sigma_i) ,f,T-\sigma_i-s)$ implies that given $X_{\theta}( \sigma_i) $ and $\sigma_i <T$, we have $$\begin{aligned}
G_\theta(X_{\theta}( \sigma_i),f,T - \sigma_i,k) &= \E\left( \int_{0}^{\delta_i} \Delta_{\zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-\sigma_i-s) ds \middle \vert X_{\theta}( \sigma_i) ,\sigma_i \right) \\
& = \E\left( \int_{ \sigma_i }^{\sigma_{i} +\delta_i } \Delta_{\zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-s) ds \middle \vert X_{\theta}(\sigma_i) ,\sigma_i \right) \\
& = \E\left( \int_{\sigma_i \wedge T}^{\sigma_{i+1} \wedge T } \Delta_{\zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-s) ds \middle \vert X_{\theta}( \sigma_i) ,\sigma_i \right).\end{aligned}$$ Substituting this in the last expression for $S_{\theta}(f,T)$ and using the fact that $X_\theta(s) = X_\theta(\sigma_i)$ for all $s \in [ \sigma_i ,\sigma_{i+1})$ we get $$\begin{aligned}
S_{\theta}(f,T)& = \sum_{k = 1}^K \E\left( \sum_{ i = 0 }^{\infty} \E\left( \int_{\sigma_i \wedge T}^{\sigma_{i+1} \wedge T } \frac{ \partial \lambda_k ( X_\theta (s) , \theta ) }{ \partial \theta } \Delta_{ \zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-s) ds \middle\vert \mathcal{F}_{\sigma_i} \right) \right) \\
& = \sum_{k = 1}^K \sum_{ i = 0 }^{\infty} \E\left( \int_{\sigma_i \wedge T}^{\sigma_{i+1} \wedge T } \frac{ \partial \lambda_k ( X_\theta (s) , \theta ) }{ \partial \theta } \Delta_{ \zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-s) ds \right) \\
& = \sum_{k = 1}^K \E\left( \int_{0}^{T } \frac{ \partial \lambda_k ( X_\theta (s) , \theta ) }{ \partial \theta } \Delta_{ \zeta_k} \Psi_{\theta}(X_\theta(\sigma_i) ,f,T-s) ds \right).\end{aligned}$$ This completes the proof of this result.
For each $k=1,\dots,K$ define $g_k,h_k$ by\
$g_k(x,t) = \partial \lambda_k(x) \Delta_{\zeta_k} \Psi(xk,f,T-t)$ and $h_k(x,t) = \partial \lambda_k(x) \Delta_{\zeta_k} \tilde{\Psi}_{\alpha_1,\beta_1(t)}(x_k,f,T-t)$. Without loss of generality, we can assume that there exists a $C>0$ such that $$\max\{\partial \lambda_k(x), f(x) \, | k =1,\dots,K \} \leq C (1 + \|x\|^p), \forall x
\in \N^d_0.$$ Then due to Lemma \[lem-phi-Ass123\] we obtain $$\label{eq-hk-gk-bnd}
\begin{aligned}
&\sup_{t \in [0,T]} |h_k(x,t) - g_k(x,t)| \\
&\leq \partial \lambda_k(x) C
C_1(p,T,\alpha_1) \left((1 + \|x\|^{\xi(p)}) + (1+\|x+\zeta_k\|^{\xi(p)}) \right)
\tau_{ \textnormal{max} }^\gamma \\
&\leq C^2 C_1(p,T,\alpha_1) (1 + \|x\|^{p}) \left((1 + \|x\|^{\xi(p)}) +
(1+\|x+\zeta_k\|^{\xi(p)}) \right) \tau_{ \textnormal{max} }^\gamma\\
&\leq c_0(p) C^2 C_1(p,T,\alpha_1) \left(1 + \|x\|^{(p+\xi(p))}\right) \tau_{ \textnormal{max} }^\gamma ,
\end{aligned}$$ where $c_0(p)$ is a constant that depends only on $p$ as well as $\zeta_1,\dots,\zeta_K$. Lemma \[lem-phi-Ass123\] also shows that $$\label{eq-hk-bnd}
\begin{aligned}
\sup_{t \in [0,T]} |h_k(x,t)| &\leq \partial \lambda_k(x) C C_3(p,T,\alpha_1) \left( (1
+ \|x\|^p) + (1+\|x+\zeta_k\|^p) \right)\\
&\leq c_1(p) C^2 C_3(p,T,\alpha_1) (1 + \|x\|^{2p})
\end{aligned}$$ and $\sup_{t \in [0,T]} |g_k(x,t)| \leq c_1(p) C^2 C_2(p,T) (1 + \|x\|^{2p})$, where $c_1(p)$ is again a constant that depends only on $p$ and $\zeta_1,\dots,\zeta_K$.
From and Lemma \[lem-phi-Ass123\] it follows that $$\begin{aligned}
\label{eq-hkZ-hkX}
&\sup_{t \in [0,T]} |\E(h_k(Z_{\alpha_0,\beta_0}(x_0,t),t)) - \E(h_k(X(t),t))| \\
& \leq c_1(p) C^2 C_3(p,T,\alpha_1) C_1(2p,T,\alpha_0) \left(1 + \|x_0\|^{\xi(2p)}\right) \tau_{ \textnormal{max} }^\gamma. \notag\end{aligned}$$ Moreover from , we get $$\E(|h_k(X(t),t)-g_k(X(t),t)|) \leq c_0(p) C^2 C_1(p,T,\alpha_1) (1 +
\E(\|X(t)\|^{(p+\xi(p))}) ) \tau_{ \textnormal{max} }^\gamma,$$ and hence using Assumption 2, we obtain $$\begin{aligned}
\label{eq-hkX-gkX}
& \sup_{t \in [0,T]} \E(|h_k(X(t),t)-g_k(X(t),t)|) \\
& \leq c_0(p) C^2 C_1(p,T,\alpha_1) C_2(p+\xi(p),T) \left(1 + \|x_0\|^{p + \xi(p)}\right) \tau_{ \textnormal{max} }^\gamma. \notag\end{aligned}$$
Note that $$\begin{aligned}
\left| \tilde{S}(f,T)-S(f,T) \right| & = \left| \sum_{k=1}^K \int_{0}^T \left( \E(h_k(Z_{\alpha_0,\beta_0}(x_0,t),t)) - \E(g_k(X(t),t)) \right)dt \right| \\
& \leq \sum_{k=1}^K \Big{|}\int_0^T \E(h_k(Z_{\alpha_0,\beta_0}(x_0,t),t)) dt - \int_0^T \E(g_k(X(t),t)) dt \Big{|} \\
& \leq \sum_{k=1}^K \int_0^T |\E(h_k(Z_{\alpha_0,\beta_0}(x_0,t),t)) - \E(h_k(X(t),t))|dt \\
&+ \sum_{k=1}^K \int_0^T |\E(h_k(X(t),t)) - \E(g_k(X(t),t))|dt.\end{aligned}$$ Using and we obtain the bound $$\begin{aligned}
\left| \tilde{S}(f,T)-S(f,T) \right| &\leq K T c_1(p) C^2 C_3(p,T,\alpha_1) C_1(2p,T,\alpha_0) \left(1 + \|x_0\|^{\xi(2p)}\right) \tau_{ \textnormal{max} }^\gamma\\
&+ K T c_0(p) C^2 C_1(p,T,\alpha_1) C_2(p+\xi(p),T) \left(1 + \|x_0\|^{p + \xi(p)}\right) \tau_{ \textnormal{max} }^\gamma,\end{aligned}$$ which proves the theorem.
Supplementary Tables and Algorithms {#sec:appB}
-----------------------------------
----- --------- --------- -------- ---------- ---------- --------- -------- ----------
$T$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
5 -90.079 0.093 0.139 0.379E-5 -90.938 0.078 0.813 0.121E-5
10 -264.5 0.309 0.099 0.97E-5 -266.34 0.243 0.793 0.247E-5
$T$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
5 -90.632 0.088 0.4746 0.078E-5 -86.456 0.089 4.155 0.033E-5
10 -268.77 0.142 1.716 0.054E-5 -268.214 0.146 1.503 0.021E-5
$T$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
5 -90.749 0.097 0.604 0.343E-5 -86.481 0.098 4.128 0.343E-5
10 -268.82 0.169 1.734 0.152E-5 -267.92 0.173 1.393 0.131E-5
----- --------- --------- -------- ---------- ---------- --------- -------- ----------
: [**Birth-death model:**]{} Sensitivity estimation results for $T = 5,10$. For all the methods, $N = 10^5$ are used to estimate the following quantities - the estimator mean , the standard deviation , the relative error (RE) percentage and the relative standard deviation adjusted computation cost (RSDCC) in seconds. The exact sensitivity values are $-90.204$ for $T = 5$ and $-264.241$ for $T = 10$.[]{data-label="bdexample_table"}
------------ -------- --------- -------- -------- --------- --------- --------- --------
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ 1.202 0.0107 0.625 0.0046 1.185 0.0131 0.822 0.0023
$\alpha_2$ -2.133 0.0132 0.663 0.0021 -2.3968 0.0148 13.087 0.0008
$\beta$ -5.924 0.0419 1.144 0.0020 -8.5372 0.0562 42.456 0.0008
$\gamma$ 54.372 0.1679 0.367 0.0009 60.156 0.191 10.232 0.0003
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ 1.053 0.11 11.883 0.1925 1.183 0.0491 1.021 0.0088
$\alpha_2$ -2.007 0.267 5.305 0.3219 -2.734 0.0991 29.011 0.0066
$\beta$ -5.865 0.4535 2.1339 0.1053 -8.787 0.1813 46.617 0.0021
$\gamma$ 54.67 1.1589 0.1794 0.0080 59.431 0.3907 8.9044 0.0002
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ 1.158 0.0793 3.13 0.0919 1.129 0.0781 5.4895 0.0562
$\alpha_2$ -1.999 0.1306 5.701 0.0823 -2.415 0.1109 13.9646 0.0254
$\beta$ -6.21 0.1777 3.625 0.0161 -8.853 0.2198 47.7203 0.0074
$\gamma$ 54.546 0.4756 0.0469 0.0015 59.807 0.4267 9.5925 0.0006
------------ -------- --------- -------- -------- --------- --------- --------- --------
: [**Genetic toggle switch:**]{} Sensitivity estimation results w.r.t. all the model parameters $\alpha_1, \alpha_2 , \beta$ and $\gamma$. For all the methods, $N = 10^5$ are used to estimate the following quantities - the estimator mean , the standard deviation , the relative error (RE) percentage and the relative standard deviation adjusted computation cost (RSDCC) in seconds. The true sensitivity values are approximately $1.195 \pm 0.009$ for $\theta = \alpha_1$, $-2.1194 \pm 0.01$ for $\theta = \alpha_2$, $ -5.9929 \pm 0.035$ for $\theta = \beta$ and $54.5721 \pm 0.133 $ for $\theta = \gamma$. These values are estimated with eIPA using $10^6$ samples and they are expressed in the form $s_0 \pm l$, which signifies that the $99\%$ confidence interval is $(s_0 - l,s_0+l)$. []{data-label="gtsexexample_table"}
------------ --------- --------- -------- -------- --------- --------- -------- --------
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ -67.73 1.17 1.31 1.6801 -65.2 0.8 5 0.2886
$\alpha_2$ -2982.2 10.6 0.078 0.0193 -2821.8 7.66 5.3 0.0053
$\alpha_3$ 145.36 1 0.22 0.2880 131.04 0.73 9.66 0.0623
$\gamma_1$ 259.45 8.86 0.92 2.0139 250.4 8.2 2.6 0.7723
$\gamma_2$ -119.38 1.01 0.13 0.4097 -90.78 0.74 24.1 0.1251
$\gamma_3$ -30.38 7.82 8.98 104.45 -23.45 2.97 15.75 11.484
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ -633.79 6.21 823.5 0.0334 -621.15 2.1 805.1 0.0017
$\alpha_2$ -2987.1 10.01 0.24 0.0039 -2891.5 7.16 2.97 0.0009
$\alpha_3$ 356.95 22.3 146.1 1.3379 206.2 5.3 42.19 0.0972
$\gamma_1$ 265.69 4.59 3.34 0.1019 250.5 1.43 2.5 0.0048
$\gamma_2$ -51.61 10.7 56.8 14.764 -22.8 1.16 80.9 0.3845
$\gamma_3$ -31.74 4.98 13.85 8.407 -24.61 1.42 11.72 0.4871
$\theta$ Mean Std Dev RE$\%$ RSDCC Mean Std Dev RE$\%$ RSDCC
$\alpha_1$ -648.1 2.38 844.3 0.0039 -620.9 2.1 804.7 0.0028
$\alpha_2$ -3076.6 10.5 3.2 0.0033 -2897.2 7.5 2.8 0.0016
$\alpha_3$ 349.55 4.18 141 0.041 216.6 5.09 49.4 0.1315
$\gamma_1$ 260.01 1.23 1.14 0.0064 251.7 1.41 2.1 0.0075
$\gamma_2$ -41.29 0.6 65.5 0.0602 -21.91 1.16 81.7 0.6639
$\gamma_3$ -33.98 0.52 21.88 0.0666 -23.78 0.91 14.7 0.3494
------------ --------- --------- -------- -------- --------- --------- -------- --------
: [**Repressilator model:**]{} Sensitivity estimation results w.r.t. model parameters $\alpha_1, \alpha_2 , \alpha_3,\gamma_1,\gamma_2$ and $\gamma_3$. For all the methods, $N = 10^5$ are used to estimate the following quantities - the estimator mean , the standard deviation , the relative error (RE) percentage and the relative standard deviation adjusted computation cost (RSDCC) in seconds. The exact sensitivity values are approximately $-68.6271 \pm 1$ for $\theta = \alpha_1$, $-2979.88 \pm 8$ for $\theta = \alpha_2$, $145.041 \pm 0.7$ for $\theta = \alpha_3$, $257.091 \pm 7.4$ for $\theta = \gamma_1$, $-119.526 \pm 0.9$ for $\theta = \gamma_2$ and $-27.8796 \pm 4.5$ for $\theta = \gamma_3$. These values are estimated with eIPA using $10^6$ samples and they are expressed in the form $s_0 \pm l$, which signifies that the $99\%$ confidence interval is $(s_0 - l,s_0+l)$[]{data-label="repressexample_table"}
Set $S = 0$ Set $z = x_0$ and $t = 0$ Calculate $\tau= \Call{GetTau}{z,t,T}$ Update $S \gets S + \tau \left|\partial \lambda_k(z) \right|$ Update $t \gets t +\tau$ Set $( \tilde{R}_1,\dots, \tilde{R}_K) = \Call{GetReactionFirings}{z,\tau }$. Set $z \gets z + \sum_{k=1}^K \zeta_k \tilde{R}_k$. $ S/(N_0 M_0) $
Set $\tau_1 = \Call{GetTau}{z_1,t,T}$, $\tau_2 = \Call{GetTau}{z_2,t,T}$ and $\tau = \tau_1 \wedge \tau_2$ Set $A_{k1} = \lambda_k(z_1) \wedge \lambda_k(z_2)$, $A_{k2} = \lambda_k(z_1) - A_{k1} $ and $A_{k3} = \lambda_k(z_2) - A_{k1} $ Set $\tilde{R}_{k i} = \Call{Poisson } {A_{ki} \tau } $ for $i=1,2,3$ Update $z_1 \gets z_1 + \tilde{R}_{k1} \zeta_{k} + \tilde{R}_{k2} \zeta_{k} $ Update $z_2 \gets z_2 + \tilde{R}_{k1} \zeta_{k} + \tilde{R}_{k3} \zeta_{k} $ Update $t \gets t+\tau$ $f(z_2)-f(z_1)$
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: In fact the cost of generating a realization of $s_\theta(f,T)$ is usually smaller for CFD in comparison to CRP (i.e. $\mathcal{C}( \textnormal{CFD}) < \mathcal{C}( \textnormal{CRP})$), because the CFD coupling is such that if $X_\theta (t) = X_{\theta +h}(t)$ for some $t < T$, then this equality will hold for the remaining time-interval $[t,T]$, allowing us to directly set $s_{\theta,h}(f,T) = 0$ without completing the simulation in the interval $[t,T]$.
[^5]: We allow the step-size selection to depend on both the current time $t$ and the final time $T$. This is especially important for simulating the auxiliary paths that are required to compute the $\hat{D}_{kij}$-s in (see Sections \[sec:tauest1\] and \[sec:tauest2\]).
[^6]: All the computations in this paper were performed using C++ programs on an Apple machine with the 2.9 GHz Intel Core i5 processor.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Silicon Photo-Multipliers (SiPM) are regarded as novel photo-detectors to replace conventional Photo-Multiplier Tubes (PMTs). However, the breakdown voltage dependence on the ambient temperature results in a gain variation of $\sim$3$\% /^{\circ} \mathrm C$. This severely limits the application of this device in experiments with wide range of operating temperature, especially in space missions. An experimental setup was established to investigate the temperature and bias voltage dependence of gain for the Multi-Pixel Photon Counter (MPPC). The gain and breakdown voltage dependence on operating temperature of an MPPC can be approximated by a linear function, which is similar to the behavior of a zener diode. The measured temperature coefficient of the breakdown voltage is $(59.4 \pm 0.4$ mV)$/^{\circ} \mathrm C$. According to this fact, an analog power supply based on two zener diodes and an operational amplifier was designed with a positive temperature coefficient. The measured temperature dependence for the designed power supply is between 63.65 to 64.61 mV/$^{\circ} \mathrm C$ at different output voltages. The designed power supply can bias the MPPC at an over-voltage with a temperature variation of $\sim$ 5 mV$/^{\circ} \mathrm C$. The gain variation of the MPPC biased at over-voltage of 2 V was reduced from 2.8$\% /^{\circ} \mathrm C$ to 0.3$\% /^{\circ} \mathrm C$ when biased the MPPC with the designed power supply for gain control. Detailed design and performance of the analog power supply in the temperature range from -42.7$^{\circ}\mathrm{C}$ to 20.9$^{\circ}\mathrm{C}$ will be discussed in this paper.'
address:
- 'Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China'
- 'University of Chinese Academy of Sciences, Beijing, China'
- 'Yunnan University, Kunming, China'
author:
- Zhengwei Li
- Congzhan Liu
- Yupeng Xu
- Bo Yan
- Yanguo Li
- Xuefeng Lu
- Xufang Li
- Shuo Zhang
- Zhi Chang
- Jicheng Li
- He Gao
- Yifei Zhang
- Jianling Zhao
bibliography:
- 'references.bib'
title: 'A novel analog power supply for gain control of the Multi-Pixel Photon Counter (MPPC)'
---
cite\#1[\[[\#1]{}\]]{}
SiPM ,MPPC ,gain control ,temperature dependence
Introduction
============
The Silicon Photomultiplier (SiPM) has become a promising device for applications in medical imaging, high energy particle physics and particle astrophysics with high Photon Detection Efficiency (PDE), high gain (up to $10^{6}$), low cost, low operating voltage ($<100\,\mathrm{V}$), excellent timing resolution ($\sim 120\,\mathrm{ps}$) and insensitivity to magnetic field [@Lu201430; @Buzhan2003; @Kovaltchouk2005; @DelGuerra2011]. The SiPM also referred to as SSPM, AMPD, MPPC, MRSAPD. It consists of an array of Avalanche Photodiodes (APDs) working in the Geiger mode. The APDs are biased above the breakdown voltage $(V_ \mathrm{BD})$ with quenching resistor in serial and connected in parallel. The difference between the bias voltage $(V_\mathrm{bias})$ and breakdown voltage is called over-voltage $(\Delta V=V_\mathrm{bias}-V_\mathrm{BD})$.
Their insensitivity to magnetic field, low operating volatge ($<100$ V), compact size and sensitivity to a small number of photons make them novel light detectors for the scintillator detectors on board the Hard X-ray Modulation telescope (HXMT) [@Li201663]. One of the most important problems for the application of SiPM onboard the HXMT is the gain dependence on the operating conditions (bias voltage, operating temperature). The gain of SiPM is a linear function of the over-voltage. The breakdown voltage has a positive temperature coefficient about $50\mathrm{mV}/^{\circ}\mathrm C$ which results in a typical gain variation of $\sim$3$\% /^{\circ} \mathrm C$. The scintillator detectors on board the HXMT will operate in the temperature range from -40$^{\circ}\mathrm{C}$ to 15$^{\circ}\mathrm{C}$. Therefore, it is essential to design a gain control and stabilization system for the SiPM to operate in a wide range of temperatures.
One way to keep the gain of a SiPM stable is to maintain the operating temperature at a constant value by using external temperature control system. This method is mainly used in ground experiments, such as the outer hadron calorimeter of the CMS detector [@Freeman2010393]. The temperature control system need a cooling and heating system to keep the temperature stable, which is not suitable for the scintillator detectors onboard HXMT. As the extra power budget for the cooling and heating system is unacceptable for the HXMT. An alternative approach to maintain a stable gain is to bias the SiPM at a constant over-voltage. The bias voltage applied to the SiPM should be adjusted according the environment temperature. This can be implemented in different ways such as closed loop feedback of the dark current [@2012lee], use of thermistors to compensate the drift of the over voltage [@Miyamoto2009], a linear feedback on operating bias voltage [@Marrocchesi2009; @Dorosz2013202; @Baszczyk201685; @Shukla] or variable gain amplifiers to compensate the variation of the SiPM gain [@Yamamoto2011]. Even the SiPM can be used as a temperature sensor according to the amplitude of dark noise pulses to control the gain. The gain of SiPM can be measured as a function of temperature according to the average amplitude of the dark noise pulses. A negative feedback loop is established to adjust the bias voltage of the SiPM according to the variation of the gain to keep the gain stable [@Licciulli2013]. However, the method that utilizing the dark current or thermistors would induce a large gain deviation from the constant value when operating in a large temperature range. This limits its application in the case of HXMT. Secondly, linear feedback or negative feedback on operating voltage need the design of a complex feedback electronic system based on digital processing unit, which limits its use in the HXMT. The method that adjust the SiPM bias return line according the temperature-to-voltage converter [@Bencardino2009] is simple enough to be used in the HXMT. This method needs a precise power source to bias the SiPM at a fixed voltage. In the case of HXMT, a space qualified precise power source is needed. This limits the use of such method onboard the HXMT.
In this paper, a simple analog power supply system for gain control and stabilization of MPPC on board HXMT was developed by using zener diode as temperature sensor. The MPPC is one type of UV sensitive SiPM with a p-on-n structure developed by Hamamatsu with relative low dark count rate [@Musienko2009a; @Danilov2009]. The type of MPPC used on board HXMT is S10362-33-050C, consisting of 3600 pixels, covering geometrically $61.5\%$ of total area of $3\mathrm{mm}\times 3\mathrm{mm}$. The breakdown voltage of zener diode has a positive/negative linear temperature coefficient according to its breakdown voltage. An analog power supply system with a linear positive temperature coefficient can be designed based on zener diodes. Different values of temperature coefficient can be obtained by using different zener diodes with different breakdown voltage. The designed analog power supply system is a closed loop feedback system to bias the SiPM at a fixed over-voltage at different environment temperature by adjusting the bias voltage automatically.
The working principle of the analog power supply system was presented in Section \[sect:principle\]. In Section \[sect:temperature\], the temperature dependence of the MPPC, zener diodes and designed analog power supply system were investigated. In Section \[sect:performance\], the performance of the MPPC biased by the designed analog power supply, has been investigated for demonstration of gain stability in the temperature range from -42.7$^{\circ}\mathrm{C}$ to 20.9$^{\circ}\mathrm{C}$. The measured results show that the designed power supply system for gain control of MPPC meets the operating temperature requirement on board HXMT. A further optimized simple analog circuit to maintain the gain of a large array of MPPC was proposed in Section \[sect:discussion\].
As the designed analog power supply system will be used in outer space, the zener diodes should be space-qualified product. However, there are no significant difference between the measured temperature coefficients of the zener voltage for the space-qualified products and industrial products. So the results of the space-qualified product is not shown in this paper.
Working principle of the analog power supply {#sect:principle}
============================================
The MPPC consists of an array of avalanche photo-diodes operating in Geiger mode with the diodes reversely biased above the electrical breakdown voltage ($V_\mathrm{BD}$). At this bias voltage, the electric field in the diode depletion region is sufficiently high for free carriers in the depletion region to produce additional carriers by impact ionization, resulting in a self-sustaining avalanche. The free carriers can be generated by incident photons or thermal emission. The total charge in the avalanche can be evaluated as $Q=C_\mathrm{pixel}\times \Delta V$, where $C_\mathrm{pixel}$ is the effective pixel capacitance. Neglecting the capacitance difference among the different pixel cells, the integrated charge is identical for different cells. The MPPC gain is defined as the charge produced in a single pixel avalanche, expressed in elementary charge unit $$\label{eq:gaindeltav}
G(\Delta V)=\frac{Q}{e}=\frac{C_\mathrm{pixel}}{e}(V_{\mathrm{bias}}-V_{\rm{BD}})=\frac{C_\mathrm{pixel}}{e}(V_{\mathrm{bias}}-V^{0}_{\rm{BD}}-\alpha T).$$
![Block diagram of the analog power supply. The bias voltage applied to the resistor R4 is used to keep the zener diodes working in avalanche mode. The output voltage of the power supply can be controlled by the input voltage $V_{\mathrm in}$.[]{data-label="fig.voltagesource"}](zenarcompensation.eps){width="0.9\linewidth"}
The parameter $\alpha$ in Equation (\[eq:gaindeltav\]) represents the temperature coefficient of the breakdown voltage of the MPPC, and the parameter $V^{0}_{\rm{BD}}$ represents breakdown voltage of MPPC at the temperature of 0$^{\circ} \mathrm C$. Therefore, the gain of MPPC could be maintained at a constant value if the MPPC is biased by a voltage source with a positive temperature coefficient of $\alpha$.
A simple analog power supply system based on two zener diodes and a dual operational amplifier is designed, as shown in Figure. \[fig.voltagesource\]. The voltage between the resistor R4 and zener diode D2 is referred as the output voltage ($V_{\mathrm out}$). The output voltage is connected to the MPPC to bias the MPPC at voltage of $V_{\mathrm out}$. A programmable voltage $V_{\mathrm in}$ was used as the input voltage to control the output voltage $V_{\mathrm out}$, which can be expressed as flowing $$V_{\mathrm out}(V_{\mathrm in},T) = A \cdot V_{\mathrm in}+V_{\mathrm zener}(\mathrm D1)+V_{\mathrm zener}(\mathrm D2).
\label{eq:outputvoltage}$$ The parameter A in Equation \[eq:outputvoltage\] can be expressed as $A= \frac{\mathrm R3+\mathrm R1}{\mathrm R3}$. When the zener diode is working in the avalanche mode, the zener voltage has a positive linear temperature coefficient. From the equation (\[eq:outputvoltage\]), it can be found that the output voltage $V_{\mathrm out}$ have a temperature coefficient which equals to the sum of that of the two zener diodes. This indicates that an analog power supply with different temperature coefficient can be obtained by using zener diodes with different temperature coefficient. To set the two zener diodes working in avalanche mode, a bias voltage should be applied to the resistor R4. The applied bias voltage should be larger than the output voltage $V_{\mathrm out}$. In this paper, the bias voltage is about 80V and the designed value of R4 is about 50 $\mathrm{k}\Omega$. The zener voltage of the D1 and D2 is approximately 33 V, so the current that flowing through D2 and D1 is larger than 200 $\mu$A. If the bias voltage changes about 2V, the current that flowing through the D2 and D1 will change less than 40uA. This change is small compared to the 200uA current. And the zener diodes is working in the avalanching mode, the change of the zener voltage that across the Zener diodes induced by the variance of flowing current could be neglected. This indicates that the $V_{\mathrm out}$ is independent on the bias voltage which is applied to the resistor R4, just as shown in equation (\[eq:outputvoltage\]).
The independence on the bias voltage reduces the requirement for a specific precise power supply system. A simple charge-pump circuit that regulate the 12 V to 80 V can meet the requirement of the power supply to bias the resistor R4 at approximately 80 V. As the $V_{\mathrm out}$ is independent on the bias voltage, the temperature dependence of the charge-pump circuit will not effect the performance of output voltage $V_{\mathrm out}$. This simplifies the design of the MPPC readout electronics.
The over-voltage of MPPC biased by the analog power supply can be expressed as flowing $$\begin{split}
& \Delta V(V_{\mathrm in},T) = A \cdot V_{\mathrm in}+V_{\mathrm zener}(\mathrm D1)+V_{\mathrm zener}(\mathrm D2)-V_{\mathrm BD} \\
& =A \cdot V_{\mathrm in}+(V^{0}_{\mathrm zener}(\mathrm D1)+V^{0}_{\mathrm zener}(\mathrm D2)-V^{0}_{\mathrm BD})+(\alpha_{1}+\alpha_{2}-\alpha)\cdot \mathrm T \\
\end{split}
\label{eq:overvoltage}$$ where $\alpha_{1}$ and $\alpha_{2}$ represents the zener voltage temperature coefficient of diode D1 and D2 respectively, $V^{0}_{\mathrm zener}$ represents the zener voltage of the diode at $0^{\circ} C$. If the sum of the zener voltage temperature coefficients of the two zener diodes equals that of the MPPC, the over-voltage of MPPC can be maintained at a constant value. This means that the gain of MPPC can be kept at a constant value at different environment temperatures.
Temperature dependence {#sect:temperature}
======================
![Experimental setup.[]{data-label="fig.setup"}](experimentsetup.eps){width="1.0\linewidth"}
An experimental setup in dark condition was established to investigate the temperature and bias voltage dependence of the gain for the MPPC. The temperature coefficient of the breakdown voltage of the MPPC could be derived from the gain dependence on operating temperature and bias voltage. The derived temperature coefficient is used as the baseline specification of the temperature coefficient for the design of the analog power supply system. According to the measured temperature coefficient of MPPC, the zener diode with a temperature coefficient of about 30 mV$/^{\circ} \mathrm C$ was chosen as the zener diode of the power supply system.
This method can be generalized to other MPPC and SiPM with different temperature coefficient. The temperature coefficient of zener diode working in different zener voltage is different, it increases with the zener voltage of the diode. The temperature coefficient of the BZV55 series zener voltage ranges from -3.5 mV$/^{\circ} \mathrm C$ to 88.6 mV$/^{\circ} \mathrm C$. So, if the temperature coefficient of the SiPM is different, different zener diodes with different coefficient could be used in series to bias the SiPM. The total temperature coefficient of the zener diodes in series is the sum of that all of the zener diodes. In this work the BZV55-C33 zener diode is chosen, with a typical temperature coefficient ranging from 23.3 mV$/^{\circ} \mathrm C$ to 33.4 mV$/^{\circ} \mathrm C$ according to the datasheet of zener diode. 14 samples of BZV55-C33 were measured in this paper. It is found that the ture temperature coefficients are almost the same for the 14 samples , ranging from 31.31 mV$/^{\circ} \mathrm C$ to 32.18 mV$/^{\circ} \mathrm C$. The detailed performance of the zener diode and designed analog power supply system were investigated.
Experimental setup {#sect:setup}
------------------
![Measured pulse-height distribution of the MPPC illuminated by the LED.[]{data-label="fig.pulseheight"}](singlephoton.eps){width="0.8\linewidth"}
The experimental setup shown in Fig. \[fig.setup\] has been developed to study the characterization of the MPPC. A fast LED driven by the generator was used as light source to illuminate the MPPC. The bias voltage of the MPPC is provided by a Keithley 6487 Picoammeter/Voltage Source. The signal from the MPPC is amplified by a fast amplifier (designed with the OPA 657, gain G=-20). And then the amplified signal is readout by a QDC (CAEN V965A) with 33ns gate width. The 33ns gate signal was the synchronizing signal of the LED drive signal which was provided by the generator. The biased circuit of the MPPC and the LED were placed in a light-tight box. The light-tight box was placed into a climatic chamber to study the performance of MPPC at different environment temperatures.
{width="0.35\linewidth"} {width="0.35\linewidth"}
Gain and breakdown voltage dependence of the MPPC
-------------------------------------------------
![Measured gain vs over-voltage for different operating temperatures. The lines shown in the figure are the linear fitting results.[]{data-label="fig.gainvsovervoltage"}](gainvsovervoltage-updated.eps){width="0.8\linewidth"}
The gain and breakdown voltage dependence of the MPPC were investigated by using the experimental setup as shown in Fig.\[fig.setup\]. The light intensity of the LED that illuminated the MPPC was tuned to make sure that the number of photons incident on to the sensitive area of the MPPC was below 10. In this way, the single equivalent photon pulse-height distribution can be measured with clearly discrete peaks, just as shown in Fig. \[fig.pulseheight\]. The distance between the different single equivalent photon peaks is linearly proportional to the gain of the MPPC. By dividing this distance with the conversion gain of the QDC and the electron charge, the gain of the MPPC can be measured.
The measured gain of the MPPC as a function of bias voltage is shown in Fig. \[fig.breakdownvoltage\]. At a fixed operating temperature, the gain of the MPPC is a linear function of bias voltage applied to the MPPC. By fitting the gain dependence on the bias voltage with linear equation (\[eq:gaindeltav\]), the breakdown voltage of the MPPC can be derived at a given operating temperature. The measured breakdown voltage of the MPPC in the temperature range from $-44^{\circ}\mathrm{C}$ to $20.9^{\circ}\mathrm{C}$ was shown in Fig. \[fig.breakdownvoltage\]. The breakdown voltage of the MPPC increases as a linear function of operating temperature. The fitted linear temperature coefficient of the breakdown voltage from Fig. \[fig.breakdownvoltage\] is $(59.4 \pm 0.4$ mV)$/^{\circ} \mathrm C$. This indicates that the analog power supply connected to the MPPC should have a positive temperature coefficient of $59.4 \mathrm mV/^{\circ} \mathrm C$ to keep the gain at a constant value at different operating temperatures.
The measured gain dependence on the over-voltage is shown in Figure. \[fig.gainvsovervoltage\]. It was derived from Figure. \[fig.breakdownvoltage\] by subtracting the breakdown voltage from the bias voltage. It is found that all of the linear curves of the gain dependence on the over-voltage passed through the zero origin point. The slope of the linear function becomes slightly smaller as temperature decreases. This will result in an deviation from constant value for the gain of MPPC even if the bias voltage of the MPPC is adjusted correctly according to the temperature coefficient of the MPPC.
Breakdown voltage dependence of the zener diode
-----------------------------------------------
{width="0.35\linewidth"} {width="0.35\linewidth"}
The zener diode working in the avalanche mode has a positive temperature coefficient and the temperature coefficient increases with the zener voltage. The zener diode with a zener voltage of 33 V is chosen as the temperature sensor for the analog power supply system. The temperature coefficient of the zener diode is about 32 mV$/^{\circ} \mathrm C$, so the sum of two such zener diodes is about 64 mV$/^{\circ} \mathrm C$ which nearly equals to that of the MPPC. The output voltage of the power supply system based on the two 33 V zener diodes can range from 66 V to 74 V near room temperature. The over-voltage applied to the MPPC when biased by the power supply system is in the range from 0 V to 3.5 V. The zener diode BZV55-C33 from NXP Semiconductors was studied as the zener diode for the power supply system, as shown in Figure. \[fig.voltagesource\]. The breakdown voltage dependence for the 14 BZV55-C33 diode samples were studied in the temperature range from $-32.3^{\circ} \mathrm C$ to $18.5^{\circ} \mathrm C$. The measured breakdown voltage for one of the zener diodes was shown in the Figure. \[fig.zenerbreakdown\]. The temperature coefficient for the zener diode was derived from the breakdown voltage dependence on the operating temperature by linear fitting. The measured temperature coefficient for the zener diode samples were ranging from 31.31 mV$/^{\circ} \mathrm C$ to 32.18 mV$/^{\circ} \mathrm C$, as shown in the right panel of Figure. \[fig.zenerbreakdown\]. The measured breakdown voltage for the samples at $8^{\circ} \mathrm C$ are in the range from 32.06 V to 32.20 V. The difference for the breakdown voltage of different zener diode samples is within 0.14 V.
Performance of the designed analog power supply
-----------------------------------------------
![Block diagram of experimental setup for the measurement of the temperature coefficient of the analog power supply.[]{data-label="fig.zenercompensation-measure"}](zenercompensation-measure.eps){width="1.0\linewidth"}
![Measured output voltage as linear function of control voltage (upper panel) and output voltage as function of operating temperature (lower panel). The solid lines in the figures show the linear fit results.[]{data-label="fig.outputvsinput"}](vcvsoutput.eps "fig:"){width="0.8\linewidth"} ![Measured output voltage as linear function of control voltage (upper panel) and output voltage as function of operating temperature (lower panel). The solid lines in the figures show the linear fit results.[]{data-label="fig.outputvsinput"}](outputvstemperature.eps "fig:"){width="0.8\linewidth"}
The measured temperature coefficient of the zener diode is approximately 32 mV$/^{\circ} \mathrm C$. This means that the temperature coefficient for the analog power supply system based on two zener diodes is about 64 mV$/^{\circ} \mathrm C$. This is larger than that of the MPPC. Two zener diodes with the minimum temperature coefficient were chosen as the D1 and D2 diode in the power supply system as shown in Fig. \[fig.voltagesource\]. The performance of the designed power supply system were measured at different environment temperatures ranging from -$38.7^{\circ} \mathrm C$ to 22.1$^{\circ} \mathrm C$. The experimental setup is shown as Figure. \[fig.zenercompensation-measure\]. The input voltage $V_{in}$ was provided by a generator to control the output voltage. The bias voltage applied to resistor R4 is provided by the Keithley 6487 with a voltage of 80 V. During the measurement, the Keithely 6487 measured the current ($I$) flowing through the resistor R4. The output voltage can be derived by subtracting the cross voltage applied to the resistor R4 from the 80 V bias, $V_{\mathrm output}=V_{\mathrm bias}-I \cdot R4$.
![Measured temperature coefficient at different control voltages for the power supply system.[]{data-label="fig.coefficient-tcs"}](coefficient-tcs.eps){width="0.8\linewidth"}
The measured output voltage at different environment temperatures are shown in the upper panel of Fig. \[fig.outputvsinput\]. It is found that the output voltage of the power supply system increases as a linear function of the control voltage ($\mathrm V_{\mathrm in}$). The measured output voltage at room temperature of about 22.1$^{\circ} \mathrm C$ ranging from 70.75 V to 75.92 V. This means that the designed power supply system can bias the MPPC with an over-voltage ranging from 0.06 V to 5.23 V. From this measurement, the output voltage dependence on the operating temperature can be derived by fixing the control voltage at a constant value. The results are shown in the lower panel of Fig. \[fig.outputvsinput\]. The output voltage increases as a linear function of the operating temperature at a given control voltage. The temperature coefficient of the output voltage can be derived by fitting the output voltage dependence on the operating temperature with a linear function. The measured temperature coefficient at different control voltages were shown in the Fig. \[fig.coefficient-tcs\]. It is found that the measured temperature coefficient is in the range from 63.65 mV/$^{\circ} \mathrm C$ to 64.61 mV/$^{\circ} \mathrm C$. The maximum temperature coefficient was measured at the control voltage of 3.6 V. The measured temperature coefficient of the power supply system is larger than that of the MPPC by a value ranging from 4.5 mV/$^{\circ} \mathrm C$ to 5.2 mV/$^{\circ} \mathrm C$. This means that if the MPPC is powered by the designed power supply system, the gain offset caused by the variation of environment temperature will be reduced from $\sim$2.9$\%$/$^{\circ} \mathrm C$ to $\sim0.25\%$/$^{\circ} \mathrm C$ when the over-voltage is about 2 V.
Gain control and stabilization of MPPC {#sect:performance}
======================================
The gain of the MPPC when biased by the designed power supply system for gain control was measured for comparison with that biased at 72.1 V by Keithley 6487 Picoammeter/Voltage Source without gain control. The measurement experimental setup is shown as in Fig. \[fig.setup\].
![Measured pulse-height distribution of the MPPC with and without the designed voltage source for gain control at 20.9$^{\circ} \mathrm C$.[]{data-label="fig.compare-noise"}](compare-noise-updated.eps){width="0.8\linewidth"}
The gain was measured at different environment temperatures ranging from -42.7$^{\circ} \mathrm C$ to +20.9$^{\circ} \mathrm C$. The gain with gain control was measured with a control voltage of 2.7 V. This means that the MPPC can be biased at 72.3 V by the designed power supply system at 20.9$^{\circ} \mathrm C$. The measured pulse-height of the MPPC at 20.9$^{\circ} \mathrm C$ was shown in Fig. \[fig.compare-noise\]. The equivalent noise charge (ENC) of the electronic readout system including the amplifier could be derived from the pedestal of the pulse-height distribution of MPPC. The derived ENC without and with the gain control were $1.16 \times 10^{5}$ and $1.05 \times 10^{5}$ respectively. These results showed that the designed voltage source for gain control do not induce extra noise to the whole electronic readout system.
The measured gain of MPPC is shown in the upper panel of Fig. \[fig.gainactivecontrol\]. It is found that the gain of the MPPC without gain control becomes smaller when the environment temperature increases, while the gain with gain control becomes larger. This is because the temperature coefficient of the designed power supply system is larger than that of the MPPC with a value of about 5mV/$^{\circ} \mathrm C$. The gain of the MPPC without gain control increased from $4.9 \times 10^{5}$ to $1.4 \times 10^{6}$ when temperature ranging from 20.9$^{\circ} \mathrm C$ to -42.7$^{\circ} \mathrm C$. The gain has been increased by a factor of 1.84 due to the 63.6$^{\circ} \mathrm C$ variation of the environment temperature, while the gain variation with gain control was reduced to 0.278 at the same temperature variation, ranging from $5.4 \times 10^{5}$ to $3.9 \times 10^{5}$. This means that the compensation for variation of breakdown voltage caused by the change of temperature is efficient. The gain dependence on the environment temperature can be fitted by a linear function $G(T)=A+B\cdot T$. The linear fitted results for the gain without gain control are $A = (8.741 \pm 0.017) \times 10^{5}$ and $B = -(0.183 \pm 0.001) \times 10^{5}$. The fitted results with gain control are $A = (4.883 \pm 0.013) \times 10^{5}$ and $B = (0.024 \pm 0.001) \times 10^{5}$. From the fitted results, it is found that the absolute slope of the gain dependence on operating temperature is reduced by an order of magnitude with the designed power supply for gain control.
![Measured gain variance vs temperature with and without the designed voltage source for gain control.[]{data-label="fig.gainactivecontrol"}](gain-72.1-updated.eps "fig:"){width="0.8\linewidth"} ![Measured gain variance vs temperature with and without the designed voltage source for gain control.[]{data-label="fig.gainactivecontrol"}](relativegain.eps "fig:"){width="0.8\linewidth"}
The relative temperature coefficient for the gain of the MPPC with and without the designed power supply system for gain control was measured for comparison. The temperature coefficient for the gain is defined as following $$A=|\frac{dG}{dT}|\cdot\frac{1}{G}$$ where $G$ represents the gain of MPPC and $T$ represents the environment temperature. The biased voltage applied to the MPPC was tuned to measure the gain at different over-voltages. The gain was measured at different environment temperatures. The measured temperature coefficient for the MPPC at different over-voltages is shown in the lower panel of Fig. \[fig.gainactivecontrol\]. It is found that the temperature coefficient for the gain is in the range from 2.8$\%$/$^{\circ} \mathrm C$ to 2.15$\%$/$^{\circ} \mathrm C$ when the MPPC was biased with an over-voltage ranging from 2 V to 2.5 V without gain control. The temperature coefficient was reduced below 0.5$\%$/$^{\circ} \mathrm C$ when the MPPC was biased by the designed power supply system for gain control. The temperature coefficient becomes smaller when the MPPC was biased with a larger over-voltage. The measured temperature coefficient for the gain of MPPC was reduced approximately by an order of magnitude when biased by the designed programmable power supply system for gain control. It was reduced from 2.8$\%$/$^{\circ} \mathrm C$ to 0.3$\%$/$^{\circ} \mathrm C$. This measured result is consistent with the prediction for the designed power supply system based on the measured temperature coefficient of the MPPC and zener diodes.
Conclusions and Discussion {#sect:discussion}
==========================
![Low power dissipation temperature compensation circuit for the MPPC.[]{data-label="fig.mppccompensation"}](mppccompensation.eps){width="0.7\linewidth"}
The gain of the MPPC was measured at different operating temperatures to determine the gain temperature coefficient. The measured temperature coefficient of the breakdown voltage was $(59.4 \pm 0.4$ mV)$/^{\circ} \mathrm C$. The zener diode working in avalanche region has a similar positive linear temperature coefficient of zener voltage. By combining two zener diodes in series, an analog power supply system can be designed to have a positive linear temperature coefficient which equals that of the MPPC. The measured temperature coefficient for the designed power supply system based on two BZV55-C33 zener diodes was in the range from 63.65 mV/$^{\circ} \mathrm C$ to 64.61 mV/$^{\circ} \mathrm C$. The relative gain temperature coefficient for the MPPC was reduced from 2.8$\%$/$^{\circ} \mathrm C$ to 0.3$\%$/$^{\circ} \mathrm C$ when biased at over-voltage of 2 V by the designed power supply system. A good stabilization of the gain for varies temperature from -42.7$^{\circ} \mathrm C$ to +20.9$^{\circ} \mathrm C$ was achieved. The designed analog power supply system has been used as the bias voltage source for the MPPC on board the HXMT. Now, the flight-model of the analog power supply is under construction.
The main drawback of the designed analog power supply was the large power dissipation which was about 50 mW. This will limit the application of the designed power supply system especially in the case with thousands of MPPCs. The designed power supply system includes an operating amplifier which is quite complex for the large detector system with thousands of MPPCs. The designed power supply system can be simplified by using negative linear temperature coefficient zener diodes, just as shown in Fig. \[fig.mppccompensation\]. By this optimization, the power consumption is expected to be reduced from 50 mW to about 5 mW. Several zener diodes which have negative linear temperature coefficient are in series with the MPPC to compensation the voltage offset caused by the temperature variation. The zener diode is biased by a precise voltage source with feedback system to eliminate the effect of temperature. As the zener diodes have a negative temperature coefficient, the voltage that applied to the MPPC will have a positive temperature coefficient with a absolute value equals to the sum of that of the zener diodes. By using proper number of zener diodes with given temperature coefficient, the temperature coefficient of the voltage applied to the MPPC can become equal to that of the MPPC. Therefore the gain of the MPPC can be maintained at a constant value at different operating temperatures. This method can be used to maintain the gain of a large array of MPPC for its simpler circuit and lower power consumption.
Acknowledgment {#sect:acknowledgement .unnumbered}
==============
This work is supported by the HXMT project, the Space Science Advance Research Program of Chinese Academy of Sciences and the National Natural Science Foundation of China under Grant No.11603025.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce a set of minimal simplified models for dark matter interactions with the Standard Model, connecting the two sectors via either a scalar or pseudoscalar particle. These models have a wider regime of validity for dark matter searches at the LHC than the effective field theory approach, while still allowing straightforward comparison to results from non-collider dark matter detection experiments. Such models also motivate dark matter searches in multiple correlated channels. In this paper, we constrain scalar and pseudoscalar simplified models with direct and indirect detection experiments, as well as from existing LHC searches with missing energy plus tops, bottoms, or jets, using the exact loop-induced coupling with gluons. This calculation significantly affects key differential cross sections at the LHC, and must be properly included. We make connections with the Higgs sector, and conclude with a discussion of future searches at the LHC.'
author:
- 'Matthew R. Buckley'
- David Feld
- Dorival Gonçalves
title: Scalar Simplified Models for Dark Matter
---
Introduction \[sec:intro\]
==========================
The case for the existence of dark matter is strong. Decades of evidence from multiple independent lines [@Zwicky:1933gu; @Rubin:1980zd; @Olive:2003iq; @Ade:2013zuv] reveal that this form of matter has a significant role in the composition and evolution of our Universe (for a review, see [*e.g.*]{}, Ref. [@Bertone:2004pz]). No particle in the Standard Model is a suitable candidate for dark matter and so we need new physics to explain it. Though we lack evidence of the nature of the dark sector, if particle dark matter has a mass at the TeV scale or lower and was ever in thermal equilibrium in the early Universe, we have good reason to expect interactions with the visible sector to be within reach of our present experiments. However, this is of course not guaranteed.
Perhaps the best known example of such dark matter is a weakly-interacting massive particle which becomes a thermal relic with the appropriate energy density after freeze-out. This type of dark matter is realized in many extensions of the Standard Model introduced to solve other problems of a theoretical nature ([*e.g.*]{} Naturalness and Hierarchy). However, looking beyond this class of dark matter, even models of non-thermal dark matter often require significant annihilation cross sections into either the Standard Model or some hidden sector, so as not to overclose the Universe [@Buckley:2011kk]. It is therefore well-motivated to search for dark sector particles in a range of experiments, including the Large Hadron Collider (LHC).
When looking for dark matter, we can cast the experimental reach in terms of specific models of dark matter which are UV-complete. These models usually have a number of additional new particles with more significant interactions with the Standard Model than the dark matter itself. The canonical example of this sort is the supersymmetric neutralino, which is accompanied by a host of new charged and colored superpartners. Despite the advantage of UV-complete models, interpreting results in this way has some drawbacks: $i$) the results may be difficult to recast for new models; $ii$) correlating results with non-collider experiments may be very dependent on UV-complete parameters; $iii$) focusing on a specific high-energy model runs the risk of overlooking other experimentally interesting channels; and $iv$) tuning the experimental selection criteria could reduce the sensitivity to other types of dark matter.
In order to approach the problem in a somewhat model-independent way while still allowing for comparison between different classes of experiments, it has been useful to present the results of experimental searches in an effective field theory (EFT) framework [@Cao:2009uw; @Goodman:2010yf; @Goodman:2010ku]. The EFT approach assumes contact term interactions between dark matter and SM particles with the particle(s) connecting the two sectors integrated out of the low-energy spectrum. The validity of the EFT approach diminishes in the regime where the momentum transfer cannot be neglected relative to the (unknown) mass of the heavy particles. For direct detection this condition is usually satisfied, as long as mediators are not extremely light, as the momentum scale is on the order of 10 keV. Indirect detection and thermal freeze-out involve the annihilation of non-relativistic dark matter and so the EFT is applicable as long as the mediator is significantly heavier than twice the dark matter mass, assuming no additional new particles in the theory [@Abdallah:2014hon].
However, when considering the production of dark matter at particle colliders through high $p_T$ visible particles recoiling against invisible dark matter [@Birkedal:2004xn; @Feng:2005gj; @Beltran:2008xg; @Konar:2009ae; @Beltran:2010ww; @Bai:2010hh; @Rajaraman:2011wf; @Fox:2011pm; @Bai:2012xg; @Fox:2012ee; @Carpenter:2012rg], the momentum transfer in dark matter pair production events is large enough to render the EFT assumption invalid for a significant range of dark matter masses, couplings, and mediator masses [@Bai:2010hh; @Fox:2011fx; @Fox:2011pm; @Shoemaker:2011vi; @Fox:2012ee; @Weiner:2012cb; @Busoni:2013lha; @Buchmueller:2013dya; @Buchmueller:2014yoa; @Busoni:2014sya; @Busoni:2014haa]. As the momentum flowing through the production diagram is proportional to both the transverse momentum of the dark matter particles ([*i.e.*]{} the missing transverse momentum, or MET) and the transverse momentum of recoiling visible particles required for the trigger, this issue will be even more pressing at the LHC Run-II, as the trigger requirements on MET and jet $p_T$ will be higher than those used in Run-I. Rather than viewing the invalidity of the EFT formalism as a drawback, it should be seen as an optimistic statement: if dark matter is being produced at colliders, it is generally the case that new mediating particles are being produced as well. As we look to interpret results from dark matter experiments and design new search strategies at the LHC, a balance should be struck between the very general (but often inapplicable) EFT approach and a full theory like supersymmetry. One solution has been found in [*Simplified Models*]{} [@Alwall:2008ag; @Alves:2011wf; @Goodman:2011jq], which resolve the contact interaction into a single exchange particle, without adding in the full complexity of a UV-complete model. By specifying the spin and gauge quantum numbers of the dark matter and the mediators, the parameter space can be made relatively small, allowing an easy conversion of bounds between experiments and theories. Previous papers have discussed colored mediators [@An:2013xka; @DiFranzo:2013vra; @Papucci:2014iwa], which result in $t$-channel production of dark matter in a manner very similar to squarks in supersymmetry. Other works have considered vector and axial vector $Z'$ models [@An:2012va; @Frandsen:2012rk; @Busoni:2014haa], which cause $s$-channel dark matter production at colliders.
In this paper we consider a class of simplified models with a spin-0 scalar or pseudoscalar mediator, which allows $s$-channel production of dark matter from Standard Model partons at the LHC. These models are attractive in their simplicity, requiring only a minimal extension of the Standard Model’s particle content. New scalars or pseudoscalars can also be easily accommodated in extended Higgs sectors, and it is not unreasonable to expect the Higgs to have contact with the dark sector. As with other simplified models, scalar mediators predict LHC signatures in a number of correlated channels; this can be used to our advantage when designing new searches.
As previous works [@Haisch:2012kf; @Haisch:2013fla; @Haisch:2013ata; @Busoni:2014sya; @Ghorbani:2014qpa; @Crivellin:2014qxa] have pointed out, scalar and pseudoscalar mediator models and EFTs face unique simulation issues at colliders. Making the well-motivated assumption that the mediator couplings to Standard Model fermions proportional to the Higgs Yukawas, the mediator is primarily produced at the LHC through a loop-induced interaction with gluons. As was noted in the context of scalar EFTs, this loop-induced coupling must be calculated assuming large momentum transfer, as the trigger requirements at the LHC for most dark matter searches require significant transverse momentum in the event. Just as large momenta requires the expansion of a point-like dark matter-Standard Model EFT interaction to include a mediator, the mediator-gluon interaction must also be resolved as the momentum transfer increases $p_{T\phi}=\mathcal{O}( 2m_t)$. A sketch of the successive levels of effective theories is shown in Figure \[fig:momentum\_cartoon\]. As we will show, the large momentum transfer at the LHC forces us to fully resolve the top-loop induced coupling, just as it forces us to resolve the mediator in the EFT.
![A heuristic diagram presenting the successive levels of effective theories that must be expanded as the momentum flow (proportional to the MET) through the interaction increases. On the left we have the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$. In the center two effective theories with either $(m_\phi \rightarrow \infty, \mbox{finite}~m_t)$ (top) or $(\mbox{finite}~m_\phi,m_t \rightarrow \infty)$ (bottom). On the right the Full Theory with finite $(m_\phi, m_t)$. \[fig:momentum\_cartoon\]](./pictorial_graphs){width="0.73\columnwidth"}
In this paper, we provide two benchmark models for scalar and pseudoscalar mediated simplified models, with a five-dimensional parameter space. We demonstrate the non-negligible effects of resolving the mediator loop-induced coupling to gluons in collider simulations, compared to the effective interactions. We derive bounds on these parameters using data from direct and indirect detection, as well as predictions assuming that the dark matter is a thermal relic. We then show the existing constraints on these benchmarks from a number of Run-I LHC searches, including – but not limited to – the MET plus jets searches that have been of primary interest previously. This comprehensive set of bounds on scalar mediators has not been previously collected, and underlines the necessity of multiple complimentary channels when searching for dark matter at the LHC [@Lin:2013sca].
In Section \[sec:models\] we set up our two benchmark models for scalar and pseudoscalar mediators. We introduce a set of parameters which describe the relevant phenomenology for current and future experimental results. In this section we also show the effects of the resolved top-loop on the distribution of transverse momentum at colliders. In Section \[sec:noncollider\] we show constraints on these models from non-collider physics: direct and indirect detection, as well relic abundance cross section. Constraints from existing LHC Run-I missing energy searches are discussed in Section \[sec:collider\_bounds\] in three channels: missing transverse energy with associated jets, with associated top quark pairs, and with associated bottom quarks. We apply our constraints to the special case of the 125 GeV Higgs as the scalar mediator in Section \[sec:higgs\]. We then conclude by outlining additional searches and improvements that could be made for future analyses.
Simplified Models {#sec:models}
=================
In this paper we consider interactions between Dirac fermion dark matter $\chi$ and Standard Model fermions mediated by either a new scalar $\phi$ or a new pseudoscalar $A$. Our choice of fermionic dark matter is somewhat arbitrary; our results would translate to the scalar dark matter case with minor modifications, though this assumption would introduce additional parameters. Our two benchmark models take the form
$$\begin{aligned}
{\cal L}_S & = & {\cal L}_\text{SM}+ \frac{1}{2} (\partial_\mu \phi)^2 - \frac{1}{2} m_\phi^2 \phi^2
+i \bar{\chi} \slashed{\partial} \chi - m_\chi \bar{\chi} \chi - g_\chi \phi \bar{\chi}\chi
- \sum_{\text{fermions}} g_v \frac{y_f}{\sqrt{2}} \phi \bar{f}f \;, \label{eq:Lphi} \\
{\cal L}_A & = & {\cal L}_\text{SM}+\frac{1}{2} (\partial_\mu A)^2 - \frac{1}{2} m_A^2 A^2
+i \bar{\chi} \slashed{\partial} \chi - m_\chi \bar{\chi} \chi - i g_\chi A \bar{\chi}\gamma^5 \chi
- \sum_{\text{fermions}} i g_v \frac{y_f}{\sqrt{2}} A \bar{f} \gamma^5f . \label{eq:LA}\end{aligned}$$
Here, ${\cal L}_\text{SM}$ is the Lagrangian of the Standard Model. Such models introduce five free parameters: dark matter mass $m_\chi$, mediator mass $m_{\phi}$ or $m_A$, the dark matter-mediator coupling $g_\chi$, the flavor-universal Standard Model-mediator coupling $g_v$, and the mediator width $\Gamma_{\phi}$ or $\Gamma_A$.[^1] Keeping the width as a free parameter leaves open the possibility that the mediator has other couplings to additional particles, perhaps in an expanded dark sector. Furthermore, as the cross section for dark matter production, annihilation, and scattering to nucleons is proportional to product of the couplings $\left( g_\chi g_v \right)^2$ and the width depends on the sum of terms proportional to $g_\chi^2$ and $g_v^2$ separately, by keeping the width as a free parameter, we can set limits on the combination $g_\chi g_v$ as a function of the width without specifying the individual couplings $g_v$ and $g_\chi$. This is how we will present our bounds in Sections \[sec:noncollider\] and \[sec:collider\_bounds\].
We set the fermion couplings proportional to the SM Yukawa couplings, using the Minimal Flavor Violating (MFV) assumption [@D'Ambrosio:2002ex]. This avoids introducing precision constraints from flavor measurements. Additionally, note that the left-handed Standard Model fermions are $SU(2)_L$ doublets and the right-handed fermions are singlets, while the dark matter cannot be primarily an $SU(2)_L$ multiplet with $Y\neq 0$, due to direct detection bounds. If $\chi$ is a complete Standard Model gauge singlet, then the mediator $\phi$ or $A$ must have some mixing with the Higgs sector to interact with both the doublet fermions and the dark matter, justifying the Yukawa-proportional coupling assumption. Another possibility is that dark matter is a doublet-singlet mixture, as in the case of a neutralino, allowing the mediator to be an $SU(2)_L$ doublet while still avoiding direct detection constraints. This again involves mass terms in the dark sector proportional to the electroweak symmetry breaking scale, which suggests (though does not require) couplings proportional to Yukawa terms.
We assume that the coupling $g_v$ is universal across all the families of quarks and leptons. One could loosen this requirement without introducing large flavor violation. Taking a cue from two-Higgs doublet models for example, the up-type and down-type couplings could be varied independently. We will not explore this possibility in detail here, but we note such deviations from the baseline model would change the ratios of expected signals in the various collider channels we consider. This again motivates a broad set of experimental searches.
As we have seen, this set of simplified models has some obvious connections with the Higgs sector [@Fox:2011pm; @Djouadi:2012zc]. As a gauge-singlet scalar, the mediator $\phi$ will generically mix with the neutral Higgs. If the SM Higgs is part of an extended Higgs sector, then the pseudoscalar $A$ would fit easily into the model (for example, as the pseudoscalar in a two-Higgs doublet model). If the models are so intimately related to Higgs physics, one might expect some coupling to $W$ and $Z$ bosons, which we do not allow in our baseline models. We justify this omission by noting that even for scenarios where the scalar and/or pseudoscalar are part of a Higgs sector, deviations from alignment in supersymmetry are constrained to be small [@Craig:2013hca; @Carena:2013ooa], which in turn implies that the coupling to $W/Z$ bosons of new scalars and pseudoscalars in the Higgs sector would likely be small compared to the $125$ GeV Higgs.
Similarly, we would expect explicit dimension-4 $\phi-h$ or $A-h$ couplings in our Lagrangians Eqs. and . In a full UV-complete theory, into which the simplified model presumably fits, these couplings would be set by some unspecified dynamics. In this work, we set them to zero for simplicity, as we did for the $W$ and $Z$ couplings.
Analogously to the production of the Higgs, the dominant form of dark matter production at the LHC would be through gluon fusion, as the tree-level couplings to the light quarks are Yukawa-suppressed. This production mode is dominantly through the loop induced $g-g-\phi (A)$ coupling. Representative diagrams for the leading-jet process are shown in Figure \[fig:loop\_feyndiagram\]. Note that in the production of the mediators in channels with associated $b$ or $t$ quarks is largely dominated by the tree-level terms, though as in Higgs production, loop effects can be important in the $\phi(A)+$ heavy flavor channels. If the external particles in the loop induced $g-g-\phi (A)$ interaction are on-shell, then it can be exactly calculated in a single coupling value, as in Higgs physics. At leading-order, the on-shell Lagrangians for our two benchmark models gain the additional terms [@toploop; @pseudoloop1; @pseudoloop2; @pseudoloop3; @Harlander:2005if] $$\begin{aligned}
{\cal L}_{S,\text{loop}} = \frac{\alpha_s}{8 \pi} \frac{g_v}{v} \tau [1+(1-\tau) f\left(\tau \right)] G^{\mu\nu}{G}_{\mu\nu}\phi
\;, \qquad \qquad
{\cal L}_{A,\text{loop}} = \frac{\alpha_s}{4 \pi} \frac{g_v}{v} \tau f\left(\tau \right) G^{\mu\nu}\tilde{G}_{\mu\nu}A
\;,
\label{eq:lag_loop}\end{aligned}$$ where $\tau = 4 m_t^2/m_{\phi (A)}^2$, $y_t$ is the top Yukawa, $v$ is the Higgs vacuum expectation value, and the function $f(\tau)$ is defined as $$\begin{aligned}
f(\tau) =
\begin{cases}
\arcsin^2 \frac{1}{\sqrt{\tau}} \;, & \tau \ge 1 \;, \\
-\frac{1}{4} \left( \log \frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}} - i \pi \right)^2\;, & \tau < 1\;.
\end{cases} \end{aligned}$$ We should emphasize that the effective coupling approximation can be accurately calculated for arbitrary top and mediator masses. However, for associated production of $\phi$ or $A$ plus jets at collider, with momenta and energy scales where the loop induced top contributions start to be resolved, that is $p_{T,\phi} = \mathcal{O}(2m_t)$, this effective operator breaks down and the one-loop dynamics should be taken into account. Also note that the scalar coupling to gluons is suppressed relative to the pseudoscalar by $\gtrsim30\%$ for mediator masses below $\sim 400$ GeV. This will result in slightly weaker bounds on the scalar model relative to pseudoscalars in channels where the gluon coupling dominates ([*i.e.*]{}, LHC monojets).
![Sample of the leading-order Feynman diagrams, in the Full Theory with finite top mass effects, contributing to the scalar plus jet production at the LHC.[]{data-label="fig:loop_feyndiagram"}](./feynman.pdf){width="\columnwidth"}
![Missing energy distribution for the process $pp\rightarrow \bar{\chi}\chi + j$ in the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$ (equivalent to the left panel of Fig. \[fig:momentum\_cartoon\]), for a finite mediator mass with an effective coupling to gluons $m_t \rightarrow \infty$ (lower center panel of Fig. \[fig:momentum\_cartoon\]) and the Full Theory including the top mass effects (right panel of Fig. \[fig:momentum\_cartoon\]). On the left panel we display the results for a light mediator and on the right for a very heavy one (equivalent to the upper center panel of Figure \[fig:momentum\_cartoon\]). These distributions were generated at the parton level with [MCFM]{} and LHC at 8 TeV.[]{data-label="fig:etmiss_intro"}](./etmiss_intro_mh100 "fig:"){width="0.4\columnwidth"} ![Missing energy distribution for the process $pp\rightarrow \bar{\chi}\chi + j$ in the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$ (equivalent to the left panel of Fig. \[fig:momentum\_cartoon\]), for a finite mediator mass with an effective coupling to gluons $m_t \rightarrow \infty$ (lower center panel of Fig. \[fig:momentum\_cartoon\]) and the Full Theory including the top mass effects (right panel of Fig. \[fig:momentum\_cartoon\]). On the left panel we display the results for a light mediator and on the right for a very heavy one (equivalent to the upper center panel of Figure \[fig:momentum\_cartoon\]). These distributions were generated at the parton level with [MCFM]{} and LHC at 8 TeV.[]{data-label="fig:etmiss_intro"}](./etmiss_intro_mh1200 "fig:"){width="0.4\columnwidth"}
In Section \[sec:collider\_bounds\] we will discuss further details of the missing transverse energy searches with associated jets used the LHC experiments. For this section, it is sufficient to state that significant transverse missing momentum is required (that is, large transverse momentum of the $\phi$ or $A$), along with large momentum of at least one jet, in order to pass the trigger and selection criteria. In events without additional heavy flavor tagging, the primary production vertex for the $\phi$ or $A$ will be through the top-loop coupling to gluons, in association with a hard emission of initial state radiation, see Figure \[fig:loop\_feyndiagram\].
In Figure \[fig:etmiss\_intro\], we show the missing transverse momentum distribution (MET or $\slashed{E}_T$) for $p p \to \bar{\chi} \chi+ j$ at the 8 TeV LHC, setting $m_\chi = 10$ GeV. Following our sketch (in Figure \[fig:momentum\_cartoon\]) of the inclusion of integrated-out particles as we resolve effective operators, we present the differential MET distribution from dark matter production for three different interaction hypothesis:
1. for the direct production through an EFT interaction with gluons, $\alpha_s/\Lambda^{3}\,\left[\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}\right]$;
2. for the production via a scalar mediator with an effective $g-g-\phi$ interaction vertex, as in Eq. . For comparison purposes, we show both a light (100 GeV) on-shell mediator and very heavy (1200 GeV) mediator which gives dark matter through off-shell production; and
3. for the production via a scalar mediator where the top-loop has been taken into account via the exact one-loop computation. We show once more a very light ($m_\phi=100$ GeV) and a very heavy ($m_\phi \rightarrow \infty$) mediator scenarios.
All these distributions were generated using [MCFMv6.8]{} [@Campbell:2010ff; @hj], where we have extended the process implementation ${pp\rightarrow H(A)+j \rightarrow \tau^+ \tau^-+j}$ in [MCFM]{} to accommodate the off-shell mediator production and decay to a dark matter pair. The hard scales are defined as $\mu_F^2=\mu_R^2=m_{\phi(A)}^2+p_{Tj}^2$. For further details on the event generation see Section \[sec:collider\_bounds\].
From Figure \[fig:etmiss\_intro\], we observe that for heavy mediators above $\mathcal{O}(1~\mbox{TeV})$ and $m_t \rightarrow \infty$ the [*Simplified Model*]{} can be well described by the EFT. However, for light mediators ($m_\phi=100$ GeV) or finite top mass we see that this approximation breaks down. Moreover, if accurate conclusions about such models are to be drawn from LHC data, it is clearly necessary to include the mediator-gluon interaction (induced by the heavy-quark loops) when the characteristic energies are above $\mathcal{O}(2m_t)$.
At every stage of returning the integrated particles to the spectrum (as pictorially presented in Figure \[fig:momentum\_cartoon\]), we see significant changes in the differential cross sections. There is a large decrease in the tail of the MET distributions as first the mediator and then the top-loop are correctly taken into account. Ignoring these effects in the simplified scalar model will lead to an over-prediction of the cross section at the LHC for a given set of parameters, and thus overly strong limits. Furthermore, when using search techniques that rely on detailed knowledge of the kinematic shape ([*e.g.*]{} razor variables [@Fox:2012ee; @Rogan:2010kb; @Chatrchyan:2011ek]), it is of course necessary to fully and correctly understand the shape of the signal distributions.
Before moving on to the bounds on the benchmark models, it is useful to consider the widths and branching ratios we might expect in our models of interest. In Figure \[fig:widths\], we show the partial widths for $\phi$ and $A$ decaying into Standard Model particles and dark matter as a function of mass $m_{\phi(A)}$, assuming $m_\chi = 10$ GeV and $g_v = g_\chi = 1$. It is straightforward to rescale the relevant widths if these assumptions are loosened. As can be seen, if $g_v \sim g_\chi$ and $m_\chi \ll m_\phi/2$, the decay of the mediator into dark matter is expected to dominate, unless the mediator is heavy enough for the top channel to open. This is a result of the small Yukawa couplings for the lighter fermions. It is also worth pointing out that differences in rate between the scalar and pseudoscalar partial decays are given by a distinct scaling pattern with the particle velocity $\beta_\chi=\sqrt{1-4m_\chi^2/m_\phi}$. Namely, the scalar presents a stronger suppression $\Gamma_{\phi\rightarrow\chi\chi} \propto \beta_\chi^3$ when compared to the pseudoscalar, $\Gamma_{A\rightarrow\chi\chi} \propto \beta_\chi$. As a result, when the dark matter mass is close to the kinematic limit $2m_\chi \sim m_{\phi(A)}$, we should expect constraints on the couplings of scalars to be weaker than those placed on the couplings to pseudoscalars. When the dark matter is much lighter than the mediator, the coupling constraints on the two models should be equivalent, as in this regime $\beta^{3} \sim \beta \sim 1$.
![The width $\Gamma$ of the scalar $\phi$ (left) and pseudoscalar $A$ (right) decaying into pairs of 10 GeV dark matter (black dotted), top quarks (green), bottom quarks (red), tau leptons (blue), $\gamma\gamma$ (black dashed), and the total width (black solid), as a function of the parent mass $m_\phi$ or $m_A$. Widths are calculated assuming $g_v = g_\chi =1$. \[fig:widths\]](./phi_width_scaled.pdf "fig:"){width="0.5\columnwidth"}![The width $\Gamma$ of the scalar $\phi$ (left) and pseudoscalar $A$ (right) decaying into pairs of 10 GeV dark matter (black dotted), top quarks (green), bottom quarks (red), tau leptons (blue), $\gamma\gamma$ (black dashed), and the total width (black solid), as a function of the parent mass $m_\phi$ or $m_A$. Widths are calculated assuming $g_v = g_\chi =1$. \[fig:widths\]](./A_width_scaled.pdf "fig:"){width="0.5\columnwidth"}
Non-Collider bounds {#sec:noncollider}
===================
In this section, we derive bounds on our benchmark model parameters, using direct and indirect detection experimental results, as well as the thermal relic abundance calculation. These bounds are complimentary to those set by colliders, which we will consider in Section \[sec:collider\_bounds\]. However, we caution that care must be taken in extrapolating bounds between different classes of experiments, as there are both particle physics and astrophysical assumptions that must be kept in mind. For example, the direct detection limits rely on an assumption about the local dark matter density and velocity distributions, the latter of which is expected to vary from the standard assumptions used in the experimental results [@Kuhlen:2009vh; @Lisanti:2010qx; @Mao:2012hf; @Mao:2013nda; @Kuhlen:2013tra; @Lee:2013wza; @Bozorgnia:2013pua]. While it is possible to some degree to disentangle the astrophysical uncertainties to place limits on the fundamental parameters [@Fox:2010bu; @Fox:2010bz; @Fairbairn:2012zs; @Pato:2012fw; @DelNobile:2013cta; @Feldstein:2014gza], we cannot lose sight of the assumptions that went into the analysis. Similarly, the parameters that are required to obtain a thermal relic abundance can be changed significantly if additional particles (beyond the minimal set in our benchmark simplified models) are present in the spectrum, or if the flavor-universal assumption for the coupling $g_v$ is lifted. Furthermore, we have no direct knowledge that the dark matter is a thermal relic. Thus, we wish to emphasize that no single result presented here should be taken as the final word on the limits for our models, since these searches – along with those of the colliders – are complimentary and approach the problem from different angles. Despite the caveats, these limits are useful in that they provide a sense of the size of the parameters which might be necessary to obtain a viable model of dark matter, and allow us to focus on regions where particular classes of experiments may dominate.
Direct Detection
----------------
Direct detection experiments measure the recoil energy from WIMP-nucleus scattering, placing an upper limit on the dark matter-nucleon elastic scattering cross section. This, like all the bounds we discuss in this paper, requires coupling the dark and visible sectors, and so limits on the scattering cross section provide a constraint on the combination of couplings $g_\chi g_v$. The pseudoscalar model has no velocity or momentum independent scattering cross section with protons and neutrons, and so has no significant limits from direct detection. However, assuming Dirac dark matter, the scalar mediator induces a spin-independent cross section and so the model parameters are constrained by a number of experiments. The strongest bounds at present come from LUX [@Akerib:2013tjd] for $m_\chi \gtrsim 6$ GeV and, at lower dark matter masses, by CDMS-lite [@Agnese:2013jaa].
The fundamental Lagrangian parameters are translated into dark matter-nucleon scattering cross sections using $$\begin{aligned}
\sigma_{\chi-p,n} & = & \frac{\mu^2}{\pi} f_{p,n}^2, \\
f_{p,n} & = & \sum_{q=u,d,s} f_q^{p,n}\frac{m_{p,n}}{m_q} \left(\frac{g_\chi g_v y_q}{\sqrt{2}m_\phi^2} \right) + \frac{2}{27} f_\text{TG}^{p,n} \sum_{q=c,b,t} \frac{m_{p,n}}{m_q} \left(\frac{g_\chi g_v y_q}{\sqrt{2}m_\phi^2} \right),\end{aligned}$$ where $\mu$ is the dark matter-nucleon reduced mass, and the parameters $f^{p,n}_q$ and $f^{p,n}_\text{TG}$ are proportional to the quark expectation operators in the nucleon. These must be extracted from lattice QCD simulations [@Belanger:2008sj; @Young:2009zb; @Toussaint:2009pz; @Giedt:2009mr; @Fitzpatrick:2010em], and we adopt the values from Ref. [@Fitzpatrick:2010em]. For the purposes of this paper, there is no significant difference between the proton and neutron $f_{p,n}$, and so our dark matter scattering is essentially isospin-conserving.
The finite width is not relevant to these constraints (barring widths of order $m_\phi$), so the bound is placed on the combination $g_\chi g_v$ as a function of dark matter and mediator masses, independent of width. In Figure \[fig:direct\_bounds\], we show the upper limits placed by LUX and CDMS-lite at the 95% confidence level (CL) on the coupling combination $g_\chi g_v$, as a function of the scalar mediator and dark matter masses. The discontinuity visible at $m_\chi \sim 6$ GeV is a result of the sharply weakening LUX bounds being overtaken by the CDMS-lite constraint. As we will continue to do throughout this paper, we include limits on the combination of couplings well above the perturbativity bound $g_\chi g_v \gtrsim4\pi$. Clearly, such enormous couplings are not part of a sensible perturbative quantum field theory. We include them for completeness, and to allow some comparison of the sensitivity of the different classes of experiments.
![Contour plot of 95% CL upper bounds on the coupling combination $g_\chi g_v$ from LUX [@Akerib:2013tjd] and CDMS-lite [@Agnese:2013jaa] direct detection searches on the scalar mediator benchmark model as a function of the mediator mass $m_\phi$ and dark matter mass $m_\chi$. \[fig:direct\_bounds\]](./dd_bounds_scaled.pdf){width="0.6\columnwidth"}
Indirect Detection
------------------
Indirect detection searches look for dark matter annihilating to Standard Model particles in the Universe today. Such processes could be seen by finding an otherwise unexplained excess of gamma rays or positrons coming from an area of expected high dark matter density. While direct detection searches place non-trivial limits on scalar mediator models, such models result in thermally averaged cross sections $\langle \sigma v\rangle$ which are proportional to $v^2$. The velocity $v$ of dark matter today is very small $\lesssim 10^{-2}c$, and so scalar mediators do not result in significant signals in indirect searches. The velocity-averaged annihilation cross section into Standard Model fermion final states for our two benchmark models are [@Buckley:2013jwa] $$\begin{aligned}
\langle\sigma v\rangle(\chi\bar{\chi} \to \phi^* \to f\bar{f}) & = & \sum_{f} N_f \frac{3g_\chi^2 g_v^2 y_f^2 (m_\chi^2-m_f^2)^{3/2}}{8\pi m_\chi^2\left[ (m_\phi^2 - 4m_\chi^2)^2 + m_\phi^2 \Gamma_\phi^2 \right]} T \\
\langle\sigma v\rangle(\chi\bar{\chi} \to A^* \to f\bar{f}) & = & \sum_{f} N_f \frac{g_\chi^2 g_v^2 y_f^2}{4\pi \left[ (m_A^2 - 4m_\chi^2)^2 + m_A^2 \Gamma_A^2 \right]} \left [m_\chi^2 \sqrt{1-\frac{m_f^2}{m_\chi^2}} +\frac{3m_f^2}{4 m_\chi \sqrt{1-\frac{m_f^2}{m_\chi^2}} } T \right]\end{aligned}$$ Here, $N_f$ is the number of colors of the fermion $f$, and $T$ is the temperature of the dark matter. As $T\propto v^2$, of our two simplified models, only the pseudoscalars have a thermal annihilation cross section with a velocity-independent term. Thus, only the pseudoscalar mediator gives significant annihilation in the Universe today with non-trivial bounds set by indirect detection.
Of particular interest, due to their sensitivity to multiple decay channels, are indirect searches for gamma-ray annihilation, either from direct annihilation (resulting in gamma rays with a characteristic energy of $E_\gamma = m_\chi$), or from a cascade of Standard Model decays after annihilation into heavy, charged, and unstable Standard Model particles, which provide a continuum of gamma rays. For gamma-ray energies (and thus dark matter masses) below approximately a TeV, the Fermi Gamma-Ray Space Telescope (FGST) provides the best bounds at present [@Abdo:2010ex; @GeringerSameth:2011iw; @Ackermann:2011wa; @Geringer-Sameth:2014qqa]. In particular in this paper we will use the bounds set by the FGST in Ref. [@Ackermann:2011wa], searching for dark matter annihilation in dwarf spheroidal galaxies orbiting the Milky Way (see also Ref. [@GeringerSameth:2011iw] for an independent analysis). At the moment these are the most constraining.
We comment that there is an excess of gamma rays from the Galactic Center reported in the FGST data-set [@Goodenough:2009gk; @Hooper:2010mq; @Hooper:2011ti; @Boyarsky:2010dr; @Abazajian:2012pn; @Hooper:2012sr; @Hooper:2013rwa; @Gordon:2013vta; @Huang:2013pda; @Abazajian:2014fta; @Daylan:2014rsa]. Though the source of these gamma rays is still uncertain [@Abazajian:2010zy; @Wharton:2011dv; @Hooper:2013nhl; @Carlson:2014cwa], if interpreted in terms of dark matter, it could be be accommodated by annihilation through a pseudoscalar mediators with Standard Model couplings proportional to Yukawas [@Abdullah:2014lla; @Basak:2014sza; @Berlin:2014pya; @Arina:2014yna; @Cheung:2014lqa; @Balazs:2014jla], as in our benchmark simplified model.
![Contour plot of 95% CL upper bounds on $g_\chi g_v$ derived from indirect detection constraints set by the FGST dwarf spheroidal analysis [@Ackermann:2011wa] in the $b\bar{b}$ channel, as a function of the pseudoscalar mediator mass $m_A$ and the dark matter mass $m_\chi$. The width is set assuming $g_v = g_\chi$, which is relevant only near resonance. \[fig:indirect\_bounds\]](./id_bounds_scaled.pdf){width="0.59\columnwidth"}
In this paper, we use only the 95% CL upper limits on the indirect annihilation cross section into pairs of $b$-quarks from the FGST dwarf analysis [@Ackermann:2011wa], converted to limits on our model parameters by calculating the velocity averaged cross section $\langle \sigma v\rangle$ (see Ref. [@Buckley:2013jwa] for details) evaluated at $v \to 0$. Constraints on $g_\chi g_v$ are shown in Figure \[fig:indirect\_bounds\]. The width $\Gamma_A$ can play an important role here near resonance, so to reduce the parameter space we choose a width under the assumption that the two couplings are equal. This has only a minor effect on the majority of the parameter space. We further assume that no other annihilation channels are present.
Thermal Relic Abundance
-----------------------
By measuring CMB anisotropies, surveys such as the Planck mission have measured the dark matter contribution to the Universe’s energy budget to be $\Omega_{\chi}h^2 = 0.1187 \pm 0.0017$ [@Ade:2013zuv]. From standard Boltzmann relic density calculations [@Kolb], this implies a thermal annihilation cross section of $\langle \sigma v\rangle \sim 3 \times 10^{-26}$ cm$^3$/s. If we assume that $\phi$ is the only connection between the dark and visible sectors, and we further assume that the dark matter is a thermal relic, we can calculate the couplings $g_\chi$ and $g_v$ necessary for the production of the observed density of dark matter.
As with indirect detection, near resonance ($m_\phi \sim 2 m_\chi$) we must assume knowledge of the mediator width $\Gamma_{\phi(A)}$. We make the same assumption as before: that the width is calculated as if $g_v = g_\chi$. Annihilation near resonance can have significant effects on the cross section during thermal freeze-out, which we take into account using the methods outlined in Ref. [@3Excpns]. Away from resonance, the thermally averaged cross section becomes identical to that calculated for the indirect detection constraints, evaluated at the freeze-out temperature $T_f = m_\chi/x_f \sim m_\chi/ 25$.
Additionally, when $m_\phi < m_\chi$, dark matter can annihilate in the process $\bar{\chi}\chi \to \phi \phi$, followed by decay of the $\phi$. Thus a thermal relic can be obtained even when $g_v \sim 0$, as long as the $\phi$ is not sufficiently long-lived as to decay after Big Bang Nucleosynthesis. Such detector-stable particles are completely consistent as a dark matter mediator, but may require searches targeted towards displaced vertices. For the purposes of this paper, will not consider these models in more detail here, though the possibility should not be ignored. The required combinations of couplings $g_\chi g_v$ in order to obtain a thermal abundance are shown in Figure \[fig:thermal\_bounds\], assuming the only open channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$. We again emphasize that the regions of mass and coupling parameter space that do not yield a correct thermal relic under our specific set of assumptions are still of great interest, and so these constraints should not be taken as the final word on dark matter physics. Recall that we are discussing a simplified scenario, which presumably fits into a larger model of the dark sector. If the couplings are too small to give the correct relic abundance, then the simplified model predicts an over-abundance of dark matter from thermal processes. However, entropy dilution could reduce the dark matter density, if the physics in the Early Universe is non-standard [@Hooper:2013nia]. Somewhat more prosaically, the full theory of the dark sector could contain additional mediating particles that increase the annihilation cross section [@ArkaniHamed:2008qn]. If the couplings under consideration are larger than required for thermal annihilation, then non-thermal models of dark matter (such as asymmetric dark matter) are an attractive possibility [@Kaplan:2009ag; @Cohen:2009fz; @Belyaev:2010kp; @Davoudiasl:2010am; @Buckley:2010ui; @Buckley:2011kk].
![Required values of $g_\chi g_v$ as a function of mediator mass $m_{\phi(A)}$ and dark matter mass $m_\chi$ assuming that dark matter is a thermal relic and the only annihilation channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$, for the scalar (left) and pseudoscalar (right) simplified models. \[fig:thermal\_bounds\]](./thermal_scalar.pdf "fig:"){width="0.42\columnwidth"} ![Required values of $g_\chi g_v$ as a function of mediator mass $m_{\phi(A)}$ and dark matter mass $m_\chi$ assuming that dark matter is a thermal relic and the only annihilation channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$, for the scalar (left) and pseudoscalar (right) simplified models. \[fig:thermal\_bounds\]](./thermal_pseudo.pdf "fig:"){width="0.42\columnwidth"}
Collider bounds {#sec:collider_bounds}
===============
Having placed bounds on our simplified models from direct detection, indirect detection, and under the assumption that the dark matter obtains the thermal relic abundance, we now turn to bounds from the LHC. The most obvious signature for dark matter at colliders is missing transverse momentum (more colloquially, missing transverse energy). When dark matter is produced it escapes the detector unseen, leaving an imbalance of momentum which can be measured in the transverse plane. This missing transverse momentum is a powerful signature for new physics models. MET signatures must be accompanied by some associated production of visible particles, both for momentum conservation and triggering. We consider three signatures in this paper: MET with associated untagged jets, MET with two associated dileptonic tops, and MET plus one or two $b$-tagged jets.
In all these searches, we follow our previous policy of setting upper bounds on the combination $g_v g_\chi $. However, unlike the previous examples, the branching ratios of the mediators $\phi$ or $A$ are integral to the bounds set. By setting the limit on the combination of couplings, the mediator width $\Gamma_{\phi(A)}$, which depends on $g_\chi^2$ and $g_v^2$ separately, must be specified as an independent parameter.
Both the simplified models and EFTs can consider scenarios where the mass hierarchy is inverted ($2m_\chi > m_{\phi(A)}$). For EFTs, this makes no difference (other than bringing into question the applicability of the effective operator approach). However, in our simplified models, if the mediator is light enough to be produced at a collider, but the dark matter is heavy enough so that it cannot be the product of on-shell decay of the mediator, then it is likely that better search strategies would be those based around the decays of the mediator into visible final states. For heavy mediators ([*i.e.*]{} $m_{\phi(A)} \gtrsim 1$ TeV at the LHC) the searches for dark matter with masses satisfying $2m_\chi < m_{\phi(A)}$ would be reliant on the off-shell mediator production. For scalars and pseudoscalar mediators, however, the current constraints in this regime from the LHC turn out to be extremely weak. As a result, in this paper, we will concentrate on the $m_{\phi(A)} > 2m_\chi$ regime, and leave the remainder of the mass plane for future work.
Considering the importance of the width on the collider constraints for much of the accessible parameter space, we chose to parametrize the derived limits on $g_\chi g_v$ at fixed dark matter and mediator masses, varying the width $\Gamma_{\phi(A)}$. We choose two mediator masses: ${m_{\phi(A)} = 100}$ GeV, and 375 GeV, and $m_\chi = 40$ GeV. For on-shell mediator production, the bounds could be easily extrapolated to other dark matter masses (up to the kinematic limit $2m_\chi = m_{\phi(A)}$) by rescaling the overall branching ratio into dark matter at a new mass point. Recall that the kinematic suppression for scalars ($\beta^3$) will be more significant than that of pseudoscalars ($\beta$) for the 100 GeV benchmark, as a 40 GeV dark matter particle is near the kinematic threshold.
Mono-jet Search
---------------
At a hadron collider, unless the mediator has large couplings to $W/Z/\gamma$ compared its coupling to the colored partons, we would expect the strongest constraints to come from the production of dark matter in association with an initial state jet radiation [@Goodman:2010ku; @Beltran:2010ww; @Fox:2011pm; @Goodman:2010yf; @Rajaraman:2011wf; @Fox:2012ee]. Both ATLAS [@ATLASmonojet] and CMS [@CMSmonojet] have performed dedicated “monojet” searches using Run-I LHC data at $\sqrt{s} = 8$ TeV. We note that the “monojet” moniker is something of a misnomer, as these analyses do allow a second high-$p_T$ jet in the sample.
We use results from CMS [@CMSmonojet] to derive bounds on couplings for our benchmark models. The CMS search used a data sample corresponding to an integrated luminosity of $19.5$ fb$^{-1}$. Events are required to have one jet with $p_{Tj} > 110$ GeV. A second jet is allowed, but no more than two jets with $p_{Tj}> 30$ GeV. Signal events are grouped into seven MET bins: $\slashed{E}_T > 250$, 300, 350, 400, 450, 500, and 550 GeV. The CMS Collaboration has provided the number of events in each bin that can be accommodated as signal at the 95% CL, which we use to place bounds on $g_\chi g_v$ as a function of $m_{\phi(A)}$, $m_\chi$, and $\Gamma_{\phi(A)}$, using the most constraining limit from any of the seven MET signal bins.
As we showed in Section \[sec:models\], the treatment of the $g$–$g$–$\phi(A)$ interaction as an effective operator would introduce significant errors in the extrapolated bounds on the model parameters. Hence, accurate distributions of MET and jet $p_T$ require simulation of $\phi$ or $A$ plus a hard parton including the exact heavy-quark loop effects. We implement this in [MCFMv6.8]{} [@Campbell:2010ff; @hj], modifying the process ${pp\rightarrow H(A)+j \rightarrow \tau^+ \tau^-+j}$ in [MCFM]{} to produce events files which can be subsequently showered and hadronized by [Pythia8]{} [@Sjostrand:2006za; @Sjostrand:2007gs], then fed into a detector simulator. Note that, while the CMS analysis allows a second jet, our [MCFM]{} simulation is limited to one hard parton, though additional jets are generated through the [Pythia8]{} parton shower. See Refs. [@hjets; @Campanario:2010mi; @Buschmann:2014twa] for issues pertaining the simulation of the second jet including the top mass effects. In addition, we generalized the [MCFM]{} implementation including the possibility of off-shell mediator production. As there are no full Next-to-Leading order (NLO) predictions including the top mass effects for this process in the literature, we include these effects via a flat correction factor $K\sim 1.6$ obtained using the infinite top mass limit [@kfactor] . Our hard scales are defined as $\mu_F^2=\mu_R^2=m_{\phi(A)}^2+p_{Tj}^2$, and we used the [CTEQ6L1]{} parton distribution functions [@cteq].
![Missing transverse momentum differential cross sections for the scalar (left panel) and pseudoscalar (right panel) mediators. The leading order effective gluon couplings are shown as dashed lines, and the exact loop-induced calculations are solid. We assume the LHC at 8 TeV. \[fig:pt\_distributions\]](./etmiss_scalar.pdf "fig:"){width="0.4\columnwidth"} ![Missing transverse momentum differential cross sections for the scalar (left panel) and pseudoscalar (right panel) mediators. The leading order effective gluon couplings are shown as dashed lines, and the exact loop-induced calculations are solid. We assume the LHC at 8 TeV. \[fig:pt\_distributions\]](./etmiss_pseudo.pdf "fig:"){width="0.4\columnwidth"}
![Lower limit on the coupling $g_\chi g_v$ set by the CMS monojet search as a function of dark matter mass $m_\chi$, assuming mediators of 100 GeV, $\Gamma_{\phi(A)}/m_{\phi(A)} = 10^{-3}$, and exclusively on-shell production of dark matter. The constraint on the scalar mediator is shown in red and pseudoscalars in blue. \[fig:dm\_dependence\]](./dm_dependence.pdf){width="0.48\columnwidth"}
Whereas the primary effect on the bounds placed on the combination $g_\chi g_v$ from varying the width $\Gamma_{\phi(A)}$ is just a rescaling of the branching ratio to dark matter, there can be small secondary effects when the width is significant compared to the mediator mass. To investigate these effects, as well as demonstrate the importance of the full simulation on the bounds, we also generate dark matter events in our two simplified models using [MadGraph5]{} [@Alwall:2011uj; @Alwall:2014hca]. This implementation starts with the inclusion of our [*Simplified Model*]{}, presented in Eq. , into [Feynrules]{} [@Alloul:2013bka] which generates a model file that is subsequently used by [MadGraph]{}. In [MadGraph]{} we produce $\phi(A)$ events matched up to two jets via the MLM scheme [@Mangano:2006rw]. We also include the detector simulation through [Delphes3]{} [@deFavereau:2013fsa]. In Figure \[fig:pt\_distributions\], we compare the distributions for the leading jet $p_T$ and the MET in the narrow width approximation generated by both [MCFM]{} and [MadGraph5]{}, after the CMS event selection criteria. As in Figure \[fig:etmiss\_intro\], the effective gluon operator overestimates the distribution tails, which would lead to an overly aggressive bound on the couplings. Notice that these differential distributions do not differ from the exact result by just a flat factor, but have different shapes. While these effects are important here, they will be even more critical in future LHC runs, where the energies will be higher and the MET cuts will be harsher. To confirm the consistency of our implementation, we have produced results in the EFT limit $(m_t\rightarrow \infty,m_\phi\rightarrow \infty)$ and validated it against the CMS EFT bounds [@CMSmonojet].
Using these simulations, we place 95% CL bounds on $g_\chi g_v$ as a function of $\Gamma_{\phi(A)}/m_{\phi(A)}$, for 100 and 375 GeV mediators and $m_\chi =40$ GeV. Our results are shown in Figure \[fig:scalar\_bounds\] for the scalar mediator and Figure \[fig:pseudo\_bounds\] for the pseudoscalar. Two points from these results should be addressed in detail.
1. The different dependence on the scalar and pseudoscalar widths on $\beta$ have an important effect on the results. For the light mediator, the scalar partial width into dark matter ($\propto\beta^{3}$) significantly reduces the total cross section when compared to the pseudoscalar ($\propto\beta$). As a result, the couplings to the scalar must be larger than the pseudoscalar for the $100$ GeV mediators. For the heavy mediator, neither scenario has a significant kinematic suppression. This is dependent on our choice of dark matter mass; as the dark matter mass increases, we expect to see the scalar bounds weaker faster than the pseudoscalar. This is explicitly an effect due to on-shell production of the mediator; if the dark matter mass was heavier than $m_{\phi(A)}/2$, then the monojet channel would only be sensitive to production of dark matter via an off-shell mediator, which does not scale with the kinematic suppression factor. In Figure \[fig:dm\_dependence\], we show the scaling of the monojet bound as a function of dark matter mass, assuming a 100 GeV scalar or pseudoscalar (and $\Gamma_{\phi(A)}/m_{\phi(A)} = 10^{-3}$).
2. In the case of the [MCFM]{} results, the changing width only causes a rescaling of the total rate of mediator production times decay into dark matter through the changing branching ratios. While this is the dominate effect for the finite width calculation, there is a subleading effect at $\Gamma_{\phi(A)}/m_{\phi(A)} \gtrsim 0.1$, where the tail of the mediator $p_T$ distribution (and thus the MET) can be increased relative to the narrow width approximation. This is a result of the mediator being able to be produced with $q^2$ very far away from the expected mass, convolved with the proton parton distribution functions. For the 100 GeV mediators, as the width is increased this secondary effect causes the bound on $g_\chi g_v$ to weaken less rapidly than one would expect from the branching ratio alone. The effect is negligible for the 375 GeV mediators.
Heavy Flavor Searches
---------------------
One would expect that the strongest constraint that the LHC can place on the dark matter decay channels of our benchmark scalar and pseudoscalar mediators comes from the general jets plus missing transverse energy search discussed previously, as the production cross section here is highest. However, channels with missing energy associated with particles other than untagged jets can have significantly lower backgrounds (and different systematics) than the monojets. Therefore, we can and should consider searches in additional channels. Though we will often find that limits placed on the couplings will be weaker than those placed by the monojet search, this approach is still critical as the LHC continues to ramp up to higher energies and luminosities. Recall that we are working with a simplified model, purposefully constructed to minimize the number of free parameters. Therefore, under these assumptions we can predict the exact ratio of signal strength in multiple channels, as the cross section for each is set by the same masses and couplings. However, we must be open to deviations from the simplified model. For example, if the couplings to up- and down-type couplings are not set by a universal coupling $g_v$, or if the loop-induced gluon coupling does not depend solely on the couplings to top and bottom quarks, then it is quite possible that the signal in the monojet channel could be suppressed relative to other production mechanisms. Discovery in more than one channel would also allow better understanding of the theoretical underpinnings of any new physics.
With that motivation in mind, it is clearly important to look for new physics in many associated channels. Even when considering modifications to the baseline models, it is still reasonable to assume that the interactions with fermions are largely MFV, and therefore that the mediator is most strongly coupled to the heaviest fermions. Therefore, we show here limits on production of the $\phi$ or $A$ in association with top and bottom quarks, followed by the invisible decay of the mediator into dark matter. Some of the main production diagrams for such processes are shown in Figure \[fig:feyn\_heavy\].
![Representative Feynman diagrams contributing to heavy quark flavor plus dark matter production at the LHC in our [*Simplified Models*]{}. \[fig:feyn\_heavy\]](./feyn_heavy.pdf){width="0.5\columnwidth"}
![95% CL upper limits on $g_\chi g_v$ for scalar mediators from collider searches as a function of $\Gamma_\phi/m_{\phi}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) scalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. The grayed-out region indicates where the minimum width consistent with $g_\chi g_v$ is greater than the assumed width. \[fig:scalar\_bounds\]](./scalar_combo_bounds_100_40_scaled.pdf "fig:"){width="0.5\columnwidth"}![95% CL upper limits on $g_\chi g_v$ for scalar mediators from collider searches as a function of $\Gamma_\phi/m_{\phi}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) scalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. The grayed-out region indicates where the minimum width consistent with $g_\chi g_v$ is greater than the assumed width. \[fig:scalar\_bounds\]](./scalar_combo_bounds_375_40_scaled.pdf "fig:"){width="0.5\columnwidth"}
We use the CMS dedicated search for dark matter produced in events with dileptonic tops [@CMSttbar], performed on 19.7 fb$^{-1}$ of integrated luminosity at the 8 TeV LHC. The analysis requires exactly two isolated leptons with individual $p_T > 20$ GeV and $\sum{p_t} > 120$ GeV, and at least two jets with $p_T > 30$ GeV. The invariant mass of the leptons must be greater than $20$ GeV, and if they are the same flavor, a $Z$-mass veto of $|m_{\ell\ell} - 91~\mbox{GeV}|>15$ GeV is applied. The two jets are required to have invariant mass of less than $400$ GeV. The signal region is $\cancel{E}_T > 320$ GeV. As with the monojet analysis described previously, we can straightforwardly recast the CMS limits to apply to our benchmark models, based on the number of events seen in their signal region. Signal was generated using [MadGraph5]{}, passed through the [Pythia6]{} and [Delphes3]{} pipeline described earlier. As in the monojet case, we validate our results using the dark matter EFT to compare with the CMS results. We show the bounds from this channel on $g_\chi g_v$ for our benchmark mediator models (for mediators of 100 and 375 GeV, and 40 GeV dark matter) as a function of mediator width in Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\].
Finally, we can consider the associated production of the mediator $\phi$ or $A$ with $b$-quarks. Until recently, no dedicated dark matter search similar to the monojet or dileptonic top plus MET analyses has been performed for the process $p p \rightarrow \chi \bar{\chi} +b \bar{b}$, and constraints could only be extracted using the sbottom searches $p p \rightarrow \tilde{b}^{*} \tilde{b} \rightarrow \chi \bar{\chi} + b \bar{b}$ from CMS [@CMSbbar] and ATLAS [@Aad:2013ija]. These searches have selection criteria which are far from ideal for the kinematics of the simplified models, but they do place relevant constraints directly on the tree-level interaction between $b$-quarks and the mediator.
Recently however, ATLAS has published a dedicated search for dark matter produced in associated with $b$-tagged jets in 20.3 fb$^{-1}$ of 8 TeV data [@Aad:2014vea]. Two signal categories in this search are relevant for our analysis here. In both, the analysis vetoes events with leptons that have $p_T > 20$ GeV and requires $\cancel{E}_T > 300$ GeV. The azimuthal angle between all jets and the MET must be $\Delta \phi >1$. Signal Region SR1 requires one or two jets, at least one of which must be $b$-tagged (at a $60\%$ efficiency) and have $p_T > 100$ GeV. Signal region SR2 requires three or four jets in the event, again requiring at least one to be $b$-tagged with $p_T > 100$ GeV. If a second $b$-tagged jet exists, it must have $p_T> 60$ GeV, and the second highest $p_T$ jet must have $p_T > 100$ GeV. ATLAS provides the 95% CL upper limit on the number of events in each signal region which can be accommodated by new physics, and we validate our simulation using the EFT results.
We again generate our signal events using [MadGraph5]{}, through the tree-level coupling of the mediator and the $b$-quarks. As with the monojet search, for each of our benchmark models, we use the strongest limit on $g_\chi g_v$ set by either of these signal regions.
![95% CL upper limits on $g_\chi g_v$ for pseudoscalar mediators from collider searches as a function of $\Gamma_A/m_{A}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) pseudoscalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the indirect detection limit in the $b\bar{b}$ channel from FGST. \[fig:pseudo\_bounds\]](./pseudoscalar_combo_bounds_100_40_scaled.pdf "fig:"){width="0.5\columnwidth"}![95% CL upper limits on $g_\chi g_v$ for pseudoscalar mediators from collider searches as a function of $\Gamma_A/m_{A}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) pseudoscalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the indirect detection limit in the $b\bar{b}$ channel from FGST. \[fig:pseudo\_bounds\]](./pseudoscalar_combo_bounds_375_40_scaled.pdf "fig:"){width="0.5\columnwidth"}
The results from this analysis are shown along with our previous limits as a function of mediator width in Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\]. Along with the bounds derived from colliders, we include the direct and indirect constraints (for scalar and pseudoscalar models, respectively) and the required value of $g_\chi g_v$ to obtain the thermal relic abundance. While it is a very useful benchmark to compare the experimental sensitivity, note that coupling values that diverge from that required for a thermal relic are still experimentally and theoretically interesting: as we consider only a [*Simplified Model*]{}, we do not attempt to specify the full theory. Further, we do not even know that dark matter is in fact a thermal relic. If dark matter was generated through some asymmetric process (like baryons), then one would not expect the low-energy annihilation channels to obtain a thermal abundance.
In Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], we also show the exclusion region of coupling-width parameter space that is theoretically inconsistent. While we cannot specify a width only from the coupling combination $g_\chi g_v$, we can calculate the minimum possible width (assuming only decays into the dark matter and the Standard Model fermions) that is consistent with a given value of $g_\chi g_v$. That is, for a given width $\Gamma_{\phi(A)}$, we find the minimum value of the product $g_\chi g_v$ which would allow $$\Gamma_{\phi(A)} > \frac{g_\chi^2 m_{\phi(A)} }{8\pi} \left(1-\frac{4m_\chi^2}{m_{\phi(A)}^2} \right)^{n/2}+\sum_f \frac{g_v^2 y_f^2 m_{\phi(A)}}{16\pi} \left(1-\frac{4m_f^2}{m_{\phi(A)}^2} \right)^{n/2},$$ for any values of that $g_\chi$ and $g_v$ which satisfy the product constraint (here $n= 1$ for pseudoscalars and 3 for scalars). We gray-out the regions of $g_\chi g_v$ parameter space where minimum width possible for any $g_\chi$ and $g_v$ is larger than the assumed $\Gamma_{\phi(A)}$.
Examining Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], it is interesting to note that the top constraints on the scalar mediator are competitive (within the accuracy of our simulated search) with those of the monojet channel at low mediator masses. This is due to the relative suppression of the scalar coupling to gluons compared to the coupling to the fermions Eq. . The pseudoscalar gluon coupling does not have the same level of suppression, leading to a larger production cross section in the monojet channel, and thus better bounds when compared to the heavy flavor channel. As the mediator mass increases, the production of a heavy particle in association with the two massive tops is suppressed, and the monojet constraint regains its preeminence for the scalar model.
The $b$-tagged channel places significantly weaker constraints on these models than the monojet or the top channels. However, as this probes directly the coupling to the down-sector, it would be sensitive to deviations the universal coupling assumption in a way that the top channel is not, as the top channel relies on the same coupling as the loop-induced monojet search, unless new colored particles coupling to the mediator exist in the spectrum. The direct detection constraints are also very powerful compared to the collider reach (though for dark matter masses less than $\sim 6$ GeV, the colliders are more constraining) for scalar mediators, while the pseudoscalars are much less constrained by the indirect searches, are comparable with the current LHC constraints. However, as we argued previously, multiple probes in multiple channels are still necessary, as simple modifications of the basic model or experiment-specific backgrounds and uncertainties could increase the sensitivity of one mode while decreasing another. In our search for new physics, we must exhaust all reasonable search strategies.
Higgs Mediators \[sec:higgs\]
=============================
As we have often mentioned throughout this work, there are obvious connections between our scalar and pseudoscalar simplified models and Higgs physics. In addition to the possible embedding of the simplified models into extended Higgs sectors, the couplings (both tree-level and loop-induced) even in the general scenarios have many similarities with Higgs physics (due in part to the MFV assumption). The correct technique for generation of high $p_T$ events through the gluon-mediator coupling was also inherited from Higgs physics. With these considerations, it is reasonable to ask what bounds can be set on the 125 GeV Higgs itself, assuming that it is the scalar mediator between the visible and the dark sector. This is the well-known “Higgs Portal” scenario for dark matter [@Burgess:2000yq; @Davoudiasl:2004be; @Patt:2006fw; @Andreas:2008xy; @Barger:2008jx; @Lerner:2009xg; @He:2009yd; @Barger:2010mc; @Djouadi:2011aa; @Kanemura:2010sh; @Mambrini:2011ik; @He:2011de; @Han:2013gba; @Greljo:2013wja; @Okada:2013bna; @Chacko:2013lna; @deSimone:2014pda; @Endo:2014cca; @Drozd:2014yla; @Englert:2013gz] (similarly, one could consider the “dilaton” portal [@Bai:2009ms; @Agashe:2009ja; @Blum:2014jca; @Efrati:2014aea]). Collider bounds on the 125 GeV Higgs decaying to dark matter can be placed in two ways. First, just as we have done previously, we can place limits on the total cross section from the monojet and heavy flavor channels, which can be translated into limits on the coupling of the Higgs to dark matter. Secondly, we can use the experimental measurements of the Higgs width to constrain the addition of new channels to Higgs decay.
![95% CL upper limits on $g_\chi g_v$ for the 125 GeV Higgs from collider searches as a function of the width $\Gamma$, assuming 40 GeV dark matter. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. Three vertical lines show experimental limits on the 125 GeV Higgs’ width assuming Standard Model couplings and an invisible branching ratio of 54% [@Chatrchyan:2014tja] (dotted purple), the upper limit on the width from interference with the $Z$ [@Khachatryan:2014iha] (dashed purple), and the maximum possible width from the $4\ell$ lineshape (solid purple) [@Chatrchyan:2013mxa]. \[fig:higgs\_bounds\]](./higgs_combo_bounds_40_scaled.pdf){width="0.5\columnwidth"}
We can extract constraints on the total width of the Higgs in three different ways. First, if we require that the coupling to the Standard Model is exactly that of [*the*]{} Standard Model Higgs, then by requiring the visible production and decay channels are consistent with observations, the total invisible branching ratio must be less than $0.54$ at 95% CL [@Chatrchyan:2014tja] (see also Ref. [@Aad:2014iia]). Given the Standard Model Higgs width of 4.1 MeV [@Heinemeyer:2013tqa], the addition of a decay to dark matter saturating this bound gives a total width of at most 8.9 MeV. Furthermore, as this assumes that $g_v = 1$, in this restricted subset of the model space, the dark matter coupling can be constrained to be less than $$g_\chi^2 \leq \frac{8\pi}{m_h} \left(1-\frac{4m_\chi^2}{m_h^2} \right)^{-3/2} \times \left( 8.9~\mbox{MeV} \times 0.54 \right).$$
This chain of logic does require that the Higgs couplings be exactly the Standard Model values. Somewhat weaker constraints can be placed on the invisible branching ratio once this assumption has been lifted. This does not extend to statements about the total width. Though perhaps unlikely from a theoretical standpoint, it is possible that a larger branching ratio to dark matter could be compensated by larger couplings for the production of the Higgs, leaving the rates for the observed channels unchanged [@Belanger:2013kya].
The second method of measuring the Higgs width relaxes the requirement that the couplings to the fermions and gauge bosons are as in the Standard Model, and places a bound on the width via the measured interference of the Higgs and the $Z$. This constrains the Higgs width to be $\Gamma_h < 17.4$ MeV [@CMS-PAS-HIG-14-002; @Khachatryan:2014iha]. However, as with the invisible Higgs decay measurement, this interference effect does make some assumptions about the production mechanism of the Higgs [@Englert:2014aca; @hjets]. The third method remains fully agnostic as to the Higgs couplings. This is the most robust, but least constraining measurement: the direct measurement from the $h \to ZZ^* \to 4\ell$ channel, which has measured $\Gamma_h < 3.4$ GeV [@Chatrchyan:2013mxa].
In Figure \[fig:higgs\_bounds\], we show the collider and direct detection constraints on the 125 GeV Higgs boson as a function of total width, assuming a coupling to dark matter $g_\chi$ (unlike Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], note that the horizontal axis is $\Gamma_h$, not $\Gamma_{\phi(A)}/m_{\phi(A)}$). As before, we parametrize the coupling to the Standard Model fermions as $g_v$. Given the present concordance between experiment and theory, the primary model-building focus for Higgs physics appears to be concentrating on scenarios with $g_v\sim 1$, and it appears to be difficult to find models where large deviations from the Standard Model prediction is consistent with all Higgs data in a realistic extension of the Standard Model [@sfitter]. As we saw in the general scalar mediator, the collider bounds are much less constraining than those set by direct detection experiments. While the collider constraints are relatively insensitive to dark matter masses below $m_h/2$, the direct detection bounds weaken significantly significantly if the dark matter is below $\sim 6$ GeV.
It is surprising to see that the associated top channel is comparable here to the monojets, given the experimental difficulties in probing Standard Model $tth$ production. However, recall that the Standard Model search in this channel is forced to rely on $h\to b\bar{b}$ decay. If we assume a significant branching ratio of the 125 GeV Higgs into invisible dark matter, the much lower backgrounds in the dileptonic top plus MET channel allow the experiments to set a bound comparable to that of the monojets.
Conclusions
===========
The next few years of data from the LHC Run-II has the potential to shed new light on the nature of dark matter. The EFT formalism has been very useful in the analysis of Tevatron and LHC data, allowing straightforward comparisons to direct and indirect searches, and moving dark matter searches in a more model-independent direction. However, the powerful bounds set by the LHC push the theory into a regime where the EFT often does not generally apply. This should be a cause for optimism: the break-down of the consistency of the EFT implies that, for much of the parameter space, if the LHC can produce dark matter then it can also produce associated particles that mediate the interaction between the dark sector and our own.
Previous works have introduced various simplified models which bridge the theoretical divide between the EFT and complete models such as supersymmetry. We add to this work by constructing two benchmark models of spin-0 mediators coupling to dark matter consisting of Dirac fermions. While such attractive models have been considered in the past, we – for the first time – provide a comprehensive set of constraints from direct detection, indirect searches, and three collider channels associated with missing transverse energy.
As previous works have noted, care must be taken when simulating scalar mediated missing energy searches at the LHC, as these are primarily produced through a top-loop induced coupling to gluons. As the transverse momentum flowing through this loop is large compared to $2m_t$ (and may be large compared to the mediator mass), it has been demonstrated that working in approximations of infinite top mass and/or on-shell gluons can incorrectly predict the MET and $p_T$ distributions. In this paper, we clearly show the impact of these effects on the distribution of jet $p_T$ and MET, which are critical to missing energy searches at the LHC, and outline appropriate techniques for simulating these models. These issues will become even more important in future LHC runs, where higher energies will force harsher MET and $p_T$ cuts, further increasing the deviation between the distributions predicted by an effective operator treatment of the loop-coupling, and the correct one.
For our benchmark models, the monojet channel remains the most constraining out of all the collider bounds. However, associated heavy flavor searches are important; associated production with tops can rival the monojet channel in the low mediator mass region. As such these additional searches should be pursued as complimentary to the monojet bounds, sensitive to different combination of couplings. Similarly, the direct detection bounds place much more powerful limits on the couplings for scalar models, assuming the dark matter mass is heavier than $\sim 6$ GeV. However, there are astrophysical uncertainties inherent to direct detection limits, and the LHC searches provide an complimentary testing ground, one that independent of the uncertainties on our local dark matter density and velocity distribution. Similar astrophysical uncertainties also relate to bounds placed by indirect detection, and further collider searches may be a key factor in resolving the active debate about claimed signals from the Galactic Center. As can be seen from Figure \[fig:pseudo\_bounds\], the current constraints already touch on the relevant parameter space for $m_\chi \sim 40-50$ GeV, and can indeed rule out simplified models with mediators much heavier than 100 GeV as the source of the anomaly. Though modifications of the benchmark simplified model can explain the Galactic Center excess with particles that have vanishing LHC cross sections [@Abdullah:2014lla; @Cheung:2014lqa], it is interesting that one of the simplest scenarios is not yet ruled out, yet lies within realistic reach of the LHC in the near future.
The searches we extracted bounds from in this paper were pre-existing and easily adapted to our simplified models. However, as should be clear, many other possible channels exist, which would place complimentary bounds on the couplings of our benchmark models. In addition to further missing energy searches in association with heavy flavor – in particular, searches with $\tau$ leptons, which would probe the mediator-lepton coupling – we suggest that future work should also consider the constraints from decays of mediators back into Standard Model particles.
Given couplings $g_\chi$ and $g_v$ which are of the same order of magnitude, one would expect decays to dark matter to dominate. However, it is possible that $g_v \gg g_\chi$, or that the dark matter itself is kinematically inaccessible as a decay product of the mediator. In this second case, though some missing energy constraints can be placed from dark matter production via off-shell mediators, the collider production cross section of the mediator itself would be far higher. Channels with decays to $b\bar{b}$, $\tau \tau$, top pairs, or the experimentally clean $\gamma\gamma$ signatures are all likely candidates for dark matter simplified models, particularly with the spin-0 mediators considered here. If the width is small, than long-lived mediators are possible, and searches for displaced decays back to visible particles could place important limits on models with small $g_v$ and $g_\chi$ which would be otherwise inaccessible. CMS, ATLAS, and Tevatron have performed searches in some of the prompt channels, though their results are typically presented in terms of two-Higgs doublet models. As we have described in this paper, such signatures can be relevant to a large range of models, and could be an important part of our search for the new physics of dark matter.
Acknowledgements {#acknowledgements .unnumbered}
================
MRB thanks Tim Tait, Dan Hooper, and Maria Spiropulu for helpful comments and suggestions.
[99]{}
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[^1]: If referring to both the scalar and pseudoscalar models simultaneously, we will use mediator mass $m_{\phi(A)}$ and mediator width $\Gamma_{\phi(A)}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A massive star undergoes a continual gravitational collapse when the pressures inside the collapsing star become insufficient to balance the pull of gravity. The Physics of gravitational collapse of stars is well studied. Using general relativistic techniques one can show that the final fate of such a catastrophic collapse can be a black hole or a naked singularity, depending upon the initial conditions of gravitational collapse. While stars are made of baryonic matter whose collapse is well studied, there is good indirect evidence that another type of matter, known as dark matter, plays an important role in the formation of large-scale structures in the universe, such as galaxies. It is estimated that some eighty-five percent of the total matter in the universe is dark matter. Since the particle constituent of dark matter is not known yet, the gravitational collapse of dark matter is less explored. Here we consider first some basic properties of baryonic matter and dark matter collapse. Then we discuss the final fate of gravitational collapse for different types of matter fields and the nature of the singularity which can be formed as an endstate of gravitational collapse. We then present a general relativistic technique to form equilibrium configurations, and argue that this can be thought of as a general relativistic analog of the standard virialization process. We suggest a modification, where the Top-Hat collapse model of primordial dark matter halo formation is modified by using the general relativistic technique of equilibrium. We also explain why this type of collapse process is more likely to happen in the dark matter fields.'
author:
- Dipanjan Dey
- 'Pankaj S. Joshi'
title: '**Gravitational Collapse of Baryonic and Dark matter**'
---
Introduction
============
Stars are born in the cloud of gas and dust which is known as nebulae. Nebulae are primarily made of hydrogen gas. Slightly overdense regions in that gas cloud become more dense due to their own gravitational pull. Now as hydrogen is baryonic matter, it slowly radiates energy as it is falling under its gravitational potential. During this time the radiation pressure is not enough to balance the gravitational pull. As time goes by, the gas becomes more dense and hotter. At last the time comes when the temperature becomes high enough to fuse two hydrogen atoms into a helium atom. This nuclear fusion releases a huge amount of radiation which resists the gas cloud to collapse further and a star is born. Stars shine as they burn their nuclear fuel mainly hydrogen, fusing it into helium and later other heavier elements. After shinning for millions of years, the time comes when the star runs out of the fuel and cannot resist its own gravitational pull, and it finally collapses under its own gravity.
The more massive a star is, the shorter the lifespan it has. For massive stars like 10 to 20 times the solar mass, they exhaust their nuclear fuel far quickly than our sun. For the sun-like stars, the hydrogen in the central region is used up first to make a core of helium, and then the hydrogen in the spherical shell around the core begins to fuse. This causes the star to grow gradually in size until the red giant phase is achieved. After that, the inner core collapses further and creates more heavy elements. The outer shells of star explode and create a planetary nebula. The inner core stops collapsing when the quantum pressure of degenerate electron gas balances the gravitational force. This central dense, compact object is known as a white dwarf. There is a mass limit of stars for which the final state of gravitational collapse is a white dwarf. The limit is approximately 1.4 times the solar mass. This mass limit was set by Subrahmanyan Chandrasekhar in 1934 [@Chandra1]. Stars having masses greater than that limit causes further collapse of the core. Generally, the outer layers of these massive stars explode as a supernova explosion. The collapsing core settles down to another equilibrium stage which is caused by quantum pressure of neutrons, and in this way, a neutron star is born [@Lieb],[@Latt],[@annu]. This neutron star is barely ten to twenty kilometers in size. For more massive stars having masses greater than about ten solar masses, the above-mentioned quantum pressures become insufficient to balance the gravity. Then the continual gravitational collapse of the star becomes inevitable.
Massive stars which have a mass of the order of about ten solar masses or more burn much faster, and they are far more luminous than our sun-like stars. The lifespan of these massive stars cannot be more than ten to twenty million years, whereas, for a sun-like star, the lifespan is in billions of years. So one of the most important questions in astronomy and astrophysics today is to investigate the final fate of these very short-lived massive stars. General relativity indicates that this type of continual gravitational collapse terminates into a space-time singularity if certain energy conditions hold throughout the collapse process. However, the presently available results so far in general relativity cannot predict the nature of such a singularity. One of the most intriguing problems in today’s gravitational physics is whether the central ultra-high density region could be visible to outside observers, or these should always be covered by an event horizon of gravity. We will discuss this further in section (\[ccc\]).
As we know, stars are made of baryonic matter. Besides gravitational interaction, baryonic matter particles can also interact with each other by means of strong, weak, and electromagnetic forces. Only with gravitational force, the stars would not form any kind of bound states. Due to these three interactions, the star can radiate away energy, and therefore it can go to the lower energy states in a gravitating system. This process is responsible for the formation of stars, planets as well. Now, the situation becomes more interesting if we give our attention to larger scales, i.e. the galactic scales and the cosmological scale, where in fact the dark matter could dominate.
A typical sun-like star has a radius from about $10^5$ to $10^6$ km. The largest stars have radii of $10^{10}$ km approximately. On the other hand, the galactic scale ranges from the order of 10 to 100 kiloparsec (kpc), which is the scale of $10^{16}$ km and higher. The cosmological scales start from 100 megaparsecs (mpc). The galaxy clusters scale falls in between the galactic scale and cosmological scale.
In the galactic scale, it is observed that stars, rotating around the center of a galaxy, have an orbital velocity which remains constant or ‘flat’, with increasing distance away from the galactic center [@Rubin],[@Rubin2]. This was a highly surprising result, as this kind of dynamics cannot be seen in a planetary system. In a planetary system, the velocities of planets decrease as the radial distance from the star increases. Flat galactic rotation curves indicate that each galaxy is surrounded by a significant amount of unknown matter which is not actually seen, known as dark matter. It is estimated that the total dark matter mass should be around three times the total mass of baryonic mass in a galaxy. It is roughly considered that dark matter has a spherical shaped halo-like structure enshrouding each galaxy. It is named as “dark matter” because it does not appear to interact with the usual standard model particles, and therefore it is very difficult to detect a dark matter particle by the modern detectors.
Therefore, Dark matter is supposed to interact with ordinary matter only through gravity. So, only by its gravitational effects, we can gather information, such as its positions, mass, density profiles, etc. The existence of dark matter also can be proved indirectly by the large-scale structure in the cosmological scale. If there was no dark matter, then it can be shown that only baryonic matter would not lead to the present large-scale structure of the universe. Due to the inert property of dark-matter, the structures made by it can be distinguished from the structures made by baryonic matter. One could ask, how did the dark-matter start to form its structures. How did it form its halo-like structure around the galaxy? What are the internal properties of the dark matter halo? As we do not know what the dark matter is, these are among the most difficult questions of cosmology.
In its cosmological scale, our universe is almost flat and homogeneous. In that scale, one can explain the dynamics of our universe by spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Perturbations in that metric are seeded by the matter density perturbations of our universe. The quantum fluctuations during the inflationary era are responsible for such density perturbations. During the inflationary era, the universe expanded exponentially, but the Hubble horizon remained constant. Therefore, quantum perturbation modes during the inflationary era left the Hubble horizon and reentered the horizon after the inflationary era. The perturbation modes became classical before they reentered the horizon. Therefore, the perturbation modes which enter the Hubble horizon at the end stage of radiation domination create density perturbation in matter [@Hogan:1985bc],[@lidd] [@sengor],[@white].
As the baryonic matter decoupled after dark matter decoupling, the perturbations in dark-matter started to grow before the growth of perturbations in baryonic matter field. The inert property of dark matter helps it to decouple first and form first primordial structures in our universe. The dynamics of density perturbations in dark matter can be explained by linear perturbation theory as long as the density contrast $\frac{\delta\rho}{\bar{\rho}}<1$, where $\delta\rho=\rho-\bar{\rho}$ and $\bar{\rho}$, $\rho$ are the background density and total density of perturbed area respectively. However, when this density contrast becomes almost one, linear perturbation theory is insufficient to explain the dynamics of the perturbation modes. That is the time when perturbation modes entered into the non-linear regime and form overdense patches in our universe. As we have mentioned, in the cosmological scale we use spatially flat FLRW metric to describe the dynamics of homogeneous, isotropic and expanding universe. Therefore, the background of the overdense regions is expanding. It is generally considered that as the background FLRW universe expands in a homogeneous and isotropic fashion, these overdense matter distributions also expand initially in an isotropic and homogeneous fashion, but with a slower rate. As the density contrast of the overdense regions increases with time, they gradually detach from the cosmic expansion. As we have mentioned earlier, dark matter forms its structure before the formation of baryonic structures. Therefore, these overdense patches can be thought of as a very early stage of primordial dark-matter halo. Initially, these hallo-like structures expand with the background, then the time comes when it stops expanding and starts collapsing due to its own gravitational pull [@Frenk],[@Cooray].
As we have discussed, except gravity, baryonic matter can interact with each other by the extra three forces. Therefore, they can form different types of complicated celestial objects. On the other hand, dark matter does not show any observational evidence of interaction with any baryonic matter, except gravity. Therefore, it is unable to radiate energy. As it cannot radiate energy, it cannot form any bound objects at stellar scale, like stars, planets, etc. Dark matter can form structures at galactic scale or higher. As we do not know the particle constituent of dark matter, it is very difficult to predict the final fate of dark matter collapse. Generally, dark matter is considered as collisionless cold matter. Here cold matter means dark matter particle has velocity much less than the velocity of light. This is known as cold dark matter (CDM) model [@DelPo]. This model achieves great success in explaining how the large-scale structure of the universe evolves with time. In cosmological scales, this model has excellent agreement with observational results. However, this model is unable to solve some problems in the galactic scale. For example, it does not have a satisfactory explanation of why the center of the galaxy is core-like, why it is not cuspy at the center. There are also other problems like: ‘missing satellite’ problem, ‘too big to fail’ problem, and others.
There is a modified version of CDM, and it is known as the $\Lambda$ CDM [@DelPo], where $\Lambda$ stands for the cosmological constant. To incorporate the accelerated expansion of the universe in the CDM model this modified version has been proposed. However, the problems in the galactic scale remain unsolved [@Bullock],[@Weinberg]. To solve these problems, self-interacting dark matter (SIDM) model is proposed [@Spergel]. In this model, the dark matter particles are considered to be self-interacting, which cannot be neglected. This model has more promising solutions for those problems in galactic scale.
All these models have a common assumption for the equilibrium state of collapsing dark matter halo. They all use the so-called virialization technique to explain the final equilibrium state of a dark matter collapse [@Merr]. The Virial theorem generally gives a relation between the average kinetic energy of a stable system consisting of $N$ number of particles, which is bound by some potential energy, with that of the total potential energy. A system is said to be virialized when the particles of the system have some dynamics or kinetic energy, but overall the system is static. It can be mathematically proved that virialization should occur after a very large time only. When the particles in a gravitating system have only gravitational interaction between each other, then at virialized state the system should follow, $$\langle T\rangle_{\tau\rightarrow\infty} = -\frac12\Bigg\langle \sum_{i=1}^{N}{\bf F}_i\cdot{\bf r}_i \Bigg\rangle_{\tau\rightarrow\infty}\, ,
\label{vir1}$$ where ${\bf F}_i$ is net the force on the $i$th particle, ${\bf r}_i$ is the position vector of $i$th particle and $T$ is the total kinetic energy of the system. Here, the time averaging is done over time $\tau$. The time $\tau$ should be very large so that the system can achieve the virialized state. For a gravitating system we can write, $$\sum_{i=1}^{N}{\bf F}_i\cdot{\bf r}_i = -V_T\, ,
\label{vir2}$$ where $V_T$ is the total potential energy of the system. Using eqs. (\[vir1\]), (\[vir2\]) we finally get the virialization equation, $$\langle T\rangle=-\frac12 \langle V_T \rangle\, .$$ For the CDM case, system virializes through different processes: phase mixing, violent relaxation [@Lynd], chaotic mixing, and the Landau damping [@Merr]. For SIDM case a system virializes through more complicated mechanism which we will not discuss here [@tulin],[@Saxton]. As we have discussed above, dark matter forms first structures in the universe. When the Baryonic matter cools down, it starts forming its structure due to the gravitational potential of overdense primordial dark matter halos [@Birnboim:2003xa], [@jing], [@Silk], [@tanu]. The virialization process, which we have discussed above, is a classical Newtonian process. Therefore, it does not incorporate any general relativistic phenomena during the collapse, e.g. the apparent horizon, singularity, black hole, event horizon, etc. One could, however, ask, if the dark matter is capable of creating these types of non-trivial celestial objects. As we discussed earlier, we do not know the particle constituent of dark matter, so it is very difficult to answer this question.
This paper is organized as follows. In section (\[ccc\]) we will discuss the causal structure of collapsing space-time. In that section, we will discuss the possible nature of singularity which can be formed as an end state of gravitational collapse. In section (\[dust\]), the final fate of the gravitational collapse of homogeneous and inhomogeneous dust will be discussed. In section (\[stable\]) we will review a work [@Joshi:2011zm], where it is shown how the presence of pressure in a collapsing system can equilibrate the system to a stable configuration. In section (\[Halo\]) we will review another work [@Bhatt], where it is shown that without using virialization technique one can describe primordial dark matter halo formation by using a general relativistic technique of equilibrium. In the last section, we summarize and discuss some future outlook.
Stellar Collapse and Cosmic Censorship Conjecture {#ccc}
=================================================
In General relativity, it is inevitable to have a space-time singularity as the end state of a gravitational collapse for a physically realistic matter cloud, that maintains certain positivity of energy conditions. At the singularity, a space-time is timelike or null geodesically incomplete. In a singular space-time, it is possible for at least one freely falling particle or photon to end its existence within a finite time (or affine parameter) or to have begun its existence a finite time ago. The singularity theorems [@Seno],[@Hawk] tell us that the gravitational collapse of a massive collapsing matter cloud would terminate into a space-time singularity if the following conditions are satisfied,
- Throughout the collapse process the following energy condition should hold, $$R_{ab}X^aX^b\geq 0\,\, ,
\label{strong1}$$ where $X^a$ is a non-spacelike geodesic and $R_{ab}$ is the Ricci tensor. For timelike geodesics, the above condition is known as [*strong energy condition*]{} [@poiss]. The strong energy condition states that gravity should be always attractive. Now, if the space-time curvature is seeded by a perfect fluid, then this condition can be written as, $$\begin{aligned}
\rho+3P\geq 0\, \, ,\,\, \rho+P\geq 0\,\, .
\label{strong2}
\end{aligned}$$ Now, for a light-like geodesic, the above condition (eq. (\[strong1\])) is known as the [*null energy condition*]{}. For a perfect fluid, this can be written as, $$\begin{aligned}
\rho+P\geq 0\, \, .
\label{null1}
\end{aligned}$$ From eqs. (\[strong2\]), (\[null1\]), it can be seen that the validity of strong energy condition throughout the collapse implies the validity of null energy condition.
- The causal structure of the collapsing matter should obey strong causality condition. There should not be any closed timelike geodesics in the space-time.
- There should be a closed future trapped surface in the collapsing matter cloud.
The final space-time becomes timelike or null geodesically incomplete when the above conditions are fulfilled. From the above discussion, it should be noted that some situations can arise where space-time is null geodesically incomplete but timelike geodesically complete, e.g., the Reissner-Nordstrom solution. Here, we should also make some comments on the weak energy condition as it is considered as a necessary condition for a realistic matter field to satisfy. The weak energy condition states that energy density should always be positive with respect to any observer. Mathematically it can be written as, $$T_{ab}u^au^b\geq 0\,\, ,
\label{weak1}$$ where $u^a$ is any timelike vector. For a perfect fluid case, one can write the above equation as, $$\begin{aligned}
\rho >0, \, \, \rho+P\geq 0,\, \, .
\label{weak2}\end{aligned}$$ Therefore, weak energy condition implies null energy condition but does not imply strong energy condition. Weak energy condition is considered to be valid for any regular point in a space-time. On the other hand, strong energy condition can be violated for some circumstances. For the vacuum energy dominated universe (e.g. inflationary era), and for the dark-energy-dominated universe, strong energy condition is violated. However, for those cases also the weak energy condition is always valid.
So, the singularity theorems tell us on the conditions to have possible singular space-times as the end state of gravitational collapse. However, the key point is, these theorems are unable to predict whether the singularity that occurs can be visible to an asymptotic observer. It is generally assumed that if the final state of a catastrophic continual gravitational collapse terminates into a singular space-time, then the final singularity should always be covered by a null hypersurface, which is known as the event horizon. On this null hypersurface, the incoming null geodesics are converging and outgoing null geodesics are parallel, that is, they have a vanishing expansion. The singularity inside this null hypersurface is known as spacelike singularity. As timelike or null geodesics are converging inside the event horizon, the three-dimensional volume, which is covered by the event horizon, becomes totally black or invisible to any outside observer. Therefore, this region is called a black hole. Any information about the singularity inside the black hole cannot reach to the outside observer.
As was stated above, it was generally considered that gravitational collapse of any realistic matter field always terminates into a black hole. However, there is no mathematical interpretation or proof behind this assumption. So this is a conjecture. In 1969 Roger Penrose gave this conjecture [@pen1], [@Pen2], which states that continual gravitational collapse of any physically realistic matter field will generically (i.e. stable under small perturbations) terminate into a black hole if certain energy conditions are obeyed throughout the collapse. So this conjecture states that every space-time singularity should be covered by an event horizon of gravity. That it should not be globally naked, visible to faraway observers, is known as the Weak Cosmic Censorship Conjecture (WCC). In 1978 Penrose suggested a stronger version of this conjecture which states that any singularities that arise from regular initial data are not even locally visible. This is known as Strong Cosmic Censorship Conjecture (SCC).
To verify these two conjectures, we have to compare the time of singularity formation and the time of apparent horizon or trapped surfaces formation. Trapped surfaces are the two-dimensional hypersurfaces on which both the incoming and outgoing null geodesics are converging [@Bizon:1988vv], [@Ellis], [@Weinberg:1972kfs]. This type of 2-surfaces can form inside a collapsing matter cloud. The apparent horizon is the boundary of these trapped surfaces. A singularity is globally visible when null geodesics starting from space-time singularity can escape the collapsing matter before the formation of the apparent horizon, and then reaches the asymptotic observer. A singularity is locally visible when space-time singularity at the center of the collapsing matter is formed before the formation of apparent horizon around the center so that singularity could be visible locally, but it is not globally visible. The global and local visibility of singularity is described diagrammatically in fig. (\[visibility\]). SCC actually avoids any kind of timelike or null singularities. As this conjecture is yet to be proven, it is one of the most challenging unsolved problems of gravitational physics.
From the above discussion, it is clear that the position and formation of apparent horizon plays a very important role in the causal structure of a collapsing space-time. The position of apparent horizon can be derived from the expression of the expansion parameter ($\theta$) of null geodesic congruences. As an example, one can examine the position of apparent horizon in the following LTB metric, $$\begin{aligned}
ds^2= -dt^2 +\frac{R^{\prime 2}(r,t)dr^2}{1+E(r)}+R^2(r,t)d\Omega^2 \, ,\nonumber
\label{LTBMetric}\end{aligned}$$ where $R(t,r)$ is the physical radius and ($t,r$) are co-moving time and radius respectively and $E(r)$ is a real valued function of $r$. Here and in general a prime over a function denotes the partial derivative with respect to $r$ and a dot over a function denotes the partial derivative with respect to $t$. Now for this space-time one can derive the following expression of expansion parameter for outgoing radial null geodesics, $$\Theta_l=\frac{2}{R}\left(\dot{R}+\sqrt{1+E(r)}\right)\, .
\label{expanout}$$ Similarly, one can get the following expression of $\Theta_n$ for incoming radial null geodesics, $$\Theta_n=\frac{2}{R}\left(\dot{R}-\sqrt{1+E(r)}\right)\, .
\label{expanin}$$ One can express $\dot{R}$ in term of Misner-Sharp mass term ($F$) [@Misner], [@May:1966zz]. Mathematically the Misner-Sharp mass can be written as, $$F(r,t)=R(r,t)\left(-E(r)+\dot{R}(r,t)^2\right)\, .
\label{Misner}$$ So, during collapse one can write the expression of $\dot{R}$ as, $$\dot{R}=-\sqrt{\frac{F}{R}+E(r)}\, ,$$ where the negative sign indicates that matter is collapsing. Using this expression of $\dot{R}$, we get the following expressions of expansion parameter for outgoing and incoming radial null geodesics, $$\Theta_l=\frac{2}{R}\left(\sqrt{1+E(r)}-\sqrt{\frac{F}{R}+E(r)}\right)\, ,\,\,\,\, \Theta_n=-\frac{2}{R}\left(\sqrt{1+E(r)}+\sqrt{\frac{F}{R}+E(r)}\right)\,\, .
\label{LTBtheta}$$ The above equations indicate that the expansion parameter for incoming radial null geodesics should always be less than zero. On the other hand, the expansion parameter for outgoing radial null geodesics could change its sign. For $R=F$, $\Theta_l$ becomes zero, for $R>F$, $\Theta_l$ becomes negative and for $R<F$, $\Theta_l$ becomes positive. Now, it is known that the expansion parameter for outgoing radial null geodesics becomes zero at the apparent horizon. Mathematically, apparent horizon can be expressed ([@Lasky:2006hq],[@Lasky:2006mg]) as the surface on which, $$\Theta_l=0\,\,,\, \Theta_n<0\, .
\label{apparent}$$ From eqs. (\[LTBtheta\]),(\[apparent\]) one can say that the condition for existence of apparent horizon is, $$F(r_c)=R(r_c,t_{AH})\, .
\label{conApp2}$$ The above condition states that when at a particular comoving radius $r_c$ and comoving time $t_{AH}$ the corresponding $R(r_c,t_{AH})$ reaches the value of $F(r_c)$, an apparent horizon will form in LTB space-time. So, throughout the collapse, the condition to avoid the apparent horizon for any co-moving radius $r$ is, $$\frac{F(r,t)}{R(r,t)}<1\, .
\label{app1}$$
We note that global and local visibility of singularity is a very big problem to investigate in gravitation theory. In many literature available [@Vaz:1995ig]-[@Joshi:2007am], [@Adler:2005vn], over the last three decades, this issue has been extensively investigated. The results show that there is always a possibility that singularity becomes globally and locally visible, at least for some time. The genericity of these naked singularities was also investigated [@Joshi:2011qq],[@Satin:2014ima] [@Joshi:2007hq], and it was found that there can be generic naked singularity solutions of Einstein equation. This does not really disprove the CCC, as there is always a debate on the physical viability of the initial conditions which are taken for a collapsing scenario. However, this also implies that the cosmic censorship, if at all valid, can hold only under very fine-tuned conditions, and is thus severely restricted in any case.
![(a) Here, light rays from the central singularity can escape the collapsing matter cloud before the apparent horizon formation, and they reach the asymptotic observer. Therefore the central singularity can be visible for faraway observers. (b) Here, light rays fall into the apparent horizon before they reach the boundary of collapsing matter cloud. However, as these propagate inside the cloud, the singularity is locally visible. The picture is taken from [@Joshi:2012mk]. []{data-label="visibility"}](visibility.pdf)
Gravitational Collapse Models {#dust}
=============================
In 1939 Oppenheimer, Snyder [@Oppen] and Datt [@Datt] gave a continual gravitational collapse model . This is homogeneous (energy density independent of spatial coordinates), spherical collapse of a pressure-less fluid (dust). This is the simplest model of continual gravitational collapse of any spherically symmetric matter field. When a star having tens of times of the solar masses, exhausts its internal nuclear fuel and goes into catastrophic gravitational collapse, it inevitably forms a space-time singularity as the final state of gravitational collapse. This type of collapse process has to be modeled using general relativity. While OSD was an attempt in that direction, as we know, any physically realistic collapsing matter has inhomogeneity in density during the collapse, and will also have non-zero internal pressures. Therefore the OSD collapse can only give a rather idealized model of the continual gravitational collapse process. This was the first general relativistic model of continual gravitational collapse which predicted a black hole as an end state of collapse.
Before we discuss the final fate of the OSD collapse, we should mention the regularity conditions for a physically realistic collapsing scenario. A collapsing system should follow the following regularity conditions,
- During the collapse every space-time point of the collapsing system must be regular so that every physical parameter of that space-time should have finite value throughout the collapse.
- Weak energy condition, which we have discussed earlier, should be maintained throughout the collapse.
- Throughout the collapse there should not be any shell-crossing singularity [@Hellaby:1985zz],[@szekeres1999shell],[@Joshi:2012ak]. The partial derivative of the physical radius $R$ with respect to comoving coordinate $r$ should always be greater than zero to avoid a shell- crossing singularity: $$R'(t,r)>0.
\label{shell1}$$
Keeping all these collapse regularity conditions in mind, we can now start to discuss the final fate of OSD collapse, which is a spherically symmetric, homogeneous, dust collapse.
Dust Collapse with Homogeneous Density
--------------------------------------
The OSD collapse can be mathematically described by spherically symmetric Lemaitre-Tolman-Bondi (LTB) metric [@LTB1], [@LTB2], [@LTB3], $$\begin{aligned}
ds^2= -dt^2 +\frac{R^{\prime 2}(r,t)dr^2}{1+E(r)}+R^2(r,t)d\Omega^2 \, .\nonumber
\label{LTBMetric}\end{aligned}$$ As said earlier, $R(t,r)$ is the physical radius and ($t,r$) are comoving time and radius. From Einstein equations, one can get, $$\begin{aligned}
\rho=\frac{F^\prime}{R^2 R^\prime}\,,\,\,\,\,
P_r =P_\perp =-\frac{\dot{F}}{R^2 \dot{R}}=0\,,\,\,\,\,
\label{p0}\end{aligned}$$ where $\rho$, $P_r$ and $P_\perp$ are the energy density, radial pressure and azimuthal pressure respectively.
![Oppenheimer-Snyder-Datt (OSD) collapse: The figure is showing dynamical evolution of a spherically symmetric, homogeneous dust cloud collapse. Here, both the strong and weak cosmic censorship conjectures are valid. The picture is taken from [@Joshi:2012mk]. \[f:one\]](bh.pdf){width="9cm"}
The expression of the Misner-Sharp mass $F$ [@Misner] is given in eq. (\[Misner\]). Using the $\rho$ expression in eq. (\[p0\]), we can write, $$F(t_i,r)=\int_0^r\rho(t_i,\tilde{r})\tilde{r}^2d\tilde{r}\,\, ,
\label{F1}$$ where we consider $R(r,t_i)=r$ and $t_i$ is the initial time. For homogeneous gravitational collapse, we have the mass function $F$ given as, $$F(r,t_i)=F_0 r^3\,\, ,
\label{F2}$$ where $F_0=\frac{\rho(t_i)}{3}$. From eq. (\[p0\]) we can see that for a dustlike collapse, $F$ becomes time-independent. Therefore, the expression of $F$ in eq. (\[F2\]) must remain unchanged throughout the collapse. Now, in the LTB metric we always have a scaling degree of freedom. Therefore we can consider the physical radius as, $$R(t,r)=rf(t,r)\,\, ,
\label{R1}$$ with $f(t_i,r)=1$. For a collapse process we should always have $\dot{f}(t,r)<0$. In a collapse process $f$ always decreases with time monotonically. Therefore, we can use $f$ as ‘time-coordinate‘ replacing $t$ and every function of $t$ can be thought of as the function of $f$. The expression of Misner-sharp mass in eq. (\[Misner\]) is the equation of motion of this system. Using that equation one can write time $t$ as a function of $r$ and $f$: $$t(f,r)=t_i+\int^1_f\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+\frac{E(r)}{r^2}}}\,\, .
\label{texp}$$ As during the collapse $E(r)$ should be regular at the center, we can write $E(r)$ as, $$\begin{aligned}
E(r)&=& r^2b(r)\nonumber\\
&=& r^2\left(b_0+b_1 r+b_2 r^2+......\right)\,\, ,
\label{E1} \end{aligned}$$ where we have expanded $b(r)$ in a Taylor series around the center. Therefore, The coefficients $b_1, b_2,...$ can be written as $b_1=\frac{db}{dr}\rvert_{r=0}$, $b_2=\frac{1}{2}\frac{d^2b}{dr^2}\rvert_{r=0}$ etc., close to the center. Using the eq. (\[E1\]), we can now rewrite the eq. (\[texp\]) as, $$t(f,r)=t_i+\int^1_f\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}\,\, .
\label{texp2}$$ Now, the condition for singularity is $$R(t_s,r)=0\,\, ,$$ where $t_s$ is the comoving time of singularity formation for the comoving radius $r$. Therefore, at the time of singularity formation we can write, $$f(t_s,r)=f_s=0\,\, .
\label{singf}$$ Using eq. (\[texp2\]) and eq. (\[singf\]), we can write the time of singularity formation as, $$t_s(r)=t_s(0,r)=t_i+\int^1_0\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}\,\, .
\label{tsing1}$$ For simplicity we can assume, $b(r)=0$, which is called a marginally bound collapse. With that assumption we can now fully integrate the above equation and get the analytic expression of the time of singularity formation: $$t_s=t_i+\frac23\frac{1}{\sqrt{F_0}}
\label{tsing2}$$ The above expression of $t_s$ indicates that every portion of the collapsing cloud shrinks into a point simultaneously. The condition for apparent horizon is $R(r,t_{AH})=F(r)$, where $t_{AH}$ is the comoving time of apparent horizon formation at comoving radius $r$. Using the eq. (\[texp2\]) we can write the time $t_{AH}$ of apparent horizon formation as, $$\begin{aligned}
t(f_{AH},r)=t_{AH}(r)&=&t_i+\int_{f_{AH}}^1\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}\,\, \nonumber\\
&=&t_i+\int_{0}^1\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}-\int^{f_{AH}}_0\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}\nonumber\\
&=&t_s-\int^{f_{AH}}_0\frac{d\tilde{f}}{\sqrt{\frac{F_0}{\tilde{f}}+b(r)}}.
\label{texp3}\end{aligned}$$ Now, from the condition of apparent horizon formation, we get: $f_{AH}=r^2F_0$. Finally, using eq. (\[texp3\]) and assuming $b(r)=0$ we get, $$t_{AH}(r)=t_s-\frac23 F_0 r^3\,\, .
\label{eqahosd}$$ As $F_{0}>0$, for this case we always have $t_{AH}<t_s$ which means, for homogeneous dust collapse the singularity cannot be locally naked and as it is not locally naked it is not globally naked also. Here we have shown the results only for the case where $b(r)=0$. However, it can be shown that for any regular value of $b(r)$, we always get $t_{AH}<t_s$. In fig. (\[f:one\]), a homogeneous, spherical dust collapse is shown diagrammatically. In that diagram one can see how apparent horizon forms before the singularity formation at $r=0$. Therefore, for spherically symmetric, homogeneous dust collapse, it is inevitable to have a singularity as a final state of gravitational collapse, and that singularity is not visible locally or globally.
![Inhomogeneous Dust Collapse: A space-time singularity, which is formed from an inhomogeneous, spherically symmetric dust collapse, can be visible to external observers in the universe, in violation to both the weak and strong cosmic censorship conjectures. The picture is taken from [@Joshi:2012mk]. \[f:two\]](ns.pdf){width="9cm"}
Dust Collapse with Inhomogeneous Density
----------------------------------------
Collapse of an inhomogeneous dust cloud is more realistic than the collapse of a totally homogeneous dust cloud [@Bondi:1947fta],[@joshibook]. In any physically realistic system, which is collapsing under its own gravitational pull, the density generally decreases as the radial distance from center increases. One can consider a simple model case where, $$\rho(t_i,r)=\rho_0+\rho_{2}r^2\,\, ,$$ where $\rho_{2}<0$ . For this energy density we have, $$F(r)=F_{0}r^3+F_{2}r^5\nonumber\,\, ,$$ where $F_{2}<0$. Using the same technique, which was discussed above, we get the following expression of $t_s(r)$ and $t_{AH}(r)$ for inhomogeneous dust collapse, $$\begin{aligned}
\label{ts2}
t_{s}(r)&=&t_s(0)-\frac{F_2}{3F_{0}^{\frac32}}r^2+....\,\, ,\\
t_{AH}(r)&=&
t_s(r)-\frac23F_{0}r^3=t_s(0)-\frac{F_2}{3F_{0}^{\frac32}}r^2-\frac23F_{0}r^3+ ....\,\, ,
\label{tah2}\end{aligned}$$ where $t_s(0)$ is the time of singularity formation for the center at $r=0$ and it can be written as, $$t_s(0)=t_i+\frac23\frac{1}{\sqrt{F_0}}\,\, .
\label{tsing2}$$ From eq. (\[ts2\]) one can see that in an inhomogeneous dust collapse, the time of singularity formation is different for different values of $r$. It can also be seen that around the center the $r^2$ term will dominate over $r^3$ term and as $F_2<0$ we have: $$t_{AH}(r)>t_s(0)\,\,.$$ So for this case, the singularity must be at-least locally naked. So the SCC is violated here. For a general form of $b(r)$ one can show that singularity can also be globally naked which is a violation of WCC [@Joshi:2004tb], [@Joshi:1993zg]. In fig. (\[f:two\]), it is shown how naked singularity, which is formed as an end state of inhomogeneous spherical dust collapse, can be globally visible. So it is thus seen that an inhomogeneity in density can change the nature of singularity.
From the above discussion, we can conclude that any spherically symmetric homogeneous or inhomogeneous dust collapse should terminate into a black hole or naked singularity, in a finite amount of proper time, depending on the initial density profile for the matter cloud. We have not shown or discussed here that these collapsing scenarios do terminate into a stable configuration as a final stage of gravitational collapse. However, in general relativity, it is possible to have collapsing solutions which can terminate into a stable equilibrium singular or regular space-time as the end state of gravitational collapse. This technique has been developed and can be found in many literature [@Cooperstock:1996yv] - [@Joshi:2011zm]. We now consider these solutions briefly in the next section, towards a possible application to collapse for larger scales other than stellar collapse.
Stable Configuration from Gravitational Collapse {#stable}
=================================================
In the previous section, we have seen that the inclusion of inhomogeneity can change the nature of singularity. In this section, we will see how the pressure inside a collapsing matter field can possibly form stable configurations at the end stage of gravitational collapse. To describe this type of gravitational collapse we have to start with a general collapsing space-time. The metric of a general spherical collapsing system can be written as, $$\begin{aligned}
ds^2 = - e^{2\nu(r,t)} dt^2 + {R'^2\over G(r,t)}dr^2 + R^2(r,t) d\Omega^2\, ,
\label{genmetric}\end{aligned}$$ where $\nu(r,t)$, $R(r,t)$ and $G(r,t)$ are real functions of $r$ and $t$. The above metric can describe a spherical symmetric gravitational collapse as those functions are independent of azimuthal coordinates. One can describe a spherically symmetric, inhomogeneous, non-dust like gravitational collapse using this metric maintaining some collapsing conditions which we have discussed before. The collapsing conditions and the initial conditions of the collapse will give some constrains on the functions of above metric which should be maintained throughout the collapse. Like LTB metric, here also we have a scaling degree of freedom due to which we can write $$\begin{aligned}
R(r,t)= r f(r,t)\,,
\label{scaling1}\end{aligned}$$ where $f(r,t)$ is a real and positive valued function of $r,t$. Now, for the present metric the Misner-Sharp mass term can be written as, $$\begin{aligned}
F = R\left(1 - G + e^{-2\nu} \dot{R}^2\right)\,.
\label{ms} \end{aligned}$$ If the general collapsing space-time is seeded by an anisotropic fluid, one can write the energy density and radial pressure as, $$\begin{aligned}
-T^0_{\,\,\,\,0}=\rho=\frac{F^\prime}{R^2 R^\prime}\,,\,\,\,\,
T^1_{\,\,\,\,1}=P_r = -\frac{\dot{F}}{R^2 \dot{R}}\,,
\label{eineqns}\end{aligned}$$ and the expression of azimuthal pressures can be derived from $G_2^2$ term and $T^{\mu\nu};\nu=0$ equation. Using the $G_2^2$ and $T^{\mu\nu};\nu=0$ one can get an equation which actually relates $\rho, p_{r}, p_{\perp}, \nu^{\prime}$ and the equation is, $$\nu'=2\frac{P_\perp-P_r}{\rho+P_r}\frac{R'}{R}-\frac{P_r'}
{\rho+P_r} \,\, .
\label{pthecon}$$ The vanishing of the non-diagonal terms in the Einstein equations gives, $$\begin{aligned}
\dot{G}=2\frac{\nu^\prime}{R^\prime}\dot{R}G\,.
\label{vanond}\end{aligned}$$ For an isotropic energy-momentum tensor where $P_r=P_\perp$, the Einstein equations give, $$\begin{aligned}
\label{P_rho}
\rho = {F'\over R^2 R'}\;, \,\; P_r = P_\perp=- {\dot F\over R^2\dot R} ~,
\label{P_rho}\end{aligned}$$ while eq. (\[vanond\]) remains the same for an isotropic fluid and eq. (\[pthecon\]) becomes, $$\nu'=-\frac{P_r'}
{\rho+P_r} \,\, .$$ Now, we are going to discuss the final state of gravitational collapse of an anisotropic fluid. For simplicity, we can consider zero radial pressure and non-zero azimuthal pressure. For this case, we have 4 equations, $$\begin{aligned}
\rho=\frac{F^\prime}{R^2 R^\prime}\,,\,\,\,\,
P_\perp = \frac12 \rho R \frac{\nu^\prime}{R^\prime}\,,\,\,\,\,
\dot{G}=2\frac{\nu^\prime}{R^\prime}\dot{R}G\,,\,\,\,\,F = R\left(1 - G + e^{-2\nu} \dot{R}^2\right)\,,
\label{eineqns2}\end{aligned}$$ and 6 unknown functions ($\rho, P_{\perp}, R, G, F, \nu$ ), so we always have the freedom to choose the functional expressions of two free functions. As we have discussed previously, for zero radial pressure, the Misner-Sharp mass term becomes time-independent. Let us consider the initial physical radius as, $$R(t_i,r)=rf(t_i,r)=r\,\,.
\label{phyr1}$$ Now, if we consider homogeneous initial density as another initial condition, then like the LTB case here also we can write the following expression of Misner-Sharp mass term, $$F(t_i,r)=\int_0^r\rho(t_i,\tilde{r})\tilde{r}^2d\tilde{r}\,\,=\frac{\rho_i}{3}r^3= M_0r^3\,\, ,
\label{misner1}$$ where $M_0=\frac{\rho_i}{3}$. As Misner-Sharp mass term is time-independent, the expression of it in eq. (\[misner1\]) remains the same throughout the collapse. The initial expression of $G(t,r)$ can be written as $G_i(r)$ which can be chosen. Let us define another variable $k(t,r)=\frac{P_{\perp}}{\rho}$. Using eqs. (\[misner1\]), (\[scaling1\]), and (\[eineqns2\]) we get, $$\begin{aligned}
\label{eq:EEsrhokappaGt}
\rho &=& \frac{3 M_0}{f^2(f + r f^{\prime})}.\label{energy1} \\
k &=& \frac{f}{2}\frac{r \nu^{\prime}}{f + r f^{\prime}}\, ,\label{kvalue1} \\
\dot{G} &=& 2\frac{r \nu^{\prime}}{f + r f^{\prime}} \dot{f}G\,\, ,\label{gvalue}\\
M=M_0 &=& f\left(\frac{1 - G}{r^2} + e^{-2\nu} \dot{f}^2\right).
\label{mvalue}\end{aligned}$$ As we know, in this collapsing system we always have two free functions to choose. The above equations also indicate the same thing: four equations and six unknowns ($\rho, G, M, k,\nu, f $). Here, we consider $M$ and $k$ as two free functions. Due to the initial homogeneous density profile, we get the expression of the first free function $M$ as $M_0$. However, for $k$ it is very difficult to choose any expression, as we have to maintain all the collapsing regularity conditions which were discussed in previous section.
As $f(t,r)$ decreases monotonically with respect to $t$, we can write any dynamic variable as a function of $(f,r)$ instead of $(t,r)$, as we did for dust collapse. Now, the solution of above equations can be written in terms of $(f,r)$ as, $$\begin{aligned}
\nu(t,r) &=& 2\int_0^r d\tilde{r}~k\frac{f + \tilde{r} f^{\prime}}{\tilde{r} f} = 2\int_0^r d\tilde{r}~\frac{k}{\tilde{r}} + 2\int_0^r d\tilde{r}~k \frac{f^{\prime}}{f}, \label{nueq1}
\\
\int \frac{dG}{G} &=& 4\int_f^1 d\tilde{f}~\frac{k}{\tilde{f}} \Rightarrow G(f,r) = G_0(r)~exp{\left[4\int_f^1 d\tilde{f}~\frac{k}{\tilde{f}}\right]}, \label{Geq1} \\
\dot{f} &=& -e^{\nu}\left[\frac{G-1}{r^2} + \frac{M_0}{f}\right]^{1/2} \Rightarrow \frac{\dot{f}}{f}= -e^{\nu}\left[\frac{G-1}{r^2f^2} + \frac{M_0}{f^3}\right]^{1/2}\,\, ,
\label{eq:GCFunctions} \end{aligned}$$ where we use $\dot{G}=G_{,f}\dot{f}$ and $G_0(r)$ is integration constant. Using the above expression of $G$ and $\nu$, we can write the general collapsing metric as, $$ds^2 = -e^{ 2\int_0^r d\tilde{r}~k\frac{f + \tilde{r} f^{\prime}}{\tilde{r} f}}~dt^2 + \frac{(f + r f^{\prime})^2}{G_0(r) e^{4\int_f^1 d\tilde{f}~\frac{k}{\tilde{f}}}}~dr^2 + r^2 f^2~d\Omega^2,$$ Now, as we have one degree of freedom left, we can choose a constant value of $k$. For the constant value of $k$, we get the following expression of $\nu$, $$\label{nu}
\nu(f,r) = 2k_c\ln{r} + 2k_c\int dr~\frac{f^{\prime}}{f}\,\, , \nonumber$$ where $k_c$ is the constant value of $k$. So, for any value of $f$, we cannot remove the logarithmic divergence part (the first part of $\nu$ expression ). Therefore, regularity condition will be violated if one considers a constant value of $k$ throughout the collapse.
Now we discuss the equilibrium conditions for the collapsing cloud and we show how the pressure in the collapsing system can equilibrate the system in an asymptotic time. In the previous section, we have seen that any spherically symmetric dust collapse terminates necessarily into a space-time singularity, where the nature of the singularity, namely visible or otherwise, depends on the initial conditions. However, in the present scenario, due to the non-zero value of pressure, we have an extra degree of freedom which can be used to balance the gravitational pull to achieve an asymptotic equilibrium configuration as an end state of gravitational collapse, as we show below.
In this type of collapsing system we always can define a potential $V(f,r)$ as, $$V(f,r) = - \dot{f}^2 = - e^{2\int_0^r d\tilde{r}~k\frac{f + \tilde{r} f^{\prime}}{\tilde{r} f}}\left[\frac{G_0 e^{4\int_f^1 d\tilde{f}~\frac{k}{\tilde{f}}}-1}{r^2} + \frac{M_0}{f}\right].
\label{pot}$$ When the collapsing system reaches the equilibrium state, we should have $f=f_e$, $\dot{f}=\ddot{f}=0$, where the subscript ‘$e$‘ specifies the equilibrium value of any quantity as, $$f_e(r) \equiv \lim_{t \to \cal{T}} f(r,t),$$ where $\cal{T}$ is very large comoving time. Using this equilibrium condition and eq. (\[pot\]), we get, $$\begin{aligned}
V(f_e, r) &=& 0 \Rightarrow \left[\frac{G_e-1}{r^2} + \frac{M_0}{f_e}\right] = 0, \nonumber \\
V_{,f}(f_e, r) &=& 0 \Rightarrow \left[\frac{(G_e)_{,f}}{r^2} - \frac{M_0}{f_e^2}\right] = 0\,\, ,\end{aligned}$$ which implies, $$\begin{aligned}
\label{eq:Ge}
G(f_e, r) &=& 1 - \frac{M _0 r^2}{f_e}, \\
G_{,f}(f_e, r) &=& \frac{M_0 r^2}{f_e^2}\,. \nonumber\end{aligned}$$ Using the above equation and eq. (\[Geq1\]), we get, $$G_e = G_0(r) = 1 - \frac{M_0 r^2}{f_e}\,\,\label{Ge1} .$$ Using eq. (\[energy1\]), the expression of energy density $\rho_e$ at equilibrium can be written as, $$\begin{aligned}
\label{eq:rhoe}
\rho_e = \frac{3M_0}{f_e^2[f_e + r f^{\prime}(f_e,r)]}\,\, .\end{aligned}$$ Now, from eq. (\[gvalue\]) and eq. (\[eineqns2\]) we can write, $$\begin{aligned}
P_{(\perp)e}=\frac12\rho_e\frac{r\nu^{\prime}_e}{f_e+rf^{\prime}_e}f_e~~ =~~ \frac{\rho_e}{4}f_e\frac{(G_{,f})_e}{G_e}\,\, .
\label{angpress}\end{aligned}$$ Using eqs. (\[Ge1\]), (\[eq:rhoe\]), and (\[angpress\]) we get, $$\begin{aligned}
k_e = \frac{P_{(\perp)e}}{\rho_e} = \frac{1}{4}\frac{M_0r^2}{f_e - M_0r^2}\,\, .
\label{eq:kappae}\end{aligned}$$ We can now demand a constant value of $k_e$. As we discussed previously, due to the logarithmic divergence, we cannot choose a constant value of $k$ throughout the collapse. We always can think that this constant $k$ value can be achieved in asymptotic time. As the collapsing system cannot reach this condition in a finite time, the regularity condition will always be maintained. So, here we always consider that the collapsing system will reach the equilibrium condition in a very large comoving time.
From eq. (\[eq:kappae\]), we can see that for $f_e=br^2$ (where $b$ is a positive constant) the value of $k_e$ becomes constant, and it can be written as, $$k_e=\frac14\frac{M_0}{b - M_0}\,\, .$$ For $f_e=br^2$, $G_e$ becomes, $$G_e=1-\frac{M_0}{b}\,\, .
\label{Gfinal}$$
![The equilibrium configuration is achieved asymptotically and settles to a final radius. The central density becomes very large and tends to singularity in infinite co-moving time.[]{data-label="equilibrium"}](equilibrium.jpg)
Using eq. (\[nueq1\]), we get the final expression of $\nu$ as, $$\begin{aligned}
\nu_e(r_b) - \nu_e(r) &=& 2k_e \int_r^{r_b} d\tilde{r}~\frac{f_e + \tilde{r} f^{\prime}(f_e,\tilde{r})}{\tilde{r} f_e} \nonumber \\
&=& 2k_e \int_r^{r_b} d\tilde{r}~\frac{(\tilde{r}^2 + 2 \tilde{r}^2)}{ \tilde{r}^3}\nonumber\\
&=& 6k_e \ln{\frac{r_b}{r}} = 2 k_e \ln{\left(\frac{R_e(r_b)}{R_e(r)}\right)}\,\, ,
\label{nub}\end{aligned}$$ where $r_b$ is the maximum radius of the final static metric. Here, we use $R_e(r)=rf_e=br^3$. From above eq. (\[nub\]), we can write the $g_{tt}(R_e)$ term of final static metric as, $$\begin{aligned}
g_{tt}(R_e)=e^{2\nu(r_b)}\left(\frac{R_e(r)}{R_e(r_b)}\right)^{4k_e}\,\, .
\label{gtte}\end{aligned}$$ From eq. (\[Gfinal\]) and eq. (\[gtte\]), we can now write the final static metric which can be formed at the end state of gravitational collapse in asymptotic time: $$ds^2=-e^{2\nu(r_b)}\left(\frac{R_e(r)}{R_e(r_b)}\right)^{\frac{M_0/b}{1-\frac{M_0}{b}}}dt^2+\frac{dR_e^2}{\left(1-\frac{M_0}{b}\right)}+ R_e^2~d\Omega^2\,\, .$$ As for the present case we do not consider any matter flow across the boundary of collapsing matter cloud, the final space-time should match with an external Schwarzschild metric: $$ds^2=-\left(1-\frac{\mathbb{M}_0 \mathbb{R}_b}{\mathbb{R}}\right)dt^2+\frac{d\mathbb{R}^2}{\left(1-\frac{\mathbb{M}_0 \mathbb{R}_b}{\mathbb{R}}\right)}+\mathbb{R}^2d\Omega^2\,\, ,$$ where $\mathbb{R}(r_b)=\mathbb{R}_b$. From induced metric matching and extrinsic curvature matching, we get the following conditions, $$\begin{aligned}
\mathbb{R}(r_b)&=&R_e(r_b)\,\,\,\,\,\,\Rightarrow\,\,\,\,\, \mathbb{R}_b=R_b\,\,\, ,\\
G_{e}(r_b)&=&1-\frac{M_0}{b}=1-\mathbb{M}_0\,\,\, ,\\
e^{2\nu(r_b)}&=&1-\mathbb{M}_0\,\,\ .\end{aligned}$$ From the matching conditions, one can see that the Schwarzschild mass $\mathbb{M}_0=\frac{M_0}{b}$. Satisfying all the above matching conditions, we get the expression of final metric: $$\begin{aligned}
ds^2&=& -\left(1-\frac{M_0}{b}\right)\left(\frac{R_e}{R_b}\right)^{\frac{M_0/b}{1-\frac{M_0}{b}}}~dt^2 + \frac{dR_e^2}{\left(1-\frac{M_0}{b}\right)}+ R_e^2~d\Omega^2\,\, .\end{aligned}$$
This space-time is called the JMN space-time [@Joshi:2011zm], which has a singularity at the center but no trapped surfaces, and which is obtained as a limiting spacetime from gravitational collapse. The collapsing space-time can equilibrate itself to this static space-time in an asymptotic time. Now, the ratio between the Misner-Sharp mass and physical radius becomes, $$\frac{F(r)}{R_e}=\frac{M_0}{b}\, .$$ As it was shown previously, for a dynamic space-time the ratio between Misner-Sharp mass and physical radius should be always less than unity to avoid apparent horizon or trapped surface formation. From the $g_{rr}$ component of JMN space-time, it can be checked that $\frac{M_0}{b}$ should always be less than one. Therefore, there is no trapped surface or apparent horizon. Light rays from the central high-density region can reach the asymptotic observer. In fig. (\[equilibrium\]) we diagrammatically describe the global visibility of ultra-high density region at the center.
From the above discussion, one can see that a collapsing system can have a stable configuration as a final state if there exists some pressure in it. The internal pressure can balance the gravitational pull to slow down the collapse and generate the possibility of an equilibrium state with a very high-density region at the center. Besides this equilibrium state, a black hole can also be the possible final state of non-dust-like collapse. The final outcome depends upon the initial conditions of the collapsing scenario. The collapsing process, which we have discussed above, is a ‘slow process’ in the sense that it takes a very large time to equilibrate the collapsing system. On the other hand, the collapse processes, which terminate into black-holes or naked singularity, generally take a finite amount of time to reach the final state. The main reason behind this may be that when trapped surfaces form at any radius of collapsing dust-like matter, they propagate all over the collapsing system or may suck up all the matter into it in a finite amount of time. However, when there is no trapped surface throughout the collapse, the presence of pressure in the matter cloud slowly balance the gravitational pull and it takes a very large amount of comoving time to equilibrate the collapsing system. If we recall, in the virialization process also, the collapsing system with gravitationally interacting $N$ number of particles can be virialized in only a very large system time.
Therefore, on the basis of both these processes, namely the general relativistic process of equilibrium, and the virialization process, we find that the equilibrium configuration is achieved only in a very large limit of the comoving time general relativistically, or the system time classically. We may therefore conclude or indicate that the above-mentioned relativistic process of equilibrium is a general relativistic analog of usual virialization. As baryonic matter can dissipate its energy, the structures of baryonic matter are much more dynamic compared to the structure of dark matter [@white]. Local dissipations can change the stable configuration of baryonic matter in a finite time and on the other hand, dissipation-less dark matter retains its stable structure for a very large time. Therefore, the above-mentioned general relativistic technique of equilibrium is suitable for describing dark matter collapse. In the next section, we will use this general relativistic technique to describe how dark matter forms its halo structures.
Primordial Dark-matter Halo Formation {#Halo}
=====================================
As we know, dark matter created the first structures in the universe. As the dark matter decoupled early, the density perturbations in dark matter field started to grow during the time when the baryonic matter was in thermal equilibrium. Gradually, the density perturbations became such that the density contrast $\Delta \rho/\bar{\rho} \sim 1$, and the dynamics of perturbation modes entered into the non-linear regime. As a consequence of the nonlinear growth of dark matter density contrast, various patches of overdense regions were formed. For simplicity, those overdense patches can be assumed as spherically symmetric overdense matter distributions. As the background, namely the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe expands in a homogeneous and isotropic fashion, these overdense regions also expand initially in an isotropic and homogeneous fashion. Gradually, each of these overdense patches detaches from the cosmic expansion and behave like a sub-universe. At a certain time, these patches stop expanding and then start to collapse under their own gravity. The total dynamics of the overdense regions can be modeled by the dynamics of a homogeneous, spherically symmetric dust cloud. Conventionally, closed FLRW metric is used to model the evolution of these overdense regions. This model is known as spherical top hat collapse [@Gunn]. We know that any homogeneous dust collapse finally terminates into a black hole. Therefore, in this model virialization technique is used to stabilize the collapsing matter.
Top-Hat Collapse Model
----------------------
In the spherical top-hat collapse model, the metric of the overdense regions is given by the closed FLRW space-time [@Fried]: $$\begin{aligned}
ds^2 = - dt^2 + {a^2(t)\over 1- r^2 }dr^2 + a^2(t) r^2 d\Omega^2\,,
\label{FLRWMetric}\end{aligned}$$ where $a(t)$ is the scale-factor of the closed FLRW space-time and the range of radial coordinate is $0\le r \le 1$. As in top-hat collapse model the matter inside the overdense regions are considered as dust, the above metric can be thought as a particular form of LTB metric, where the physical radius $R(t,r)= r a(t)$, the Misner-Sharp mass $F(r)= F_0 r^3$ and $E(r)=-r^2$. Here $F_0$ is related with initial density of overdense region. From the expression of Misner-Sharp mass in eq. (\[Misner\]), we can write, $$\frac{F_0}{a(t)}=1+\dot{a}^2
\label{topmisner}$$ The above equation of motion can be written in a more compact form as, $$\begin{aligned}
\frac{H^2}{H_0^2}=\Omega_{m0}\left(\frac{a_0}{a}\right)^3 + (1-\Omega_{m0})\left(\frac{a_0}{a}\right)^2 \, ,
\label{fried}\end{aligned}$$ The above differential equation is known as the Friedmann equation, where $H=\dot{a}/a$ is the Hubble parameter for the overdense sub-universe and $H_0,\,a_0$ are the initial values of $H$ and $a$. Here, the initial values correspond to the time when the overdense regions start to evolve independently and detach from the background cosmic expansion. Here, $\Omega_{m0}=\rho_0/\rho_{c0}$, where $\rho_{c0} =
3H_0^2$, and $\rho_0$ is the initial homogeneous matter density. So from eq. (\[F1\]), we can write: $F_0=\frac{\rho_0}{3}$. The solution of eq. (\[fried\]) can be written in a parametric form : $$\begin{aligned}
a = \frac{a_m}{2}(1- \cos\theta),~~~
t = \frac{t_m}{\pi}(\theta-\sin\theta)\, ,
\label{at}\end{aligned}$$ where $t_m$ is the time when the scale factor $a(t)$ reaches its maximum limit $a_m$. When $\theta=\pi$, we get the maximum value of $a(t)$. We can write $a_m$,$t_m$ in terms of $\Omega_{m0}$, $H_0$ and $a_0$ as, $$a_{m}={a_0 \Omega_{m0}\over (\Omega_{m0}-1)}\,,\,\,\,\, t_{m}={\pi \Omega_{m0}\over 2 H_0 (\Omega_{m0}-1)^{3/2}}\,,$$ where we always have $\Omega_{m0}>1$ for the overdense region. In top-hat collapse model, the overdense regions are considered as sub-universes. These sub-universes, which are described by closed FLRW metric, expand first and reach a maximum scale factor limit at time $t_m$. At this point the spherical overdense regions start collapsing. One can calculate the ratio between the the density of overdense region and the density of background at the turn-around point $t=t_m$: $$\begin{aligned}
\frac{\rho(t_{m})}{\bar{\rho}(t_{m})} = \frac{9\pi^2}{16} \sim 5.55\,,
\label{impfac}\end{aligned}$$ which implies that the spherically symmetric overdense region starts collapsing when its density becomes $5.55$ times higher than the density of the background. As we know from the LTB collapse analysis, any spherical dust collapse inevitably terminates into a space-time singularity. For the present case, the ratio between Misner-Sharp mass and physical radius is: $$\begin{aligned}
\frac{F(r)}{R(r,t)}=r^2\frac{F_0}{a(t)}=r^2(1+\dot{a}^2)\,\, ,
\label{apparent1}\end{aligned}$$ where we use eq. (\[topmisner\]). From the above equation one can check that trapped surfaces form in this type of collapse and as time goes by, they gradually propagate from the edge ($r=1$) of the sub-universe to the center ($r=0$). So, the final singularity at $\theta= 2\pi$ must be covered by an event horizon, and therefore a Black hole should be the final state for this type of collapse. To avoid the Black hole at the end of this kind of gravitational collapse, the virialization technique is used to equilibrate the collapsing system. As we know, a gravitating system virializes when the following condition is full-filled, $$\langle T \rangle=-\frac12 \langle V_T \rangle \, ,$$ where $ \langle T \rangle$ and $\langle V_T \rangle$ are the average kinetic and potential energy respectively.
Now, in spherical top-hat collapse model when the spherically symmetric overdense region having total mass $M$ reaches its maximum scale factor limit, it momentarily stands still before starting to collapse. At that moment kinetic energy is zero, and all the energy is potential energy, $$V_T=-\frac{3M^2}{5R_m}\, ,$$ where $R_m$ is the maximum physical radius of the overdense region at time $t_m$. When it has collapsed to $R=\frac{R_m}{2}$, then using the conservation of total energy one can derive the expressions of potential energy and kinetic energy, $$\begin{aligned}
V_T&=& -\frac{6M^2}{5R_m}\, ,\nonumber\\
T&=& \frac{3M^2}{5R_m} = -\frac{V_T}{2}\, .\end{aligned}$$ At this point, it is considered that the collapsing system virializes itself to a static configuration. So, in top-hat collapse model the overdense regions virialize when the scale factor becomes $a_{\rm vir}=\frac12 a_{m}$. From eq. (\[at\]), one can show that this virialization will happen when $\theta = \frac{3\pi}{2}$. At this stage the ratio is, $$\begin{aligned}
\frac{\rho(t_{\rm vir})}{\bar{\rho}(t_{\rm vir})} \sim 145\,.
\label{ratio2}\end{aligned}$$ So, the top-hat collapse model tells us that at the time $t_{\rm vir}$, when the dark matter virializes and forms its stable halo structure, the density of the halo becomes $145$ times larger than the background. Though the top-hat collapse is a very simple model to describe the non-linear evolution of density perturbation modes, it gives numbers which are very important for astrophysics.
We note, however, that there are many problems with this model as it is a totally homogeneous, spherical dust collapse. The collapsing system, which is described by the top-hat model, starts collapsing with a homogeneous matter field and remains homogeneous throughout the collapse, which is unrealistic. The second issue with this model is that here a general relativistic technique is used to describe the dynamics of the overdense regions, however, to describe a stable configuration of the collapsing system a Newtonian virialization technique is used. Therefore it follows that these two techniques are not glued together properly, neither they are compatible with each other. As we know, virialization happens in gravitating systems through different processes for which it takes a large amount of system time. It just cannot happen in a short finite amount of time or a moment of time. Before $t=t_{\rm vir}$ there is no sign of stabilizing process. Before that time the system undergoes a catastrophic gravitational collapse which could lead to a space-time singularity, but suddenly for the sake of stabilization, the virialization process is used. There is no doubt that the collapsing system would virialize when the system radius $R\sim\frac{R_m}{2}$. However, the technique, which is used in the top-hat collapse model, cannot describe the equilibrium process properly.
Our purpose here is to point out that, one can generalize the top-hat collapse model, retaining some of its simplicities and attractive features. In the next section, we are going to discuss a modification of the top-hat collapse model where we do not need to introduce a Newtonian virialization technique during the collapse to describe a stable final state, but we can use a general relativity model. In this sense, we can say that this offers a general relativistic alternative to the Newtonian virialization technique. While the model presented here is still very much a toy model which may not be physically realistic, we believe it provides several useful insights towards achieving a general relativistic description for structure formation in the universe. We also indicate here a few possibilities towards the further generalization of this scenario presented.
Modification of Top-Hat collapse model
--------------------------------------
In this section, we will briefly review the literature [@Bhatt], where a special technique is used to modify the top-hat collapse model. As we know, in the top-hat collapse model the collapsing matter field is homogeneous and pressureless throughout the whole collapse process, and therefore we cannot achieve any equilibrium configuration general relativistically. Therefore, in that model, an ad-hoc input of virialization is introduced and needed to stabilize the collapsing system. On the other hand, in the previous section, we have discussed a collapsing scenario which leads to a stable configuration at the end stage of gravitational collapse. The dynamics of that scenario is governed by the general collapsing metric, $$\begin{aligned}
ds^2 = - e^{2\nu(r,t)} dt^2 + {R'^2\over G(r,t)}dr^2 + R^2(r,t) d\Omega^2\, .
\label{genmetric2}\end{aligned}$$ With this metric, one can describe a matter field which has non-zero pressure in it. The pressure term inside the matter field is responsible for the final equilibrium configuration. However, it is very difficult to describe the total evolution of spherical overdense regions using the general collapsing metric, as we need the initial functional expressions for all the functions $\nu(r,t)$, $R(r,t)$ and $G(r,t)$. In the cosmological scenario, when the overdense regions detach from background expansion, it becomes a non-trivial task to set the initial form of those functions.
One can simplify the situation by considering the general collapsing metric in the collapsing phase only. This will help us to find the initial conditions. In this technique, the initial expansion phase of homogeneous, pressureless fluid remains as it was in the top-hat collapse model. However, from the turn-around point, the collapsing phase is described by the general collapsing metric. During the expansion phase, the matter field is homogeneous and pressureless carrying some properties of background expansion which is described by a flat FLRW metric. Now, from the starting point of the collapsing phase, inhomogeneity and pressure start to grow gradually. This pressure term finally equilibrates the collapsing system in the same way as it is described in the previous section. To pursue this total evolution of the overdense matter field, we need to glue closed FLRW metric with general collapsing metric at the turn-around point. This can be done by using Darmois-Israel junction conditions [@poiss], [@Israel:1966rt], [@Darmois1927] on the spacelike hypersurface, namely, $\Phi(t)=t-t_{m}=0$. Therefore, from the junction conditions we can get the initial functional expressions of the unknown functions in the general collapsing metric, and then we can use the general relativistic equilibrium technique to investigate the final outcome. So, in the new model of spherical collapse, the overdense sub-universe initially expands in an isotropic and homogeneous fashion (with zero pressure), which is described by closed FLRW metric. Then from the turn-around point, the non-zero pressure and inhomogeneity develop gradually in the matter field, and general collapsing metric is then used to describe this later phase.
In the top-hat collapse model, as the collapsing matter is pressureless, the mathematical technique for describing the dynamics of the overdense sub-universe is very simple. However, as the collapsing matter field is homogeneous and pressureless throughout the collapse, the model becomes physically unrealistic and an ad-hoc situation and virialization technique is used to stabilize the collapsing system. In the modified version of the top-hat model, the matter is more realistic and a new general relativistic approach is used to describe the total evolution of overdense sub-universes.
Inhomogeneous and Non-dust like Spherical Collapse {#nondust}
--------------------------------------------------
Here we discuss the modified top-hat model in some detail. To achieve a stable configuration general relativistically, we will glue the closed FLRW metric with general collapsing metric at the turn-around point by using Darmois- Israel junction conditions. These conditions will give us some initial conditions which will be used to investigate the final fate of spherical gravitational collapse of inhomogeneous, non-dustlike fluid.
![Here $g_{\mu\nu}^+$ is contracting space-time and $g_{\mu\nu}^-$ is the closed FLRW metric.[]{data-label="hypersurfaces1"}](hypersurfacemodi.jpg){width="2.5in"}
### Matching conditions
We have to match closed FLRW metric with a general collapsing metric at the spacelike hypersurface: $\Phi(t)=t-t_m=0$. In this case, for smooth matching, according to the junction conditions, we need to match induced metrics and extrinsic curvatures at the constant time spacelike hypersurface. Here the two induced metrics will be denoted as $g_{\mu\nu}^+$ and $g_{\mu\nu}^-$, where ‘+’ and ‘-’ are related with general collapsing metric and closed FLRW metric respectively. In fig. (\[hypersurfaces1\]) we schematically describe the space-time structure. Now, using eq. (\[genmetric2\]) and eq. (\[FLRWMetric\]), we can write the induced metrics as, $$\begin{aligned}
ds^2_{+}=\left(\frac{R^{\prime 2}(r,t_m)dr^2}{G(r,t_m)}+R^2(r,t_m)d\Omega^2\right),~~
ds^2_{-}=\frac{a_m^2dr^2}{1-r^2}+r^2 a_m^2d\Omega^2~.
\label{nondustmatching}\end{aligned}$$ From the first condition of smooth matching, we have $$\begin{aligned}
g_{\mu\nu}^{+}(t\rightarrow t_{m})&=&g_{\mu\nu}^{-}(t\rightarrow t_{m})\,\, ,\end{aligned}$$ and this will give us the initial expressions of $G(r,t)$ and $R(r,t)$ as, $$\begin{aligned}
R(r,t_{m})=rf(r,t_{m}) = ra_{m},~~ G(r,t_{m}) = 1-r^2\, .
\label{nmatch}\end{aligned}$$ The first initial condition implies, $f(r,t_m)=a_m$. The second condition for smooth matching is $$\begin{aligned}
K^{+}_{\mu\nu}(t\rightarrow t_{m})&=&K^{-}_{\mu\nu}(t\rightarrow t_{m})\,\, ,\end{aligned}$$ where $K_{\mu\nu}$ is the extrinsic curvature, which can be mathematically written as, $ K_{ab} = {\eta_{\alpha;}}_\beta e^\alpha_a e^\beta_b\,\, , $ where semicolon indicates co-variant derivative. Here, $\eta$ is a vector perpendicular to the $\Phi(x_\alpha)=0$ spacelike hypersurface, and it is defined as $$\begin{aligned}
\eta_\mu = {\epsilon\Phi,_\mu\over |g^{\alpha\beta}\Phi,_\alpha\Phi,_\beta|^{1/2}}\,
\label{norm}\end{aligned}$$ where $e^\alpha_a$ are the tangents to that spacelike hypersurface. Comma in the above expression indicates ordinary partial derivative, and $\epsilon\equiv +1$ is used for timelike hypersurface and $\epsilon\equiv -1$ is for spacelike hypersurface. Here, $e^\alpha_a$ is defined as, $e^\alpha_a\equiv\frac{\partial x^{\alpha}}{\partial l^a}$, where $l^a$ is the induced coordinate system on the hypersurface. As mentioned, in our case the hypersurface is $\Phi(t)=t-t_{m} =0$. On this spacelike hypersurface the induced coordinate system is, $l^a=\left\lbrace r,\theta,\phi\right\rbrace$. So, the components of tangent vectors from both sides can be written as, $$e_r^{\alpha\pm}=\left\lbrace 0,1,0,0\right\rbrace\,\, ,\,\,e_{\theta}^{\alpha\pm}=\left\lbrace 0,0,1,0\right\rbrace\,\, ,\,\,e_{\phi}^{\alpha\pm}=\left\lbrace 0,0,0,1\right\rbrace\,\, .$$ The normal vectors to the spacelike hypersurface from the both sides can be written as, $$\begin{aligned}
\eta^+_{\mu}=\left\lbrace -e^{\nu(r,t)},0,0,0\right\rbrace\,\,\, ,\,\,\eta^-_{\mu}=\left\lbrace -1,0,0,0\right\rbrace\,\, .\end{aligned}$$ Using the above expressions of tangents and normals to the spacelike hypersurface $\Phi(t)$, we can now write the expressions of non-zero extrinsic curvature components for the both sides. For closed FLRW metric we get, $$\begin{aligned}
K_{rr}^{-}&=&\frac{a(t)\dot{a}(t)}{1-r^2}\,\, ,\nonumber\\
K_{\theta\theta}^{-}&=&\frac{K_{\phi\phi}^{-}}{\sin^2\theta}=r^2a(t)\dot{a}(t)\,\, .
\label{extrin1}\end{aligned}$$ As the scale factor $a(t)$ of closed FLRW metric reaches its maximum limit $a_{m}$ at $t = t_{m}$, $\dot{a}(t_{m})=0$. Therefore, on the spacelike hypersurface all the extrinsic curvature components for closed FLRW metric become zero. So we can write, $$\begin{aligned}
K_{rr}^{-}|_{t=t_m}=K_{\theta\theta}^{-}|_{t=t_m}=K_{\phi\phi}^{-}|_{t=t_m}=0.
\label{extrin2}\end{aligned}$$ Now, for the general collapsing metric we get the following non-zero components of extrinsic curvature, $$\begin{aligned}
K_{rr}^+ &=&\frac{e^{-\nu (r,t)}}{2 G(r,t)^2}\left(r f'(r,t)+f(r,t)\right)\left\lbrace-\left(r
f'(r,t)+f(r,t)\right) {\dot G}(r,t)+2 \left({\dot f}(r,t)+r{\dot f}'(r,t)\right) G(r,t)\right\rbrace\,\, ,\nonumber\\
K_{\theta\theta}^+ &=& +r^4 f(r,t) {\dot f(r,t)} \left(e^{-\nu (r,t)}\right) = \frac{K_{\phi\phi}^+}{\sin^2\theta}\,\, .\end{aligned}$$ Now, from the induced metric matching we get $f(r,t_m)=a_m$, which indicates that $f(r,t)$ becomes $r$ independent at time $t=t_m$. From the matching of azimuthal component of extrinsic curvature we can get the condition: $\dot{f}(r,t_m)=0$. Using these two conditions, and from the matching of radial component of extrinsic curvature, we get $\dot{G}(r,t_m)=0$. So, from the matching conditions we get the following initial conditions for collapse, $$\begin{aligned}
f(r,t_m)=a_m\,\, ,\,\,\, G(r,t_{m}) = 1-r^2\,\, ,\,\,\, \dot{f}(r,t_m)=0\,\, ,\,\,\,\dot{G}(r,t_m)=0\,\, .
\label{initial1}\end{aligned}$$
### Inhomogeneous Anisotropic Collapse
Now, using the above initial conditions, we can investigate the final equilibrium metric, using the technique which we have described in the previous section. Here, the collapsing matter is considered as inhomogeneous, non-dustlike. For a simple case, one can consider dustlike inhomogeneous fluid during collapse after the expansion of the homogeneous, dustlike fluid. For this simple case we need, to match closed FLRW metric with LTB metric on the spacelike hypersurface $\Phi=t-t_m=0$. However, it can be shown [@Bhatt] that if the fluid remains dustlike during the expansion as well as the collapsing phase, and if the fluid is homogeneous during the expansion phase, then the fluid has to be homogeneous during the collapsing phase also. Therefore, inhomogeneous dust collapse after homogeneous dust expansion is not possible. Now, we are going to discuss the collapsing scenario where we consider the collapse of an inhomogeneous, anisotropic fluid after the expansion of the homogeneous dustlike fluid. For the anisotropic collapse, we will first consider one of the simplest examples where the fluid has zero radial pressure throughout the collapse. As we know, zero radial pressure gives us the time-independent Misner-Sharp mass term, and that solution can be derived from the initial conditions. Using the Misner-Sharp mass expression in eq. (\[ms\]) and the initial conditions in eq. (\[initial1\]), we can get following functional form of Misner-Sharp mass, $$\begin{aligned}
F(r) \equiv F(r,t_{m}) = rf(r,t_{m})\left[1- G(r,t_{m}) +
r^2 e^{-2\nu(r,t_{m})} \dot{f}^2(r,t_{m})\right]= a_{m} r^3\,,
\label{msm}\end{aligned}$$
As was found previously, the equilibrium condition is achieved when the following condition holds, $$\begin{aligned}
\dot{f}_e(r) = \ddot{f}_e(r) =0\,.
\label{stabc2}\end{aligned}$$ This condition is achieved in a large coordinate time $\mathcal{T}$. As we mentioned previously, in this collapsing system, we have two degrees of freedom to choose two unknown free functions. This is true for the equilibrium state also. Now, in this case, the Misner-Sharp mass term $F$ is time-independent, and therefore the functional expression of $F$ is fixed from the initial condition of collapse. So, we need to choose the expression of one unknown function. In the previous section, we choose the functional expression of $k_e(r)$ which is the ratio between azimuthal pressure and energy density. Here, we choose the functional expression of the physical radius as $$R_e(r)=br^{\alpha +1}\, .
\label{afer}$$ With this consideration, the system becomes totally solvable and one can derive the expressions of other functions.
In this case, for $\alpha=2$, the final metric will not be the same as it was in the previous section. In the previous section, we saw that $\alpha=2$ gives us constant value of $k_e(r)$. In the top-hat collapse model, the overdense region is considered as an isolated universe, and the comoving radius ranges from zero to one. This type of restriction on comoving radius was not there in the previous case. Scaling $a_{m}$ to unity, we first note that as it is a collapsing process, $R(r,t)$ should always be decreasing with time. Therefore, we always have $R_{m} > R_e$ which implies that for any value of $r$, $$\frac{R_{m}}{R_e} = \frac{1}{br^{\alpha}} > 1 \Rightarrow b < 1~.
\label{ns1}$$ As the radial coordinate $r$ can have values very close to zero, the second inequality in the above equation is necessary. Next, if we demand no apparent horizon at the starting point of the collapse, then the following inequality should hold, $$\frac{F(r,t_{m})}{R_{m}} = \frac{r^3 a_{m}}{r a_{m}} < 1~,$$ which is always satisfied in the range of $r$. As we do not know the analytic solution of total gravitational collapse, we cannot say anything about $\frac{F(r)}{R(r,t)}$ during the collapse. However, it is interesting to investigate whether there is a possibility of trapped surfaces at the equilibrium state of collapse. When the collapsing matter cloud achieves the equilibrium configuration, the ratio between the Misner-Sharp mass and physical radius becomes, $$\frac{F(r)}{R_e} = \frac{r^2}{br^{\alpha}}=1 \Rightarrow r = b^{\frac{1}{2-\alpha}}~.
\label{aphor1}$$ From the above equation, one can see that $\alpha\geq 2$ must be ruled out to avoid apparent horizon and trapped surfaces. So, we can say that for a physically realistic gravitational collapse, which has no apparent horizon at the final equilibrium state, the following inequality should hold, $$0\leq\alpha < 2,~~0<b<1~.$$ Therefore, we will use these inequalities in the derivation of the final stable metric. Now, as we know the equilibrium expressions of $F(r)$ and $R(r,t)$, we can get the final equilibrium space-time using the technique which has been discussed in the previous section. The final static metric is, $$ds_e^2 = -{\mathcal A}\left(b-r^{2-\alpha }\right)^{\frac{\alpha +1}{\alpha -2}}dt^2 + \frac{(\alpha +1)^2 b^3 r^{2 \alpha }}{b- r^{2-\alpha }}dr^2
+ b^2 r^{2\left(\alpha +1\right)}d\Omega^2~.
\label{meteq}$$ Now, if we write the above metric in terms of physical radius, we get, $$ds_e^2 = -\frac{R_b - \left(\frac{R_b}{b}\right)^{\frac{3}{1+\alpha}}}{R_b}
\left(\frac{1 - \frac{1}{b}\left(\frac{R_b}{b}\right)^{\frac{2-\alpha}{1+\alpha}}}{1 -\frac{1}{b} \left(\frac{R_e}{b}\right)^{\frac{2-\alpha}{1+\alpha}}}\right)^{\frac{1+\alpha}{2-\alpha}}dt^2
+ \frac{dR_e^2}{1-\frac{1}{b}\left(\frac{R_e}{b}\right)^{\frac{2-\alpha}{1+\alpha}}} + R_e^2d\Omega^2~,
\label{dsfinal}$$ where the above metric is matched with Schwarzschild metric at the physical radius $R_b$. From eq. (\[meteq\]), it can be seen that the metric is valid only in the range: $0<r < b^{1/(2-\alpha)}$. Therefore, we need to match this space-time with the Schwarzschild metric in that range. If we set $\alpha=0$, we get the following space-time, $$ds_e^2 = -\frac{\left(1 - \frac{R_b^2}{b^3}\right)^{\frac{3}{2}}}{\sqrt{1 - \frac{R_e^2}{b^3}}}dt^2 + \frac{dR_e^2}{1-\frac{R_e^2}{b^3}} + R_e^2d\Omega^2~.
\label{Flo}$$ The above space-time is known as Florides space-time [@Florides]. It can be seen that there is no curvature singularity at $R_e = r = 0$. At $R_e = b^{3/2}$ this space-time has a curvature singularity. Therefore, this solution is totally regular in the range $0\leq R_e<b^{3/2}$. The energy density is homogeneous in this case. In eq. (\[dsfinal\]) we present a new class of space-times which can be formed as an end state of gravitational collapse of inhomogeneous, anisotropic fluid. As we have discussed, this type of equilibrium configuration can develop from a gravitational collapse in very large co-moving time. Therefore, when the overdense regions of dark-matter start collapsing under its own gravity, the non-zero pressure within can stabilize the collapsing system in a very large comoving time. As we know, the dark matter is considered as a pressureless fluid in cosmological scale. However, in the halo scale, it need not be always pressureless during the collapse. If the general relativistic technique of equilibrium is considered as a relativistic analog of virialization, then one can always use this technique to explain the total evolution of overdense dark matter regions without using an ad-hoc virialization technique. It can also be shown that the presence of isotropic pressure can lead to a homogeneous, regular space-time as a final state of gravitational collapse. In [@Bhatt], a full analytic solution of this type of gravitational collapse is presented.
conclusion
==========
We have explored here the general relativistic possibility which can replace the usual virialization technique that is normally used in the so-called top-hat collapse models in the structure formation scenarios in astrophysics and cosmology. We show that while making the departures from the homogeneous dust form of matter, and when we consider the more realistic collapse with non-zero pressures, it naturally follows that we could generate equilibrium configurations which may represent larger structures.
As we discussed in the previous section, baryonic matter can create stable structures and one can always describe the collapse process by using the above-mentioned general relativistic technique. However, usually, the stable structures of baryonic matter are not stable forever. Baryonic matter can dissipate its energy and can accumulate into the central region of a self-gravitating system. Therefore, the structures of baryonic matter are far more dynamic than dark matter structures [@nfwprof],[@Navarro:1996gj],[@Jenkins:2000bv], [@white]-[@Birnboim:2003xa]. Baryonic matter can form compact objects like stars, planets and also can form black holes, naked singularities in a finite amount of time. On the other hand, dark matter cannot dissipate its energy, and therefore it can form its structure at larger scales like halo scale, galaxy cluster scale etc. Therefore, one can use the general relativistic technique of equilibrium more efficiently to describe dark matter collapse. A small pressure in the dark-matter field can create stable structures in asymptotic time as we showed.
Our model presented here is very much of a toy model in that we have, for the sake of simplicity, chosen the radial pressures to be vanishing, which will be typically non-zero for any physically realistic gravitational systems. We note that collapse scenarios that result into equilibrium configurations, and which have non-zero radial pressures, have been developed in [@Joshi:2013dva]. These may be explored towards astrophysically more realistic models [@Sayan]. Also, we have not examined here the comparison of actual astrophysics observations with our model, and in terms of numbers. We plan to take up these issues in future work. The advantage of our model, however, is that we have presented here a fully consistent general relativistic model to develop equilibrium structures such as galaxies, rather than using a patchwork of Newtonian and relativity techniques.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review three Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fourth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These four temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.
**Keywords:** Non-negative matrix factorization, Time series data, Gamma Markov chains, MAP estimation
author:
- |
Louis Filstroff$^{1}$ Olivier Gouvert$^1$ Cédric Févotte$^1$ Olivier Cappé$^2$\
$^1$ IRIT, Université de Toulouse, CNRS, France\
$^2$ DI ENS, CNRS, INRIA, Université PSL
bibliography:
- 'articlebib.bib'
title: 'A Comparative Study of Temporal Non-Negative Matrix Factorization with Gamma Markov Chains'
---
Introduction {#sec-intro}
============
Non-negative matrix factorization
---------------------------------
Non-negative matrix factorization (NMF) [@paatero1994positive; @lee1999learning] has become a widely used class of methods for analyzing non-negative data. Let us consider $N$ samples in $\mathbb{R}^{F}_{+}$. We can store these samples column-wise in a matrix, which we denote by $\mathbf{V}$ (therefore of size $F \times N$). Broadly speaking, NMF aims at finding an approximation of $\mathbf{V}$ as the product of two non-negative matrices: $$\mathbf{V} \simeq \mathbf{WH},
\label{eq-nmf-approx}$$ where $\mathbf{W}$ is of size $F \times K$, and $\mathbf{H}$ is of size $K \times N$. $\mathbf{W}$ and $\mathbf{H}$ are referred to as the dictionary and the activation matrix, respectively. $K$ is usually chosen such that $K \ll \min(F,N)$, hence producing a low-rank approximation of $\mathbf{V}$. This factorization is often retrieved as the solution of an optimization problem, which we can write as: $$\min_{\mathbf{W} \geq 0,~\mathbf{H} \geq 0} D(\mathbf{V}|\mathbf{WH}),
\label{eq-nmf-min}$$ where $D$ is a measure of fit between $\mathbf{V}$ and its approximation $\mathbf{WH}$, and the notation $\mathbf{A} \geq 0$ denotes the non-negativity of the entries of the matrix $\mathbf{A}$. One of the key aspects to the success of NMF is that the non-negativity of the factors $\mathbf{W}$ and $\mathbf{H}$ yields an interpretable, part-based representation of each sample: $\mathbf{v}_n \simeq \mathbf{Wh}_n$ [@lee1999learning].
Various measures of fit have been considered in the literature, for instance the family of $\beta$-divergences [@fevotte2011algorithms], which includes some of the most popular cost functions in NMF, such as the squared Euclidian distance, the generalized Kullback-Leibler divergence, or the Itakura-Saito divergence. As it turns out, for many of these cost functions, the optimization problem described in Eq. can be shown to be equivalent to the joint maximum likelihood estimation of the factors $\mathbf{W}$ and $\mathbf{H}$ in a statistical model, that is: $$\max_{\mathbf{W},\mathbf{H}} p(\mathbf{V}|\mathbf{W},\mathbf{H}).
\label{eq-nmf-max}$$ This leads the way to so-called *probabilistic* NMF, i.e., estimation or inference tasks in probabilistic models whose observation distribution may be written as: $$\mathbf{v}_{n} \sim p(~.~;\mathbf{Wh}_{n}, \boldsymbol{\Theta}), \quad \mathbf{W} \geq 0, \quad \mathbf{H} \geq 0,
\label{eq-prob-nmf}$$ that is to say that the distribution of $\mathbf{v}_n$ is parametrized by the dot product of the factors $\mathbf{W}$ and $\mathbf{h}_n$. Other potential parameters of the distribution are generically denoted by $\boldsymbol{\Theta}$. Most of the time these distributions are such that $\mathbb{E}(\mathbf{v}_n) = \mathbf{Wh}_n$.
This large family encompasses many well-known models of the literature, for example models based on the Gaussian likelihood [@schmidt2009bayesian] or the exponential likelihood [@fevotte2009nonnegative; @hoffman2010bayesian]. It also includes factorization models for count data, which are most of the time based on the Poisson distribution[^1] [@canny2004gap; @cemgil2009bayesian; @zhou2012beta; @gopalan2015scalable], but can also make use of distributions with a larger tail, e.g., the negative binomial distribution [@zhou2018nonparametric]. Finally, more complex models using the compound Poisson distribution have been considered [@simsekli2013learning; @basbug2016hierarchical; @gouvert2019recommendation], allowing to extend the use of the Poisson distribution to various supports $(\mathbb{N}, \mathbb{R}_+, \mathbb{R}, ...)$.
In the vast majority of the aforementioned works, prior distributions are assumed on the factors $\mathbf{W}$ and $\mathbf{H}$. This is sometimes referred to as *Bayesian* NMF. In this case, the columns of $\mathbf{H}$ are most of the time assumed to be independent: $$p(\mathbf{H}) = \prod_{n=1}^{N} p(\mathbf{h}_n).
\label{eq-h-idp}$$ The factors being non-negative, a standard choice is the Gamma distribution, which can be sparsity-inducing if the shape parameter is chosen to be lower than one. The inverse Gamma distribution has also been considered.
Temporal structure of the activation coefficients
-------------------------------------------------
In this work, we are interested in the analysis of specific matrices $\mathbf{V}$ whose columns cannot be treated as exchangeable, because the samples $\mathbf{v}_n$ are correlated. Such a scenario arises in particular when the columns of $\mathbf{V}$ describe the evolution of a process over time.
From a modeling perspective, this means that correlation should be introduced in the statistical model between successive columns of $\mathbf{V}$. This can be achieved by lifting the prior independence assumption of Eq. , thus introducing correlation between successive columns of $\mathbf{H}$. In this paper, we consider a Markov structure on the columns of $\mathbf{H}$: $$p(\mathbf{H}) = p(\mathbf{h}_1) \prod_{n \geq 2} p(\mathbf{h}_n | \mathbf{h}_{n-1}).
\label{eq-h-markov}$$ We will refer to such a model as a $\textit{dynamical}$ NMF model. Note that very recent works go beyond the Markovian assumption, i.e., assume dependency with multiple past time steps, and are labeled as “deep” [@gong2017deep; @guo2018deep].
Several works [@fevotte2013non; @schein2016poisson] assume that the transition distribution $p(\mathbf{h}_n | \mathbf{h}_{n-1})$ makes use of a transition matrix $\boldsymbol{\Pi}$ of size $K \times K$ to capture relationships between the different components. In this case, the distribution of $h_{kn}$ depends on a linear combination of all the components at the previous time step: $$p(\mathbf{h}_n | \mathbf{h}_{n-1}) = \prod_{k} p(h_{kn}|\sum_l \pi_{kl} h_{l(n-1)}).
\label{eq-trans-h-matrix}$$
In this work, we will restrict ourselves to $\boldsymbol{\Pi} = \mathbf{I}_K$. Equivalently, this amounts to assuming that the $K$ rows of $\mathbf{H}$ are a priori independent, and we have $$p(\mathbf{H}) = \prod_{k} p(h_{k1}) \prod_{n \geq 2}p(h_{kn}|h_{k(n-1)}).
\label{eq-h-markov-idp}$$ We will refer to such a model as a *temporal* NMF model.
A first way of dealing with the temporal evolution of a non-negative variable is to map it to $\mathbb{R}_+$. It is then commonly assumed that this variable evolves in Gaussian noise. This is for example exploited in the seminal work of @blei2006dynamic on the extension of latent Dirichlet allocation to allow for topic evolution[^2]. A similar assumption is made in @charlin2015dynamic, which introduces dynamics in the context of a Poisson likelihood (factorizing the user-item-time tensor). Gaussian assumptions allow to use well-known computational techniques, such as Kalman filtering, but result in loss of interpretability.
We will focus in this paper on naturally non-negative Markov chains. Various non-negative Markov chains have been proposed in the NMF literature [@cemgil2007conjugate; @fevotte2009nonnegative; @acharya15nonparametric]. They are all built in relation with the Gamma (or inverse Gamma) distribution. As a matter of fact, these models exhibit the same drawback: the chains all have a degenerate stationary distribution. This can lead to undesirable behaviors, such as the instability or the degeneracy of realizations of the chains. We emphasize that this is problematic from the probabilistic perspective only, since these prior distributions may still represent an appropriate regularization in a MAP setting.
Contributions and organization of the paper
-------------------------------------------
The contributions of this paper are 4-fold:
- We review the existing non-negative Markov chains of the NMF literature and discuss some of their limitations. In particular we show that these chains all have a degenerate stationary distribution;
- We present an overlooked non-negative Markov chain from the time series literature, the first-order autoregressive Beta-Gamma process, denoted as BGAR(1) @lewis1989gamma, whose stationary distribution is Gamma. To the best of our knowledge, this particular chain has never been considered to model temporal dependencies in matrix factorization problems;
- We derive majorization-minimization-based algorithms for maximum a posteriori (MAP) estimation in the NMF models with all presented prior structures on $\mathbf{H}$, including BGAR(1);
- We compare the performance of all these models on a prediction task on three real-world datasets.
The paper is organized as follows. Section \[sec-study\] introduces and compares non-negative Markov chains from the literature. Section \[sec-map\] presents MAP estimation in temporal NMF models. Experimental work is conducted in Section \[sec-exp\], before concluding in Section \[sec-ccl\].
Comparative study of Gamma Markov chains {#sec-study}
========================================
This section reviews existing models of Gamma Markov chains, i.e., Markov chains which evolve in $\mathbb{R}_{+}$ in relation with the Gamma distribution. We have identified three different models in the NMF literature:
1. Chaining on the rate parameter of a Gamma distribution (Section \[sec-rate\]);
2. Chaining on the rate parameter of a Gamma distribution with an auxiliary variable (Section \[sec-cd-rate\]);
3. Chaining on the shape parameter of a Gamma distribution (Section \[sec-shape\]).
As shall be discussed in these subsections, these three models are all built around the assumption $\mathbb{E}(h_{kn}|h_{k(n-1)}) \propto h_{k(n-1)}$ (which roughly means that the chain should not drift too far away from its previous value), but lack a well-defined stationary distribution, which leads to the degeneracy of the realizations of the chains. A fourth model from the time series literature, called BGAR(1), is presented in Section \[sec-bgar\]. It is built to have a well-defined stationary distribution (it is marginally Gamma distributed), but does not share the property $\mathbb{E}(h_{kn}|h_{k(n-1)}) \propto h_{k(n-1)}$. The realizations of the chain are not degenerate and exhibit some interesting properties. To the best of our knowledge, this kind of process has never been used in a probabilistic NMF problem to model temporal evolution.
Throughout the section, $(h_n)_{n \geq 1}$ denotes the (scalar) Markov chain of interest, where the index $k$ as in Eq. has been dropped for enhanced readability. It is further assumed that $h_1$ is set to a fixed, deterministic value.
Chaining on the rate parameter {#sec-rate}
------------------------------
### Model
Let us consider a general Gamma Markov chain model with a chaining on the rate parameter: $$h_{n}|h_{n-1} \sim \text{Gamma} \left( \alpha, \frac{\beta}{h_{n-1}} \right).
\label{eq-gmc-rate-def}$$ As it turns out, Eq. can be rewritten as a multiplicative noise model: $$h_{n} = h_{n-1} \times \phi_n,
\label{eq-gmc-rate-noise}$$ where $\phi_n$ are i.i.d. Gamma random variables with parameters $(\alpha, \beta)$. We have $$\mathbb{E}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta}h_{n-1}, \quad
\text{var}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta^2}h^2_{n-1}.$$
This model was introduced in @fevotte2009nonnegative to add smoothness to the activation coefficients in the context of audio signal processing. The parameters were set to $\alpha > 1$ and $\beta = \alpha - 1$, such that the mode would be located at $h_{n} = h_{n-1}$. A similar inverse Gamma Markov chain was also considered in @fevotte2009nonnegative and in @fevotte2011majorization.
### Analysis
From Eq. we can write: $$h_{n} = h_1 \prod_{i=2}^{n} \phi_i.$$
The independence of the $\phi_i$ yields: $$\begin{aligned}
\mathbb{E}(h_{n}) & = h_1 \left( \frac{\alpha}{\beta} \right)^{n-1}, \\
\text{var}(h_{n}) & = h_1^2 \left[ \left( \frac{\alpha^2}{\beta^2} + \frac{\alpha}{\beta^2} \right)^{n-1} - \left( \frac{\alpha^2}{\beta^2} \right)^{n-1} \right].
\label{eq-gmc-rate-moments}\end{aligned}$$
We enumerate all the possible regimes ($n \rightarrow +\infty)$, which all give rise to degenerate stationary distributions for different reasons:
- $\beta > \sqrt{\alpha(\alpha+1)}$: both mean and variance go to zero;
- $\beta = \sqrt{\alpha(\alpha+1)}$: variance converges to 1, however the mean goes to zero;
- $\beta \in \left]\alpha;\sqrt{\alpha(\alpha+1)}\right[$: variance goes to infinity, mean goes to zero;
- $\beta = \alpha$: mean is equal to 1, but the variance goes to infinity;
- $\beta < \alpha$: both mean and variance go to infinity.
Each subplot of Figure \[fig-ch4-gmc-rate\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha,\beta)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
Hierarchical chaining with an auxiliary variable {#sec-cd-rate}
------------------------------------------------
### Model
Let us consider the following Gamma Markov chain model introduced in @cemgil2007conjugate: $$\begin{aligned}
z_{n}|h_{n-1} & \sim \text{Gamma}(\alpha_z, \beta_z h_{n-1}), \label{eq-gmc-cd-def1} \\
h_{n}|z_{n} & \sim \text{Gamma}(\alpha_h, \beta_h z_{n}). \label{eq-gmc-cd-def2}\end{aligned}$$ As it turns out, this model can also be rewritten as a multiplicative noise model: $$h_{n} = h_{n-1} \times \tilde{\phi}_n,
\label{eq-gmc-cd-noise}$$ where $\tilde{\phi}_n$ are i.i.d. random variables defined as the ratio of two independent Gamma random variables with parameters $(\alpha_h, \beta_h)$ and $(\alpha_z, \beta_z)$. The distribution of $\tilde{\phi}_n$ is actually known in closed form, namely $$\tilde{\phi}_n \sim \text{BetaPrime} \left( \alpha_h, \alpha_z, 1, \tilde{\beta} \right),
\label{eq-gmc-cd-betaprime}$$ with $\tilde{\beta} = \frac{\beta_z}{\beta_h}$ (see Appendix \[app-a\] for a definition). We have $$\begin{aligned}
\mathbb{E}(h_{n}|h_{n-1}) & = \tilde{\beta} \frac{\alpha_h}{\alpha_z-1} h_{n-1} & \text{for~} \alpha_z > 1, \\
\text{var}(h_{n}|h_{n-1}) & = \tilde{\beta}^2 \frac{\alpha_h ( \alpha_h + \alpha_z - 1)}{(\alpha_z-1)^2 (\alpha_z-2)} h^2_{n-1} & \text{for~} \alpha_z > 2.\end{aligned}$$
This model is less straightforward in its construction than the previous one, as it makes use of an auxiliary variable $z_n$ (note that a similar inverse Gamma construction was proposed as well in @cemgil2007conjugate). There are two motivations behind the introduction of this auxiliary variable:
1. Firstly, it ensures what is referred to as “positive correlation” in @cemgil2007conjugate, i.e., $\mathbb{E}(h_n|h_{n-1}) \propto h_{n-1}$ (something the model described by Eq. does as well).
2. Secondly, it ensures the so-called conjugacy of the model, i.e., the conditional distributions $p(z_{n}|h_{n-1},h_{n})$ and $p(h_{n}|z_{n},z_{n+1})$ remain Gamma distributions. Indeed, these are the distributions of interest when considering Gibbs sampling or variational inference. This property is not satistfied by the model described by Eq. (i.e., $p(h_n|h_{n-1},h_{n+1})$ is neither Gamma, nor a known distribution).
This particular chain has been used in the context of audio signal processing in @virtanen2008bayesian (under the assumption of a Poisson likelihood, which does not fit the nature of the data), and also to model the evolution of user and item preferences in the context of recommender systems [@jerfel17dynamic; @do2018gamma].
### Analysis
From Eq. , we can write: $$h_{n} = h_1 \prod_{i=2}^{n} \tilde{\phi}_{i}.$$
We have by independence of the $\tilde{\phi}_i$: $$\begin{aligned}
\mathbb{E}(h_{n}) & = h_{1} \left( \tilde{\beta} \frac{\alpha_h}{\alpha_z - 1} \right)^{n-1} \qquad \qquad \qquad \text{for~}\alpha_z > 1, \\
\text{var}(h_{n}) & = h_1^2 \tilde{\beta}^{2(n-1)} \left[
\left( \frac{\alpha_h^2}{(\alpha_z - 1)^2} + \frac{\alpha_h ( \alpha_h + \alpha_z - 1)}{(\alpha_z-1)^2 (\alpha_z-2)} \right)^{n-1} \right. \notag \\
& \left. \qquad \qquad \qquad - \left( \frac{\alpha_h^2}{(\alpha_z - 1)^2} \right)^{n-1}
\right]~\text{for~} \alpha_z > 2.\end{aligned}$$
As in the previous model, we can show that either the expectation or the variance diverges or collapses as $n \rightarrow \infty$ for every possible choice of parameters, which means that they all give rise to a degenerate stationary distribution of the chain. Each subplot of Figure \[fig-ch4-gmc-cd\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha_z,\beta_z,\alpha_h,\beta_h)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
Chaining on the shape parameter {#sec-shape}
-------------------------------
### Model
Let us consider a general Gamma Markov chain model with a chaining on the shape parameter: $$h_{n}|h_{n-1} \sim \text{Gamma}(\alpha h_{n-1}, \beta).
\label{eq-gmc-shape-def}$$ We have $$\mathbb{E}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta}h_{n-1}, \quad
\text{var}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta^2}h_{n-1}.$$
In contrast with the two models presented previously, this model cannot be rewritten as a multiplicative noise model. This model is therefore more intricate to interpret. It was introduced in @acharya15nonparametric in the context of Poisson factorization. It is mainly motivated by a computational trick that can be used when working with a Poisson likelihood, hence making a Gibbs sampling feasible in the model. The authors set the value of $\alpha$ to 1 (though the same trick can be applied for any value of $\alpha$).
### Analysis
Using the law of total expectation and total variance, it can be shown that $$\mathbb{E}(h_{n}) = h_1 \left( \frac{\alpha}{\beta} \right)^{n-1}, \quad
\text{var}(h_{n}) = h_{1} \frac{1}{\beta} \left( \frac{\alpha}{\beta} \right)^{n-1} \sum_{i=1}^{n-1} \left( \frac{\alpha}{\beta} \right)^i.$$
The discussion is hence driven by the value of $r = \frac{\alpha}{\beta}$.
- If $r < 1$, mean and variance go to zero;
- If $r = 1$, mean is fixed but variance goes to infinity (linearly);
- If $r > 1$, mean and variance go to infinity.
This chain only exhibits degenerate stationary distributions. Each subplot of Figure \[fig-ch4-gmc-shape\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha,\beta)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
BGAR(1) {#sec-bgar}
-------
We now discuss the first order autoregressive Beta-Gamma process of @lewis1989gamma, a stochastic process which is marginally Gamma distributed. The authors referred to the process as “BGAR(1)”. However, to the best of our knowledge, no extension to higher-order autoregressive processes exists in the time series literature. As such, from now on, we will simply refer to it as “BGAR”.
### Model
Consider $\alpha > 0$, $\beta > 0$, $\rho \in [0,1[$. The BGAR process is defined as: $$\begin{aligned}
h_1 & \sim \text{Gamma}(\alpha,\beta), \label{eq-gmc-bgar-def1} \\
h_n & = b_n h_{n-1} + \epsilon_n \qquad \text{for~} n \geq 2 \label{eq-gmc-bgar-def2},\end{aligned}$$ where $b_n$ and $\epsilon_n$ are i.i.d. random variables distributed as: $$\begin{aligned}
b_n & \sim \text{Beta}(\alpha \rho, \alpha(1-\rho)), \label{eq-gmc-bgar-def3} \\
\epsilon_n & \sim \text{Gamma}(\alpha(1-\rho), \beta) \label{eq-gmc-bgar-def4}.\end{aligned}$$ $(h_n)_{n \geq 0}$ is called the BGAR process. It is parametrized by $\alpha$, $\beta$ and $\rho$. We emphasize that the distribution $p(h_n|h_{n-1})$ is not known in closed form. Only $p(h_n|h_{n-1}, b_n)$ is known; it is a shifted Gamma distribution. The generative model may therefore be rewritten as $$\begin{aligned}
h_1 & \sim \text{Gamma}(\alpha,\beta), \label{eq-b1} \\
b_n & \sim \text{Beta}(\alpha \rho, \alpha(1-\rho)) \quad \text{for~}n \geq 2, \label{eq-b2} \\
h_n | b_n, h_{n-1} & \sim \text{Gamma}(\alpha(1-\rho), \beta, \text{loc} = b_n h_{n-1}) \label{eq-b3} \\
& \qquad \qquad \qquad \qquad \qquad~\text{for~}n \geq 2, \notag\end{aligned}$$ where the distribution in Eq. is a shifted Gamma distribution with a location parameter “loc”.
We have $$\begin{aligned}
\mathbb{E}(h_n|h_{n-1}) & = \rho h_{n-1} + \frac{\alpha(1-\rho)}{\beta}, \label{eq-gmc-bgar-cond-moment1} \\
\text{var}(h_n|h_{n-1}) & = \frac{\rho(1-\rho)}{\alpha + 1}h^2_{n-1} + \frac{\alpha(1-\rho)}{\beta^2}. \label{eq-gmc-bgar-cond-moment2}\end{aligned}$$ As we can see, BGAR(1) already differs from the three previously presented models because the conditional expectation $\mathbb{E}(h_n|h_{n-1})$ is not proportional to $h_{n-1}$ (it is an affine transformation).
### Analysis
To study the marginal distribution of the process, we recall the following lemma.
If $X \sim \text{Beta}(a,b)$ and $Y \sim \text{Gamma}(a+b,c)$ are independent random variables, then $Z = XY$ is $\text{Gamma}(a,c)$ distributed. \[lemma-ch4\]
Let $(h_n)_{n \geq 1}$ be a BGAR process. Then $h_n$ is marginally $\text{Gamma}(\alpha,\beta)$ distributed.
Follows by induction. Consider $n$ such that $h_n$ is $\text{Gamma}(\alpha,\beta)$ distributed. Then, $\epsilon_{n+1}h_n$ is $\text{Gamma}(\alpha \rho,\beta)$ distributed (Lemma \[lemma-ch4\]). Finally, $h_{n+1} = \epsilon_{n+1}h_n + b_{n+1}$ is $\text{Gamma}(\alpha,\beta)$ distributed (sum of independent Gamma random variables), which concludes the proof.
Therefore the parameters $\alpha$ and $\beta$ control the marginal distribution. The parameter $\rho$ controls the correlation between successive values, as discussed in the following proposition.
Let $(h_n)_{n \geq 1}$ be a BGAR process. Let $n$ and $r$ be two integers such that $r > 1$. We have $\text{corr}(h_n, h_{n+r}) = \rho^{r}$. \[prp2\]
See Appendix \[app-b\] for $r=1$.
Proposition \[prp2\] implies that the BGAR(1) process admits a (second order) AR(1) representation. Two limit cases of BGAR can be exhibited:
- When $\rho = 0$, the $h_n$ are i.i.d. random variables;
- When $\rho \rightarrow 1$, the process is not random anymore, and $h_n = h_1$ for all $n$ (note that $\rho = 1$ is not an admissible value).
Finally, from Eq. , we have $$\bigg( \mathbb{E}(h_n|h_{n-1}) > h_{n-1} \bigg) \Leftrightarrow \bigg( h_{n-1} < \frac{\alpha}{\beta} \bigg).$$ If $h_{n-1}$ is below the mean of the marginal distribution ($\frac{\alpha}{\beta}$), then $h_n$ will be in expectation above $h_{n-1}$, and vice-versa.
Note that BGAR is not the only Markovian process with a marginal Gamma distribution considered in the literature. We mention the GAR(1) process (first-order autoregressive Gamma process) of @gaver1980first, which is also marginally Gamma distributed. However, this particular process is piecewise deterministic, and its parameters are “coupled”: the parameters of the marginal distribution also have an influence on other properties of the model. As such, it is less suited to our problem, and will not be considered here.
Figure \[fig-ch4-bgar\] displays three realizations of the BGAR process, with parameters fixed to $\alpha = 2$ and $\beta = 1$, and a different parameter $\rho$ in each subplot. The mean of the marginal distribution is displayed in red. When $\rho = 0.5$, the correlation is weak, and no particular structure is observed. However, as $\rho$ goes to 1, the correlation becomes stronger, and we typically observe piecewise constant trajectories.
MAP estimation in temporal NMF models {#sec-map}
=====================================
We now turn to the problem of maximum a posteriori (MAP) estimation in temporal NMF models. More precisely, we assume a Poisson likelihood, that is $$v_{fn} \sim \text{Poisson}([\mathbf{WH}]_{fn}),$$ and we also assume that $\mathbf{W}$ is a deterministic variable. We consider four different models corresponding to the four temporal structures on $\mathbf{H}$ presented in Section \[sec-study\]. As such, $\mathbf{V}$ and $\mathbf{H}$ define a hidden Markov model, as displayed on Figure \[figure-hmm\].
(0,0) circle (0.5); (1.5,0) circle (0.5); (3,0) circle (0.5); (0,-1.5) circle (0.5); (1.5,-1.5) circle (0.5); (3,-1.5) circle (0.5); (0.5,0) – (1,0); (2,0) – (2.5,0); (0, -1) – (0, -0.5); (1.5, -1) – (1.5, -0.5); (3, -1) – (3, -0.5); (3.5, 0) – (3.95, 0); (-0.5, 0) – (-0.95, 0); (0,0) node[[$\mathbf{h}_{n-1}$]{}]{}; (1.5,0) node[[$\mathbf{h}_{n}$]{}]{}; (3,0) node[[$\mathbf{h}_{n+1}$]{}]{}; (0,-1.5) node[[$\mathbf{v}_{n-1}$]{}]{}; (1.5,-1.5) node[[$\mathbf{v}_{n}$]{}]{}; (3,-1.5) node[[$\mathbf{v}_{n+1}$]{}]{};
(1.5,-2.5) node [$\bullet$]{}; (1.5,-2.8) node [[$\mathbf{W}$]{}]{}; (1.5,-2.5) – (1.5,-2); (1.5,-2.5) – (0,-2); (1.5,-2.5) – (3,-2); (1.5, -2.5) – (-0.75, -2.125); (1.5, -2.5) – (3.75, -2.125);
Generally speaking, joint MAP estimation in such models amounts to minimizing the following criterion $$\begin{aligned}
C(\mathbf{W},\mathbf{H}, \boldsymbol{\beta}) & = -\log p(\mathbf{V},\mathbf{H};\mathbf{W},\boldsymbol{\beta}) \label{eq-ccc} \\
& = -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \left( \log p(h_{k1}) + \sum_{n \geq 2} \log p(h_{kn}|h_{k(n-1)};\beta_k) \right),\end{aligned}$$ that is to say that the factors $\mathbf{W}$ and $\mathbf{H}$, as well as the scale hyperparameters $\boldsymbol{\beta} = [\beta_1, \dotsc, \beta_K]^{\text{T}}$, are going to be estimated (shape hyperparameters $\alpha$ or $\rho$ will be treated as fixed).
The optimization of the function $C$ is carried out with a block coordinate descent scheme over the variables $\mathbf{W}$, $\mathbf{H}$, and $\boldsymbol{\beta}$.
For the first two steps, we resort to a majorization-minimization (MM) scheme, which consists in iteratively majorizing the function $C$ (by a so-called auxiliary function, tight for some $\tilde{\mathbf{W}}$ or $\tilde{\mathbf{H}}$), and minimizing this auxiliary function instead. We refer the reader to @hunter2004tutorial for a detailed tutorial. Under this scheme, the function $C$ is non-increasing. As it turns out, only the Poisson likelihood term $-\log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ needs to be majorized. This is a well-studied issue in the NMF literature. As stated in @lee2000algorithms [@fevotte2011algorithms], the function $$G_1(\mathbf{H};\tilde{\mathbf{H}}) = - \sum_{k,n} {p}_{kn} \log (h_{kn}) + \sum_{k,n} q_{k} h_{kn},
\label{eq-g1}$$ with the notations $$p_{kn} = \tilde{h}_{kn} \sum_f w_{fk} \frac{v_{fn}}{[\mathbf{W\tilde{H}}]_{fn}}, \quad q_k = \sum_f w_{fk},$$ is a tight auxiliary function of $- \log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ at $\mathbf{H} = \tilde{\mathbf{H}}$. Similarly the function $$G_2(\mathbf{W};\tilde{\mathbf{W}}) = - \sum_{f,k} p'_{fk} \log (w_{fk}) + \sum_{f,k} q'_{k} w_{fk},
\label{eq-g2}$$ with the notations $$p'_{fk} = \tilde{w}_{fk} \sum_n h_{kn} \frac{v_{fn}}{[\mathbf{\tilde{W}H}]_{fn}}, \quad q'_k = \sum_n h_{kn},$$ is a tight auxiliary function of $- \log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ at $\mathbf{W} = \tilde{\mathbf{W}}$.
Finally, for all considered models the function $C$ can be minimized in closed form w.r.t. the variable $\beta_k$.
Minimization w.r.t. **W**
-------------------------
The optimization w.r.t. $\mathbf{W}$ is common to all algorithms, and amounts to minimizing $G_2(\mathbf{W};\tilde{\mathbf{W}})$ only. The scale of $\mathbf{W}$ must be however be fixed in order to prevent potential degenerate solutions such that $\mathbf{W} \rightarrow + \infty$ and $\mathbf{H} \rightarrow 0$. Indeed, consider $\mathbf{W}^{\star}$ and $\mathbf{H}^{\star}$ minimizers of Eq. , and let $\boldsymbol{\Lambda}$ be a diagonal matrix with non-negative entries. Then $$\begin{aligned}
C(\mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}, \boldsymbol{\Lambda} \mathbf{H}^{\star}) & = - \log p(\mathbf{V}|\boldsymbol{\Lambda} \mathbf{H}^{\star}; \mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}) - \log p(\boldsymbol{\Lambda} \mathbf{H}^{\star}) \\
& = - \log p(\mathbf{V}|\mathbf{H}^{\star}; \mathbf{W}^{\star}) - \log p(\boldsymbol{\Lambda} \mathbf{H}^{\star}),\end{aligned}$$ and depending on the choice of the prior distribution $p(\mathbf{H})$, we may obtain $C(\mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}, \boldsymbol{\Lambda} \mathbf{H}^{\star}) < C(\mathbf{W}^{\star}, \mathbf{H}^{\star})$, i.e., a contradiction. Therefore, in the following we impose that $||\mathbf{w}_k||_1 = 1$.
The constrained optimization is performed with the following update rule $$w_{fk} = \frac{p'_{fk}}{\sum_f p'_{fk}}, \label{eq-upw}$$ see Appendix \[app-c\] for the proof.
The following subsections detail the optimization w.r.t. $\mathbf{H}$ (and other variables when necessary), which amounts to the minimization of $G_1(\mathbf{H};\tilde{\mathbf{H}}) - \log p(\mathbf{H})$, as well as the minimization of $C$ w.r.t. $\beta_k$, for each considered model
Chaining on the rate parameter {#chaining-on-the-rate-parameter}
------------------------------
The transition distribution $p(h_{kn}|h_{k(n-1)})$ is given by Eq. . The optimization w.r.t. $h_{kn}$ amounts to solving an order-2 polynomial equation $$a_{2,{kn}} h_{kn}^2 + a_{1,{kn}} h_{kn} + a_{0,{kn}} = 0.
\label{eq-poly-1}$$ The coefficients of the polynomial equation are given in Table \[table-1\]. This bears resemblance with the methodology described in @fevotte2009nonnegative, where the authors aimed at retrieving MAP estimates with a EM-like algorithm (with an exponential likelihood).
[cccc]{} $n$ & $a_{2,{kn}}$ & $a_{1,{kn}}$ & $a_{0,{kn}}$\
$1$ & $q_{k}$ & $\alpha - p_{1k}$ & $-\beta_k h_{k2}$\
\
$2,\dotsc,N-1$ & $q_k + \frac{\beta_k}{h_{k(n-1)}}$ & $1 - p_{kn}$ & $- \beta_k h_{k(n+1)}$\
\
$N$ & 0 & $q_k$ + $\frac{\beta}{h_{k(N-1)}}$ & $1- \alpha - p_N$\
The update for the hyperparameter $\beta_k$ is given by $$\beta_k = \frac{\alpha (N-1)}{\sum_{n \geq 2} \frac{h_{kn}}{h_{k(n-1)}}}.$$
Hierarchical chaining with an auxiliary variable {#hierarchical-chaining-with-an-auxiliary-variable}
------------------------------------------------
In this case, since the transition distribution $p(h_{kn}|h_{k(n-1)})$ is not known in closed form, we resort to optimizing the slighlty more involved following criterion $$\begin{aligned}
& C(\mathbf{W},\mathbf{H},\mathbf{Z}, \boldsymbol{\beta}_h, \boldsymbol{\beta}_z) = \notag \\
& -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \bigg[ \log p(h_{k1}) + \sum_{n \geq 2} \left( \log p(z_{kn}|h_{k(n-1)};\beta_{z,k}) + \log p(h_{kn}|z_{kn};\beta_{h,k}) \right) \bigg],\end{aligned}$$ where $\boldsymbol{\beta}_h = [\beta_{h,1}, \dotsc, \beta_{h,K}]^{\text{T}}$ and $\boldsymbol{\beta}_z = [\beta_{z,1}, \dotsc, \beta_{z,K}]^{\text{T}}$. We recall that $p(z_{kn}|h_{k(n-1)})$ and $p(h_{kn}|z_{kn})$ are given by Eq. and Eq. , respectively. Note that @cemgil2007conjugate proposed a Gibbs sampler and variational inference, and as such the development of the MAP algorithm is novel.
The update for $z_{kn}$ is given by $$z_{kn} = \frac{\alpha_{z} + \alpha_{h} - 1}{\beta_{z,k} h_{k(n-1)} + \beta_{h,k} h_{kn}}.$$ As for the updates for $h_{kn}$, they are given by $$\begin{aligned}
h_{k1} & = \frac{p_{k1} + \alpha_{z}}{q_k + \beta_{z,k} z_{k2}}, \\
h_{kn} & = \frac{p_{kn} + \alpha_{h} + \alpha_{z} - 1}{q_k + \beta_{h,k} z_{kn} + \beta_{z,k} z_{k(n+1)}} \qquad n \in \{ 2,\dotsc, N \}, \\
h_{kN} & = \frac{p_{kN} + \alpha_{h} - 1}{q_k + \beta_{h,k} z_{kN}}.\end{aligned}$$
As such, imposing $\alpha_{k} \geq 1$ is a sufficient condition to preserve the non-negativity of all the updates. Finally, the update for the parameters $\beta_{z,k}$ and $\beta_{h,k}$ are given by $$\beta_{z,k} = \frac{(N-1)\alpha_{z}}{\sum_{n \geq 2} h_{k(n-1)} z_{kn}}$$ and $$\beta_{h,k} = \frac{(N-1)\alpha_h}{\sum_{n \geq 2} h_{kn} z_{kn}}.$$
Chaining on the shape parameter {#chaining-on-the-shape-parameter}
-------------------------------
The transition distribution $p(h_{kn}|h_{k(n-1)})$ is given by Eq. . The optimization w.r.t. $h_{kn}$ amounts to solving the following equations $$-p_{k1} + (q_k - \alpha \log(\beta_k h_{k2}) + \alpha \Psi(\alpha h_{k1}) )h_{k1} = 0,$$ $$\begin{aligned}
(1-\alpha h_{k(n-1)}-p_{kn}) & + (q_k + \beta_k - \alpha \log(\beta_k h_{k(n+1)})) h_{kn} + \alpha \Psi(\alpha h_{kn}) h_{kn} = 0,\end{aligned}$$ where $\Psi$ denotes the digamma function. Solving such equations can be done numerically with Newton’s method. Finally the update for $h_{kN}$ is given by $$h_{kN} = \frac{p_{kn} + \alpha h_{k(N-1)} -1}{q_k + \beta_k}.$$ The update for $\beta_k$ is given by $$\beta_k = \alpha \frac{\sum_{n \geq 2} h_{k(n-1)}}{\sum_{n \geq 2} h_{kn}}.$$
Note that a Gibbs sampling procedure is proposed in @acharya15nonparametric [@schein2016poisson], and as such the development of the MAP algorithm is novel.
BGAR(1) {#alg-map-bgar}
-------
In this case, since the transition distribution $p(h_{kn}|h_{k(n-1)})$ is not known in closed form, we resort to optimizing the slightly more involved following criterion $$\begin{aligned}
& C(\mathbf{W},\mathbf{H},\mathbf{B}, \boldsymbol{\beta}) = \notag \\
& -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \bigg( \log p(h_{k1})~+ \sum_{n \geq 2} \left( \log p(h_{kn}|h_{k(n-1)},b_{kn};\beta_k) + \log p(b_{kn}) \right) \bigg) \label{eq-map-bgar}.\end{aligned}$$
In the following, we will use the notations $\gamma_k = \alpha_k(1-\rho_k)$ and $\eta_k = \alpha_k \rho_k$.
### Constraints
By construction, the variables $h_{kn}$ and $b_{kn}$ must lie in a specific interval given the values of all the other variables. Indeed, as $h_{kn} = b_{kn} h_{k(n-1)} + \epsilon_{kn}$ (Eq. ), where $\epsilon_{kn}$ is a non-negative random variable, we obtain $h_{kn} \geq b_{kn} h_{k(n-1)}$, $b_{kn} \leq \frac{h_{kn}}{h_{k(n-1)}}$, and $h_{kn} \leq \frac{h_{k(n+1)}}{b_{k(n+1)}}$.
This leads to the following constraints $$\begin{aligned}
0 & \leq h_{k1} \leq \frac{h_{k2}}{b_{k2}}, \\
b_{kn} h_{k(n-1)} & \leq h_{kn} \leq \frac{h_{k(n+1)}}{b_{k(n+1)}} \qquad 2 \leq n < N, \\
b_{kN} h_{k(N-1)} & \leq h_{kN},\end{aligned}$$ and $$0 \leq b_{kn} \leq \min \left( 1, \frac{h_{kn}}{h_{k(n-1)}} \right).$$
We therefore introduce the notations $$\begin{aligned}
c_{kn} = b_{kn} h_{k(n-1)}, \quad d_{kn} = \frac{h_{k(n+1)}}{b_{k(n+1)}}, \quad x_{kn} = \frac{h_{kn}}{h_{k(n-1)}},\end{aligned}$$ as these quantities arise naturally in our derivations.
### Minimization w.r.t. $h_{kn}$
The optimization of Eq. w.r.t. $h_{kn}$ may give rise to intractable problems, due to the logarithmic terms in the objective function. To alleviate this issue, we propose to control the limit values of the auxiliary function, by restricting ourselves to certain values of the hyperparameters. In particular, choosing $(1-\gamma_k) < 0$ ensures the existence of at least one minimizer.
In all sub-cases, the optimization w.r.t. $h_{kn}$ amounts to solving an order-3 polynomial equation $$a_{3,{kn}} h_{kn}^3 + a_{2,{kn}} h_{kn}^2 + a_{1,{kn}} h_{kn} + a_{0,{kn}} = 0.
\label{eq-poly-2}$$ The coefficients of the polynomial equation are given in Table \[table-2\]. If several roots belong to the definition interval, we simply choose the root which gives the lowest objective value.
### Minimization w.r.t. $b_{kn}$
Similarly, logarithmic terms of the objective function may give rise to degenerate solutions. Using the same reasoning, we choose to impose $(1-\gamma_k) < 0$ and $(1-\eta_k) < 0$ to ensure the existence of at least one minimizer.
The minimization of the auxiliary function w.r.t. $b_{kn}$ amounts to solving the following order 3 polynomial over the interval $[0, \min(1, x_{kn})]$ $$a_{3,kn} b_{kn}^3 + a_{2,kn} b_{kn}^2 + a_{1,kn} b_{kn} + a_{0,kn} d_{kn},$$ where $$\begin{aligned}
a_{3,kn} & = -\beta_k h_{k(n-1)}, \\
a_{2,kn} & = 2(1-\gamma_k) + (1-\eta_k) + \beta_k h_{k(n-1)}(x_{kn}+1), \\
a_{1,kn} & = -(1-\gamma_k)(x_{kn}+1) - (1-\eta_k)(x_{kn} + 1) - \beta_k h_{k(n-1)} x_{kn}, \\
a_{0,kn} & = (1-\eta_k) x_{kn}.\end{aligned}$$
### Minization w.r.t. $\beta_k$
The minimization of $C$ w.r.t. $\beta_k$ can be done in closed form and results in the following update rule $$\beta_k = \frac{(N-1) \alpha (1-\rho)}{\sum_{n \geq 2} (h_{kn} - b_{kn}h_{k(n-1)})}.$$
### Admissible values of hyperparameters
To recap the discussion on admissible values of hyperparameters, to ensure the existence of minimizers of the auxiliary function, we have restricted ourselves to $$\left\{
\begin{array}{l}
\alpha_k(1-\rho_k) > 1 \\
\alpha_k \rho_k > 1
\end{array}
\right.$$ This set is graphically displayed on Figure \[fig-admv\]. As we can see, choosing the value of $\rho_k$ to be close to one (to ensure correlation) leads to high values of $\alpha_k$.
Experimental work {#sec-exp}
=================
We now compare the performance of all considered temporal NMF models on a prediction task on three real datasets. This task will consist in hiding random columns of the considered datasets and predicting those missing values. We will also include the performance of a naive baseline, which we detail in the following subsection. Adapting the MAP algorithms presented in Section \[sec-study\] in a setting with a mask of missing values only consist in a slight modification, presented in Appendix \[app-d\]. Python code will be made available upon acceptance.
Experimental protocol
---------------------
For each considered dataset, the experimental protocol is as follows.
First of all, a value of the factorization rank $K$ (which will be used for all considered methods) must be selected. To do so, we apply the standard KL-NMF algorithm [@lee2000algorithms; @fevotte2011algorithms] on 10 random training sets, which consist of $80 \%$ of the original data, with a pre-defined grid of values for $K$. We then select the value of $K$ which yields the lowest generalized Kullback-Leibler error (KLE) (see definition below) on the remaining $20 \%$ of the data.
For the prediction experiment itself, we create 5 random splits of the data matrix, where $80 \%$ corresponds to the training set, $10 \%$ to the validation set, and the remaining $10 \%$ to the test set. To do so, we randomly select non-adjacent columns of the data matrix (excluding the first one and the last one), half of which will make up the validation set and the other half the test set. We also consider 5 different random initializations.
Thus, for each split-initialization pair, all the algorithms are run from this initialization point on the training set until convergence (the algorithms are stopped when the relative decrease of the objection function falls under $10^{-5}$). For each method, a grid of shape hyperparameters is considered, and the selection of this parameter is based on the lowest KLE on the validation set. The predictive performance of each method is then computed on the test set by comparing the original value $v_{fn}$ and its associated estimate $\hat{v}_{fn} = [\mathbf{WH}]_{fn}$ with two different metrics. Denoting by $\mathcal{T}$ the test set, we consider
- the generalized Kullback-Leibler error (KLE) $$\text{KLE} = \sum_{(f,n) \in \mathcal{T}} \left[ v_{fn} \log \left( \frac{v_{fn}}{\hat{v}_{fn}} \right) - v_{fn} + \hat{v}_{fn} \right];$$
- the relative error, as in @schein2016poisson $$\text{RE} = \sum_{(f,n) \in \mathcal{T}} \frac{|v_{fn} - \hat{v}_{fn}|}{v_{fn} + 1}.$$
Finally, we compare the NMF-based approaches to the following naive baseline, based on a random guess. In this case, the values of the missing columns are simply estimated by drawing $\hat{v}_{fn}$ from the empirical distribution of the observed data coefficient in every row $f$.
Datasets
--------
The following datasets are considered
- The `NIPS` dataset[^3], which contains word counts (with stop words removed) of all the articles published at the NIPS[^4] conference between 1987 and 2015. We grouped the articles per year, yielding an observation matrix of size $11463 \times 29$. We obtained $K = 3$.
- The `ICEWS` dataset[^5], an international relations dataset, which contains the number of interactions between two countries for each day of the year 2003. The matrix is of size $6197 \times 365$. We obtained $K = 15$.
- The `last.fm` dataset, based on the so-called “last.fm 1K” users[^6], which contains the listening history of users with timestamps information. We preprocessed this dataset to obtain the monthly evolution of the listening counts of artists with at least 20 different listeners. This yields a dataset of size $7017 \times 53$. We obtained $K = 6$.
Experimental results
--------------------
The averaged KLE and RE over the 25 split-initialization pairs are reported on Table \[table-nips\] for the `NIPS` dataset, on Table \[table-icews\] for the `ICEWS` dataset, and on Table \[table-last\] for the `last.fm` dataset.
First of all, on all the considered datasets, the naive baseline yields the worst performance results, for all metrics, as expected. Moreover, all temporal models achieve comparable predictive performance, and the slight advantage of some methods over the others being data-dependent.
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $14.8 \times 10^5 \pm 57.4 \times 10^4$ $7.45 \times 10^4 \pm 17.9 \times 10^3$
Rate (II.A) $\mathbf{1.07 \times 10^5 \pm 2.01 \times 10^4}$ $\mathbf{2.98 \times 10^4 \pm 2.53 \times 10^3}$
Hierarchical (II.B) $\mathbf{1.07 \times 10^5 \pm 2.02 \times 10^4}$ $\mathbf{2.98 \times 10^4 \pm 2.45 \times 10^3}$
Shape (II.C) $1.29 \times 10^5 \pm 2.75 \times 10^4$ $\mathbf{3.07 \times 10^4 \pm 4.07 \times 10^3}$
BGAR (II.D) $\mathbf{1.05 \times 10^5 \pm 1.74 \times 10^4}$ $\mathbf{3.02 \times 10^4 \pm 2.00 \times 10^3}$
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $99.1 \times 10^4 \pm 44.2 \times 10^3$ $3.33 \times 10^4 \pm 4.06 \times 10^2$
Rate (II.A) $\mathbf{8.68 \times 10^4 \pm 2.95 \times 10^3}$ $\mathbf{2.65 \times 10^4 \pm 5.93 \times 10^2}$
Hierarchical (II.B) $\mathbf{8.80 \times 10^4 \pm 3.39 \times 10^3}$ $\mathbf{2.62 \times 10^4 \pm 7.80 \times 10^2}$
Shape (II.C) $\mathbf{8.79 \times 10^4 \pm 3.31 \times 10^3}$ $\mathbf{2.61 \times 10^4 \pm 6.90 \times 10^2}$
BGAR (II.D) $9.09 \times 10^4 \pm 4.16 \times 10^3$ $\mathbf{2.54 \times 10^4 \pm 5.30 \times 10^2}$
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $66.5 \times 10^4 \pm 1790 \times 10^2$ $3.40 \times 10^4 \pm 89.9 \times 10^2$
Rate (II.A) $\mathbf{1.55 \times 10^4 \pm 7.13 \times 10^2}$ $\mathbf{1.18 \times 10^4 \pm 5.55 \times 10^2}$
Hierarchical (II.B) $\mathbf{1.57 \times 10^4 \pm 6.18 \times 10^2}$ $\mathbf{1.17 \times 10^4 \pm 6.06 \times 10^2}$
Shape (II.C) $2.01 \times 10^4 \pm 51.9 \times 10^2$ $\mathbf{1.19 \times 10^4 \pm 7.16 \times 10^2}$
BGAR (II.D) $\mathbf{1.59 \times 10^4 \pm 9.10 \times 10^2}$ $\mathbf{1.20 \times 10^4 \pm 6.86 \times 10^2}$
Conclusion {#sec-ccl}
==========
In this paper, we have reviewed existing temporal NMF models in a unified MAP framework and introduced a new one. These models differ by the choice of the Markov chain structure used on the activation coefficients to induce temporal correlation. We began by studying the previously proposed Gamma Markov chains of the NMF literature, only to find that they all share the same drawback, namely the absence of a well-defined stationary distribution. This leads to problematic behaviors from the generative perspective, because the realizations of the chains are degenerate (although this is not necessarily a problem in MAP estimation). We then introduced a Markovian process from the time series literature, called BGAR(1), which overcomes this limitation, and which, to the best of our knowledge, had never been exploited for learning tasks.
We then derived MAP estimation algorithms for all these models, in the context of a Poisson likelihood, which allowed for a comprehensive comparison on a prediction task on real datasets. As it turns out, we cannot claim that there is a single model which outperforms all the others. Their strengths and weaknesses appear to depend on the nature of the data at hand, which is reasonable.
Future work will focus on finding a way to perform inference with the BGAR prior for a less restrictive set of hyperparameters, which might increase the performance of this particular model. We will also work to derive similar algorithms in the context of different likelihoods, such as an exponential likelihood [@fevotte2009nonnegative], which can be easily done thanks to the MM framework.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under grant agreement No 681839 (project FACTORY).
The Beta-Prime distribution {#app-a}
===========================
Distribution for a continuous random variable in $[0,+\infty[$, with parameters $\alpha > 0$, $\beta > 0$, $p > 0$ and $q > 0$. Its p.d.f. writes, for $x \geq 0$: $$f(x;\alpha,\beta,p,q) = \frac{p \left( \frac{x}{q} \right)^{\alpha p -1} \left(1 + \left(\frac{x}{q}\right)^p \right)^{-\alpha - \beta}}{q \text{B}(\alpha, \beta)}.$$
BGAR(1) linear correlation {#app-b}
==========================
We have between two successive values $h_n$ and $h_{n+1}$: $$\begin{aligned}
& \text{corr}(h_n, h_{n+1}) \\
& = \frac{\mathbb{E}(h_n h_{n+1}) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\mathbb{E}(h_n(b_{n+1}h_n + \epsilon_{n+1})) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\mathbb{E}(b_{n+1})\mathbb{E}(h_n^2) + \mathbb{E}(h_n)\mathbb{E}(\epsilon_{n+1}) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\frac{\alpha \rho}{\alpha \rho + \alpha(1-\rho)} \frac{\alpha(\alpha+1)}{\beta^2} + \frac{\alpha}{\beta} \frac{\alpha(1-\rho)}{\beta} - \frac{\alpha}{\beta} \frac{\alpha}{\beta} }{\frac{\alpha}{\beta^2}} \\
& = \rho.\end{aligned}$$
Constrained optimization {#app-c}
========================
We want to optimize $G_2(\mathbf{W};\tilde{\mathbf{W}})$ w.r.t. $\mathbf{W}$ s.t. $\sum_f{w_{fk}} = 1$. Rewriting this with Lagrange multipliers $\boldsymbol{\lambda} = [\lambda_1,\dotsc,\lambda_{K}]^{\text{T}}$, this is tantamount to $$\min_{\mathbf{W}, \boldsymbol{\lambda}} G_2(\mathbf{W};\tilde{\mathbf{W}}) + \sum_k \lambda_k (||\mathbf{w}_k||_1 - 1).$$ Deriving w.r.t $w_{fk}$ yields $$w_{fk} = \frac{p'_{fk}}{q'_k + \lambda_k}. \label{eq-lagr}$$ We retrieve the constraint by summing this expression over $f$. This gives the expression of the Lagrange multiplier: $\lambda_k = \sum_f p'_{fk} - q'_k$. Substituting this expression into Eq. , we obtain the following update rule $$w_{fk} = \frac{p'_{fk}}{\sum_f p'_{fk}}.$$
Algorithms with missing values {#app-d}
==============================
In the context of missing values, let us consider a mask matrix $\mathbf{M}$ of size $F \times N$ such that $m_{fn} = 1$ if the entry $v_{fn}$ is observed and 0 otherwise. The likelihood term can then be written as $$-\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) = - \sum_{f,n} m_{fn} \log p(v_{fn}|[\mathbf{WH}]_{fn}).$$ The auxiliary function $G_1$ of Eq. and $G_2$ of Eq. can then be written is the same way, with $$p_{kn} = \tilde{h}_{kn} \sum_f w_{fk} \frac{m_{fn} v_{fn}}{[\mathbf{W\tilde{H}}]_{fn}}, \quad q_{kn} = \sum_f m_{fn} w_{fk},$$ for $G_1$, and $$p'_{fk} = \tilde{w}_{fk} \sum_n h_{kn} \frac{m_{fn} v_{fn}}{[\mathbf{\tilde{W}H}]_{fn}}, \quad q'_{kn} = \sum_n m_{fn} h_{kn},$$ for $G_2$.
[^1]: These models are sometimes generically referred to as “Poisson factorization” or “Poisson factor analysis”.
[^2]: Note that this particular mapping is actually slightly more complex, as the $K$-dimensional real vector must be mapped to the $(K-1)$ simplex due to further constraints in the model.
[^3]: <https://archive.ics.uci.edu/ml/datasets/NIPS+Conference+Papers+1987-2015>
[^4]: Now called NeurIPS.
[^5]: <https://github.com/aschein/pgds>
[^6]: <http://ocelma.net/MusicRecommendationDataset/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'How to leverage cross-document interactions to improve ranking performance is an important topic in information retrieval (IR) research. However, this topic has not been well-studied in the learning-to-rank setting and most of the existing work still treats each document independently while scoring. The recent development of deep learning shows strength in modeling complex relationships across sequences and sets. It thus motivates us to study how to leverage cross-document interactions for learning-to-rank in the deep learning framework. In this paper, we formally define the permutation-equivariance requirement for a scoring function that captures cross-document interactions. We then propose a self-attention based document interaction network and show that it satisfies the permutation-equivariant requirement, and can generate scores for document sets of varying sizes. Our proposed methods can automatically learn to capture document interactions without any auxiliary information, and can scale across large document sets. We conduct experiments on three ranking datasets: the benchmark Web30k, a Gmail search, and a Google Drive Quick Access dataset. Experimental results show that our proposed methods are both more effective and efficient than baselines.'
author:
- 'Rama Kumar Pasumarthi, Xuanhui Wang, Michael Bendersky, Marc Najork'
bibliography:
- 'main.bib'
title: 'Self-Attentive Document Interaction Networks for Permutation Equivariant Ranking'
---
=1
| {
"pile_set_name": "ArXiv"
} |
Introduction {#intro}
============
One-dimensional or quasi-one-dimensional magnetic systems show many fascinating properties which continue to attract an intense theoretical activity. One of these properties is the presence of a spin gap in antiferromagnetic Heisenberg chains with integer spin[@haldane] and in ladders.[@ladders] Another, particularly complex, system which presents a spin gap is the spin-Peierls (SP) system. In this system a Heisenberg chain coupled to the lattice presents an instability at a critical temperature, $T_{SP}$, below which a dimerized lattice pattern appears and a spin gap opens in the excitation spectrum.[@pytte]
The interest in the spin-Peierls phenomena was recently revived after the first inorganic SP compound, CuGeO$_3$, was found.[@hase] This inorganic material allows the preparation of better samples than the organic SP compounds and hence several experimental techniques can be applied to characterize the properties of this system.[@regnault] Besides, this compound can be easily doped with magnetic and non-magnetic impurities, leading to a better understanding of its ground state and excitations.[@lussier]
Spin-Peierls systems present also a very rich and interesting behavior in the presence of an external magnetic field. Below the spin-Peierls transition temperature, and for magnetic fields $H$ smaller than a critical value $H_{cr}(T)$, the system is in its spin-Peierls phase, characterized by a gapped nonmagnetic ($S^z=0$) ground state with a dimerized pattern or alternating nearest-neighbor (NN) interactions. For $T < T_{tc} < T_{SP}$, at $H=H_{cr}(T)$ a transition occurs from the dimerized phase to a gapless incommensurate (IC) state characterized by a finite magnetization, $S^z > 0$. $T_{tc}$ is the temperature of the point at which the dimerized, incommensurate and uniform phases meet. The dimerized-IC transition was predicted by some theories[@cross2] to be of first order at low temperatures, and this is the behavior found in experimental studies[@hase2; @loosd]. Other theories predict that this transition is a second order one.[@fujita]
A simple picture of the dimerized-IC transition can be obtained by mapping the Heisenberg spin chain to a spinless fermion system by a Jordan-Wigner transformation. The effect of the magnetic field favoring a nonzero $S^z$ due to the Zeeman energy can be interpreted as a change in the band filling of the equivalent spinless fermion system. As a result, the momentum of the lattice distortion moves away from $\pi$ as $\tilde{q} = (1-S^z/N) \pi$, where $N$ is the number of sites on the chain. However, since [*umklapp*]{} processes pin the momentum at $\pi$ up to a critical field $H_{cr}(T)$, the lattice distortion will remain a simple dimerization and the magnetic ground state will remain a singlet.[@crossfisher] Theoretical[@fujita; @nakano; @buzdin] and experimental[@kiry; @fagot] studies indicate that the lattice distortion pattern in the IC phase corresponds to an array of solitons. A complementary picture indicating how a soliton lattice could appear as a consequence of the finite magnetization in the IC phase is the following. Let’s assume that the dominant contribution to the magnetic ground state comes from a state of NN singlets or dimers. An up spin replacing a down spin destroys a singlet and gives rise to two domain-walls or solitons separating regions of dimerized order which are shifted in one lattice spacing with respect to each other. Each soliton carries a spin-1/2. Due to the spin-lattice coupling it is expected that the lattice solitons are driven by these magnetic solitons.
The soliton formation in spin-Peierls systems has been studied analytically by bosonization techniques applied to the spinless fermion model.[@affleck] The coupling to the lattice is treated usually in the adiabatic approximation. The resulting field-theory formalism has lead to important results, the most remarkable being the relation between the soliton width and the spin-Peierls gap, $\xi \sim \Delta^{-1}$.[@nakano] Although this formalism has been extended to a Heisenberg model with competing NN and next-nearest-neighbor (NNN) antiferromagnetic interactions[@zang2; @dobryriera], it presents some unsatisfactory features.
In the first place, there are some recent experimental results[@kiry] for the soliton width in the IC phase in CuGeO$_3$ indicating a disagreement with the theoretical prediction. Although there might be a contribution to the soliton width coming from magnetic[@zang2] or elastic[@dobryriera] interchain couplings which would explain at least partially this disagreement, it is also possible that the differences could be due to several approximations involved in the bosonized field theory. One should take into account that these theories are valid in principle in the long wave-length limit, and the applicability of their results to real materials can not be internally assessed. Then, our first motivation to start a numerical study of the IC phase in spin-Peierls systems is to measure the importance of these approximations in the analytical approach.
In the second place, the field theory approach does not provide a detailed dependence of the magnitudes involved in terms of the original parameters of the microscopical models. For example, even for the simplest case[@nakano] the expression obtained for the spin-wave velocity must be replaced by the exact one known from Bethe’s exact solution of the Heisenberg chain. In this sense, numerical studies could give information about how the relevant magnitudes depend on the original parameters without further approximations.
With these motivations, in this article we want to initiate the study of the incommensurate phase in SP systems using numerical methods. These methods give essentially exact results for finite clusters, and they can be used to check various approximations required by the analytical approaches and the validity of their predictions. Besides, the numerical simulations provide a detailed information of the dominant magnetic and lattice states. In Section \[lanczos\] we present the model considered and we study several features of the soliton formation in the IC phase using the Lanczos algorithm. In particular we analyze the effect of NNN interactions on the soliton width. In Section \[monte\] we perform Monte Carlo simulations using the world line algorithm –which allows us to study larger chains than the ones accessible to the Lanczos algorithm– in order to reduce finite size effects.
Exact diagonalization study {#lanczos}
===========================
The one-dimensional model which contains both the antiferromagnetic Heisenberg interactions and the coupling to the lattice is:
$$\begin{aligned}
{\cal H} &=& J \sum_{i = 1}^N (1 + (u_{i+1}- u_{i}))\;
{\bf S}_i \cdot {\bf S}_{i+1} \nonumber \\
&+& J_2 \sum_{i = 1}^N {\bf S}_i \cdot {\bf S}_{i+2}
+ \frac{K}{2} \sum_{i=1}^N (u_{i+1}- u_{i})^2
\label{hamtot}\end{aligned}$$
where ${\bf S}_i$ are the spin-1/2 operators and $u_i$ is the displacement of magnetic ion $i$ with respect to its equilibrium position. Periodic boundary conditions are imposed. The first term, which corresponds to the nearest neighbor (NN) interactions, contains the spin-lattice coupling in the adiabatic approximation. The second term contains the AF NNN interactions, which were proposed in Refs. \[\] to fit the experimental magnetic susceptibility data in CuGeO$_3$. Several other properties of this material have been reasonably described using this model.[@haas; @rierakoval; @poilblanc] As in Ref. \[\], we assume for simplicity that the lattice distortion does not affect the second neighbor interactions. In principle, the NNN interactions should be corrected by a term proportional to $(u_{i+2}- u_{i})$ which vanishes in the dimerized phase but not necessarily in the incommensurate phase. This correction should be important precisely in the region around a soliton. It is customary to introduce the frustration constant $\alpha= J_2/J$. The estimated value of $\alpha$ in CuGeO$_3$ varies between 0.24 (Ref. \[\]) and 0.36 (Ref. \[\]). In this second case, $\alpha$ is larger than the critical value $\alpha_c \approx 0.2411$ above which in the absence of dimerization a gap opens in the excitation spectrum.[@okamoto]
Our purpose is to study numerically Hamiltonian (\[hamtot\]) with exact diagonalization (Lanczos) techniques and by Monte Carlo simulations. In this latter case, in order to avoid the well-known sign problem due to the frustration, we will consider only the diagonal second neighbor interaction
$$\begin{aligned}
{\cal H}_2^{zz} =
J_2^{z} \sum_{i = 1}^N S_i^{z} S_{i+2}^{z},
\label{h2n-zz}\end{aligned}$$
instead of the isotropic NNN interactions (second term of Eq. \[hamtot\]).
It is quite apparent that the main numerical difficulty is related to the handling of the set of displacements $\{u_i\}$, which in principle can take arbitrary values to describe the various distortion patterns present in the dimerized and IC phases of the system. These displacements are calculated self-consistently by the following iterative procedure. First, we introduce the bond distortions defined as $\delta_i = (u_{i+1}- u_{i})$. Then, the equilibrium conditions for the phononic degrees of freedom:
$$\begin{aligned}
\frac{\partial \langle {\cal H} \rangle}{\partial \delta_i} +
\lambda = 0
\label{eqcond}\end{aligned}$$
lead to the set of equations: $$\begin{aligned}
J \langle {\bf S}_i \cdot {\bf S}_{i+1} \rangle + K \delta_i -
{J \over N} \sum_{i=1}^N \langle {\bf S}_i \cdot {\bf S}_{i+1}
\rangle = 0 ,
\label{distor-eqn}\end{aligned}$$ which satifies the constraint $\sum_i \delta_i =0$. This constraint has been included in Eq. (\[eqcond\]) through the corresponding Lagrange multiplier $\lambda$. The expectation values are taken with respect to the ground state of the system. The iterative procedure starts with an initial distortion pattern $\{ \delta_i^{(0)} \}$, which in general we choose at random. At the step $n$, with a distortion pattern $\{ \delta_i^{(n-1)} \}$, we diagonalize Hamiltonian (\[hamtot\]) using the Lanczos algorithm and compute the correlations $\langle {\bf S}_i \cdot {\bf S}_{i+1} \rangle$. We replace these correlations in Eq. (\[distor-eqn\]) and the new set $\{ \delta_i^{(n)} \}$ is obtained. We repeat this iteration until convergence. Essentially the same procedure is followed in the quantum Monte Carlo algorithm, as it is discussed in Section \[monte\].
We have applied this exact diagonalization procedure to determine the distortion patterns in the 20 site chain at $T=0$. In the first place we consider the case of $S^z=0$. As mentioned above, this corresponds to a dimerized lattice, i.e. $\delta_i = (-1)^i \delta_0$. Notice that for this simple case, the equilibrium distortion amplitude $\delta_0$ could be determined in an easier way by computing the energies of the spin part of Hamiltonian for a set of values of $\delta_0$. Then, adding the elastic energy and interpolating one obtains the minimum total energy. We have performed this calculation in order to check our iterative algorithm.
The results for $\delta_0$ vs. $K$, for $S^z=0$, are shown in Fig.1 for $\alpha = 0.0$, 0.2 and 0.4, and $J_2^{z} = 0.2$, and 0.4. It can be seen that, as expected, for $\alpha >0$ the dimerized state is more favorable and this leads to a larger $\delta_0$ for a given $K$. To a lesser extent this trend is also present for $J_2^{z} > 0$.
The dependence of $\delta_0$ with $K$ can be inferred from the scaling relation between the energy and the dimerization, $E_0(\delta_0)-E_0(0) \sim \delta_0^{2\nu}$ (plus logarithmic corrections) with $\nu=2/3$, in principle valid for $\alpha < \alpha_c$ and small $\delta_0$.[@crossfisher; @spronken; @laukamp] Then, it is easy to obtain $\delta_0 \sim K^{-3/2}$, a relation which is approximately satisfied by our numerical data. The fact that $\delta_0$ vanishes at a finite value $\hat{K}$ of the elastic constant, is just a finite size effect. By diagonalizing chains of $N=12$, 16 and 20 sites, for $\alpha=0$, we have verified that $\hat{K}$ increases with the lattice size, as it can be seen in Fig. \[fig1.5\], and it should eventually diverge in the bulk limit.
Once we have determined the equilibrium distortion as a function of $K$, we are able to compute the singlet-triplet spin gap, defined as the following difference of ground state energies: $$\begin{aligned}
\Delta = E_{0,dim}(S^z=1)-E_0(S^z=0)
\label{spingap}\end{aligned}$$ It is worth to emphasize that $E_{0,dim}$ is the ground state energy of the system for $S^z=1$ with the dimerization obtained for $S^z=0$ and the same set of parameters. The results of this calculation are shown in Fig. \[fig2\]. Consistently with the larger $\delta_0$ shown in Fig. \[fig1\], the gap increases with $\alpha$. The effect of $J_2^{z}$ is much weaker than that of the isotropic second neighbor interaction which is not surprising since the 1D ground state magnetic structure, with a dominant dimerized state, has essentially a quantum (off-diagonal) origin. This small increase in $\Delta$ for a given $K$ is consistent with the small increase in $\delta_0$ shown in Fig. \[fig1\]. The corresponding scaling relation, $\Delta \sim K^{-1}$, obtained from the relation between the singlet-triplet gap and the dimerization, $\Delta \sim \delta_0^{2/3}$, is again reasonably satisfied by our numerical data.
. \[fig2\]
We now consider the case of $S^z=1$, which corresponds to the incommensurate region just above the dimerized-incommensurate transition. We have determined the distortion pattern for a 20 site chain using the iterative procedure described above. As discussed at the beginning of this section, the two solitons or domain walls separating dimerized regions are clearly distinguishable. (A typical pattern can be seen in Fig. \[soliton\].) The maximum distortion $\delta_0$, shown in Fig. \[fig3\], presents similar behavior as the one shown in Fig. \[fig1\] corresponding to $S^z=0$. In particular, the fact that $\delta_0$ vanishes at a finite $K$ is again due to finite size effects.
In order to compute the soliton width, we use the following form to fit the numerically obtained distortion patterns:
$$\begin{aligned}
\delta_i = (-1)^i \tilde{\delta} \tanh \left(\frac{i-i_0 -
{\frac d 2}}{\xi}
\right)\tanh \left(\frac{i-i_0+{\frac d 2}}{\xi}\right),
\label{fittanh}\end{aligned}$$
which corresponds to modeling each soliton as an hyperbolic tangent, as obtained in the analytical approach to this problem.[@nakano] The amplitude $\tilde{\delta}$, the soliton width $\xi $, and the soliton-antisoliton distance $d$, are the parameters determined by the numerical fitting. The amplitude $\tilde{\delta}$ should be equal to the maximum distortion $\delta_0$ defined above for well separated solitons, i.e. $d \gg \xi$. The main limitation of this calculation arises in the region where, for a given $\alpha$, $K$ is so large that the solitons have a substantial overlap in the 20 site chain, and the fitting function (\[fittanh\]) is no longer appropriate. In this case, the elliptic sine should be used to describe the soliton lattice. This is the region where finite size effects are important, as it was discussed above with respect to Figs. \[fig1\] and \[fig3\]. However, this situation is not directly relevant to experiment since in real materials the solitons are well-separated.[@kiry] We show in Fig. \[fig4\] the soliton width as a function of the gap $\Delta$ for the 20 site chain, for the same values of $\alpha$ and $J_2^{z}$ as before. It can be seen that the there is a linear dependence of the soliton width with the inverse of the gap. This behavior is consistent with the theoretical prediction:[@nakano]
$$\begin{aligned}
\xi =v_s/\Delta,
\label{width-gap}\end{aligned}$$
where $v_s$ is the spin-wave velocity for $\alpha < \alpha_c$. It was recently shown that the relation (\[width-gap\]), originally obtained for the unfrustrated chain,[@nakano] is also valid in the presence of frustration.[@dobryriera] For $\alpha > \alpha_c$, $\Delta$ contains a contribution from the frustration due to the presence of a gap even in the absence of dimerization.
A linear fitting of these curves in the region $\xi > 2.5$ gives the slopes 1.87, 1.70 and 1.63 , for $\alpha =0.0,\;0.2$ and $0.4$ respectively. Recently, a numerical study[@fledder] has proposed the law: $v_s=\frac \pi 2 (1-1.12 \alpha )$ in the bulk limit for $\alpha < \alpha_c$, From this law one gets $v_s =$ 1.57, 1.22, for $\alpha =0.0$ and $0.2$ respectively. We can observe that the slopes obtained by fitting the curves shown in Fig. (\[fig4\]) are systematically larger than these values of $v_s$. Besides, the effect of $\alpha$ is weaker in the numerical data than that predicted by Eq. (\[width-gap\]). For $\alpha= 0.4 > \alpha_c \approx 0.2411$, we have estimated $v_s$ by fitting the excitation dispersion relation $\varepsilon(k) = E_{0,dim}(S^z=1,k) - E_0(S^z=0,k=0)$ with the law $\varepsilon(k)^2 = \Delta^2 + v_s^2 k^2 + c k^4$ around $k=0$ and $\delta_i =0$. For $L=20$ we obtained $v_s = 0.707$, a value which is also smaller than the slope of the curve $\xi$ vs. $1/\Delta$ for $\alpha=0.4$ in Fig. (\[fig4\]). This disagreement between the prediction obtained by the continuum bosonized theory and the numerical results could be due to the approximations involved in the former or to finite size effects present in the latter. The study of much larger lattices than those considered in this section will be done in the following section using quantum Monte Carlo simulations. On the other hand, for the case of $J_2^z = 0.4$ the slope is actually [*larger*]{} ($\approx 2.1$) than the value obtained for the Heisenberg chain with NN interactions only. This effect is opposite to that of the isotropic NNN interactions and it will be further discussed in the next section.
Monte Carlo simulations {#monte}
=======================
In order to treat longer chains than those considered in the Lanczos diagonalization study of the previous section, we have implemented a world-line Monte Carlo algorithm[@WLMC] suited to this problem. The partition function is re-expressed as a functional integral over wordline configurations, where the contribution on each imaginary-time slice is given by the product of the two-site evolution matrix elements,
$$\begin{aligned}
W_{i,i+1}(\tau )=\langle S_{i,\tau }^zS_{i+1,\tau }^z\left| {\rm e}^{-
\Delta \tau J_i {\bf S}_i \cdot {\bf S}_{i+1}}\right| S_{i,\tau
+\Delta \tau }^zS_{i+1,\tau +\Delta \tau }^z\rangle
\nonumber\end{aligned}$$
where $J_i = J(1 + \delta_i)$. These matrix elements are the Boltzmann weights associated with a bond ($ i,i+1$) in a time step $\Delta \tau =1/mT$ in the Trotter direction, where $ T $ is the temperature and $m$ is the Trotter number. Since the exchange couplings depend on the lattice displacements, these matrix elements are site dependent.
We implemented the algorithm with the addition of a dynamic minimization of the free energy with respect to the lattice displacements. Starting from a given initial configuration (random distribution of spins and a dimerized pattern for the lattice displacements) we typically considered $2\times 10^3$ sweeps for thermalization. During the next $4\times 10^3$ sweeps we measured the derivative of the magnetic free energy, which, in the limit of $T \rightarrow 0$, is given by
$$\frac{\partial {\cal F}_M}{\partial \delta _i}=J\langle \langle
{\bf S}_i {\bf \cdot S}_{i+1}\rangle \rangle _T \ .
\label{DF}$$
Leaving 3 sweeps between each measurement for de-correlation this produces $10^3$ independent values to obtain the thermal average. With this free-energy gradient we corrected the displacements according to (\[distor-eqn\]) and repeated the procedure, including the $2\times 10^3$ sweeps for thermalization since the spins have to accommodate to the new lattice distorsions. Once the displacement pattern is stabilized within statistical fluctuations —we typically considered $\sim $150 iterations, see Fig. \[150\]— we performed measurements of several quantities. For this we obtained 100 independent groups of $10^3$ measurements each, following the same procedure as described above, [*i.e.*]{}, i) thermalization, ii) measurements of $\frac{\partial {\cal F}_M}{\partial \delta _i}$ and observables, and iii) correction of the displacement pattern due to statistical fluctuations.
In our calculations we considered chains of 64 sites with periodic boundary conditions and a temperature $T=0.05J$. We checked that this value is low enough to study ground-state properties by comparison with measurements at even lower temperatures. On the other hand, at higher temperatures the soliton is not observed and there is no definite pattern of lattice displacements. We took $m=80$ for the Trotter number, which is large enough to reproduce the Lanczos results on smaller chains (see Fig. \[fig1.5\]). For some particular quantities like the energy gap, which require more precision, we considered also $m=160.$ In addition, comparison with results for a longer chain with $N=128$ indicates that in the parameter range of our calculations the Monte Carlo results have no sizeable finite-size effects.
In Fig. \[fig1.5\] we show the Monte Carlo results for the homogeneous dimerization of the 64 site chain in the $S^z=0$ subspace as a function of the elastic constant $K$, together with the Lanczos results for smaller chains. Notice that in the parameter range considered the 64 site chain does not have the finite size effects present for smaller chains, namely, the vanishing of $\delta_0$ for finite values of $K$. The inset shows the expected scaling behavior $\delta \propto K^{-3/2}$ discussed in the previous section. As a further check, we have also reproduced the scaling behavior of the energy gain $E_0(\delta_0)-E_0(0)$ and gap with $\delta_0$ with a measured exponent $\nu =2/3$ within statistical errors.
=9.00cm
The soliton structure in the subspace with $S^z=1$ is given in Fig. \[soliton\], where we plot the displacement envelope $\widetilde{\delta }_i=(-)^i\delta_i$ and the local magnetization $\langle S_i^z\rangle $ , for different values of the elastic constant $K.$ Notice that the displacements are normalized by their maximum values (shown in Fig. \[fig1.5\]) and the local magnetization by the classical value $S=1/2.$ Consequently, the size of lattice distortions in different panels cannot be directly compared. For small values of $K$ there is a well defined soliton-antisoliton structure in the distortion pattern, with the associated local magnetization following a staggered order. There is a net $1/2$ spin density near each domain wall, which makes the excess $S^z=1$. As in the previous section, we fitted a two-soliton solution (\[fittanh\]), with $\tilde{\delta }_0=1$ because of the normalization adopted. The results for the soliton width $\xi$ are shown in Fig. \[xi-k\]. For increasing values of $K$ the soliton width grows until the displacement profile resembles a sine law (see Fig. \[soliton\]). This sinusoidal pattern is typical of the soliton lattice, observed for large values of $S^z$. It can be seen that the scaling $\xi \sim K$ obtained in \[\] is well reproduced in the whole parameter range considered, as indicated by the linear fit to the data (dashed line). This figure shows that the soliton width for $J_2^z = 0.3$ also presents a linear dependence with $K$. These features observed in the 64 site chain are qualitatively similar to those present in the 20 site chain as determined by exact diagonalization. Besides, it can seen in this figure that the reduction of $\xi$ is much stronger when the isotropic NNN is taking into account.
We have performed a simple study on the soliton-antisoliton interaction. For this study we fixed the distortion pattern to the law (\[fittanh\]) with the previously fitted value of $\xi ,$ and considered increasing values of $d.$ For small $K$ ($\leq 2J$) we found that the total energy becomes a constant (within statistical fluctuations) when $ d\geq 4\xi ,$ which implies that the soliton-antisoliton pairs shown in the left panels of Fig. 4 are not interacting. This was confirmed by allowing the lattice distortion to evolve starting from a pattern like (\[fittanh\]) with an initial separation larger than $d,$ which produces the same result for $\xi $ and the total energy.
Next, we study the behavior of the soliton width $\xi $ with the spin-Peierls gap $\Delta $. That is, we compare the quantity $\xi $ that characterizes the $S^z=1$ soliton state, with the singlet-triplet excitation gap $\Delta $ above the dimerized $S^z=0$ ground state. As shown in Fig. \[xi-gap\], these two quantities are inversely related to each other, as discussed in the previous section. The slope of the linear fit is $1.9$, very close to the value 1.87 obtained by exact diagonalization of the 20 site chain in the previous section. This result confirms the disagreement between the numerical results with the analytical prediction pointed out in Section \[lanczos\]. Also shown in Fig. \[xi-gap\] are the results for $J_2^z = 0.3$. A linear fit to these results leads to a slope $\approx 2.3$ , i.e. larger than the value corresponding to $J_2^z = 0.0$. This increase of the slope between $\xi$ and $\Delta^{-1}$ is consistent with the result obtained for the 20 site lattice by exact diagonalization and $J_2^z = 0.4$. This behavior should be contrasted with the [*reduction*]{} of the slope found for the isotropic NNN interaction. A possible explanation of this behavior could be the following. As discussed in the previous section, the term ${\cal H}_2^{zz}$ leads to a smaller increase of the spin gap than the fully isotropic NNN interaction. On the other hand, the Ising interaction could be more effective in punishing the excess $\langle S^z \rangle$ which appears around a soliton leading to a smaller reduction of the soliton width than the one caused by the isotropic term, as it can be seen in Fig. \[xi-k\]. A more detailed study of the Hamiltonian in the presence of the term of ${\cal H}_2^{zz}$ is clearly necessary to fully understand this behavior.
Finally, it is possible to estimate the critical value of the magnetic field at zero temperature. By adding a Zeeman term to the Hamiltonian (\[hamtot\]), $-g \mu_B S^z H$ ($\mu_B$: Bohr’s magneton), $H_{cr}$ may be calculated as: $$\begin{aligned}
H_{cr} = E_0(S^z=1)-E_0(S^z=0)
\label{hmagcrit}\end{aligned}$$ in units of $g \mu_B$. $E_0(S^z=1)$ is the ground state energy of (\[hamtot\]), and then $H_{cr} < \Delta$, which is the value expected of a gapped system in the absence of magneto-elastic coupling. The behavior of $H_{cr}$ as a function of $\Delta$ is shown in Fig. \[hcritvsgap\] for the 64 site chain, $\alpha=J_2^z=0.0$, and for the 20 site chain, $\alpha=J_2^z=0.4$. It is apparent a linear dependence over all the range studied, which is in agreement with the mean-field prediction[@pytte; @crossfisher], $H_{cr} \approx 0.84 \Delta$. However, we obtain a coefficient considerable smaller, $H_{cr}/\Delta \approx 0.47$, almost independent of $\alpha$. This value is also smaller than twice the soliton formation energy calculated in Ref. \[\]. The finite value at the origin of the curves corresponding to $\alpha=J_2^z=0.4$ is a finite size effect.
Conclusions {#conclu}
===========
In this article we have analyzed the magnetic soliton lattice in the incommensurate phase of spin-Peierls systems using numerical methods. There is a remarkable agreement between the results obtained by exact diagonalization using the Lanczos algorithm and those obtained by quantum Monte Carlo with the world-line algorithm. The relations among various features of the solitons and magnetic properties of the system have been determined and compared with analytical results. Our starting point is a microscopical model proposed to describe several properties of CuGeO$_3$, consisting of a 1D AF Heisenberg model with nearest and next-nearest neighbor interactions.
In the first place we have not detected any crossover in the behavior of the quantities examined as $\alpha$, the ratio of NNN to NN interactions, becomes greater than $\alpha_c$ at least for the small chains considered. That is, there are only smooth changes as $\alpha$ varies between 0.0 and 0.4. The most important effect of the competing NNN interaction is a [*reduction*]{} of the soliton width $\xi$ as a function of the inverse of the singlet-triplet spin gap $\Delta$. Furthermore, the effect of the diagonal term (\[h2n-zz\]) is much less important and in some cases even qualitatively different to that of the isotropic NNN term.
Although several functional forms predicted by continuum analytical theories have been confirmed by our numerical data, there are some important quantitative differences. The most important disagreement between our numerical results and the analytical predictions is related to the coefficient in the relation $\xi \sim \Delta^{-1}$, i.e. we have obtained a systematically higher value than the theoretical value which is the spin-wave velocity. The estimated value of $H_{cr}/\Delta$ is also noticeable smaller than the mean-field result and slighly smaller than the prediction of bosonized field theory. The relevance of these numerical results to real SP materials, such as CuGeO$_3$ and the recently discovered NaV$_2$O$_5$,[@navo] has to be determined experimentally.
The numerical procedures developed in this article could be applied to the study of several other properties of the incommensurate phase of spin-Peierls systems, such as the static magnetization as a function of the magnetic field (recently measured in CuGeO$_3$ by Fagot-Revurat [*et al.*]{}[@fagot]) and the order of the transition from the dimerized to the incommensurate phases.[@loosd]
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Numerical calculations indicate that this relation still holds with an exponent $\nu$ close to 2/3, for $\alpha > \alpha_c$, at least for not too small $\delta_0$ and $\alpha < 1/2$; M. Laukamp and J. Riera, (to be published).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This article reviews recent studies of mean-field and one dimensional quantum disordered spin systems coupled to different types of dissipative environments. The main issues discussed are: (i) The real-time dynamics in the glassy phase and how they compare to the behaviour of the same models in their classical limit. (ii) The phase transition separating the ordered – glassy – phase from the disordered phase that, for some long-range interactions, is of second order at high temperatures and of first order close to the quantum critical point (similarly to what has been observed in random dipolar magnets). (iii) The static properties of the Griffiths phase in random Ising chains. (iv) The dependence of all these properties on the environment. The analytic and numeric techniques used to derive these results are briefly mentioned.'
address: |
Laboratoire de Physique Théorique et Hautes Energies, Jussieu, France and\
Laboratoire de Physique Théorique, Ecole Normale Supérieure, Paris, France\
[email protected]
author:
- 'Leticia F. Cugliandolo'
title:
- Dissipative quantum disordered models
- 'Dissipative quantum disordered models[^1]'
---
Introduction {#intro}
============
Glasses slowly evolve towards equilibrium though never reaching it in observable times scales. Scientific research in this area started more than a century ago. The amount of experimental data gathered is huge. Systems that would reach equilibrium in observable time-scales, such as weakly sheared complex liquids, can be driven out of equilibrium by external perturbations and still evolve slowl.y Powders stay in static metastable states unless externally tapped or sheared: these non-equilibrium perturbations slowly drive them towards more compact configurations. Even if [*a priori*]{} very different, these systems share many dynamic properties.
In several cases of practical interest quantum effects play an important role. On the experimental side spin-glass phases have been identified in many condensed matter systems at very low temperature. Among them we can cite the bi-layer Kagome system[@Kagome] S$_r$Cr$_s$Ga$_4$O$_{19}$, the polychlore structure[@pyrochlore] Li$_x$Zn$_{1-x}$V$_2$O$_4$, the dipolar magnet[@aeppli; @aeppli-first] LiHo$_x$Y$_{1-x}$F$_4$ and the high $T_c$ compound[@highTc] La$_{1-x}$Sr$_2$Cu$_2$O$_4$. Quantum glassy phases exit also in electronic systems[@Zvi] and structural glasses[@Osheroff] such as Mylar. The driven case is also very important in quantum systems, think of an electronic device driven by an external current.
The non-equilibrium dynamics of classical glasses, weakly driven complex liquids and granular matter have been rationalized within a theoretical approach that is based on the solution of mean-field simple models.[@LesHouches] How much of the classical glassy phenomenology survives at very low temperatures where quantum effects are important is a question that deserves careful theoretical and experimental analysis. The impossibility of simulating the real-time evolution of quantum systems of moderate size enhances the importance of [*solving*]{} simple mean-field or low dimensional models.
On the other hand, peculiar phenomena in quantum phase transitions have been signaled analytically and experimentally in systems with and without quenched disorder.[@Sachdev-book; @Senthil] For instance, at low temperatures and intermediate dipole concentration, the dipolar-coupled Ising magnet LiHo$_x$Y$_{1-x}$F$_4$ in a transverse field exhibits a spin-glass-like phase.[@aeppli; @aeppli-first] The phase transition is of second order at low transverse field but becomes first order close to the quantum critical point.[@aeppli-first] Morever, the relaxation in the glassy phase is extremely slow and has very strong memory effects.
Another hallmark of finite dimensional disordered quantum spin models are Griffiths-McCoy singularities, that lead to a highly non trivial paramagnetic phase and critical behaviour.
In this contribution we summarize the results of recent studies of the statics, dynamics and critical properties of mean-field and one dimensional quantum disordered spin systems coupled to an environment.[@Culo-Culolo] We also briefly mention related studies on a driven mesoscopic ring,[@Lili] dilute antiferromagnets,[@Malcolm] and a manifold in an infinite dimensional quenched random potential.[@Pierre] In Sect. \[questions\] we review the main questions that we attempted to answer in these papers. In Sect. \[models\] we recall the definition of the models that we studied. The basic techniques used to study classical glassy models with or without disorder are well documented in the literature (the replica trick, scaling arguments and droplet theories, the dynamic functional method used to derive macroscopic equations from the microscopic Langevin dynamics, functional renormalization, Montecarlo and molecular dynamic numerical methods). On the contrary, the techniques needed to deal with the statics and dynamics of quantum macroscopic systems are much less known in general. We briefly mention the ones that we used to study dissipative disordered quantum models.[@Boulder] Finally, in Sect. \[perspectives\] we list some projects for future research.
Questions and main results {#questions}
==========================
Effect of quantum fluctuations on glassiness
--------------------------------------------
Since glasses are not expected to reach equilibrium in experimentally accessible times, it is important to device a method to understand the influence of quantum fluctuations on their trully nonequilibrium [*real-time*]{} dynamics. Intuitively, one expects quantum fluctuations to only affect the short-time dynamics; however, they are also expected to act as thermal fluctuations. It is then not clear [*a priori*]{} whether quantum fluctuations would tend to destroy glassiness or modify it drastically.
The usual methods of equilibrium quantum statistical mechanics are inappropriate to describe this nonstationary situation. We presented a formalism suited to study the real-time dynamics of a general nonlinear, possibly disordered, model in contact with a bath that can also be applied to glassy models.[@Culo] The method is a combination of the Schwinger-Keldysh or closed-time path technique to study real-time phenomena, with the Feynman-Vernon approach to dissipation that consists in modelling the coupling to the environment with an ensemble of quantum harmonic oscillators. As a particular case we studied the relaxation of the spherical version of the quantum $p$ spin fully-connected disordered model. We analyzed the relaxation of ‘random initial conditions’ in the limit of vanishing coupling strength taken after the long waiting-time ($t_w$) limit. The same technique was applied to other quantum problems[@Lev]$^-$[@Premi] and related studies appeared.[@other-quantum]$^-$[@Galitsky] We later studied the effect of a strong coupling to the environment[@bath-spherical] ([*i.e.*]{} $t_w\to\infty$ with $\alpha$ finite) as discussed below.
In the disordered phase the dynamics is fast and occurs in equilibrium. The correlation and linear response are stationary, [*i.e.*]{} $C(t + t_w,t_w)=C(t)$ and $R(t+t_w,t_w)=R(t)$. They both oscillate with a $\Gamma$-dependent frequency that also depends on the characteristics of the bath $(\alpha,s)$ if $\alpha$ is not taken to zero. Correlations and responses are linked by the quantum fluctuation dissipation theorem ([fdt]{}). At high temperatures and after a short transient the system decoheres and the dynamics becomes classical ([*e.g.*]{} responses and correlations are related by the classical [fdt]{}).
In the ordered phase the glassy dynamics persists asymptotically if the thermodynamic limit has been taken at the outset. The dynamics of glassy systems occurs out of equilibrium and the correlations and responses loose time translation invariance. If $t_w$ denotes the time elapsed since a quench into the [sg]{} phase, $C(t + t_w,t_w)$ and $R(t+t_w,t_w)$ depend on both $t$ and $t_w$. The order in which the limits $t_w\to\infty$ and $t\to\infty$ are taken is very important. For sufficiently long $t$ and $t_w$ but in the regime $t
\ll t_w$, the dynamics is stationary and the correlation reaches a plateau $q_{\sc ea}$. A few quantum oscillations exist at very short $t$ (and arbitrarily large value of $t_w$) and they later disappear. For times $t > t_w$, the system enters an [*aging*]{} regime where the correlation function depends on $t_w$ explicitly. In this regime, the correlation vanishes at long times, $\lim_{t \to \infty} C(t + t_w,t_w) = 0$, at a rate that depends on $t_w$. The behaviour is thus qualitative similar to what observed classically, even if the scaling laws are modified by the quantum fluctuations. One checks that the relaxation of typical (highly energetical) initial conditions approaches a threshold level in phase space with a higher energy-density than the equilibrium one.
The comparison of responses and correlations in the ordered phase is particularly interesting. The quantum [fdt]{} is a complicated integral relation between the correlation and the linear response. This relation holds for $t\ll t_w$ when the correlation decays to the plateau. In the second regime the quantum [fdt]{} is no longer verified, much as it happens in the classical problem. A comparison of the integrated responses and the symmetrized correlation in a parametric manner[@LesHouches] shows that the two quantities are related by a [*classical*]{} [fdt]{} with an effective temperature[@Cukupe] $T_{\sc eff}$ that depends on the parameters in the problem $(T, \Gamma)$ \[and the characteristics of the bath, $\alpha, s$\]. We proved that $T_{\sc eff}>T$ and it is different from zero even when the environment is at $T=0$. $T_{eff}$ drives the dynamics at late epochs and it makes the dynamics appear classical in that two-time regime. $T_{eff} > 0$ even when the temperature of the bath is zero. The generation of a non-trivial $T_{eff}$ for the slow part of the decay gives support to the “decoherent” effect observed in the decay of correlations.
The similarity between the second decay in the classical and quantum problem can be argued as follows. The responses decay rather fast to zero when the time difference increases (though integrated over a time-interval of very long length does not vanish.[@Cuku]) The explicit dependence on $\hbar$ in the regime of widely separated times comes from factors with higher powers of $R$ that vanish[@Culo]. The effect of quantum fluctuations on the slow time-difference regime is simply to renormalize[@Kech] certain parameters in the equations that otherwise look classical.
Models with $p\geq 3$ interactions have different static and dynamic phase transitions, with the latter surrounding a larger region of the $(T, \Gamma)$ phase diagram. The static and dynamic ordered-disorder phase transition present a second-to-first order transition.[@Niri; @Cugrsa1] Close to the classical critical temperature quantum effects are small and the phase transition is discontinuous but of second order, as in the classical case. There are no discontinuities in the thermodynamic quantities but there is a plateau that develops in the correlation function when the transition is approached from the disordered side. This is the behaviour expected in classical glasses. Spin-glasses instead have continuous transitions (without precursors). Conversely, close to the quantum critical point quantum fluctuations drive the transition first order thermodynamically. Across the first order line the susceptibility is discontinuous and shows hysteresis. This is similar to what has been observed in the dipolar-coupled Ising magnet LiHo$_x$Y$_{1-x}$F$_4$ in a transverse field.[@aeppli-first]
We also adapted the [*Ansatz*]{} of marginal stability[@Cugrsa1; @ams; @Niri] to identify the dynamic critical line that is consistent with the one found using the Schwinger-Keldysh formalism. The analytic continuation of the imaginary-time dependent correlation computed with the [ams]{} in the absence of the bath is identical to the [*stationary*]{} part of the non-equilibrium correlation function ($C>q_{\sc ea}$) when one takes the long-time limit first and the limit in which the coupling to the bath goes to zero next.[@Cugrsa1; @Pierre]
In the classical case, the study of the Thouless-Anderson-Palmer ([tap]{}) free energy landscape has been very useful to understand the behaviour of these systems.[@LesHouches] A [tap]{} approach can also be developed for quantum problems.[@Bicu] It helps understanding the existence of a dynamic and a static critical line as well as the change in nature of the transition close to the quantum critical point.
Effect of the bath: decoherence and localization
------------------------------------------------
The quantum systems mentioned in Sect. \[intro\] are not totally isolated but in contact with environments of different type.
The low-energy physics of many tunneling systems is well described by the spin-boson model in which the two equivalent degenerate states are represented by the eigenstates $\sigma_z = \pm 1$ of an Ising pseudo-spin. A transverse field coupled to $\sigma_x$ (say) represents the tunneling matrix element. The coupling to the environment is given in terms of its spectral density $I(\omega)\propto \alpha\,\omega^s$ for $\omega \ll \omega_c$, where $\alpha$ is a dimensionless coupling constant and $\omega_c$ a high frequency cutoff. The exponent $s$ characterizes different types of environment. The Ohmic case ($s =
1)$ is quite generally encountered but superOhmic ($s>1$) and subOhmic ($s<1$) baths also occur in, [*e.g.*]{}, the Kondo effect in unconventional hosts.
The coupling of quantum two-level systems ([tls]{}) to a dissipative environment has decisive effects on their dynamical properties. The dilute case, in which interactions between the [tls]{} can be neglected, has been extensively investigated.[@Standard; @Leggett-review] This problem, is related the $1d$ Ising model with inverse squared interactions and the anisotropic Kondo model. In the Ohmic case, at zero temperature, there is a phase transition[@Bray-Moore] at $\alpha=1$. For $\alpha<1$ there is tunneling and two distinct regimes develop. If $\alpha<1/2$ the system relaxes with damped coherent oscillations; in the intermediate region $1/2<\alpha<1$ the system relaxes incoherently. For $\alpha>1$ quantum tunneling is suppressed and $\langle \hat \sigma_z\rangle \neq 0$ signalling that the system remains localized in the state in which it was prepared. These results also hold for sub-Ohmic baths while weakly damped oscillations persist for super-Ohmic baths. At finite temperatures (but low enough such that thermal activation can be neglected), there is no localization but the probability of finding the system in the state it was prepared decreases slowly with time for $\alpha>\alpha^{\sc
c}$.
The effect of dissipation on the phase transition, critical behaviour, ordered phase, localization and decoherence properties of macrocopic interacting systems is only now starting to be analyzed.[@bath-spherical; @bath-SU2; @Troyer] In thermodynamic equilibrium, in the absence of the bath, the interactions between the [tls]{} lead to the appearance of an ordered state at low enough temperature. If the interactions are of random sign, as in the models we considered, the latter will be a glassy state. In this phase the symmetry between the states $\sigma_i^z = \pm 1$ at any particular site is broken but there is no global magnetization, $\sum_i \langle \hat \sigma_i^z \rangle = 0$. The coupling to the bath also competes with the tunneling term. We thus expect the presence of noise to increase the stability of the glassy state. The consequences of this fact are particularly interesting when there is localization at some $\alpha=\alpha^{\sc c}$: a quantum critical point at ${J}=0$, $\alpha=\alpha^{\sc c}$ separates the disordered and the ordered state such that, for $\alpha > \alpha^{\sc c}$, the glassy phase survives down to ${J}=0$.
A system of non-interacting localized [tls]{} and a glassy state [*in equilibrium*]{} are in some way similar However, this resemblance is only superficial. The details of the dynamics of the two systems are expected to be quite different, with $C$ saturating at a finite value in the localized state, and $C$ decaying down to zero in the glassy phase.
Effect of the bath on interacting mean-field models
---------------------------------------------------
The problem of a single [tls]{} being a difficult one, that of an infinite set of interacting [tls]{} seems hardly solvable. Therefore, as a first step, we focused on the effect of the reservoir on the $p$-spin [*spherical*]{} model,[@bath-spherical] a problem that we studied with the real-time approach and the replica Matsubara technique. The position of the critical lines strongly depends on the strength of the coupling to the bath and the type of bath (Ohmic, subOhmic, superOhmic). For a given type of bath, the ordered glassy phase is favored by a stronger coupling. The classical static and dynamic critical temperatures remain unchanged by the coupling to the environment. The identity between the analytic continuation of the imaginary-time correlation to real-time and the correlation in the Schwinger-Keldysh approach also holds if the strength of the bath is finite.
The spherical model localizes in the absence of interactions when coupled to a subOhmic bath: $C(t+t_w,t_w)$ reaches, for any waiting-time $t_w$ and long enough $t$, a plateau that it never leaves. When interactions are switched on localization disappears and the system undergoes a phase transition towards a glassy phase.
Similar results were found for the $SU(N)$ random Heisenberg model in the limit $N\to\infty$[@Bipa], the $p=2$ spherical model[@Premi] and the SU(2) $p$-spin model.[@bath-SU2]
Effect of disorder: Griffiths singularities
-------------------------------------------
Griffiths singularities[@Griffiths]$^-$[@McCoy] in classical finite dimensional random systems are so weak that their consequences have not been clearly observed neither experimentally nor numerically. Instead, when quantum fluctuations are introduced they are much stronger and rare regions completely determine the static and dynamic behaviour in the Griffiths phase. The isolated quantum random Ising chain has been studied in great detail with a decimation technique. [@dfisher] It undergoes a quantum phase transition from a [pm]{} to a ferromagnet for a special relation between the distribution of exchanges and transverse fields. The quantum phase transition is of second order with the correlation time scaling exponentially with the spatial correlation length ([*activated*]{} scaling) Within the renormalization group procedure this is a characteristic of an “infinitely strong disorder” fixed point.[@dfisher] Within the [pm]{} Griffiths phase the distributions of local linear and non-linear magnetic susceptibilities are large and typical and average values are very different, with the latter being dominated by rare regions. The behaviour is higher spatial dimensions is similar. The analysis of Montecarlo simulations of the equivalent $d+1$ classical Ising model is quite tricky. Initially, it was claimed that there was conventional scaling in $d=1$[@1dMC] as well as in $d>1$[@2dsim] but a more careful analysis of the numerical data confirmed the activated scaling in both cases.[@1dfermions]
On the real-time dynamic side, there have been some studies of the relaxation of special initial conditions of the isolated random Ising chain at constant energy.[@IgloiRieger]
Disorder and dissipation: fate of Griffiths phase?
--------------------------------------------------
If the interactions between two-level systems placed in a finite dimensional space are random one may wonder what is the effect of the bath on the Griffiths phase. The answer to this question has been debated over the last years.[@Antonio-etal]$^-$[@Miranda] We addressed this problem using Montecarlo simulations of the equivalent $2d$ classical system. [@Culolo] Preliminary results for rather small systems ($N_x\leq 32$, $N_\tau\leq 256$) show that an Ohmic bath favors the glassy phase. Our results are compatible with (but we do not prove) activated scaling at criticality, at least for small values of $\alpha$.
The models {#models}
==========
Disordered quantum spin-$\frac12$ models with two-body interactions are defined by $$H_S = - \sum_{ij} J_{ij} \hat \sigma^z_i \hat \sigma^z_j
+ \sum_i \Gamma_i \hat \sigma_i^x + \sum_i h_i \hat \sigma_i^z
\; .
\label{HS}$$ $i=1,\dots, N$ labels the spins that lie on the vertices of a cubic $d$ dimensional lattice and are represented by Pauli matrices. The interaction strengths $J_{ij}$ couple near-neighbours only and are chosen from a probability distribution, $P(J)$. The average and variance are defined as $[J_{ij}]=J_o$ and $[J_{ij}^2]=J^2/(2c)$, where $J_o$ and $J$ are $O(1)$ and $c=2d$ is the connectivity of the lattice. The next-to-last term is a coupling to a random quenched local transverse field $\Gamma_i$. The last term is the coupling to a longitudinal field that serves to compute local susceptibilities.
Several generalizations that render the model easier to treat analytically are:
– [*Fully-connected limit.*]{} One allows each spin to interact with all others, $c\to N-1$. – [*Multi-spin interactions.*]{} In the fully-connected case one can considers $$H_S = - \sum_{i_1\dots i_p} J_{i_1\dots i_p}
\hat \sigma^z_{i_1} \dots \hat \sigma^z_{i_p}
+ \sum_i \Gamma_i \hat \sigma_i^x
+ \sum_i h_i \hat \sigma_i^z
\; .$$ where the sum runs over all $p$-uplets,[@pspin; @Niri; @Cugrsa1; @bath-SU2] with $p$ an integer parameter, $p \geq 2$. The exchanges are random independent variables with variance $p!{J}^2/(2 N^{p-1})$. This model provides a mean-field description of the structural glass transition and glassy physics that is also intimately related to the mode-coupling approach.[@LesHouches] In its dilute limit, with no geometry but finite connectivity on each site, this model is related to the $K$-sat optimization problem. [@Zecchina]
– [*Spherical variables – a particle in a random potential.*]{} One considers the spherical limit, $\sum_{i=1}^N \langle \hat \sigma_i^2\rangle =N$, in which the $\hat \sigma_i$ may be interpreted as the coordinates of a particle moving on an $N$-dimensional sphere. A kinetic term, $K = \sum_{i=1}^N \hat P_i^2/(2M)$, is then included in the Hamiltonian, with $\hat P_i$ the conjugated momentum satisfying the commutation rules $[\hat P_i, \hat P_j] = 0$, $[\hat P_i, \hat \sigma_j] = - i \hbar \delta_{ij}
$. Other spherical models have been discussed.[@Theo-spherical]
The coupling to the environment is modelled by $H = H_S + H_B + H_I + H_{CT}$, where $H_B$ is the Hamiltonian of the bath, $H_I$ represents the interaction between the system and the bath and $H_{CT}=\sum_{l=1}^{\tilde{N}}(2 m_l \omega_{l}^2)^{-1}
(\sum_{i=1}^{N} c_{i l} \hat{\sigma}_i^z )^2$ is a counter-term that is usually added to eliminate an undesired mass renormalization induced by the coupling.[@Standard] We assume that each spin is coupled to its own set of $\tilde{N}/N$ independent harmonic oscillators with $\tilde N$ the total number of them. For simplicity we consider the bilinear coupling, $H_{I} = -\sum_{i=1}^N \hat{\sigma}_i^{z} \sum_{l=1}^{\tilde{N}}
c_{i l} \hat{x}_{l} $. For $p=2$ the fully-connected limit with two-body interactions models metallic spin-glasses.[@Kondo]
Perspectives
============
In this article we reviewed recent studies of insulating disordered magnets.
The principal merit of the fully-connected models is that they are simple enough to be studied in detail. Yet, many of their properties are generic and expected to hold at least qualitatively for more realistic cases.[@LesHouches] The analysis of the fully-connected models is by now quite complete, having applied the replica theory, the real-time dynamics approach, and the investigation of the [tap]{} free-energy landscape. A problem that remains not fully developed though is the treatment of the relaxation of initial conditions that are correlated with disorder. [@Cugrsa2]
Recently, much progress has been done in the study of the statics of classical dilute spin models, that is to say, models defined on random hyper graphs with finite connectivity. The statics of these models encode problems in combinatorial opimization such as $K$-sat.[@Zecchina] An interesting mapping relates (isolated) dilute quantum disordered spin systems to the dynamics of special purpose algorithms used in combinatorial optimization – such as Walk-Sat.[@walk-sat] It would be interesting to study the latter using tools developed for the former and [*vise versa*]{}.
The great challenge remains to understand the behaviour of glassy systems with and without disorder in finite dimensions. In particular, one could try to adapt the decimation technique[@dfisher] to study disordered spin$-\frac12$ models in contact with an environment. One can also envisage applications of the ideas described in this paper to other quantum systems evolving out of equilibrium. In this respect, we have studied the generation of an effective temperature in a conducting ring threaded by time-dependent magnetic field and coupled to a reservoir;[@Lili] and we plan to analyse quantum dilute antiferromagnets[@Malcolm] and the low-temperature dynamics of the Bragg glass,[@Pierre] as well as other related physical problems.
Acknowledgements {#acknowledgements .unnumbered}
================
This contribution summarizes work done in collaboration with G. Biroli, D. Grempel, G. Lozano, H. Lozza and C. da Silva Santos. I also wish to thank L. Arrachea, C. Chamon, T. Giamarchi, L. Ioffe, M. Kennett, J. Kurchan, P. Le Doussal, R. Monasson, M. J. Rozenberg and G. Semerjian for collaborations on related problems. I acknowledge financial support from an Ecos-Sud travel grant, an ACI project, the J. S. Guggenheim Foundation and ICTP-Trieste, as well as hospitality from the Universidad de Buenos Aires and Universidad Nacional de La Plata, ICTP, and the KITP.
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[^1]: Contribution to the 12th International Conference on Recent Progress in Many-Body Theories. Santa Fe, New Mexico, 23-27 August 2004
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In a recent paper, Mohan et al. \[Can. J. Phys. [**95**]{} (2017) 173\] have reported results for collision strengths ($\Omega$) and effective collision strengths ($\Upsilon$) for transitions from the ground to higher 51 excited levels of F-like Ba XLVIII. For the calculations of $\Omega$, the Dirac atomic $R$-matrix code (DARC) and the flexible atomic code (FAC) have been adopted, in order to facilitate a direct comparison. However, for the subsequent calculations of $\Upsilon$, DARC alone has been employed. In this comment, we demonstrate that while their limited results for $\Omega$ are comparatively reliable, for $\Upsilon$ are not, particularly for the allowed transitions and at lower temperatures. Apart from the non expected behaviour, their $\Upsilon$ values are overestimated for several transitions, by about a factor of two.'
---
[ ]{}\
[**[Kanti M Aggarwal]{}**]{}\
Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast,\
Belfast BT7 1NN, Northern Ireland, UK\
e-mail: [email protected]\
Received: 26 October 2017; Accepted: 23 May 2018
[**Keywords:**]{} Oscillator strengths, collision strengths, effective collision strengths\
PACS numbers: 32.70.Cs, 34.80.Dp
------------------------------------------------------------------------
Introduction
============
In a recent paper, Mohan et al. [@mm1] have reported results for collision strengths ($\Omega$) and effective collision strengths ($\Upsilon$) for transitions from the 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ ground to 51 excited levels of F-like Ba XLVIII, which belong to the 2s$^2$2p$^5$, 2s2p$^6$, and 2s$^2$2p$^4$3$\ell$ configurations. For the calculations of $\Omega$, they have adopted two independent codes, namely the Dirac atomic $R$-matrix code (DARC) and the flexible atomic code (FAC). These codes are freely available on the websites\
[http://amdpp.phys.strath.ac.uk/UK\_APAP/codes.html]{} and [https://www-amdis.iaea.org/FAC/]{}, respectively. By making comparisons between the two calculations for $\Omega$, although for only limited ($<$4%) transitions, they have concluded a good agreement, for most transitions. However, for the subsequent calculations of $\Upsilon$, DARC alone has been employed. In the absence of any other existing similar results, no direct comparisons were possible, but they speculated their data to be ‘reliable, authentic, and accurate’. However, we find exactly the opposite of this, because their $\Upsilon$ results are neither correct in behaviour nor in magnitude.
Collision strengths and effective collision strengths
=====================================================
For the construction of wavefunctions, Mohan et al. [@mm1] adopted the general-purpose relativistic atomic structure package (GRASP). Several versions of this code are currently in use by many workers, but the one adopted by them is the original one, i.e. GRASP0, but considerably modified by P.H. Norrington and I.P. Grant. This is available at the same website as DARC, and can be directly linked to the latter. They included a large CI (configuration interaction) among 431 levels of 29 configurations, namely 2s$^2$2p$^5$, 2s2p$^6$, 2s2p$^5$3$\ell$, 2p$^6$3$\ell$, 2s$^2$2p$^4$3$\ell$, 2s$^2$2p$^4$4$\ell$, 2s2p$^5$4$\ell$, 2s$^2$2p$^4$5$\ell$, and 2s2p$^5$5$\ell$. Energies for these levels and oscillator strengths (f-values) for transitions among these have already been reported by them in a separate paper [@mm2]. Since inclusion of such a large number of levels in a scattering calculation requires significantly large computational resources, they restricted their work to the lowest 52 levels, which belong to the 2s$^2$2p$^5$, 2s2p$^6$, and 2s$^2$2p$^4$3$\ell$ configurations. In fact, these configurations generate 60 levels in total, but they have preferred to ignore the remaining 8. Nevertheless, CI for F-like ions is not very important as may be noted from our work on 17 ions with 37 $\le$ Z $\le$ 53 [@kma1]. Therefore, we have performed simple calculations among these 60 levels, which will facilitate a direct comparison with their results, and hence some assessment of the accuracy and reliability of their data.
For the calculations of $\Omega$, Mohan et al. [@mm1] adopted the DARC code, included a wide range of partial waves with angular momentum $J \le$ 41, considered a wide range of energy (up to 1000 Ryd), and resolved resonances to determine $\Upsilon$ at temperatures below 10$^6$ K. For our calculations, the FAC is employed which is based on distorted-wave (DW) approximation, and generally provides comparable results of $\Omega$ with $R$-matrix, as may also be noted from Table 2 of [@mm1]. Similarly, the energies obtained with this code are comparable with those of Mohan et al., and there are no appreciable discrepancies, in magnitude or orderings, for any level – see also Table 1 of [@mm1]. For this reason we are not listing our energy levels for Ba XLVIII. However, in our subsequent calculations of $\Upsilon$, resonances are not included, which mainly affect the forbidden transitions, not the allowed ones. For this reason we will focus our comparisons on some allowed transitions alone.
I J Transition GRASP FAC
--- ---- ------------------------------------------------------------------------ ------- ------- --
1 22 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – 2s$^2$2p$^4$($^3$P$_2$)3d $^2$P$_{1/2}$ 0.120 0.119
1 23 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – 2s$^2$2p$^4$($^3$P$_2$)3d $^2$D$_{5/2}$ 0.624 0.636
1 24 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – 2s$^2$2p$^4$($^3$P$_2$)3d $^2$P$_{3/2}$ 0.463 0.437
1 25 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – 2s$^2$2p$^4$($^1$S$_0$)3d $^2$D$_{5/2}$ 0.464 0.455
: Comparison of oscillator strengths (f-values) for some transitions of Ba XLVIII from ground to higher excited levels.
[GRASP: calculations of Mohan et al. [@mm1],[@mm2] with GRASP\
FAC: present calculations with FAC\
]{}
In Table 1 we compare f-values between our work and that of Mohan et al. [@mm1] for four transitions, namely 1–22/23/24/25, i.e. 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – (2s$^2$2p$^4$)\[$^3$P$_2$\]3d $^2$P$^o_{1/2}$, \[$^3$P$_2$\]3d $^2$D$^o_{5/2}$, \[$^3$P$_2$\]3d $^2$P$^o_{3/2}$, and \[$^1$S$_0$\]3d $^2$D$^o_{5/2}$ – see Table 1 of [@mm1] for the definition of all levels. These transitions are allowed, comparatively stronger, and have no discrepancies in their f-values. Therefore, the corresponding results for $\Omega$ are also expected to be comparable (within a few percents), which indeed is the case as shown in Fig. 1. In Fig. 2 we make similar comparisons for $\Upsilon$, but the differences are striking between the two independent calculations, particularly at temperatures below 10$^5$ K. Since resonances (if any) do not make a significant contribution to the determination of $\Upsilon$, the reported results of Mohan et al. are not only abnormal in behaviour towards the lower end of the temperature range, but are also overestimated, by up to a factor of two.
Another example for the incorrect $\Upsilon$ results of Mohan et al. [@mm1] is the 1–3 transition, i.e. 2s$^2$2p$^5$ $^2$P$^o_{3/2}$ – 2s2p$^6$ $^2$S$_{1/2}$, for which they have shown $\Omega$ in their Fig. 2. It is clear that $\Omega$ values rise gradually over a wide energy range of about 100 Ryd, without any resonances. Therefore, corresponding $\Upsilon$ results should also either rise gradually, or remain nearly constant, over the small temperature range below 10$^6$ K, equivalent to $\sim$6.3 Ryd. This indeed is the case in our calculations as $\Upsilon$ is 0.053, whereas the corresponding results of Mohan et al. are 0.0767, 0.0516, and 0.0587 at the respective temperatures of 10$^4$, 10$^5$, and 10$^6$ K. The same conclusion of invariable $\Upsilon$, for this transition at temperatures below 10$^6$ K, was noted earlier for another F-like ion, namely Kr XXVIII [@kr28]. Therefore, for the 1–3 transition the $\Upsilon$ value of Mohan et al. is clearly overestimated by 50% at T$_e$ = 10$^4$ K.
The reason for the anomalous behaviour of $\Upsilon$ results by Mohan et al. [@mm1], particularly at lower temperatures, is not difficult to diagnose. This is primarily due to their choice of energy mesh ($\delta$E) for the resolution of resonances and the determination of $\Upsilon$. Their $\delta$E is 0.065 Ryd equivalent to $\sim$10 260 K, i.e. greater than the lowest T$_e$ (10 000 K) they considered. For an accurate performance of the integral in Eq. 13 of [@mm1], this energy mesh is (very) very coarse, and hence has greatly affected the $\Upsilon$ results. Although we have shown comparisons for allowed transitions alone, it is clear that their results will be in even greater error for the forbidden transitions, for which the resonances are numerous, as shown in their Figs. 1 and 3. An indication of that is visible in their results for the 1–7, 1–8, and 1–39 transitions, listed in their Table 3.
Conclusions
===========
By performing simple calculations in this short paper, we have demonstrated that the earlier reported $\Omega$ results of Mohan et al. [@mm1] are largely error free, but for $\Upsilon$ are overestimated by up to a factor of two, for several allowed transitions, mainly at lower temperatures. In addition, the behaviour of their $\Upsilon$ is not correct. Furthermore, they have reported $\Upsilon$ for less than 4% of the transitions (among 52 levels) of Ba XLVIII, whereas a complete set of data for all transitions is preferred in any diagnostic or modelling applications of plasmas. Similarly, in fusion plasmas for which such collisional data for this ion may be required, the prevailing temperatures are far higher, up to $\sim$10$^8$ K. Therefore, it is recommended that fresh calculations for this ion should be performed, for a much wider range of transitions and temperatures, for the reliable adoption of collisional data for this ion.
We will like to (re)emphasise again that the reliability of any calculation does not (much) depend on the (in)accuracy of the code adopted, but on its implementation. If a code is incorrectly and/or non-judicially applied then large discrepancies may occur, as demonstrated in this paper. A number of large discrepancies, for several atomic parameters, have been noted earlier for many ions, and these have been highlighted in our recent paper [@atom], along with suggestions for their resolutions.
[999]{} M. Mohan, A. Goyal, I. Khatri, S.S. Singh, and A.K. Singh. Can. J. Phys. [**95**]{}, 173 (2017). I. Khatri, A. Goyal, S. Aggarwal, A.K. Singh, and M. Mohan. At. Data Nucl. Data Tables [**107**]{}, 367 (2016). K.M. Aggarwal and F.P. Keenan. At. Data Nucl. Data Tables [**109-110**]{}, 205 (2016). K.M. Aggarwal, F.P. Keenan, and K.D. Lawson. At. Data Nucl. Data Tables [**97**]{}, 225 (2011). K.M. Aggarwal. Atoms [**5**]{}, 5040037 (2017).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'While quark-hadron duality is well-established experimentally, the current theoretical understanding of this important phenomenon is quite limited. To expose the essential features of the dynamics behind duality, we use a simple model in which the hadronic spectrum is dominated by narrow resonances made of valence quarks. We qualitatively reproduce the features of duality as seen in electron scattering data within our model. We show that in order to observe duality, it is essential to use the appropriate scaling variable and scaling function. In addition to its great intrinsic interest in connecting the quark-gluon and hadronic pictures, an understanding of quark-hadron duality could lead to important benefits in extending the applicability of scaling into previously inaccessible regions.'
address:
- '$^{(1)}$ Jefferson Lab, 12000 Jefferson Ave, Newport News, VA 23606'
- |
$^{(2)}$ Special Research Centre for the Subatomic Structure of Matter,\
Adelaide University, Adelaide 5005, Australia
- '$^{(3)}$ Department of Physics, Old Dominion University, Norfolk, VA 23529'
author:
- 'Nathan Isgur$^{(1)}$, Sabine Jeschonnek$^{(1)}$, W. Melnitchouk$^{(1,2)}$, and J. W. Van Orden$^{(1,3)}$'
title: 'Quark-Hadron Duality in Structure Functions'
---
Introduction
============
Background
----------
Duality is a much used and much abused concept. In some cases it is used to describe an equivalence between quark- and hadron-based pictures which is trivial; in others an equivalence which is impossible. In almost all cases, the conceptual framework in which duality is discussed and used is either hopelessly muddled or hopelessly abstract. Nevertheless, the data indicate that some extremely interesting and potentially very important “duality” phenomena are occurring at low energy.
We begin by making the trivial observation that any hadronic process can be correctly described in terms of quarks and gluons, assuming that Quantum Chromodynamics (QCD) is the correct theory for strong interactions. While this statement is obvious, it rarely has practical value, since in most cases we can neither perform nor interpret a full QCD calculation. We will refer to the above statement that any hadronic process can be described by a full QCD calculation as “degrees of freedom duality”: if one could perform and interpret the calculations, it would not matter at all which set of states — hadronic states or quark and gluon states — was used.
On the other hand, there are rare cases where the average of hadronic observables is described by a perturbative QCD (pQCD) calculation. We reserve the use of the term “duality” to describe these rare correspondences, in contrast to the trivial “degrees of freedom duality” described above. In these rare cases, a quark-gluon calculation leads to a very simple description of some phenomenon even though this phenomenon “materializes” in the form of hadrons. Deep inelastic scattering is the prototypical example, and the one on which we focus here. These rare examples are all characterized by a special choice of kinematic conditions which serve to expose the “bare” quarks and gluons of the QCD Lagrangian. In the case of deep inelastic scattering, the kinematics are such that the struck quark receives so much energy over such a small space-time region that it behaves like a free particle during the essential part of its interaction. This leads to the compellingly simple picture that the electron-nucleon cross section is determined in this kinematic region by free electron-quark scattering, i.e. duality is exact for this process in this kinematic regime.
For inclusive inelastic electron scattering from a proton in the scaling region, the cross section is determined by the convolution of a non-perturbative and currently difficult to calculate parton distribution function with an electron-quark scattering cross section determined by perturbative QCD (pQCD). For semileptonic decays of heavy quarks, [*e.g.*]{} $\bar{B} \rightarrow X_c l
\bar \nu_l$, one can prove using pQCD that the decay rate is determined by that of the underlying heavy quark, in this case obtained from the process $b \rightarrow c l \bar \nu_l$ [@isgurwise]. In $e^+e^- \rightarrow hadrons$, it is the underlying $e^+e^- \rightarrow q \bar q$ process that applies because of pQCD. However, while duality applies to all of these phenomena, we will see that even in these special processes we must invoke an averaging procedure to identify the hadronic results with the quark-gluon predictions.
In addition to its need of an averaging procedure, it is easy to see that the pQCD picture of inelastic electron scattering must fail for $Q^2 \rightarrow 0$. For duality to hold for the nucleon structure functions in this case, the elastic electric proton and neutron form factors, which take the value of the nucleon charges for $Q^2
\rightarrow 0$, would have to be reproduced by electron scattering off the corresponding $u$ and $d$ quarks. This is possible for the proton since the squares of the charges of two $u$ quarks and one $d$ quark add up to 1 [@gottfried]. However, for the neutron, the squared quark charges cannot add up to 0, so it is clear that local duality in inclusive inelastic electron scattering from a neutron must fail for $Q^2 \rightarrow 0$. Also, we know that duality must fail for polarized structure functions at low $Q^2$, as the Ellis-Jaffe sum rule and the Gerassimov-Drell-Hearn sum rule, which can be written as integrals over $g_1(\nu,Q^2)$ at different $Q^2$, are negative (GDH sum rule for $Q^2 = 0$) and positive (Ellis-Jaffe sum rule at $Q^2$ of several GeV), respectively [@g1dual].
Thus duality in inelastic electron scattering has to hold in the scaling regime and must in general break down at low energy. Obviously, a very interesting question is what happens in between these regimes, [*i.e.*]{} how does duality break down? This paper answers this question, which is not only interesting in itself, but also crucial for practical, quantitative applications of duality.
Introducing Local Averaging and Our Model
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We begin by discussing the issue of averaging. If duality is relevant at all at low energy, then it is quite obvious that we need to perform some sort of average: the smooth, analytic pQCD prediction cannot in general correspond exactly to the generally highly structured hadronic data. For low energies this requirement is universally accepted; however, even in the “scaling” region one must average in principle. To see this, consider QCD in the large-$N_c$ limit [@largenc]. We can do this because no element of the pQCD results for deep inelastic scattering depends on the number of colors. However, in this limit the hadronic spectrum consists entirely of infinitely narrow noninteracting resonances [@baryoncaveat], [*i.e.*]{}, there are only infinitely narrow spikes in the $N_c \rightarrow \infty$ hadronic world. Since the quark level calculation still yields a smooth scaling curve, and the kinematic conditions for being in the scaling region are unchanged as $N_c \rightarrow \infty$, we see that we must average even in the scaling region. While in Nature, the resonances have fairly broad decay widths so that the averaging takes place automatically in the data, the large $N_c$ limit shows us that averaging is always required in principle. It is thus clearly important to be able to define this averaging procedure, [*[*e.g.*]{}*]{}, how large the intervals must be and which resonances have to be included.
It is easy to see that this procedure will not be universal, and will certainly not simply be that the resonances one-by-one locally average the pQCD-derived scaling curve: the averaging method will depend on the process and on the target. Consider, as an illustration of these points, the case of a spinless quark and antiquark with charges $e_1$ and $e_2$ and equal masses bound into a nonrelativistic $q_1 \bar q_2$ system. The inelastic electron scattering rate calculated at the quark level in leading twist will then be proportional to $e^2_1+e^2_2$. Since the elastic state will be produced with a rate proportional to $(e_1+e_2)^2$, it clearly cannot in general be locally dual to the scaling curve [@schmidt]. How then is duality realized in this system? Consider the charge operator $\sum_i e_i e^{i\vec q \cdot \vec
r_i}$: from the ground state it excites even partial wave states with an amplitude proportional to $e_1+e_2$ and odd ones with an amplitude proportional to $e_1-e_2$. Thus the resonances build up a cross section of the form $\alpha_1 (e_1+e_2)^2 + \alpha_2 (e_1-e_2)^2 +
\alpha_3 (e_1+e_2)^2 + \cdots$ and one can see by explicit calculations in models that (up to phase space factors) the cross terms in this sum will cancel to give a cross section proportional to $e^2_1+e^2_2$ once averaged over nearby even and odd parity resonances. It is clear that such target- and process-dependence is worthy of study. However, in this paper we will restrict ourselves to a model with $e_2=0$ so that local duality might apply [@CloseIsgur].
The question of the validity of low energy duality, [*i.e.*]{}, duality in electron scattering at finite beam energies in inelastic electron scattering after suitable averaging, is as old as the first inclusive electron scattering experiments themselves. It begins with the seminal paper of Bloom and Gilman [@bgduality], which made the observation that the inclusive $F_2$ structure function in the resonance region at low $Q^2$ generally oscillates about and averages to a global scaling curve which describes high $Q^2$ data. More recently, interest in Bloom-Gilman duality has been revived with the collection of high precision data on the $F_2$ structure function from Jefferson Lab [@JLAB]. These data not only confirmed the existence of Bloom-Gilman duality to rather low values of $Q^2$, but also seem to demonstrate that for the proton the equivalence of the averaged resonance and scaling structure functions holds also for each resonance so that duality also exists locally.
Here we present a model for the study of quark-hadron duality in electron scattering that uses only a few basic ingredients. Namely, in addition to requiring that our model be relativistic, we assume confinement and that it is sufficient to consider only valence quarks (this latter simplification being underwritten, as mentioned previously, by the large $N_c$ limit). In addition, since our model is designed to explore conceptual issues and not to be compared to data, and since we postpone addressing spin-dependent issues to later work, for simplicity we also take the quarks, electrons and photons to be scalars. A model with these features will not give a realistic description of any data, but it should allow us to study the critical questions of when and why duality holds. While this model is extremely simple, we see no impediment to extending it to describe a more realistic situation since we find that duality arises from the most basic properties of our model.
We make several more convenient simplifications. Although it is our aim to study duality in electron scattering from the nucleon, [*i.e.*]{} from a three-quark-system, as a first step we study these issues in what is effectively a one quark system by considering such a quark to be confined to an infinitely massive antiquark. In the case of scalar quarks considered here, we can therefore describe the system by the Klein-Gordon equation. We also select for our confining potential one which is linear in $r$, namely $V^2 (\vec r \,) = \alpha \, r^2$, where $\alpha$ is a generalized, relativistic string constant. This choice allows us to obtain analytic solutions, without which the required numerical work for this study would be daunting. Indeed, the energy eigenvalues, $E_N = \sqrt{ 2 \sqrt{\alpha}(N + 3/2) + m^2 }$, where $m$ is the mass of interacting quark, can be readily obtained by noting the similarity to the Schrödinger equation for a non-relativistic harmonic oscillator potential: the solutions for the wave functions are the same as for the non-relativistic case.
In the next Section we construct the structure function out of resonances described by form factors, each of which individually gives vanishing contributions at large momenta, and show that it both scales and, when suitably averaged, is equal to the “free quark” result. An analysis in terms of structure function moments is presented in Section III. In Section IV we examine the onset of scaling, and the appearance of Bloom-Gilman duality, while in Section V we discuss the connection of Bloom-Gilman duality with duality in heavy quark systems. Finally, in Section VI we summarize our results and mention some possible directions for future research.
Quark-Hadron Duality in the Scaling Limit
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The differential cross section for inclusive inelastic scattering of a “scalar electron” [*via*]{} the exchange of a “scalar photon” is $$\frac{d \sigma} {dE_f d\Omega_f} = \frac{g^4}{16\pi^2}
\frac{E_f}{ E_i} \frac{1}{Q^4}\ {\cal W} \,,
\label{defscalw}$$ where the scalar coupling constant $g$ carries the dimension of a mass, and the factor multiplying the scalar structure function ${\cal
W}$ corresponds to the Mott cross section. In a model where the only excited states are infinitely narrow resonances, ${\cal W}$ is given entirely by a sum of squares of transition form factors weighted by appropriate kinematic factors: $$% {\cal W} = \sum_{N=0}^{\infty} \, \frac{1}{4 E_0 E_N} \, \,
{\cal W}(\nu,\vec q \, ^2)
= \sum_{N=0}^{N_{\rm max}} \, \frac{1}{4 E_0 E_N}\, \,
\left| F_{0N}(\vec q) \right|^2 \, \, \delta(E_N - E_0 - \nu)\, ,
\label{wscalho}$$ where $\vec q \equiv \vec p_i - \vec p_f$, the form factor $F_{0N}$ represents a transition from the ground state to a state characterized by the principal quantum number $N$, and the sum over states $N$ goes up to the maximum $N_{\rm max}$ allowed kinematically. Note that for fixed, positive $Q^2 \equiv \vec q \, ^2 - \nu^2$, $N_{max} = \infty$.
The excitation form factors can be derived using the recurrence relations of the Hermite polynomials. One finds: $$\label{eqff}
F_{0N} (\vec q \, ^2) = \frac{1}{\sqrt{N!}} \, i^N \,
\left ( \frac{|\vec q|}{\sqrt{2} \, \beta} \right ) ^N
\exp (- \vec q \, ^2 / 4 \, \beta^2) \, ,$$ where $\beta = \alpha^{1/4}$. This form factor is in fact the sum of all form factors for excitations from the ground state to degenerate states with the same principal quantum number $N$. As a precursor to our discussion of duality, we note that it will be a necessary condition for duality that these form factors (or more generally those corresponding to some other model potential) can represent the pointlike free quark. It is in fact the case that $\sum_{N=0}^{N_{\rm max}} |F_{0N} (\vec q \,)|^2 \rightarrow 1$ as $N_{\rm max} \rightarrow \infty$, a relation which follows from the completeness of the confined wave functions. Incidentally, an examination of the convergence of this sum as a function of $\vert
\vec q \vert^2$ is sufficient to make the point that reproducing the behavior of a free quark requires more and more resonances as $\vert \vec q \vert^2$ increases (details of this will be discussed in a forthcoming publication).
Scaling in the presence of confining final state interactions has previously been investigated in Refs. [@ioffe; @gurvitzrinat; @greenberg; @psl], where similar conclusions are reached. This suggests that scaling may indeed be a trivial feature of a large class of simple quantum mechanical models. Some sense of how this can occur can be obtained by considering some of the properties of the relativistic oscillator model used in this paper. In particular, consider the properties of the square of the form factors. For a fixed principal quantum number, $N$, the form factor has a maximum in $|\vec q|$ at $\vec q \,
^2_N=2\beta^2\,N$. Using $\nu_N=E_N-E_0$ and $E_N=\sqrt{2\beta^2\,N+E_0^2}$, it can be shown that $$\nu_N=\frac{Q^2_N}{2E_0}$$ where $Q^2_N=\vec q \, ^2_N - \nu_N^2$. So the position of the peak in the averaged structure function occurs at $u_{Bj}={m/E_0}$ where $u_{Bj}={Q^2/2m\nu}$ is a scaled Bjorken scaling variable $u_{Bj}\equiv\frac{M}{m}x_{Bj}$ which takes into account that as the mass of the antiquark $M_{\bar Q}\rightarrow\infty$, the constituent quark will carry only a fraction of order $m/E_0$ of the hadron’s infinite-momentum-frame momentum. Furthermore, for fixed $\vec q$ the structure function falls off smoothly for energy transfers away from the peak value. The width of this peak as a function of energy transfer also becomes constant for large $|\vec q|$.
Now consider the integral of the structure function $$\Sigma(\vec q \, ^2)=\int_0^\infty d\nu\ {\cal W}(\nu,\vec q \,
^2) =\sum_{nlm}\frac{1}{4E_0E_N}<\psi_{000}|\rho(-\vec
q)|\psi_{nlm}> <\psi_{nlm}|\rho(\vec q)|\psi_{000}>$$ where $N=2(n-1)+l$ with $n=1,2,3,\cdots$, and where $\rho(\vec
q)=e^{i\vec q\cdot\vec x}$. Since the form factor sum for a fixed $\vec q$ peaks about $E_{N_{max}}=\sqrt{\vec q \, ^2+E_0^2}$, we can substitute $E_N\rightarrow E_{N_{max}}$ and then sum over the complete set of final states to give $$\Sigma(\vec q \, ^2)\cong\frac{1}{4E_0E_{N_{max}}}\cong\frac{1}{4E_0q}$$ for large momentum transfer. Therefore, if we define the scaling function as ${\cal S}\equiv |\vec q|\ {\cal W}$, as will be done below, the area of the scaling function becomes constant at large momentum transfer.
Since the scaling function peaks at fixed $u_{Bj}$, smoothly falls about the peak, has fixed width and constant area at large momentum transfer, the model scales. It is a common misconception that the presence of scaling implies that the final states must become plane waves. In fact, the argument above makes it clear that scaling occurs when the structure function becomes independent of the final states as in the closure approximation used here.
To see duality clearly both experimentally and theoretically, one needs to go beyond the Bjorken scaling variable $x_{Bj}$ and the scaling function ${\cal{S}}_{Bj} = \nu \cal{W}$ that goes with it. This is because in deriving Bjorken’s variable and scaling function, one not only assumes $Q^2$ to be larger than any mass scale in the problem, but also that high $Q^2$ (pQCD) dynamics controls the interactions. However, duality has its onset in the region of low to moderate $Q^2$, and there masses and violations of asymptotic freedom do play a role. Bloom and Gilman used a new, [*ad hoc*]{} scaling variable $\omega'$ [@bgduality] in an attempt to deal with this fact. In most contemporary data analyses, the Nachtmann variable [@greenbergb; @nachtmann] is used together with ${\cal{S}}_{Bj}$. Nachtmann’s variable contains the target mass as a scale, but neglects quark masses. For our model, the constituent quark mass (assumed to arise as a result of spontaneous chiral symmetry breaking) is vital at low energy, and a scaling variable that treats both target and quark masses is desirable. Such a variable was derived more than twenty years ago by Barbieri [*et al.*]{} [@barbieri] to take into account the masses of heavy quarks; we use it here given that after spontaneous chiral symmetry breaking the nearly massless light quarks have become massive constituent quarks, calling it $x_{cq}$: $$% x_{cq} = \frac{1}{2 M} \left ( \sqrt{\nu^2 + Q^2} - \nu \right )
x_{cq} = \frac{1}{2 M} \left ( \sqrt{\nu^2 + Q^2} - \nu \right )
\left ( 1 + \sqrt{1 + \frac{4 m^2}{Q^2}} \right ) \, .
\label{defxdis}$$ The scaling function associated with this variable is given by: $$\label{S}
{\cal{S}}_{cq} \equiv |\vec q|\ {\cal W} = \sqrt{\nu^2 + Q^2}\ {\cal W}\,.$$ This scaling function and variable were derived for scalar quarks which are free, but have a momentum distribution. The derivation of a new scaling variable and function for bound quarks will be published elsewhere. Numerically, this scaling variable does not differ very much from the one in Eq. (\[defxdis\]). Of course all versions of the scaling variable must converge to $x_{Bj}$ and all versions of the scaling function must converge towards ${\cal{S}}_{Bj}$ for large enough $Q^2$. One can also easily verify that in the limit $m \to 0$ one obtains from (\[defxdis\]) the Nachtmann scaling variable. In the following, we use the variable $x_{cq}$ and the scaling function ${\cal{S}}_{cq}$.
We are now ready to look at scaling and duality in our model. Since the target has mass $M
\rightarrow
\infty$, it is convenient to rescale the scaling variable $x_{cq}$ by a factor $M/m$: $$\begin{aligned}
u & \equiv & {M \over m}\ x_{cq}\ ~~.\end{aligned}$$ The variable $u$ takes values from 0 to a maximal, $Q^2$ dependent value, which can go to infinity. The high energy scaling behavior of the appropriately rescaled structure function ${\cal {S}}_{cq}$ is illustrated in Fig. 1.
The structure function has been evaluated using the phenomenologically reasonable parameters $m = 0.33$ GeV and $\alpha = (0.4 ~{\rm GeV})^{1/4}$, though we remind the reader not to compare our results, which might resemble electron scattering from a $B$ meson, to nucleon data! To display it in a visually meaningful manner, the energy-dependent $\delta$-function has been smoothed out by introducing an unphysical Breit-Wigner shape with an arbitrary but small width, $\Gamma$ chosen for purposes of illustration: $$\delta(E_N - E_0 - \nu) \rightarrow \frac{\Gamma}{2 \pi} \, \,
\frac{f}{(E_N - E_0 - \nu)^2 + (\Gamma/2)^2}\, ,$$ where the factor $
f = {\pi}/[{\frac{\pi}{2}
+ \arctan {2 (E_N - E_0) \over \Gamma}] }
$ ensures that the integral over the $\delta$-function is identical to that over the Breit-Wigner shape. The curves in Fig. 1 show that scaling sets in rather rapidly. The resonances show up as bumpy structures in the low $Q^2$ region (which will be discussed in Section IV below), a trace of which is visible for the $Q^2 = 5$ GeV$^2$ curve.
By taking the continuum limit for the energy and applying Stirling’s formula, one can obtain an analytic expression for the scaling curve, valid in the scaling region, for the transition of the quark from the ground state to the sum of all excited states: $$% {\cal{S}}(u) = m^2 \, u^2 \,\frac{1}{\pi^\frac{1}{2}\beta E_0}
{\cal{S}}_{cq}(u) = \frac{E_0}{ \sqrt{\pi} \beta }
\exp{\frac{(E_0-m u)^2}{\beta^2}} \, .
\label{sfanalyt}$$ Of course we still need to verify that this scaling curve as seen in Fig. 1 found by summing over hadrons is the same as the one which we would obtain from deep inelastic scattering off the quark, [*[*i.e.*]{}*]{}, if we were to switch off the potential in the final state. In this case, the tower of hadronic states is replaced by the free quark continuum. Duality predicts that the results should be the same in the scaling limit, and by direct calculation we confirm this.
Moments of Structure Functions {#secglo}
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Bloom-Gilman duality relates structure functions at low and high $Q^2$ averaged over appropriate intervals of the hadronic mass $W$. As a quantitative measure of this feature of the data, one conventionally examines the $Q^2$-dependence of moments of structure functions. The moments offer the cleanest connection with the operator product expansion of QCD, and provide a natural connection between duality in the high- and low-$Q^2$ regions. By considering the moments, we also remove artifacts introduced through the smoothing procedure described above for the structure function itself.
The moments of the structure function ${\cal{S}}_{cq}(u,Q^2)$ are defined as: $$\label{moment}
M_n(Q^2) = \int_0^{u_{\rm max}} du \, \, u^{n-2} \,
{\cal{S}}_{cq}(u,Q^2)\, ,$$ where $u_{\rm max}$ corresponds to the maximum value of $u$ which is kinematically accessible at a given $Q^2$. Evaluating the moments of the structure function (\[S\]) explicitly one has (provided the kinematics allow us to access all excited states): $$\begin{aligned}
M_n(Q^2) &=&
\left ( \frac{r}{2 m} \right )^{n-1} \,
\sum_{N=0}^{\infty} \,
\left( \sqrt{\nu_N^2 + Q^2} - \nu_N \right)^{n-1}
\, \, \frac{E_0}{ E_N} \,
\left|F_{0N} \left(\sqrt{\nu_N^2 + Q^2} \right) \right|^2 ~~ ,\end{aligned}$$ where $\nu_N = E_N - E_0$ and $r = 1 + \sqrt{1 + 4 m^2/Q^2}$. The elastic contribution to the moments is $$M_n^{\rm elastic} (Q^2) = \left ( \frac{r}{2 m} \right )^{n-1}\,
Q^{n-1} \, \, \left| F_{00} (Q^2) \right|^2\
= u^{n-1}_0 \left| F_{00} (Q^2) \right|^2\, ~~,$$ where $u_0(Q^2)$ is the position in $u$ of the ground state. Note that $M_n^{\rm elastic} (Q^2)$ becomes independent of $n$ in the limit $Q^2
\rightarrow 0$, approaching unity and that the inelastic contributions to the moments vanish for vanishing $Q^2$.
In Fig. 2 we show the $n=2$, 4, 6 and 8 moments $M_n$ as a function of $Q^2$. All the moments appear qualitatively similar, rising to within about 10% of their asymptotic values by $Q^2=1$ GeV$^2$. Also evident is the fact that the lower moments reach their asymptotic values earlier than the higher moments. This is qualitatively consistent with the expectation from the operator product expansion discussed in [@DGP], where it was argued that the effective expansion parameter in the twist expansion $\sim n/Q^2$, so that for higher moments, $n$, the higher twist terms survive to larger values of $Q^2$.
Unfortunately, these moments do not have such useful interpretations here as they do in real deep inelastic scattering. For example, the analog of the Gross-Llewellyn Smith sum rule is not applicable here because the scalar current which couples to our quark is not conserved. Nonetheless, the moments in Fig. \[figmom1dist\] do serve to demonstrate that scaling is a natural consequence of our model, and illustrate the relative onset of scaling for different moments.
Onset of Scaling and Bloom-Gilman Duality
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After studying the scaling behavior of the structure functions in our model at high $Q^2$ and the moments over a range of four-momentum transfers, we now study the structure functions at low $Q^2$ where not only in the large $N_C$ limit but also in nature resonances are visibly dominant over a wide range in the scaling variable. Here, we consider a target where only one quark carries all the charge of the system, so there is no forced breakdown of duality at $Q^2 = 0$ of the type noted earlier for the neutron. Still, one cannot expect that the perturbative QCD result will describe even averaged hadronic observables well at very low $Q^2$: these are after all strong interactions!
If local duality holds, one might expect the resonance “spikes" to oscillate around the scaling curve and to average to it, once $Q^2$ is large enough. (We remind the reader that while scaling in deep-inelastic electron scattering from the nucleon is known from experiment to set in by $Q^2 \sim 2$ GeV$^2$, the target considered here corresponds to an infinitely heavy “meson” composed of scalar quarks interacting with a scalar current, so one should not expect numerically realistic results, only qualitative ones.) Figure 3 shows the onset of scaling for the structure function ${\cal{S}}_{cq}$ as a function of $u$, as $Q^2$ varies from 0.5 GeV$^2$ to 2 GeV$^2$. As in Fig. 1, for each of the resonances (excluding the elastic peak) the energy $\delta$-function has been smoothed out using the Breit-Wigner method with a width $\Gamma=100$ MeV. With increasing $Q^2$, each of the resonances moves out towards higher $u$, as dictated by kinematics. At $Q^2 = 0$, the elastic peak is the only allowed state and contributes about 44% of the asymptotic value of $M_2$. It remains rather prominent for $Q^2$ = 0.5 GeV$^2$, though most of $M_2$ is by this point built up of excited states, and it becomes negligible for $Q^2 \geq$ 2.0 GeV$^2$. Remarkably, the curves at lower $Q^2$ do tend to oscillate (at least qualitatively) around the scaling curve, as is observed in proton data. Note that these curves are at fixed $Q^2$, but sweep over all $\nu$. In a typical low energy experiment, $\nu $ will also be limited; in such circumstances these curves still apply, but they get cut off at the minimum value of $u$ that is kinematically allowed. For another perspective on these curves, note that $\vert \vec q \vert^2 = Q^2+\nu^2$ so for fixed $Q^2$, as $\nu$ is increased so that more and more highly excited states are created, the struck quark is being hit harder and harder.
In contrast, the structure function ${\cal{S}}_{Bj}$ when plotted as a function of the scaled Bjorken variable $u_{Bj}$ shows very poor duality between its low- and high-$Q^2$ behaviors, as seen in Fig. 4. One of the reasons for this failure is that $x_{Bj}$ and ${\cal{S}}_{Bj}$ know nothing about the constituent quark mass, while low energy free quark scattering certainly does, so the corresponding pQCD cross section calculated neglecting the quark mass is simply wrong at low energy.
Duality in Semileptonic Decays of Heavy Quarks
===============================================
We have seen that low-energy (Bloom-Gilman) duality is displayed by our model in terms of the appropriate low-energy variable $u$ and described some of the physics behind this duality (completeness of the bound state wave functions to expand a plane wave and an approximate closure based on the required expansion states being in a narrow band of $\nu$ relative to those that are kinematically allowed). To obtain a deeper understanding of the physics behind low energy duality, it is instructive to compare and contrast duality in electron scattering with that in heavy quark decays. We will begin by carefully examining duality in heavy-light systems, where it is exact in the heavy quark limit even down to zero recoil, and where the mechanisms behind this exact duality are very clear.
Duality in heavy quark systems is easily understood intuitively. Consider a $Q^* \bar q$ system where $m_Q^* >> \Lambda_{QCD}$, and imagine that $Q^*$ can decay to $Q$ by emitting a scalar particle $\phi$ of mass $\mu$: $Q^*
\rightarrow Q+\phi$. (Note that in this case it is the heavy quark that interacts with the current and not the light quark as in our model!) At the free quark level, the decay of $Q^*$ at rest will produce the $\phi$ with a single sharp kinetic energy $T_{free}$ and corresponding $Q$ recoil velocity $\vec v$. (We use the standard variables $T_{free}$ and $\vec v$, but others, like the $\phi$ recoil momentum, could be chosen.) In reality, since the heavy quarks are bound into mesons, $\phi$ will (in the narrow resonance approximation) emerge from the decay at rest of the initial meson’s ground state $(Q^* \bar q)_0$ with any of the sharp kinetic energies allowed by the processes $(Q^*
\bar q)_0 \rightarrow (Q \bar q)_n + \phi$ as determined by the strong interaction spectra of these two mesonic systems. Since in the heavy quark limit $m_{(Q^* \bar q)_n}-m_{(Q \bar q)_n} \simeq m_{Q^*}-m_Q$, $m_{(Q^*
\bar q)_n} \simeq m_{Q^*}$, and $m_{(Q
\bar q)_n} \simeq m_{Q}$, the hadronic spectral lines are guaranteed to cluster around $T_{free}$, and to coincide with it exactly as $m_Q
\rightarrow \infty$. Moreover, since $m_Q^*,m_Q >>
\Lambda_{QCD}$, one can show using an analog of the operator product expansion [@inclorig] that the strong interactions can be neglected in calculating the total decay rate ([*[*i.e.*]{}*]{}, the heavy quarks $Q^*$ and $Q$ are so heavy that the decay proceeds as though it were free.) Thus the sum of the strengths of the spectral lines clustering around $T_{free}$ is the free quark strength: there is perfect low energy duality as $m_Q^*, m_Q
\rightarrow \infty$.
What is now especially interesting is to unravel this duality to understand how the required “conspiracy” of spectral line strengths arises physically. Because the heavy quark is so massive, if it would as a free particle recoil with a velocity $\vec v$, then this velocity would be changed only negligibly by the strong interaction since in the heavy quark limit it carries off a negligible kinetic energy, but a momentum much larger than $\Lambda_{QCD}$. In the rest frame of the recoiling meson, this configuration requires that the two constituents have a [*relative*]{} momentum $\vec q$ which grows with $\vec v$. [*Thus the strong interaction dynamics is identical to that of our model in which the relative momentum $\vec q$ is supplied by the scattered electron.*]{} Moreover, in this case, with duality exact at all energies, we can reconstruct exactly how it arises. What one sees is remarkably simple [@BjSumRule; @IWonBj]. At low $\vec v$ corresponding to low $\vec q$, only the ground state process $(Q^*
\bar q)_0 \rightarrow (Q \bar q)_0 + \phi$ occurs. Since the masses and matrix elements for the transitions $(Q^*
\bar q)_0 \rightarrow (Q \bar q)_0 + \phi$ and $Q^*
\rightarrow Q + \phi$ are identical (the elastic form factor goes identically to unity as $\vec q \rightarrow 0$), the hadronic and quark spectral lines and strengths are also identical and duality is valid at $\vert \vec q
\vert^2=0$! Next consider duality at a different kinematic point (which one might reach by choosing a smaller $\phi$ mass) where $\vec v$ and therefore $\vec q$ have increased. The elastic form factor will fall, so its spectral line (which is still found at exactly the new value of $T_{free}$ in the heavy quark limit) will carry less strength. However, once $\vec q $ differs from zero, excited states $(Q \bar q)_n$ can be created, and indeed are created with a strength that exactly compensates for the loss of elastic rate. These excited state spectral lines also coincide with $T_{free}$ and duality is once again exact. Indeed, no matter how large $|\vec q|^2$ becomes, all of the excited states produce spectral lines at $T_{free}$ with strengths that sum to that of the free quark spectral line.
Heavy quark theory also allows one to go beyond the heavy quark limit to the case of quarks of finite mass. In this case one of course finds that duality-violation occurs, but that it is formally suppressed by two powers of $\Lambda_{QCD}/m_Q$ [@inclorig; @DualityControversy], with the spectral lines now clustered about $T_{free}$ but not coinciding with it. A remarkable feature of this duality violation is that the spectral line strengths differ from those of the heavy quark limit in ways that tend to compensate for the duality-violating phase space effects from the spread of spectral lines around $T_{free}$. An additional source of duality-violation is that some of the high mass resonances that are required for exact duality are kinematically forbidden since for finite heavy quark masses $m_{Q^*}-m_Q$ is finite.
From this discussion it is clear that the strong interaction dynamics of heavy-light decays is the same as that of scattering a probe off of the $Q$ of a $Q \bar q$ system [@XpQCD]: what is relevant is that the system must in each case respond to a relative momentum kick $\vec q$. Needless to say, one must still carefully organize the kinematics to expose duality: in a decay to a fixed mass $\phi$ only a single magnitude $\vert \vec q
\vert^2 $ is produced at the quark level, while in electron scattering a large range of $\vert \vec q
\vert^2 $ and $\nu$ is produced by a given electron beam.
Given these connections, it is relevant to note that in addition to the obvious conceptual relevance of heavy-light systems, model studies indicate that in these systems heavy quark behavior continues to hold qualitatively even for $m_Q \sim m$. These models are, as one might expect, similar to ours which displays the same clustering of spectral lines, the same tendency for excited state spectral lines to compensate for the fall with $\vert \vec q
\vert^2 $ of lighter states, and the same sources of duality violation such as kinematically forbidden states and mismatches between the mass of the recoiling hadrons and the struck quark. We believe that these elements of the dynamics are clearly in operation and that we have understood through our model that the qualitative applicability of duality for real systems should indeed extend all of the way down to zero recoil as seen in Nature.
Summary and Outlook
===================
We have presented a simple, quantum-mechanical model in which we were able to qualitatively reproduce the features of Bloom-Gilman duality. The model assumptions we made are the most basic ones possible: we assumed relativistic, confined, valence scalar quarks and treated the hadrons in the infinitely narrow resonance approximation. To further simplify the situation, we did not consider a three quark “nucleon” target, but a target made up by an infinitely heavy antiquark and a light quark. The present work does not attempt to quantitatively describe any data, but to give qualitative insight into the physics of duality.
Our work complements previous work on duality, where the experimental data were analyzed in terms of the operator product expansion (OPE) [@JI; @DGP]. There, it was observed that at moderate $Q^2$, the higher twist corrections to the lower moments of the structure function are small. The higher twist corrections arise due to initial and final state interactions of the quarks and gluons. Hence, the average value of the structure function at moderate $Q^2$ is not very different from its value in the scaling region. While true, this statement is merely a rephrasing in the language of the operator product expansion of the experimentally observed fact that the resonance curve averages to the scaling curve. However, the operator product expansion does not explain why a certain correction is small or why there are cancellations: the expansion coefficients which determine this behavior are not predicted. The numerical confirmation of these coefficients will eventually come from a numerical solution of QCD on the lattice, but an [*understanding*]{} of the physical mechanism that leads to the small values of the expansion coefficient will almost certainly only be found in the framework of a model like ours.
For example, one clear lesson from our study of duality is that the commonly made sharp distinction between the “resonance region”, corresponding to an invariant mass $W < 2 $ GeV for scattering from a proton, and the deep inelastic region, where $W > 2 $ GeV, is completely artificial.
Finally, we remind the reader that our model, with all the charge on a single quark, with scalar currents, and with no spin degrees of freedom, leaves much to be done in model-building. The next step is to use more realistic currents. While making the calculations more complicated, coupling to the conserved quark current will allow one to study the $Q^2$-evolution of the Gross-Llewellyn Smith and momentum sum rules. To use a spin-${1\over 2}$ target will also be a useful step forward, but it may require foregoing the great advantages of the analytic solutions of the Klein-Gordon equation. As we have emphasized, the local duality seen here [*cannot*]{} be expected for more complicated targets and processes, and pursuing this issue is also clearly very important [@CloseIsgur]. Here we have taken a first small step which nevertheless has been enough to strongly suggest that for these more realistic models and more general processes there will be a generalization of local averaging — a theoretically well-defined procedure for integrating over regions of $x_{cq}$ — which will also display low energy duality. If so, we will not only have understood quark-hadron duality. We will also have opened the door to extending studies of a variety of structure functions into previously unreachable kinematic regimes.
We thank C. Carlson, F. Close, R. Ent, J. Goity, C. Keppel, R. Lebed, I. Niculescu and S. Liuti for stimulating discussions, and N.N. Nikolaev and S. Simula for pointing out several references. This work was supported in part by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab), and by the Australian Research Council.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Coherent nonlinear multi-pulse processes, nonlinear waves and echo effects in resonant media are the topical problems of modern optics and important tools of coherent spectroscopy and quantum information science. We generalize the McCall-Hahn area theorem to the formation of an arbitrary photon echo generated during the multi-pulse excitation of the optically dense resonant media. The derived theorem made it possible to reveal the nonlinear mechanism of generation and evolution of the photon echo signals inside the media after a two-pulse excitation. We find that a series of self-reviving echo signals with total area of $2\pi$ or $0\pi$ is excited and propagates in the media depth, with each pulse having an individual area less than $\pi$. The resulting echo pulse train is a new alternative to the well-known soliton or breather. The developed pulse-area approach paves the way for more precise coherent spectroscopy, studies of different photon echo signals and quantum control of light pulses in the optically dense media.'
author:
- 'Sergey A. Moiseev$^{1}$'
- 'Mahmood Sabooni$^{2,3}$'
- 'Ravil V. Urmancheev$^1$'
bibliography:
- 'main.bib'
title: Photon echoes in optically dense media
---
Studies of coherent multi-pulse nonlinear effects like photon echo and four-wave mixing open wide opportunities for understating of light-atom interactions, fundamental processes of nonlinear and quantum optics, provide powerful techniques for spectroscopic investigation of atoms and molecules and are considered as a principal tool for implementation of basic processes in practical quantum information science [@Yetzbacher2007; @Christensson2008; @Dorfman2016; @Pezz2018; @Mourou2019]. Herein, photon echo technique [@KopvillemPhEcho1963; @Kurnit1964] attracts an especial everlasting attention in coherent spectroscopy [@Kurnit1964] and light pulse storage [@HEER197749; @Samartsev1980; @1981JEPTStyrkov; @Mossberg:82; @Carlson:831]. Recently, the photon echo in optically dense media opened promising opportunities for quantum storage of a large number of light pulses [@MoiseevKroll2001; @Tittel2009; @Lvovsky2009; @SparkesNatComm2011; @Usmani2010] and quantum processing [@OpticalQMapplicationReview] that determined a steady interest and elaboration of numerous protocols of photon echo based quantum memory [@Tittel2009; @Hosseini2011; @Rani2017; @Minnegaliev_2018; @Saglamyurek2018; @Guo2019; @Mazelanik2019], which are important for the creation of quantum repeater [@RevModPhys.83.33], microwave quantum memory [@PhysRevLett.105.140503; @Moiseev2018], etc.
The study of the properties of a two- and three-pulse photon echoes in optically dense media is the main task in the development of the multi-pulse spectroscopy and photon echo quantum memory schemes in such media. The most general theoretical description of the coherent resonant interaction of multi-pulse light fields with resonant atoms can be provided by the pulse area theorem [@McCallHahn1969; @Lamb1971; @allen1975optical; @Eberly:98; @PhysRevLett.88.243604; @Chaneliere:14; @PhysRevA.92.063815; @ThreePulseAreaTheorem]. In early works on the two-pulse (primary) photon echo, it was found that the initial excitation could result in the generation of multiple echo signals [@HAHN1971265; @allen1975optical] followed by a long-term investigation of the underlying mechanism [@FRIEDBERG1971285; @Lamb1971; @allen1975optical; @Moiseev1987; @1998-Azadeh-PRA; @1998-Wang-OC; @1999-Wang-PRA; @PhysRevA.79.053851; @Li2010; @Tsang:03]. Quite early an analytic solution for total area of all the echoes was obtained [@HAHN1971265; @allen1975optical; @1998-Azadeh-PRA], that proved that the total pulse area can tend asymptotically towards $2\pi$ in the media depth if the initial pulse area of two exciting laser pulses exceeds $\pi$. However, this solution does not allow one to describe the behavior of each individual echo pulse.
Previously acquired solution for the primary echo pulse area predicted that the echo pulse area never exceeds $\pi$ and generally decays in the depth of the media [@Moiseev1987] . This finding again stressed the ambiguity of the known physical picture behind the formation of the total nonlinear response to the multi-pulse excitation. In the recent years the stakes were raised by the demand for an efficient optical solid-state quantum memory and the noted interest in coherent multi-pulse interactions in the optically dense media.
In this Rapid Communication we find an analytical solution of the photon echo pulse area theorem posed in [@HAHN1971265; @FRIEDBERG1971285; @Lamb1971] in 1971. By analysing the solution we for the first time discover the mechanism of self-induced transparency [@McCallHahn1969] for two- and many-pulse excitation of the atomic media leading to the formation of many echo pulses. To do that we find the general analytic solution for the pulse area of an arbitrary secondary photon echo signal. The found solutions show that the echo signals are excited coherently one after another in a certain area of the medium and then disappear, generating new echo signals and creating a self-reviving echo sequence. We show that depending on the input pulse areas this echo pulse train forms a multi-pulse analogue to the well-known single pulse $2\pi$ optical soliton or a $0\pi$ optical breather despite each individual echo pulse area never exceeding $\pi$. Herein, by using the highly non-linear nature of the light-atom interaction we can control the total response of the media. Being near the thresh-old, when the incoming area of the second pulse is close to $\pi$, and by slightly changing it to being $<\pi$ or $> \pi$ one can initiate a huge change in the outcome from an optical soliton to an optical breather, respectively. This also demonstrates the potential of the pulse area approach for coherent spectroscopy of the optically dense media.
First we reproduce the McCall-Hahn area theorem and derive the general equation for the pulse area of an arbitrary echo pulse starting with the usual reduced set of Maxwell-Bloch equations [@allen1975optical] for the light field and atomic system: $$\begin{aligned}
\begin{split}
[ \partial_z + c^{-1}\partial_t ] \Omega & = i \frac{\mu}{2} \langle P\rangle,
\\
\partial_t u & = - \Delta v - \gamma u,
\\
\partial_t v & = \Delta u - \gamma v + \Omega w,
\\
\partial_t w & = -\Omega v,\\
\end{split}
\label{eq:mb_set}
\end{aligned}$$ where $\vec{r} = \vec{r}(t,z,\Delta) = (u,v,w)^T$ is the Bloch vector, each component depending on time $t$, spatial coordinate $z$ and atomic detuning $\Delta$; $P = u-iv$ - atomic polarization, electric field $E(t,z) = \varepsilon(t,z) \exp[i(kz-\omega t)] + c.c.$ is described by a complex light field envelope $\varepsilon(t,z)$ with corresponding Rabi frequency $\Omega (t,z) = (2 d/\hbar) \varepsilon (t,z)$; $\mu = 4\pi N d^2\omega/\hbar c$, $\gamma = 1/T_2$, $T_2$ is the coherence lifetime of the atomic transition and $\langle...\rangle\equiv \int_{-\infty}^{\infty} G(\Delta) ...d\Delta$ is the averaging over the inhomogeneous broadening. From now on for simplicity, we do not denote the existing dependence on $z$ in atomic and field variables $\vec{r}$ and $\Omega$.
By transferring to the pulse area $\theta = \int_{-\infty}^\infty dt \Omega (t)$ and follow [@McCallHahn1969; @Eberly:98] to find that incoming pulse areas $\theta_1, \theta_2$ satisfy the well-known pulse area theorem: $$\partial_z \theta = \tfrac{1}{2}\alpha w_0(z)\, \sin \theta(z),
\label{eq:area1}$$ where $w_0$ is the initial inversion of the atomic system, $\alpha$ is the resonant absorption coefficient [@allen1975optical]. The first pulse propagates in the undisturbed media, with $w_0=-1$ and partially inverts for the second pulse, so $w_0= -\cos\theta_1$. Substituting $w_0$ into Eq. we get the well-known solutions [@allen1975optical]: $$\begin{aligned}
\begin{split}
\theta_1(z) & = 2 \arctan \left[e^{-\alpha z/2} \tan \dfrac{\theta_1(0)}{2}\right],
\\
\theta_2(z) & = 2\arctan \left[ \kappa ~\mathrm{sech} \left( \beta - \frac{\alpha}{2}z \right)\right],
\end{split}
\label{eq:th_two_solution}
\end{aligned}$$ where $\beta=\ln\{\tan[\frac{\theta_1(0)}{2}]\}$ and $\kappa=\tan[\frac{\theta_2(0)}{2}]/\sin[\theta_1(0)]$.
Eqs. and can be used to find the total area of all excited photon echoes [@HAHN1971265; @FRIEDBERG1971285; @allen1975optical; @1998-Azadeh-PRA]: $$\theta_{\Sigma e}(z) = 2 \arctan \left[e^{-\alpha z/2} \tan \tfrac{\theta_{1}(0)+\theta_{2}(0)}{2}\right]-\theta_2(z)-\theta_1(z).
\label{eq:sum_area}$$
This solution predicts that if $\theta_2(0)<\pi, \theta_1(0)+\theta_2(0) > \pi$, the total area of all echo pulses asymptotically tends to $2\pi$ [@HAHN1971265]. It leaves however a lot of uncertainty about the mechanism and physics of the photon echo generation, since any information about the particular photon echo signals remains hidden. How exactly different echoes combine into $2\pi$ pulse area? What is the contribution of an individual echo? Moreover, if input pulse areas $\theta_1(0)<\pi/2, \theta_2(0) >\pi$, Eq. predicts the sum of all echoes to be $0$. What happens with the different echo signals in this case and does that mean that there will be no echoes? To answer all these questions, we have to analyze the generation of each echo signal individually.
To find the area theorem for an arbitrary individual photon echo signal we integrate the first of Eqs. over time around the time of echo emission $t_e$, from $t_0 = t_e-\tau/2$ to $t_1=t_e+\tau/2$, where $\tau$ is the delay between the pulses. We should also clarify the time scales assumed for the following derivation. Firstly, we assume non-overlapping pulses $\tau \gg \delta t_{1,2}$ with pulse duration being much smaller than coherence time $\delta t_i \ll T_2$, $i=1,2,e1,...$, to neglect the relaxation during the pulses. Secondly, inhomogeneous broadening of the atomic system is much larger that the pulse spectrum $\Delta_{in} > 1/\delta t_{1,2}$. Thirdly, for simplicity we consider a solid state system, meaning $T_1\gg T_2$ and thus we can neglect the population decay between the pulses. In short, $1/\Delta_{in} < \delta t_{1,2} \ll \tau \lesssim T_2$.
The expressions under the integrals, $P_0(z,\Delta)$ and $w(t,z,\Delta)$ are complex expressions consisting of several oscillating components. However most of these components will give $0$ after averaging over $\Delta$ in Eq. . To find the proper expression for the echo area we need to only take into account the phasing components of polarization and inversion that contribute to the echo formation. The details of the integration and equation handling can be found in the Supplemental material.
As a result we obtain the general equation for an arbitrary echo pulse area: $$\partial_z \theta (z)= \frac{1}{2}\alpha [ 2 v_0(z) \cos^2\frac{\theta(z)}{2} + w_0(z)\sin\theta(z)],
\label{eq:area_general}$$ where $w_0(z)$ and $v_0(z)$ are the initial values ($t=t_e-\tau/2$) of the Bloch vector resonance components with $\Delta = 0$ which only give nonzero response in the field equation in (1). After transition to $\eta = \tan\tfrac{\theta(z)}{2}$ we get a linear equation $\partial_z \eta(z) = \frac{\alpha}{2}[v_0(z)+w_0(z) \eta(z) ]$ with clear solution.
Equation describes the pulse area of a chosen echo signal given the phasing coherence $v_0$ in a presence of spectral uniform inversion $w_0$ and Eq. comes down to finding $v_0(z)$ and $w_0(z)$ for each echo signal. In Supplemental material we describe the algorithm that allows to find the $v_0, w_0$ for an arbitrary echo. But whatever they may be, we note that $|\theta|$ never exceeds $\pi$. Below we investigate the analytic solutions for the pulse areas of all the echo signals.
For primary echo we have $\vec{r}(t)=U(t-\tau)T(\theta_2)U(\tau)T(\theta_1)\vec{r}(0)$, $t_0 = 3\tau/2$ and the correct phasing components of $\tilde{v}_{0}(3\tau/2), \tilde{w}_{0}(3\tau/2)$ [@Moiseev1987; @2019-OptExpress]: $$\begin{aligned}
\begin{split}
v_{0}(3\tau/2,z) & = \Gamma_{\tau}^2 \sin\theta_1(z)\sin^2\tfrac{\theta_2(z)}{2},
\\
\tilde{w}_{0}(3\tau/2,z) & = -\cos\theta_1(z)\cos\theta_2(z),
\end{split}
\label{eq:primary_vw}
\end{aligned}$$ where $\Gamma_{\tau} = e^{-\gamma\tau}$ is the relaxation term. Corresponding Eq. gives primary photon echo pulse area: $$\theta_{e1}(z) =2\arctan\left[\Gamma_{\tau}^2 \sin\theta_1(0) \sin^2 {\tfrac{\theta_2(z)}{2}}\sinh\tfrac{\alpha z}{2} \right].
\label{eq:echo_area_solution}$$
After the incoming pulses and the primary echo pulse we have $\vec{r}(t)=U(t-2\tau)T(\theta_{e1})U(\tau)T(\theta_2)U(\tau)T(\theta_1) \vec{r}(0),~ t_0=5\tau/2$ and the phasing components $v_{0}(5\tau/2,z), w_{0}(5\tau/2,z)$ are: $$\begin{aligned}
\begin{split}
v_{0} = v_{01} & + v_{02} =
\tfrac{1}{2} \Gamma_\tau^2 \sin\theta_1(z) \sin\theta_{e1}(z) \sin\theta_2(z)
\\
+ & \Gamma_\tau^2 \cos\theta_1(z)\sin^2\tfrac{\theta_{e1}(z)}{2} \sin\theta_2(z),
\\
w_{0} = w_{01} & + w_{02} = -\Gamma_\tau^2 \sin\theta_1(z)\sin^2\tfrac{\theta_2(z)}{2} \sin\theta_{e1}(z)
\\
- & \cos\theta_1(z) \cos\theta_2(z)\cos\theta_{e1}(z).
\end{split}
\label{eq:2nd_echo_v_w}\end{aligned}$$
The first terms in both equations $v_{01} (z)= \tfrac{1}{2}\Gamma_\tau^2 \sin\theta_1\sin\theta_2\sin\theta_{e1}$ and $w_{01}(z) = - \Gamma^2_\tau \sin\theta_1\sin^2\tfrac{\theta_2}{2}\sin\theta_{e1}$ are proportional to $\sin\theta_1(z)$ and vanish when the first pulse is absorbed. They are responsible for stimulated photon echo generated by incoming pulses and the primary echo pulse. The other two components $v_{02}(z) = \Gamma_\tau^2\cos\theta_1\sin\theta_2\sin^2\tfrac{\theta_{e1}}{2}$ and $w_{02}(z) = -\cos\theta_1\cos\theta_2\cos\theta_{e1}$ are proportional to $\cos\theta_1$ are correspond to the secondary two-pulse photon echo created by the second pulse and the primary echo pulse.
Analysis of the successive echoes follows the same procedure but requires more calculations since $v_{0}$ and $w_{0}$ have more terms with each step. In the Supplemental material we introduce the phasing polarization and inversion components for the third and the fourth echoes and discuss the physical meaning of different contributions. It is obvious that the described procedure can be applied for the case with comparable transverse and longitudinal relaxations and for other light-atom equations.
We will now proceed to clarify the mechanism of the total $2\pi$ pulse area formation when $\theta_1(0)+\theta_2(0)>\pi$. Figure \[fig:echo\_areas\_999\] shows the spatial behavior of the area of incoming pulses, echo pulses and the total area depending on the optical density of the medium for $\theta_1(0)=0.1\pi, \theta_2(0)= 0.999\pi$. We see that incoming pulses excite primary and secondary echoes that in turn excite subsequent echos. Each echo pulse is born, propagates and eventually dies out within a finite spatial interval. However the total area of all existing pulses behaves strictly in accordance with McCall-Hahn area theorem Eq. and remains close to $2\pi$. This is realized due to the precise spatial consistency of all the echoes involved.
The case of $\theta_2(0)>\pi$ really helps to highlight the benefits of looking at an individual echo signal rather than at the sum of all echo signals. The second incoming pulse is big enough to form a $2\pi$-soliton on its own, and McCall-Hahn area theorem predicts that the sum of all echoes will equal $0\pi$. The impression could be that after some point in the medium there are no echoes at all. The real picture however is much more vivid, there are many hidden echoes with nontrivial areas working together to comply with the McCall-Hahn area theorem. Figure \[fig:echo\_areas\_1001\] showcases this echo pulses’ behavior for $\theta_1(0)=0.1\pi, \theta_2(0)=1.001\pi$. Each two of the subsequent echoes have opposite phases, so they are canceling each other in a dynamical equilibrium, resulting in $0\pi$ total pulse area at any point of the medium. Figure \[fig:echo\_areas\_1001\] also shows that the primary echo assists the formation of the $2\pi$ total area, which would otherwise happen much further into the medium.
We note that the echo areas in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\] behave very similar, differing only in their spatial delays. This is the case, when we can neglect the stimulated echo terms in Eqs. and find a highly accurate approximate analytic solution for each pulse area. For example, we write for the secondary echo area ($z>z_1$): $$\tan \frac{\theta_{e2}}{2} =\Gamma_{\tau} \sin\theta_2(z_1) \sin^2 {\frac{\theta_{e1}(z)}{2}}\sinh\tfrac{\alpha}{2} (z-z_1),
\label{eq:approx_sol}$$ where $\theta_{e1}$ is given in Eq. with the initial pulse areas taken at the transition point $z_1$: $(\theta_1(0),\theta_2(0)) \rightarrow (\theta_2(z_1),\theta_{e1}(z_1))$. By doing so we assume that at $z=z_1$ the first pulse was successfully absorbed by the media and neglect polarization and inversion components acquired at $z<z_1$. solution for $\theta_{e2}$ is shown with dashed lines in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\].
Equations and describe the pulse area at the output of the optically dense media. Moreover, given $\delta t_1 > \delta t_2$ they can also accurately describe the peak energy of the echo pulse [@1999-Wang-PRA; @2019-OptExpress]. This easy to measure quantity can be used for coherent multi-pulse spectroscopy of the optically dense media, where usual spectroscopy is complicated due to strong nonlinear light-atoms interaction. In this highly nonlinear regime the conventional Beer law $I_{echo} = I_0 \Gamma_\tau^2$ is not valid while Eq. can be used to measure $\Gamma_\tau$ dependence.
It also is interesting to discuss the experimental detection of photon echo train generation and what it can lead to. As it is seen in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\], one can experimentally observe only $2$ or $3$ light pulses at the output of the optical density medium, while other pulses will be highly suppressed. Herein in media with higher optical densities, we will see only higher order echo pulses, characterized experimentally by later arrival times. The photon echo experiments in such media are quite typical for many quantum memory protocols. In particular, interesting opportunity is to try detecting the spatial evolution of photon echo inside such media, for example in the rare-earth ions doped crystals [@Tittel2009; @CHANELIERE201877; @Hua_2018].
One possible candidate for high optical density and large Rabi frequency is [$^4\!I_{9/2}-\,^4F_{3/2}\,$]{}transition of [Nd$^{3+}$:YVO$_4$]{}$\:$ at $897.705\: \text{nm}$ with dipole moment $d = 9.16\times 10^{-32}\: \text{C.m}$. Considering $P=100\: mW$ and beam radius of $r=1\mu$m one could reach up to $\Omega \sim 250\: \text{MHz}$. The $\pi$-pulses can be as brief as several nanoseconds which is much shorter than $T_2$. These pulses are spatially squeezed in the medium up to 4 orders of magnitude by the group velocity reduction in the presence of a spectral hole in the optical transition [@Sabooni2013b], this would allow to observe spatial evolution of the solitons and echo pulses inside the medium.
It is worth noting that only soliton-like pulses can propagate through the medium without changing their temporal form and transferring atoms to their initial state. Accordingly, the photon echo pulses in the generated train will be stretch in time and ultimately overlap with each other deep in the medium forming a single $2\pi$-soliton in case of Fig. \[fig:echo\_areas\_999\]. Similarly the stretching echo pulses will asymptotically form a $0\pi$-breather, for the case of Fig. \[fig:echo\_areas\_1001\]. In the core of these transformations lies conservation laws of Maxwell-Bloch equations [@Lamb1972].
Finally, we summarize and conclude the long-lasting derivation of the two-pulse photon echo area theorem started over 45 years ago in [@HAHN1971265; @FRIEDBERG1971285; @Lamb1971], providing an analytic solution for the pulse area of any desired photon echo signal. We showcase the power of the pulse area approach by exploring the rich physics behind the two-pulse echo excitation of an optically dense medium in two previously understudied cases: $\theta_1(0) < \pi, \theta_2(0) \lesssim(\gtrsim) \, \pi$. For the first time we demonstrate that in both these cases a self-reviving echo train is excited deep in the medium with total pulse area $2\pi$ in the first case and $0\pi$ in the second previously unknown case. Thus a slight change in the second pulse area can lead to the dramatic change in the nonlinear multi-pulse media response: an optical soliton in one case or a soliton followed by a breather in the other case. At the same time the complex spatial dynamic of the total nonlinear media response after the two-pulse excitation is precisely aligned with the general McCall-Hahn area theorem prediction.
The developed approach of photon echo pulse area theorem can provide new insights in general analysis of coherent multi-pulse interactions with various photon echo experiments. Although the two-pulse photon echo itself cannot be used for quantum storage [@PhysRevA.79.053851], the developed pulse area approach provides intensity independent universal tool for deeper studies of quantum memory (especially for intensive light pulses and cavity assisted storage), coherent spectroscopy and generation of nonlinear waves in optically dense media. It could also be used in both optical and microwave wavelength regions, for two- and three-level atomic ensembles with arbitrary transverse and longitudinal relaxation times, etc. Next important analytic step could be to generalize and extend the results acquired here for multi-pulse excitation using inverse scattering transform, as was done in [@Kaup1977] for McCall-Hahn area theorem.
The reported study was funded by Russian Foundation for Basic Research, research project no.17-52-560009.
Supplemental material {#supplemental-material .unnumbered}
=====================
Arbitrary echo pulse area
-------------------------
Here we derive the general equation for an arbitrary individual echo pulse area. To do so we integrate the first of Eqs. (1) over time around the time of echo emission $t_e$, from $t_0 = t_e-\tau/2$ to $t_1=t_e+\tau/2$, where $\tau$ is the delay between the pulses. By assuming that $\tau \gg \delta t$, $\delta t$ being the pulse duration, we arrive to the equation for pulse area where we substitute the formal solution for $P$ from Eqs. (1): $$\begin{gathered}
\partial_z \theta = i\frac{\mu}{2}\langle \int_{t_0}^{t_1} dt \Big[ P_0(\Delta) e^{-\gamma t_e-i \Delta (t-t_e)}
\\
-i \int_{t_0}^{t} dt' \Omega (t') w(t',\Delta) e^{-(i \Delta + \gamma) (t-t')} \Big] \rangle,
\label{eq:area_1}\end{gathered}$$ where we introduced $P_0(\Delta)e^{-\gamma t_0} = P(t_0,\Delta)e^{-i \Delta \tau/2}$.
The key to finding the correct solution is proper handling of the integrals over $t$ in these two terms. One can show that $P_0(\Delta)$ and $w(t,\Delta)$ can be presented as a sum of several components $P_0 = P_0^{(0)}+P_0^{(1)}+...$ and $w = w^{(0)}+w^{(1)}+w^{(2)}+...$ with the total number of the components depending on the echo signal of interest (see Eqs. and and the following discussion). These components have a from $P_0^{(n)} \sim \exp[-i\Delta (t-t_e) -i n\Delta\tau + \varphi_n]$, $w^{(n)} \sim \cos[n\Delta\tau+\varphi_n], \text{ where } n\in \mathbb{Z},$ the phase $\varphi_n$ is either $0$ or $\pi/2$.
For $n \neq 0$, $P_0^{(n)}$ and $w^{(n)}$ are rapidly oscillating functions of $\Delta$ near the echo pulse emission time $t_e$ since $\tau\gg \delta t$. Averaging over $\Delta$ leads to that only the slowly varying terms $P^{(0)}_0$ and $w^{(0)}$ contribute to the echo pulse area in Eq. . After using $P_0(\Delta) = P_0^{(0)}$, we simply integrate the first term by taking into account: $\int_{t_0}^{t_1} dt e^{-i \Delta (t-t_e)} \rightarrow 2\pi \delta(\Delta)$ (this limit is valid assuming no temporal overlapping between the light pulses). In the second term we switch the order of temporal integrals, similar to [@allen1975optical; @Eberly:98], and arrive to the integral: $$\begin{aligned}
\begin{split}
& \langle \int_{t_0}^{t_1} dt' \Omega (t') w(t',\Delta) \int_{t'}^{t_1} dt e^{-(i \Delta + \gamma) (t-t')} \rangle =
\\
& \langle \int_{t_0}^{t_1} dt \Omega (t)\tfrac{ w^{(0)}(t,\Delta)}{\gamma+i \Delta } \rangle = \pi G(0)\int_{t_0}^{t_1} dt\Omega (t)w^{(0)}(t,0),
\end{split}\end{aligned}$$ where we have also taken into account that $w^{(0)}(t',\Delta)$ and $G(\Delta)$ are even functions of $\Delta$. Thus Eq. comes to: $$\partial_z \theta = \frac{\alpha}{2} \left[ 2 \tilde{v}_0
+\int_{t_0}^{t_1} dt \Omega(t) \tilde{w}(t) \right],
\label{eq:area_3}$$ where $\alpha = \mu\pi G(0)$ is the resonant absorption coefficient, $ \tilde{v}_0=iP_0^{(0)}(0) e^{-\frac{1}{2}\gamma\tau}$ is the resonant component of the phased coherence, $\tilde{w}(t) = w^{(0)}(t,0)$ is the resonant component of the atomic inversion. To find $\tilde{w}(t)$ and to integrate Eq. , we write the Bloch equation set for the case $\Delta = 0$, ignoring relaxation during the pulses, since $\gamma \delta t\ll 1,$: $$\begin{aligned}
\begin{split}
& \tilde{v}(t) = \tilde{v}_0 \cos \theta(t) + \tilde{w}_0 \sin \theta (t),
\\
& \tilde{w}(t) = \tilde{w}_0 \cos \theta(t) - \tilde{v}_0 \sin \theta (t),
\end{split}
\label{eq:delta_sol}\end{aligned}$$ where $\theta(t) = \int_{t_0}^t \Omega(t) dt$, and $\tilde{v}(t)$ is a resonant part of the phased coherence, $\tilde{w}_0 = \tilde{w}(t_0)$.
Equation can now be integrated, and after reassigning $\tilde{v}_0 \rightarrow v_0,~\tilde{w}_0 \rightarrow w_0 $we obtain Eq. (5): $$\partial_z \theta (z)= \frac{1}{2}\alpha [ 2 v_0(z) \cos^2\frac{\theta(z)}{2} + w_0(z)\sin\theta(z)],
\label{eq:area_general}$$
Phasing components of polarization and inversion
------------------------------------------------
Here we show in detail the calculation of $v_0$ and $w_0$ for the secondary echo and give the expressions for the third and the fourth echoes. We assume that the medium is excited by two incoming pulses having pulse areas $\theta_1, \theta_2$, that give rise to multiple photon echoes having pulse areas $\theta_{ei}$.
Under a multi-pulse excitation a two level system engages in two processes: it is either interacting with the electric field of the applied pulse, or it is left to its own devices and experiences free oscillations decaying as $e^{-\gamma t}$. In the assumed timescales of these processes the influence of the pulse with area $\theta$ can be written as a rotation of the Bloch vector around $u$-axis: $$T(\theta)\vec{r}=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos \theta & \sin\theta \\
0 & -\sin\theta & \cos\theta
\end{pmatrix}
\begin{pmatrix}
u \\
v \\
w
\end{pmatrix}.$$ And free nutation is described with another rotation matrix, this time around $w$-axis: $$U(t)\vec{r}=
\begin{pmatrix}
e^{-\gamma t} \cos \Delta t & - e^{-\gamma t} \sin \Delta t & 0 \\
e^{-\gamma t} \sin \Delta t & e^{-\gamma t} \cos \Delta t & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
u \\
v \\
w
\end{pmatrix},$$
The secondary echo is emitted at the time $t=3\tau$, and to find the phasing parts of the coherence and inversion we write the Bloch vector $\vec{r}(t)=U(t-2\tau)T(\theta_{e1})U(\tau)T(\theta_2)U(\tau)T(\theta_1) \vec{r}(0), t_0=5\tau/2$. The calculation gives: $$\begin{aligned}
\begin{split}
v(t) = & - \Gamma_\tau c_1 c_2 s_{e1} c_t
\\
& - \Gamma_\tau^2 [c_1 s_2 c_{e1} c_\tau c_t + c_1 s_2 s_\tau s_t - s_1 s_2 s_{e1} c_t c_\tau]
\\
+ \Gamma^3_\tau s_1 [ & c_\tau s_\tau s_t + c_2 c_\tau s_\tau s_t - c_2 c_{e1} c_\tau^2 c_t + c_{e1} s_\tau^2 s_t],
\end{split}
\label{eq:App_v_All}
\\
\begin{split}
w(t) = & - c_1 c_2 c_{e1} + \Gamma_\tau [s_1 s_2 c_{e1} c_\tau+c_1 s_2 s_{e1} c_\tau ]
\\
& -\Gamma_\tau^2 [s_1 c_2 s_{e1} c^2_\tau - s_1 s_{e1} s_\tau^2]
\end{split}
\label{eq:App_w_All}
\end{aligned}$$ here we use a short notation for trigonometric functions: $s_i = \sin \theta_i, c_i = \cos \theta_i, i=1,2,e1,$ $s_\tau = \sin \Delta\tau,$ $c_\tau = \cos \Delta\tau, s_t = \sin \Delta(t-\tau), c_t = \cos \Delta(t-\tau)$.
This includes the phasing components, responsible for the echo generation and that are proportional to $ \cos [\Delta (t-2\tau)]$ and non phasing components. For example the first term of $v(t)$ contains only $c_t = \cos \Delta (t-\tau)$ and is non phasing, while the second term contains $c_\tau c_t = \cos \Delta\tau \cos \Delta(t-\tau) = \frac{1}{2}[\cos\Delta(t-2\tau)+\cos\Delta t] = \frac{1}{2}\cos\Delta(t-t_e)+\frac{1}{2}\cos(\Delta (t-t_e)+2\Delta\tau)$, so we get a phasing term $-\frac{1}{2}\Gamma_\tau^2 c_1 s_2 c_{e1} \cos\Delta(t-2\tau)$ that contributes to $P^0$. For $w(t)$ it is similar, except we are now interested in the time independent terms, like the first term in Eq. . The terms with $c_\tau^2$ or $s_\tau^2$ also contribute since $c_\tau^2 (s_\tau^2) = \frac{1}{2}(1 \pm \cos 2\Delta\tau),$ where the second term will vanish after averaging over $\Delta$.
We now leave only the terms that contribute to the echo: $$\begin{aligned}
v(t) = & \frac{1}{2}\Gamma_\tau^2 [-(c_{e1} +1)c_1 s_2 + s_1 s_2 s_{e1}] \cos \Delta (t-2\tau),
\\
w(t) = & - c_1 c_2 c_{e1} + \frac{1}{2}\Gamma_\tau^2 [1-c_2 ] s_1 s_{e1},
\end{aligned}$$ and we get for $\tilde{v}_0(3/2\tau)$ and $\tilde{w}_0(3/2\tau)$: $$\begin{aligned}
\begin{split}
v_0(3/2\tau,z) = & -\Gamma^2_\tau \cos\theta_1 \sin\theta_2 \cos^2\tfrac{\theta_{e1}}{2}
\\
& + \tfrac{1}{2} \Gamma^2_\tau \sin\theta_1 \sin\theta_2 \sin\theta_{e1},
\end{split}
\label{eq:App_Sec_V}
\\
\begin{split}
w_0(3/2\tau,z) = & -\cos\theta_1 \cos\theta_2 \cos\theta_{eq}
\\
& + \Gamma^2_\tau \sin\theta_1 \cos^2\tfrac{\theta_2}{2} \sin\theta_{e1}.
\end{split}
\label{eq:App_Sec_W}
\end{aligned}$$ The first terms in Eqs. and are very similar to those of primary echo pulse and correspond to the two-pulse echo generation by the $\theta_2(z), \theta_{e1}(z)$. This contribution to the secondary echo is presented as color yellow in Fig. \[fig:App\_Pulse\_Sequences\]. The second terms in Eqs. , correspond to the stimulated echo generation and are presented by the color blue in Fig. \[fig:App\_Pulse\_Sequences\].
In the same fashion we can write the phasing coherence and inversion after four pulses, two incoming pulses and two echo pulses: $$\begin{aligned}
\begin{split}
v_0 (7\tau/2,z) = & \tfrac{1}{2} \Gamma_\tau^2 \times
\\
& \big[\sin\theta_1\sin\theta_2\cos\theta_{e1}\sin\theta_{e2} \\
& +\cos\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2} \\
& +2\cos\theta_1\cos\theta_2\sin\theta_{e1}\sin^2\tfrac{\theta_{e2}}{2}\big]
\\
& + \Gamma_\tau^4 \big[\sin\theta_1\cos^2\tfrac{\theta_2}{2}\sin^2\tfrac{\theta_{e1}}{2}\cos^2\tfrac{\theta_{e2}}{2} \\
& -\sin\theta_1 \sin^2\tfrac{\theta_2}{2}\cos^2\tfrac{\theta_{e1}}{2}\sin^2\tfrac{\theta_{e2}}{2} \big],
\end{split}
\label{eq:App_3rd_Echo_V}
\\
\begin{split}
w_0(7\tau/2,z) = & -\cos\theta_1\cos\theta_2\cos\theta_{e1}\cos\theta_{e2} \\
& - \Gamma_\tau^2 \times [ \sin\theta_1\sin^2\frac{\theta_2}{2}\sin\theta_{e1}\cos\theta_{e2} \\
& + \cos\theta_1\sin\theta_2\sin^2\frac{\theta_{e1}}{2}\sin\theta_{e2} \\
& + \frac{1}{2}\sin\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2} ] = \\
& -v_4 \sin\theta_{e2} + w_4 \cos\theta_{e2}.
\end{split}
\label{eq:App_3rd_Echo_W}
\end{aligned}$$
The first term $\sim \sin\theta_1\sin\theta_2\cos\theta_{e1}\sin\theta_{e2}$ in $ v_0 (7\tau/2,z)$ is the stimulated echo from the two incoming and the second echo pulses naturally proportional to $\Gamma_\tau^2$. The second term $\sim \cos\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2}$ is another stimulated echo generated by the second incoming and the first two echo pulses. The third term $\sim\cos\theta_1\cos\theta_2\sin\theta_{e1}\sin^2\tfrac{\theta_{e2}}{2}$ represents the contribution of the two-pulse echo from the two echo pulses. The next term $\sim \sin\theta_1\cos^2\tfrac{\theta_2}{2}\sin^2\tfrac{\theta_{e1}}{2}\cos^2\tfrac{\theta_{e2}}{2}$ is the two-pulse echo generated by the first incoming and primary echo pulses. The last term, $\sim \sin\theta_1 \sin^2\tfrac{\theta_2}{2}\cos^2\tfrac{\theta_{e1}}{2}\sin^2\tfrac{\theta_{e2}}{2}$ is the revived primary echo, generated by the first two incoming pulses and recovered by the second echo pulse (first echo pulse just suppresses its amplitude by the factor $\cos^2\tfrac{\theta_{e1}}{2}$).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $M$ be a smooth manifold, and let $\mathcal{O}(M)$ be the poset of open subsets of $M$. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) $F \colon {\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ from ${\mathcal{O}(M)}$ to the category of spaces. Weiss showed that polynomial cofunctors of degree $\leq k$ are determined by their values on ${\mathcal{O}_k(M)}$, where ${\mathcal{O}_k(M)}$ is the full subposet of ${\mathcal{O}(M)}$ whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Afterwards Pryor showed that one can replace ${\mathcal{O}_k(M)}$ by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from ${\mathcal{O}(M)}$ to any simplicial model category ${\mathcal{M}}$. If $F_k(M)$ stands for the unordered configuration space of $k$ points in $M$, we also show that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$ provided that ${\mathcal{M}}$ has a zero object. Using a new approach, we also show that if ${\mathcal{M}}$ is a general model category and $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor, then the homotopy right Kan extension of $F$ along the inclusion ${\mathcal{O}_k(M)}{\hookrightarrow}{\mathcal{O}(M)}$ is also an isotopy cofunctor.'
author:
- |
Paul Arnaud Songhafouo Tsopméné\
Donald Stanley
title: ' **Polynomial functors in manifold calculus**'
---
Introduction
============
Let $M$ be a smooth manifold, and let $\mathcal{O}(M)$ be the poset of open subsets of $M$. *Manifold calculus* is a calculus of functors suitable for studying cofunctors [^1] $F : {\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ from ${\mathcal{O}(M)}$ to the category of spaces (of which the embedding functor $\text{Emb}(-, W)$ for a fixed manifold $W$ is a prime example). So manifold calculus belongs to the world of calculus of functors, and therefore it definitely has a notion of polynomial cofunctor. Roughly speaking, a *polynomial cofunctor* is a contravariant functor ${\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ that satisfies an appropriate higher-order excision property, similar to the case of [@good03] (see Definition \[poly\_defn\]). In [@wei99 Theorems 4.1, 5.1] Weiss characterizes polynomial cofunctors. More precisely, he shows that polynomial cofunctors of degree $\leq k$ are determined (up to equivalence of course) by their values on ${\mathcal{O}_k(M)}$, the full subposet of ${\mathcal{O}(M)}$ whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Many examples of polynomial and homogeneous cofunctors are also provided in [@wei99]. Another good reference where the reader can find an introduction to manifold calculus is [@mun10].
Weiss’ characterization of polynomial cofunctors was generalized by Pryor in [@pryor15] as follows. Let ${\mathcal{B}}$ be a basis for the topology of $M$. We assume that ${\mathcal{B}}$ is *good*, that is, every element of ${\mathcal{B}}$ is a subset of $M$ diffeomorphic to an open ball. For instance, if $M = {\mathbb{R}}^m$, we can take ${\mathcal{B}}$ to be the collection of genuine open balls (with respect to the euclidean metric), or cubes, or simplices, or convex $d$-bodies more generally. For $k \geq 0$, we let ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}_k(M)}$ denote the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$. So one possible choice of ${\mathcal{B}_k(M)}$ is ${\mathcal{O}_k(M)}$ itself. In [@pryor15 Theorem 6.12] Pryor shows, in the same spirit as Weiss, that any polynomial cofunctor ${\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ of degree $\leq k$ is determined by its restriction to ${\mathcal{B}_k(M)}$. So one can replace ${\mathcal{O}_k(M)}$ by ${\mathcal{B}_k(M)}$ without losing any homotopy theoretic information when forming the polynomial approximation to a cofunctor.
In this paper we generalize the aforementioned results of Weiss-Pryor to cofunctors from ${\mathcal{O}(M)}$ to any simplicial model category ${\mathcal{M}}$. Specifically we have the following theorem, which is our first result.
\[main\_thm\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$, and let ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}(M)}$ be the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$. Consider a simplicial model category ${\mathcal{M}}$ and a cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$. Then $F$ is good (see Definition \[good\_defn\]) and polynomial of degree $\leq k$ (see Definition \[poly\_defn\]) if and only if the restriction $F|{{\mathcal{B}}_k}(M)$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]) and the canonical map $F {\longrightarrow}(F|{{\mathcal{B}}_k}(M))^!$ is a weak equivalence. Here $(F|{{\mathcal{B}}_k}(M))^! \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is the cofunctor defined as $$(F|{{\mathcal{B}}_k}(M))^!(U):= \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V).$$
Notice that Theorem \[main\_thm\] implies that the category of good polynomial cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $\leq k$ is *weakly equivalent*, in the sense of Definition \[we\_defn\], to the category of isotopy cofunctors ${{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{M}}$. Also notice that our definition of *good cofunctor* is slightly different from the classical one (see [@wei99 Page 71] or [@mun10 Definition 1.3.4] or [@pryor15 Definition 3.1]) as we add an extra axiom: our cofunctors are required to be also objectwise fibrant. We need that extra axiom to be able to use the homotopy invariance theorem (see Theorem \[fib\_cofib\_thm\]) and the cofinality result (see Theorem \[htpy\_cofinal\_thm\]). If one works with a category ${\mathcal{M}}$ in which every object is fibrant, the extra axiom becomes a tautology. This is the case in Weiss’ paper [@wei99] where ${\mathcal{M}}=$ Spaces. For the main ingredients in the proof of Theorem \[main\_thm\], see Outline of the paper below.
As mentioned earlier, our result generalizes those of Weiss. In fact, from Theorem \[main\_thm\] with ${\mathcal{M}}=\text{Spaces}$ and ${\mathcal{B}}= {\mathcal{O}}$, the maximal good basis, one can easily deduce the main results of [@wei99], which are Theorems 4.1, 5.1 and 6.1.
The following conjecture says that Theorem \[main\_thm\] still holds when ${\mathcal{M}}$ is replaced by a general model category. We believe in that conjecture, which could be handled by using the same approach as that we use to show Theorem \[main\_thm\]. The issue with that approach is the fact that some important results/properties regarding homotopy limits in a general model category (for example Theorem \[holim\_tot\_thm\] and Proposition \[comm\_prop\]) are available nowhere in the literature. So the proof of the conjecture may turn into a matter of homotopy limits. A good reference, where the reader can find the definition and several useful properties of homotopy limits (in a general model category of course), is [@hir03 Chapter 19]. Another good reference is [@dhks04].
\[main\_conj\] Theorem \[main\_thm\] remains true if one replaces ${\mathcal{M}}$ by a general model category.
Now we state our second result. Given a good cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ one can define its $k$th polynomial approximation, denoted $T_kF \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$, as the homotopy right Kan extension of the restriction $F|{\mathcal{O}_k(M)}$ along the inclusion ${\mathcal{O}_k(M)}{\hookrightarrow}{\mathcal{O}(M)}$. In order words $T_kF(U) := \underset{V \in {\mathcal{O}_k(M)}}{\text{holim}} \; F(V)$. The difference between $T_kF$ and $T_{k-1}F$ belongs to a nice class of cofunctors called *homogeneous cofunctors* of degree $k$ (see Definition \[hc\_defn\]). When $k =1$ we talk about *linear cofunctors*. Thanks to the fact that Theorem \[main\_thm\] holds for any good basis ${\mathcal{B}}$ we choose, one can prove the following result, which roughly states that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. Here $F_k(M)$ stands for the unordered configuration space of $k$ points in $M$.
\[main2\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Assume that ${\mathcal{M}}$ has a zero object (that is, an object which is both terminal an initial).
1. Then the category ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ (see Definition \[hc\_defn\]) is weakly equivalent (in the sense of Definition \[we\_defn\]) to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. That is, $${\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_1 ({\mathcal{O}}(F_k(M)); {\mathcal{M}}).$$
2. For $A \in {\mathcal{M}}$ we have the weak equivalence $${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_{1A} ({\mathcal{O}}(F_k(M)); {\mathcal{M}}),$$ where ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}})$ stands for the category of homogeneous cofunctors $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ such that $F(U) \simeq A$ for any $U$ diffeomorphic to the disjoint union of exactly $k$ open balls.
The second part of this result will be used in [@paul_don17-3]. A similar result (but with a different approach) to Theorem \[main2\_thm\] was obtained by the authors in [@paul_don17 Corollary 3.31] for very good homogeneous functors. Note that neither [@paul_don17 Corollary 3.31] nor Theorem \[main2\_thm\] was known before, even for ${\mathcal{M}}= \text{Spaces}$. Theorem \[main2\_thm\] is interesting in the sense that it reduces the study of homogeneous cofunctors of degree $k$ to the study of linear cofunctors, which are easier to handle. In [@paul_don17-3] we use it (Theorem \[main2\_thm\]) as the starting point in the classification of homogeneous cofunctors of degree $k$.
Our third result is a partial answer to Conjecture \[main\_conj\].
\[iso\_cof\_thm\] Let ${\mathcal{M}}$ be a model category. Let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor $F^{!} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $$\begin{aligned}
\label{fsrik_defn}
F^{!}(U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V)\end{aligned}$$ is an isotopy cofunctor as well.
The method we use to prove Theorem \[iso\_cof\_thm\] is completely different from that we use to prove Theorem \[main\_thm\] essentially because of the following. First note that in Theorem \[main\_thm\] ${\mathcal{M}}$ is a simplicial model category, while in Theorem \[iso\_cof\_thm\] ${\mathcal{M}}$ is a general model category. To prove Theorem \[main\_thm\], we use several results/properties of homotopy limits in simplicial model categories such as the Fubini theorem (see Theorem \[fubini\_thm\]), and Proposition \[comm\_prop\]. However, Proposition \[comm\_prop\] involves the notion of totalization of a cosimplicial object, which a priori does not make sense in a general model category.
The key concept we introduce to prove Theorem \[iso\_cof\_thm\] is called *admissible family* of open subsets. Roughly speaking, a sequence $B = B_0, \cdots, B_n$ of open balls is said to be *admissible* if $B_i \cap B_{i+1} \neq \emptyset$ for all $i$. One can extend that definition to sequences $V=V_0, \cdots, V_n$ of objects of ${\mathcal{O}_k(M)}$ (see Definition \[pw\_adm\_defn\]). Such sequences yield zigzags of isotopy equivalences of ${\mathcal{O}_k(M)}$ between $V$ and $V_n$, and the collection of those form a category denoted ${{\mathcal{D}}(V)}$ (see Definition \[dv\_defn\]). This latter category plays a crucial role in Section \[iso\_cof\_section\]. Indeed, one can deduce Theorem \[iso\_cof\_thm\] by applying the homotopy limit functor to appropriate diagrams in ${\mathcal{M}}$ indexed by ${{\mathcal{D}}(V)}$.
**Outline of the paper**
This paper is subdivided into two detailed and almost disconnected parts. The first one covers Sections \[notation\_section\], \[holim\_simpl\_section\], \[sos\_good\_section\], \[poly\_section\] and \[hc\_section\] where we prove Theorems \[main\_thm\] and \[main2\_thm\], while the second covers Section \[iso\_cof\_section\] where we prove Theorem \[iso\_cof\_thm\].
1. In Section \[notation\_section\] we fix some notation. We also give a table that plays the role of a dictionary between our notation and that of Weiss-Pryor. The purpose of that table is to help the exposition of certain proofs, especially in Subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\].
2. Section \[holim\_simpl\_section\] deals with homotopy limits in simplicial model categories. We follow Hirschhorn’s style [@hir03 Chapters 18-19]. Since the homotopy limit is so ubiquitous in this work, we first give its definition in Subsection \[holim\_subsection\]. Next, in the same subsection, we recall some of its basic properties including the homotopy invariance (see Theorem \[fib\_cofib\_thm\]), the cofinality theorem (see Theorem \[htpy\_cofinal\_thm\]), and the Fubini theorem (see Theorem \[fubini\_thm\]). All these properties are indeed used in many places in this work. Subsection \[cr\_subsection\] deals with cosimplicial replacement of a diagram. We prove Proposition \[comm\_prop\], which is the main new result of the section. It says that the canonical isomorphism $\underset{{\mathcal{D}}}{\text{holim}} \; F \cong {\text{Tot}}\; {\Pi^{\bullet}}F$ between the homotopy limit of a diagram $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ and the totalization of its cosimplicial replacement is natural in the following sense. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is a functor between small categories, then the obvious square involving the isomorphisms $\underset{{\mathcal{D}}}{\text{holim}} \; F \cong {\text{Tot}}\; {\Pi^{\bullet}}F$ and $\underset{{\mathcal{C}}}{\text{holim}} \; F \theta \cong {\text{Tot}}\; {\Pi^{\bullet}}(F \theta)$ must commutes. Proposition \[comm\_prop\] will be used in the proof of Theorem \[sos\_thm\].
3. Section \[sos\_good\_section\] proves two important results: Theorem \[sos\_thm\] and Theorem \[good\_thm\]. The first, which is the crucial ingredient in the proof of Theorem \[main\_thm\], roughly says that the homotopy limit ${F^{!}_{{\mathcal{B}}}}(U) := \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ does not depend on the choice of the basis ${\mathcal{B}}$. Specifically, it says that for any good basis ${\mathcal{B}}$ for the topology of $M$, for any isotopy cofunctor $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$, the canonical map $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ is a weak equivalence for all $U \in {\mathcal{O}(M)}$. The proof of Theorem \[sos\_thm\] goes through two big steps. The first step (see Subsection \[cof\_fsp\_subsection\]) consists of splitting ${F^{!}_{{\mathcal{B}}}}$ into smaller pieces ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}= \{{\tilde{F}^{!p}_{{\mathcal{B}}}}\}_{p \geq 0}$, and show that ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ is independent of the choice of the basis ${\mathcal{B}}$ for all $p$ (see Proposition \[sosp\_prop\]). This idea of splitting comes from the paper of Weiss [@wei99], and the nice thing is that the collection ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}$ turns out to be a cosimplicial object in the category of cofunctors from ${\mathcal{O}(M)}$ to ${\mathcal{M}}$. The second step, inspired by Pryor’s work [@pryor15], is to connect $\underset{[p] \in \Delta}{\text{holim}} \; {\tilde{F}^{!p}_{{\mathcal{B}}}}(U)$ and ${F^{!}_{{\mathcal{B}}}}(U)$ by a zigzag of natural weak equivalences (see Subsection \[sos\_good\_subsection\]). It is very important that every map of that zigzag is natural in both variables $U$ and ${\mathcal{B}}$. This is one of the reasons we really need Section \[holim\_simpl\_section\] where all those maps are carefully inspected, especially the map that appears in Theorem \[holim\_tot\_thm\]. Regarding Theorem \[good\_thm\], it says that ${F^{!}_{{\mathcal{B}}}}$ is good provided that $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor. This result is a part of the proof of Theorem \[main\_thm\], and its proof is based on Theorem \[sos\_thm\] and the Grothendieck construction (see Subsection \[gro\_const\_subsection\]).
4. In Section \[poly\_section\] we prove the main result of the first part: Theorem \[main\_thm\]. To do this we use Theorem \[sos\_thm\] and Theorem \[good\_thm\] as mentioned earlier. We also use Lemmas \[cofinal\_lem\], \[poly\_lem\], \[charac\_lem\]. The first lemma says that a certain functor is right cofinal. The second (which is a generalization of [@wei99 Theorem 4.1]) states that if $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor, then ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. The proof of this result also uses the Grothendieck construction. The third lemma (which is a generalization of [@wei99 Theorem 5.1]) is a characterization of polynomial cofunctors. Note that Lemma \[poly\_lem\] and Lemma \[charac\_lem\] are important themselves.
5. Section \[hc\_section\] deals with homogeneous cofunctors, and is devoted to the proof of Theorem \[main2\_thm\]. The key ingredient we need is Lemma \[homo\_lem\], which roughly says that homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ are determined by their values on subsets diffeomorphic to the disjoint union of exactly $k$ open balls provided that ${\mathcal{M}}$ has a zero object. So Lemma \[homo\_lem\] is also a useful result in its own right since it characterizes homogeneous cofunctors. Note that the proof of Lemma \[homo\_lem\] is based on the results we obtained in Section \[sos\_good\_section\] and Section \[poly\_section\].
6. Section \[iso\_cof\_section\] proves Theorem \[iso\_cof\_thm\]. To do this we use a completely different method (but rather lengthy) from that we used in previous sections. As mentioned earlier, the key concept here is that of *admissible family* (see Definition \[pw\_adm\_defn\]) introduced in [@paul_don17]. In Subsection \[holim\_subsectiong\] we recall some useful properties for homotopy limits in general model categories. Subsections \[dv\_subsection\], \[hp\_subsection\] are preparatory subsections dealing with technical tools needed for the proof of Theorem \[iso\_cof\_thm\]. Lastly, Subsection \[iso\_cof\_subsection\] proves Theorem \[iso\_cof\_thm\].
**Acknowledgements.** This work has been supported by Pacific Institute for the Mathematical Sciences (PIMS) and the University of Regina, that the authors acknowledge. We are also grateful to P. Hirschhorn, J. Scherer, and W. Chacholski for helpful conversations (by emails) about homotopy limits and homotopy colimits.
Notation {#notation_section}
========
In this section we fix some notation.
1. We let $M$ denote a smooth manifold. If $U$ is a subset of $M$, we let ${\mathcal{O}}(U)$ denote the poset of open subsets of $U$, morphisms being inclusions of course. In particular one has the poset ${\mathcal{O}(M)}$.
2. For $k \geq 0$, and $U \in {\mathcal{O}(M)}$, we let ${\mathcal{O}_k}(U) \subseteq {\mathcal{O}}(U)$ denote the full subposet whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. In particular one has the poset ${\mathcal{O}_k(M)}$.
3. We write ${\mathcal{O}}$ for the collection of all subsets of $M$ diffeomorphic to an open ball. Certainly ${\mathcal{O}}$ is a full subposet of ${\mathcal{O}(M)}$.
4. We let ${\mathcal{B}}$ denote a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Clearly, one has ${\mathcal{B}}\subseteq {\mathcal{O}}$ for any good basis ${\mathcal{B}}$.
5. We write ${\mathcal{M}}$ for a simplicial model category unless stated otherwise.
6. If $\beta \colon F {\longrightarrow}G$ is a natural transformation, the component of $\beta$ at $x$ will be denoted $\beta[x] \colon F(x) {\longrightarrow}G(x)$.
7. We use the notation $x := \text{def}$ to state that the left hand side is defined by the right hand side.
Since the proofs of some important results in this paper are based on [@pryor15] and [@wei99], we need a dictionary of notations which is provided by the following table. The purpose of that table is then to help the exposition of certain proofs, especially in Subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\] as we said before. The first column gives the notation that we use in this paper, while the second and the third regard the notation used in [@pryor15] and [@wei99] respectively. The notations that appear in the same row have the same meaning. The word nothing means that there is no notation with the same meaning in the corresponding paper. For instance, in the first row we have the notation ${\mathcal{O}}$ in this paper, which stands for the maximal good basis for the topology of $M$. However there is no notation in [@pryor15] and [@wei99] that has the same meaning as ${\mathcal{O}}$.
In this paper In Pryor’s paper [@pryor15] In Weiss’ paper [@wei99]
------------------------------------------------------------------------ --------------------------------------------- --------------------------------------------
${\mathcal{O}}$ nothing nothing
${\mathcal{O}(M)}$ ${\mathcal{O}(M)}$ or just ${\mathcal{O}}$ ${\mathcal{O}(M)}$ or just ${\mathcal{O}}$
${\mathcal{O}_k(M)}$ ${\mathcal{O}_k}$ ${\mathcal{O}}k$
${\mathcal{B}_k(M)}$ (see Definition \[fsb\_defn\]) ${{\mathcal{B}}_k}$ nothing
${\widetilde{{\mathcal{O}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) nothing ${\mathcal{I}}k {\mathcal{O}}k_p (M)$
${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) ${\mathcal{A}}_k ({{\mathcal{B}}_k})_p (M)$ nothing
${\widehat{{\mathcal{B}}}_{k, q}}(M)$ $({\mathcal{A}}_k)_q {{\mathcal{B}}_k}(M)$ nothing
${F^{!}_{{\mathcal{O}}}}$ (see Example \[fso\_expl\]) nothing $E^{!}$
${F^{!}_{{\mathcal{B}}}}$ (see Definition \[fsb\_defn\]) $F^{!}$ nothing
${\tilde{F}_{{\mathcal{B}}}}^p$ (see Definition \[fsbp\_defn\]) $F_p$ nothing
${\tilde{F}_{{\mathcal{B}}}}^{!p}$ (see Definition \[fsbp\_defn\]) $F^{!}_p$ nothing
${\tilde{F}_{{\mathcal{O}}}}^{!p}$ (see Definition \[fsbp\_defn\]) nothing $E^{!}_p$
${\hat{F}_{{\mathcal{B}}}}^q$ (see (\[fhq\])) $\widehat{F}_q$ nothing
${\hat{F}_{{\mathcal{B}}}}^{!q}$ (see (\[fhsq\])) $\widehat{F}^{!}_q$ nothing
For the meaning of ${\widehat{{\mathcal{B}}}_{k, q}}(M)$ we refer the reader to the beginning of Subsection \[sos\_good\_subsection\].
Homotopy limits in simplicial model categories {#holim_simpl_section}
==============================================
In this section we recall some useful definitions and results about homotopy limits in simplicial model categories. We also prove Corollary \[hir\_coro\] and Proposition \[comm\_prop\], which will be used in Section \[sos\_good\_section\]. The main reference here is Hirschhorn’s book [@hir03 Chapter 18].
Let us begin with the following remark and notation.
For the sake of simplicity, all the functors $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ in this section are covariant unless stated otherwise. However, in next sections our functors will be contravariant since manifold calculus deals with contravariant functors. This is not an issue of course since all the statements of this section hold for contravariant functors as well: it suffices to replace everywhere ${\mathcal{C}}$ by its opposite category ${{\mathcal{C}}^{\text{op}}}$.
The following standard notations will be used only in this section.
1. We let $\Delta$ denote the category whose objects are $[n] = \{0, \cdots, n\}, n \geq 0$, and whose morphisms are non-decreasing maps. For $n \geq 0$, we let $\Delta[n]$ denote the simplicial set defined as $(\Delta[n])_p = \underset{\Delta}{\text{hom}} ([p], [n])$.
2. If ${\mathcal{C}}$ is a category, we write $N({\mathcal{C}})$ for the nerve of ${\mathcal{C}}$. If $c \in {\mathcal{C}}$, we let ${\mathcal{C}}\downarrow c$ denote the over category. An object of ${\mathcal{C}}\downarrow c$ consists of a pair $(x, f)$, where $x \in {\mathcal{C}}$ and $f \colon x {\longrightarrow}c$ is a morphism of ${\mathcal{C}}$. A morphism from $(x, f)$ to $(x', f')$ consists of a morphism $g \colon x {\longrightarrow}x'$ of ${\mathcal{C}}$ such that the obvious triangle commutes.
Homotopy limits {#holim_subsection}
---------------
Here we recall the definition of the homotopy limit of a diagram in a simplicial model category. Next we recall some useful results due to P. Hirschhorn [@hir03].
\[holim\_defn\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category. Consider a covariant functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$. The *homotopy limit* of $F$, denoted $\underset{{\mathcal{C}}}{\text{holim}} \; F$, is the object of ${\mathcal{M}}$ defined to be the equalizer of the maps $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar@<1ex>[r]^-{\phi} \ar@<-1ex>[r]_-{\psi} & \underset{(f \colon c {\rightarrow}c') \in {\mathcal{C}}}{\prod} (F(c'))^{N({\mathcal{C}}\downarrow c)}. }$$ [^2] Here $\phi$ and $\psi$ are defined as follows. Let $f \colon c {\longrightarrow}c'$ be a morphism of ${\mathcal{C}}$.
1. The projection of $\phi$ on the factor indexed by $f$ is the following composition where the first map is a projection $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[rr]^-{(F(f))^{N({\mathcal{C}}\downarrow c)}} & & (F(c'))^{N({\mathcal{C}}\downarrow c)}.}$$
2. The projection of $\psi$ on the factor indexed by $f$ is the following composition where the first map is again a projection. $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c'))^{N({\mathcal{C}}\downarrow c')} \ar[rr]^-{(F(c'))^{N({\mathcal{C}}\downarrow f)}} & & (F(c'))^{N({\mathcal{C}}\downarrow c)}.}$$
Let ${\mathcal{M}}$ be a category. Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram in ${\mathcal{M}}$, then the composition $F \circ \theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called the *${\mathcal{C}}$-diagram in ${\mathcal{M}}$ induced by $F$*, and it is denoted $\theta^* F$. That is, $$\begin{aligned}
\label{theta_starx}
\theta^* F := F \circ \theta. \end{aligned}$$
The following proposition will be used in many places in this paper. Especially, we will use it to define morphisms between homotopy limits of diagrams of different shape. Also it will be used to show that certain diagrams commute. That proposition regards the change of the indexing category of a homotopy limit.
\[induced\_holim\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between two small categories. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram, then there is a canonical map $$\begin{aligned}
\label{thetax}
[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F.\end{aligned}$$ Furthermore, this map is natural in both variables $\theta$ and $F$. The naturality in $\theta$ says that if $\beta \colon \theta {\longrightarrow}\theta'$ is a natural transformation, then the following square commutes. $$\begin{aligned}
\label{nat_theta}
\xymatrix{\underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]^-{[\theta; F]} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F \\
\underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]_-{[\theta'; F]} \ar[u]^-{id} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta'^{*}F \ar[u]_-{\text{holim}(F\beta)} }\end{aligned}$$ Here we have assumed $F$ contravariant (in the covariant case, one has to reverse the righthand vertical map). Regarding the naturality in $F$, it says that if $\eta \colon F {\longrightarrow}F'$ is a natural transformation, then the following square commutes. $$\begin{aligned}
\label{theta_square}
\xymatrix{ \underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]^-{[\theta; F]} \ar[d]_-{\text{holim}(\eta)} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F \ar[d]^-{\text{holim}(\theta^*\eta)} \\
\underset{{\mathcal{D}}}{\text{holim}}\; F' \ar[rr]_-{[\theta; F']} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F'.}\end{aligned}$$
The construction of $[\theta; F]$ comes from the following observation, which provides a nice way to define a map between two equalizers. This observation will be also used in Subsection \[cr\_subsection\].
Consider the following diagrams in ${\mathcal{M}}$. $$\xymatrix{A \ar@<1ex>[r]^-{\alpha} \ar@<-1ex>[r]_-{\beta} & B} \qquad \xymatrix{A' \ar@<1ex>[r]^-{\alpha'} \ar@<-1ex>[r]_-{\beta'} & B'}.$$ If $\Psi \colon A {\longrightarrow}A'$ is a map that satisfies the property $$\begin{aligned}
\label{eq_cond}
(\text{for all $g \colon E {\longrightarrow}A$}) \left((\alpha g = \beta g) \Rightarrow (\alpha'\Psi g = \beta' \Psi g)\right),\end{aligned}$$ then we have an induced map $$\widetilde{\Psi} \colon \text{eq}\left(\xymatrix{A \ar@<1ex>[r]^-{\alpha} \ar@<-1ex>[r]_-{\beta} & B}\right) {\longrightarrow}\text{eq}\left(\xymatrix{A' \ar@<1ex>[r]^-{\alpha'} \ar@<-1ex>[r]_-{\beta'} & B'} \right).$$ Now we define $[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F$. By Definition \[holim\_defn\], the homotopy limit of $\theta^* X$ is the equalizer of the maps $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)} \ar@<1ex>[r] \ar@<-1ex>[r] & \underset{(c {\longrightarrow}c') \in {\mathcal{C}}}{\prod} (F(\theta(c')))^{N({\mathcal{C}}\downarrow c)}. }$$ Define $$\Psi \colon \underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)} {\longrightarrow}\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)}$$ as follows. For $c \in {\mathcal{C}}$ the map from $\underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)}$ to the factor indexed by $c$ is defined to be the composition $$\begin{aligned}
\label{fd_fc}
\xymatrix{\underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)} \ar[r] & (F(\theta(c)))^{N({\mathcal{D}}\downarrow \theta(c))} \ar[r] & (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)}},\end{aligned}$$ where the first map is the projection onto the factor indexed by $\theta(c)$, and the second one is induced by the canonical functor ${\mathcal{C}}\downarrow c {\longrightarrow}{\mathcal{D}}\downarrow \theta(c)$. It is straightforward to see that $\Psi$ satisfies condition (\[eq\_cond\]). We thus obtain $[\theta; F] := {\widetilde{\Psi}}$.
It is also straightforward to check that the squares (\[nat\_theta\]) and (\[theta\_square\]) commute.
We end this subsection with the following three important properties of homotopy limits. The first (see Theorem \[fib\_cofib\_thm\]) is known as the homotopy invariance for homotopy limits. The second (see Theorem \[htpy\_cofinal\_thm\]) is the cofinality theorem. And the last (see Theorem \[fubini\_thm\]) is the so-called Fubini Theorem for homotopy limits. Before we state those properties, we need the recall the following three definitions.
\[uc\_defn\] Let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor, and let $d \in {\mathcal{D}}$. The *under category* $d \downarrow \theta$ is defined as follows. An object of $d \downarrow \theta$ is a pair $(c, f)$ where $c$ is an object of ${\mathcal{C}}$ and $f$ is a morphism of ${\mathcal{D}}$ from $d$ to $\theta(c)$ . A morphism from $(c, f)$ to $(c', f')$ consists of a morphism $g \colon c {\longrightarrow}c'$ of ${\mathcal{C}}$ such that the obvious triangle commutes (that is, $\theta(g) f = f'$). In similar fashion, one has the *over category* $\theta \downarrow d$.
[@hir03 Definition 19.6.1] \[cofinal\_defn\] A functor $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is *homotopy right cofinal* (respectively *homotopy left cofinal*) if for every $d \in {\mathcal{D}}$, the under category $d \downarrow \theta$ (respectively the over category $\theta {\downarrow}d$) (see Definition \[uc\_defn\]) is contractible.
\[owf\_defn\] If ${\mathcal{C}}$ is a category and ${\mathcal{M}}$ is a model category, a functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is said to be an *objectwise fibrant* functor if the image of every object under $F$ is fibrant.
[@hir03 Theorems 18.5.2, 18.5.3] \[fib\_cofib\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category.
1. If a functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is objectwise fibrant (see Definition \[owf\_defn\]), then the homotopy limit $\underset{{\mathcal{C}}}{\text{holim}} \; F$ is a fibrant object of ${\mathcal{M}}$.
2. Let $\eta \colon F {\longrightarrow}G$ be a map of ${\mathcal{C}}$-diagrams in ${\mathcal{M}}$. Assume that both $F$ and $G$ are objectwise fibrant. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence, then the induced map of homotopy limits $\underset{{\mathcal{C}}}{\text{holim}} \; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}} \; G$ is a weak equivalence of ${\mathcal{M}}$.
[@hir03 Theorem 19.6.7] \[htpy\_cofinal\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is homotopy right cofinal (respectively homotopy left cofinal), then for every objectwise fibrant contravariant (respectively covariant) functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, the natural map $[\theta; F]$ from Proposition \[induced\_holim\_prop\] is a weak equivalence.
\[fubini\_thm\] Let ${\mathcal{M}}$ be simplicial model category, and let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories. Let $F \colon {\mathcal{C}}\times {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ be a bifunctor. Then there exists a natural weak equivalence $$\underset{{\mathcal{C}}}{\text{holim}} \; \underset{{\mathcal{D}}}{\text{holim}}\; F \stackrel{\sim}{{\longrightarrow}} \underset{{\mathcal{D}}}{\text{holim}} \; \underset{{\mathcal{C}}}{\text{holim}}\; F.$$
This is the dual of [@cha_sch01 Theorem 24.9].
Cosimplicial replacement of a diagram {#cr_subsection}
-------------------------------------
The goal of this subsection is to prove Corollary \[hir\_coro\] and Proposition \[comm\_prop\]. As we said before those two results will be used in Section \[sos\_good\_section\].
\[cr\_defn\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category. For a covariant functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$, we define its *cosimplicial replacement*, denoted ${\Pi^{\bullet}}F \colon \Delta {\longrightarrow}{\mathcal{M}}$, as $$\Pi^n F := \prod_{(c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})} F ({c_n}).$$ For $0 \leq j \leq n-1$ the codegeneracy map $s^j \colon \Pi^n F {\longrightarrow}\Pi^{n-1} F$ is defined as follows. The projection of $s^j$ onto the factor indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_{n-1}$ is defined to be the projection of $\Pi^n F$ onto the factor indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_j \stackrel{id}{{\longrightarrow}} c_j {\rightarrow}\cdots {\rightarrow}c_{n-1}$. Cofaces are defined in a similar way.
\[betax\_rmk\] Let ${\mathcal{M}}$ be a model category, and let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories. Consider a covariant functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$. Also consider a functor $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$. Recall the notation $\theta^*(-)$ from (\[theta\_starx\]). Then there exists a canonical map $$\beta^{\bullet}_F \colon \Pi^{\bullet} F {\longrightarrow}\Pi^{\bullet} (\theta^* F)$$ defined as follows. The map from $\Pi^n F$ to the factor of $\Pi^n (\theta^* F)$ indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_n$ is just the projection of $\Pi^n F$ onto the factor $F(\theta(c_n))$ indexed by $\theta(c_0) {\rightarrow}\cdots {\rightarrow}\theta(c_n)$.
The following proposition is stated (without any proof) in [@bous_kan72 Chapter XI, Section 5] for diagrams of simplicial sets.
\[rf\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant covariant functor. Then the cosimplicial replacement $\Pi^{\bullet} F$ is *Reedy fibrant* (see the definition of Reedy fibrant in the proof).
First we recall the definition of *Reedy fibrant*. Let ${Z^{\bullet}}\colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. For $n \geq 0$, we let ${\mathcal{E}}_n$ denote the category whose objects are maps $[n] {\longrightarrow}[p]$ of $\Delta$ such that $p < n$. A morphism from $[n] {\longrightarrow}[p]$ to $[n] {\longrightarrow}[q]$ consists of a map $[p] {\longrightarrow}[q]$ of $\Delta$ such that the obvious triangle commutes. The *matching object* of ${Z^{\bullet}}$ at $[n] \in \Delta$, denoted $M^n {Z^{\bullet}}$, is defined to be the limit of the ${\mathcal{E}}_n$-diagram that sends $[n] {\longrightarrow}[p]$ to $Z^p$. That is, $$M^n {Z^{\bullet}}:= \underset{([n] {\rightarrow}[p]) \in {\mathcal{E}}_n}{\text{lim}} \; Z^p.$$ By the universal property, there exists a unique map $\alpha^n \colon Z^n {\longrightarrow}M^n {Z^{\bullet}}$ that makes certain triangles commutative. That map is induced by all codegeneracies $s^j \colon Z^n {\longrightarrow}Z^{n-1}, 0 \leq j \leq n-1$. We say that ${Z^{\bullet}}$ is *Reedy fibrant* if $\alpha^n$ is a fibration for all $n \geq 0$.
We come back to the proof of the proposition. Let $n \geq 0$. Since each codegeneracy map $s^j \colon \Pi^n F {\longrightarrow}\Pi^{n-1} F$ is a projection (see Definition \[cr\_defn\]), it follows that $\alpha^n$ is also a projection. This implies (by the assumption that $F(c)$ is fibrant for any $c \in {\mathcal{C}}$) that $\alpha^n$ is a fibration, which completes the proof.
\[tot\_defn\] Let ${\mathcal{M}}$ be a simplicial model category. Let $Z^{\bullet} \colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. The *totalization* of $Z^{\bullet}$, denoted $\text{Tot} \; Z^{\bullet}$, is defined to be the equalizer of the maps $$\xymatrix{\underset{[n] \in \Delta}{\prod} (Z^n)^{\Delta[n]} \ar@<1ex>[r]^-{\phi'} \ar@<-1ex>[r]_-{\psi'} & \underset{(f \colon [n] {\rightarrow}[p]) \in \Delta}{\prod} (Z^p)^{\Delta[n]}. }$$ Here the maps $\phi'$ and $\psi'$ are defined in the similar way as the maps $\phi$ and $\psi$ from Definition \[holim\_defn\].
[@hir14 Theorem 12.5] \[holim\_tot\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be a covariant functor. Then there exists an isomorphism $$\begin{aligned}
\label{phic_map}
\Phi_{{\mathcal{C}}} \colon \underset{{\mathcal{C}}}{\text{holim}} \; F \stackrel{\cong}{{\longrightarrow}} \text{Tot} \; \Pi^{\bullet} F,\end{aligned}$$ which is natural in $F$.
This is well detailed in [@hir14 Theorem 12.5]. However, for our purposes, specifically for the proof of Proposition \[comm\_prop\] below, we will recall only the construction of $\Phi_{{\mathcal{C}}}$. The map $\Phi_{{\mathcal{C}}}$ is in fact the composition of three isomorphisms (each obtained by using the observation we made at the beginning of the proof of Proposition \[induced\_holim\_prop\]): $$\xymatrix{\underset{{\mathcal{C}}}{\text{holim}} \; F \ar[rr]^-{{\widetilde{\Psi}}_{1{\mathcal{C}}}}_-{\cong} & & X \ar[rr]^-{{\widetilde{\Psi}}_{2{\mathcal{C}}}}_-{\cong} & & X' \ar[rr]^-{{\widetilde{\Psi}}_{3{\mathcal{C}}}}_-{\cong} & & {\text{Tot}}\; {\Pi^{\bullet}}F, }$$ where
1. $X$ is the equalizer of a diagram $$\xymatrix{ \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y }$$
2. $X'$ is the equalizer of a diagram $$\xymatrix{ \underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y' }.$$
3. By the definition of the cosimplicial replacement (see Definition \[cr\_defn\]), and by the the definition of the totalization (see Definition \[tot\_defn\]), one can easily see that ${\text{Tot}}\; {\Pi^{\bullet}}F$ is the equalizer of a diagram $$\xymatrix{ \underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y''}.$$ Since we are only interested in the definition of maps, it is not important here to know the definition of $Y$, $Y'$, and $Y''$.
Recalling $\underset{{\mathcal{C}}}{\text{holim}} \; F$ from Definition \[holim\_defn\], the map ${\widetilde{\Psi}}_{1{\mathcal{C}}}$ is induced by the map $$\begin{aligned}
\Psi_{1 {\mathcal{C}}} \colon \underset{ c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} {\longrightarrow}\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} , \end{aligned}$$ which is defined as follows. The projection of $\Psi_{1 {\mathcal{C}}}$ onto the factor indexed by $(c \in {\mathcal{C}}, n \geq 0, \Delta[n] \stackrel{\sigma}{{\longrightarrow}} N({\mathcal{C}}\downarrow c))$ is the composition $$\xymatrix{ \underset{ c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{\Delta[n]},}$$ where the first map is the projection onto the factor indexed by $c$, and the second is the canonical map induced by $\sigma \colon \Delta[n] {\longrightarrow}N({\mathcal{C}}\downarrow c)$.
Regarding the map ${\widetilde{\Psi}}_{2{\mathcal{C}}}$, it is induced by the map $$\begin{aligned}
\Psi_{2 {\mathcal{C}}} \colon \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} {\longrightarrow}\underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]},\end{aligned}$$ which is defined as follows. The projection of $\Psi_{2 {\mathcal{C}}}$ onto the factor indexed by $(n \geq 0, c_0 {\rightarrow}\cdots {\rightarrow}c_n)$ is just the projection $$\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} {\longrightarrow}(F(c_n))^{\Delta[n]}$$ onto the factor indexed by $(c_n, c_0 {\rightarrow}\cdots {\rightarrow}c_n, c_n \stackrel{id}{{\longrightarrow}} c_n)$.
Lastly, the map ${\widetilde{\Psi}}_{3 {\mathcal{C}}}$ is induced by the identity map. So $$\begin{aligned}
\label{psi3_eq}
\Psi_{3{\mathcal{C}}} := id.\end{aligned}$$
[@hir03 Theorem 19.8.7] \[hir\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let $Z^{\bullet} \colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. If $Z^{\bullet}$ is Reedy fibrant then the Bousfield-Kan map ${\text{Tot}}\; Z^{\bullet} {\longrightarrow}\underset{\Delta}{\text{holim}} \; Z^{\bullet}$ (see [@hir03 Definition 19.8.6]) is a weak equivalence, which is natural in $Z^{\bullet}$.
We end this section with the following corollary and proposition. These results will be used in the course of the proof of Theorem \[sos\_thm\] and Theorem \[good\_thm\], which will be done at the end of Subsection \[sos\_good\_subsection\]
\[hir\_coro\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant covariant functor. Then the Bousfield-Kan map $\text{Tot} \; \Pi^{\bullet} F {\longrightarrow}\underset{\Delta}{\text{holim}} \; \Pi^{\bullet} F$ (see [@hir03 Definition 19.8.6]) is a weak equivalence, which is natural in ${\Pi^{\bullet}}X$.
This follows directly from Proposition \[rf\_thm\] and Theorem \[hir\_thm\].
\[comm\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between small categories. Consider a covariant functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$. Also consider the maps $[\theta; F], \beta^{\bullet}_F$, and $\Phi_{{\mathcal{C}}}$ from Proposition \[induced\_holim\_prop\], Remark \[betax\_rmk\], and Theorem \[holim\_tot\_thm\] respectively. Then the following square commutes. $$\xymatrix{ \underset{{\mathcal{D}}}{\text{holim}} \; F \ar[rr]^-{\Phi_{{\mathcal{D}}}}_-{\cong} \ar[d]_-{[\theta; F]} & & \text{Tot} \; {\Pi^{\bullet}}F \ar[d]^-{\text{Tot}\; \beta^{\bullet}_F} \\
\underset{{\mathcal{C}}}{\text{holim}} \; \theta^* F \ar[rr]_-{\Phi_{{\mathcal{C}}}}^-{\cong} & & \text{Tot} \; {\Pi^{\bullet}}(\theta^*F).}$$
Warning! Proposition \[comm\_prop\] does not follow from the naturality of the map $\Phi_{{\mathcal{C}}}$ from Theorem \[holim\_tot\_thm\]. This is because $F$ and $\theta^* F$ does not have the same domain. So to prove Proposition \[comm\_prop\] we really have to use the definition of $\Phi_{{\mathcal{C}}}$.
Recall the maps $\Psi_{i (-)}, 1 \leq i \leq 3,$ from the proof of Theorem \[holim\_tot\_thm\]. To prove the proposition, it suffices to see that the three squares induced by the pairs $(\Psi_{i {\mathcal{D}}}, \Psi_{i {\mathcal{C}}}), 1 \leq i \leq 3,$ are all commutative. Let us begin with the following square induced by $(\Psi_{1 {\mathcal{D}}}, \Psi_{1 {\mathcal{C}}})$. $$\begin{aligned}
\label{sq1_eqn}
\xymatrix{ \underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}{\downarrow}d)} \ar[rr]^-{\Psi_{1{\mathcal{D}}}} \ar[d]_-{\alpha} & & \underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} \ar[d]^-{\lambda} \\
\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}{\downarrow}c)} \ar[rr]_-{\Psi_{1{\mathcal{C}}}} & & \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(\theta(c)))^{\Delta[n]}. }\end{aligned}$$ Here $\alpha$ is the composition from (\[fd\_fc\]), while the projection of $\lambda$ onto the factor indexed by $(c, n, \sigma \colon \Delta[n] {\rightarrow}N({\mathcal{C}}{\downarrow}c))$ is the projection $$\underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} {\longrightarrow}F(\theta(c))^{\Delta[n]}$$ onto the factor indexed by $(\theta(c), n, \Delta[n] \stackrel{\sigma}{{\longrightarrow}} N({\mathcal{C}}{\downarrow}c) \stackrel{f}{{\longrightarrow}} N({\mathcal{D}}{\downarrow}\theta(c)))$, where $f$ is induced by the obvious functor ${\mathcal{C}}{\downarrow}c {\longrightarrow}{\mathcal{D}}{\downarrow}\theta(c)$. Using the definitions, it is straightforward to check that the square (\[sq1\_eqn\]) commutes.
It is also straightforward to see that the following square, induced by the pair $(\Psi_{2{\mathcal{D}}}, \Psi_{2{\mathcal{C}}})$, is commutative. $$\xymatrix{ \underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} \ar[d]_-{\lambda} \ar[rr]^-{\Psi_{2{\mathcal{D}}}} & & \underset{n\geq 0, (d_0 {\rightarrow}\cdots {\rightarrow}d_n) \in N_n({\mathcal{D}})}{\prod} (F(d_n))^{\Delta[n]} \ar[d] \\
\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow d)}{\prod} (F(\theta(c)))^{\Delta[n]} \ar[rr]_-{\Psi_{2 {\mathcal{C}}}} & & \underset{n\geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(\theta(c_n)))^{\Delta[n]} }$$ Lastly, the square induced by $(\Psi_{3{\mathcal{D}}}, \Psi_{3{\mathcal{C}}})$ is clearly commutative since $\Psi_{3{\mathcal{D}}} = id$ and $\Psi_{3{\mathcal{C}}} = id$ by (\[psi3\_eq\]).
Special open sets and good cofunctors {#sos_good_section}
=====================================
The goal of this section is to prove Theorem \[sos\_thm\] and Theorem \[good\_thm\] below. The first result is a key ingredient, which will be used in many places throughout Sections \[sos\_good\_section\], \[poly\_section\], \[hc\_section\]. It roughly says that a certain homotopy limit $\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ is independent of the choice of the basis ${\mathcal{B}}$. The second theorem is a part of the proof of the main result of this paper (Theorem \[main\_thm\]). Note that the subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\] are influenced by the work of Pryor [@pryor15].
From now on, we let $M$ denote a smooth manifold, and we let ${\mathcal{O}(M)}$ to be the poset of open subsets of $M$. Also, for $k \geq 0$, we let ${\mathcal{O}_k(M)}\subseteq {\mathcal{O}(M)}$ to be the full subposet whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Below (see Example \[fso\_expl\]) we will see that ${\mathcal{O}_k(M)}$ can be obtained in another way. In [@wei99] Weiss calls objects of ${\mathcal{O}_k(M)}$ *special open sets*.
\[sos\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{B}}$ and ${\mathcal{B}}'$ be good bases (see Definition \[gb\_defn\]) for the topology of $M$ such that ${\mathcal{B}}\subseteq {\mathcal{B}}'$. Let ${\mathcal{B}}'_k(M) \subseteq {\mathcal{O}_k(M)}$ denote the full subcategory whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}'$. Consider an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]) $F \colon {\mathcal{B}}'_k(M) {\longrightarrow}{\mathcal{M}}$. Also consider the cofunctors ${F^{!}_{{\mathcal{B}}}}, {F^{!}_{{\mathcal{B}}'}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ from Definition \[fsb\_defn\]. Then the natural map $$[\theta; F] \colon {F^{!}_{{\mathcal{B}}'}}{\longrightarrow}{F^{!}_{{\mathcal{B}}}}$$ induced by the inclusion functor $\theta \colon {{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{B}}'_k(M)$ is a weak equivalence. Here the notation comes from Proposition \[induced\_holim\_prop\].
\[good\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Consider the poset ${\mathcal{B}_k(M)}$ from Definition \[fsb\_defn\], and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor ${F^{!}_{{\mathcal{B}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ from Definition \[fsb\_defn\] is good (see Definition \[good\_defn\]).
The proof of Theorem \[sos\_thm\] and Theorem \[good\_thm\] will be done in Subsection \[sos\_good\_subsection\] after some preliminaries results.
Isotopy cofunctors
------------------
The goal of this subsection is to prove Proposition \[fsrik\_prop\], which will be used in Sections \[poly\_section\], \[hc\_section\]. Note that this result is well known in the context of topological spaces.
We begin with several definitions. The first one is the notion of isotopy equivalence, which is well known in differential topology, manifold calculus, and other areas. Nevertheless we need to recall it for our purposes in Section \[iso\_cof\_section\].
\[iso\_eq\_defn\] A morphism $U {\hookrightarrow}U'$ of ${\mathcal{O}(M)}$ is said to be an *isotopy equivalence* if there exists a continuous map $$L \colon U \times [0, 1] {\longrightarrow}U', \quad (x, t) \mapsto L_t(x) := L(x, t)$$ that satisfies the following three conditions:
1. $L_0 \colon U {\hookrightarrow}U'$ is the inclusion map;
2. $L_1(U) = U'$;
3. for all $t$, $L_t \colon U {\longrightarrow}U'$ is a smooth embedding.
Such a map $L$ is called an *isotopy from $U$ to $U'$*.
\[isotopy\_cof\_defn\] Let ${\mathcal{C}}\subseteq {\mathcal{O}(M)}$ be a subcategory of ${\mathcal{O}(M)}$, and let ${\mathcal{M}}$ be a model category. A cofunctor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called *isotopy cofunctor* if it satisfies the following two conditions:
1. $F$ is objectwise fibrant (see Definition \[owf\_defn\]);
2. $F$ sends isotopy equivalences (see Definition \[iso\_eq\_defn\]) to weak equivalences.
\[good\_defn\] Let ${\mathcal{M}}$ be a simplicial model category. A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *good* if it satisfies the following two conditions:
1. $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]);
2. For any string $U_0 {\rightarrow}U_1 {\rightarrow}\cdots$ of inclusions of ${\mathcal{O}(M)}$, the natural map $$F\left(\bigcup_{i=0}^{\infty} U_i\right) {\longrightarrow}\underset{i}{\text{holim}} \; F(U_i)$$ is a weak equivalence.
In order words, a cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is *good* if it satisfies three conditions: (a), (b) from Definition \[isotopy\_cof\_defn\], and (b) from Definition \[good\_defn\]. As we said in the introduction, this definition is slightly different from the classical one [@wei99 Page 71] (see the comment we made right after Theorem \[main\_thm\]).
\[gb\_defn\] A basis for the topology of $M$ is called *good* if each element in there is diffeomorphic to an open ball.
\[fsb\_defn\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$.
1. For $k \geq 0$, we define ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}(M)}$ to be the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$.
2. If $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is a cofunctor, we define $F^{!}_{{\mathcal{B}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $${F^{!}_{{\mathcal{B}}}}(U) := \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V).$$
\[fso\_expl\] Let ${\mathcal{O}}$ be the collection of all subsets of $M$ diffeomorphic to an open ball. Certainly this is a good basis (see Definition \[gb\_defn\]) for the topology of $M$. So, by Definition \[fsb\_defn\], one has the poset ${\mathcal{O}_k(M)}$, which is exactly the same as the poset ${\mathcal{O}_k(M)}$ we defined just before Theorem \[sos\_thm\]. If $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is a cofunctor, one also has the cofunctor ${F^{!}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $${F^{!}_{{\mathcal{O}}}}(U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V).$$ Clearly ${\mathcal{O}}$ is the biggest (with respect to the inclusion) good basis for the topology of $M$.
As we said before, the following proposition will be used in Section \[poly\_section\] and Section \[hc\_section\].
\[fsrik\_prop\] Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor.
1. Then there is a natural transformation $\eta$ from $F$ to the restriction ${F^{!}_{{\mathcal{B}}}}| {\mathcal{B}_k(M)}$, which is an objectwise weak equivalence.
2. If in addition $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), then so is the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{B}_k(M)}$.
<!-- -->
1. Let $U \in {\mathcal{B}_k(M)}$, and let $\theta, \theta' \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{{\mathcal{B}}_k}(U)$ be functors defined as $\theta(V) = V$ and $\theta'(V) = U$. Certainly there is a natural transformation $\beta \colon \theta {\longrightarrow}\theta'$. This induces by (\[nat\_theta\]) the following commutative square. $$\xymatrix{\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \ar[rr]^-{[\theta; F]} & & \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \\
\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \ar[rr]_-{[\theta'; F]} \ar[u]^-{id} & & \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(U). \ar[u]_-{\text{holim}(F\beta)} }$$ Clearly one has $F(U) \simeq \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(U)$. This allows us to define $\eta[U] := \text{holim} (F \beta)$. Since $\theta$ is the identity functor, it follows that the map $[\theta; F]$ is a weak equivalence (in fact it is the identity functor as well). The map $[\theta'; F]$ is also a weak equivalence (by Theorem \[htpy\_cofinal\_thm\]) since $\theta'$ is homotopy right cofinal. Indeed, for every $V \in {{\mathcal{B}}_k}(U)$ the under category (see Definition \[uc\_defn\]) $V \downarrow \theta'$ has a terminal object, namely $(U, V {\hookrightarrow}U)$. Now, applying the two-out-of-three axiom we deduce that the map $\text{holim} (F\beta)$ is a weak equivalence. Regarding the naturality of $\eta[U]$ in $U$, it follows easily from (\[theta\_square\]).
2. Certainly the functor ${F^{!}_{{\mathcal{B}}}}|{{\mathcal{B}}_k}(M)$ satisfies condition (a) from Definition \[isotopy\_cof\_defn\] because of Theorem \[fib\_cofib\_thm\]. Condition (b) from the same definition is also satisfied by ${F^{!}_{{\mathcal{B}}}}|{{\mathcal{B}}_k}(M)$ (this follows directly from part (i)).
The cofunctors $F^{!p}$ {#cof_fsp_subsection}
-----------------------
In this subsection we consider a good basis ${\mathcal{B}}$ and the poset ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Also we consider the basis ${\mathcal{O}}$ from Example \[fso\_expl\]. The main results here are Proposition \[sosp\_prop\] and Proposition \[isop\_prop\] whose proofs are inspired by Pryor’s work [@pryor15]. Those propositions are one of the key ingredients in proving Theorem \[sos\_thm\] and Theorem \[good\_thm\].
\[bkp\_defn\] Let $k, p\geq 0$.
1. Define ${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ to be the poset whose objects are strings $V_0 {\rightarrow}\cdots {\rightarrow}V_p$ of $p$ composable morphisms in ${\mathcal{B}_k(M)}$. A morphism from $V_0 {\rightarrow}\cdots {\rightarrow}V_p$ to $W_0 {\rightarrow}\cdots {\rightarrow}W_p$ consists of a collection $\{f_i \colon V_i {\hookrightarrow}W_i\}_{i=0}^p$ of isotopy equivalences such that all the obvious squares commute.
2. Taking ${\mathcal{B}}$ to be ${\mathcal{O}}$, we have the poset ${\widetilde{{\mathcal{O}}}_{k, p}}(M)$.
The following remark claims that the collection ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(M) = \{{\widetilde{{\mathcal{B}}}_{k, p}}(M)\}_{p \geq 0}$ is equipped with a canonical simplicial object structure.
\[bkp\_rmk\] Let $U \in {\mathcal{O}(M)}$. For $0 \leq i \leq p+1$ define $d_i \colon \widetilde{B}_{k, p+1}(U) {\longrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U)$ as $$d_i(V_0 {\rightarrow}\cdots {\rightarrow}V_{p+1}) = \left\{ \begin{array}{ccc}
V_1 {\rightarrow}\cdots {\rightarrow}V_{p+1} & \text{if} & i =0 \\
V_0 {\rightarrow}\cdots V_{i-1} {\rightarrow}V_{i+1} {\rightarrow}\cdots {\rightarrow}V_{p+1} & \text{if} & 1 \leq i \leq p \\
V_0 {\rightarrow}\cdots {\rightarrow}V_p & \text{if} & i = p+1.
\end{array} \right.$$ For $0 \leq j \leq p$ define $s_j \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\longrightarrow}\widetilde{B}_{k, p+1}(U)$ as $$s_j(V_0 {\rightarrow}\cdots {\rightarrow}V_p) = V_0 {\rightarrow}\cdots {\rightarrow}V_j \stackrel{\text{id}}{{\rightarrow}} V_j {\rightarrow}\cdots {\rightarrow}V_p.$$ One can easily check that $d_i$ and $s_j$ satisfy the simplicial relation. So ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$ is a simplicial object in Cat, the category of small categories.
\[fsbp\_defn\] Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be a cofunctor.
1. Define a cofunctor ${\tilde{F}^{!p}_{{\mathcal{B}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $${\tilde{F}^{!p}_{{\mathcal{B}}}}(U) := \underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}} \; {\tilde{F}^p_{{\mathcal{B}}}},$$ where $${\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}, \quad V_0 {\rightarrow}\cdots {\rightarrow}V_p \mapsto F(V_0).$$
2. Taking again ${\mathcal{B}}$ to be ${\mathcal{O}}$, we have the cofunctor ${\tilde{F}^{!p}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$.
The following remark will be used in Subsection \[sos\_good\_subsection\].
\[fsp\_rmk\] Let $U \in {\mathcal{O}(M)}$, and let ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$ be the simplicial object from Remark \[bkp\_rmk\]. Using the simplicial structure on ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$, one can endow the collection ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}(U) = \{{\tilde{F}^{!p}_{{\mathcal{B}}}}(U) \}_{p \geq 0}$ with a canonical cosimplicial structure as follows. First recall the notation $\theta^{*}(-)$ from (\[theta\_starx\]). Also recall the notation $[-;-]$ introduced in Proposition \[induced\_holim\_prop\]. Let $d_i$ and $s_j$ as in Remark \[bkp\_rmk\], and consider $d_0 \colon \widetilde{{\mathcal{B}}}_{k, p+1}(U) {\longrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U)$. Also consider the natural transformation $\beta \colon d_0^* {\tilde{F}^p_{{\mathcal{B}}}}{\longrightarrow}\tilde{F}^{p+1}_{{\mathcal{B}}}$ defined as $$\beta[V_0 \stackrel{f}{{\rightarrow}} V_1 {\rightarrow}\cdots {\rightarrow}V_{p+1}] := F(V_1) \stackrel{F(f)}{{\longrightarrow}} F(V_0).$$ Now define $d^i \colon {\tilde{F}^{!p}_{{\mathcal{B}}}}(U) {\longrightarrow}\tilde{F}^{!(p+1)}_{{\mathcal{B}}} (U)$ as $$d^i = \left\{ \begin{array}{ccc}
\text{holim}(\beta) \circ [d_0; {\tilde{F}^p_{{\mathcal{B}}}}] & \text{if} & i =0 \\
\left[d_i; {\tilde{F}^p_{{\mathcal{B}}}}\right] & \text{if} & 1 \leq i \leq p+1.
\end{array} \right.$$ Also define $s^j \colon \tilde{F}^{!(p+1)}_{{\mathcal{B}}} (U) {\longrightarrow}{\tilde{F}^{!p}_{{\mathcal{B}}}}(U), 0 \leq j \leq p+1$, as $$s^j = [s_j; \tilde{F}^{p+1}_{{\mathcal{B}}}].$$ Certainly the maps $d^i$ and $s^j$ satisfy the cosimplicial relations. So ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}(U)$ is a cosimplicial object in ${\mathcal{M}}$ for any $U \in {\mathcal{O}(M)}$.
\[loc\_defn\] Let ${\mathcal{M}}$ be a category with a class of weak equivalences, and let ${\mathcal{C}}$ be any other category. A functor ${\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called *locally constant* if it sends every morphism of ${\mathcal{C}}$ to a weak equivalence.
\[loc\_expl\] Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor ${\tilde{F}^p_{{\mathcal{B}}}}$ from Definition \[fsbp\_defn\] is locally constant. This follows directly from the definition of a morphism of ${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) and the definition of an isotopy cofunctor.
Now we state and prove the main results of this subsection (Proposition \[sosp\_prop\] and Proposition \[isop\_prop\]). First we need three preparatory lemmas.
\[cisinski\_lem\] Let ${\mathcal{M}}$ be a simplicial model category. Let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between small categories. Consider a functor $G \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, and assume that it is locally constant (see Definition \[loc\_defn\]). Also assume that the nerve of $\theta$ is a weak equivalence. Then the canonical map $$[\theta; G] \colon \underset{{\mathcal{D}}}{\text{holim}} \; G {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}} \; \theta^{*} G$$ (see Proposition \[induced\_holim\_prop\]) is a weak equivalence.
This is just the dual of Proposition 1.17 from [@cis09].
\[pryor1\_lem\] For any $U \in {\mathcal{O}(M)}$ the nerve of the inclusion functor ${\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$ is a homotopy equivalence for all $k, p \geq 0$.
This is done in the course of the proof of Theorem 6.12 from [@pryor15].
[@pryor15 Lemma 6.8] \[pryor2\_lem\] Let $f \colon U {\hookrightarrow}U'$ be a morphism of ${\mathcal{O}(M)}$. If $f$ is an isotopy equivalence, then the nerve of the inclusion functor ${\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U')$ is a homotopy equivalence.
\[sosp\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Let $U$ be an object of ${\mathcal{O}(M)}$. Consider $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$, the inclusion functor. Also consider ${\tilde{F}^p_{{\mathcal{O}}}}\colon {\widetilde{{\mathcal{O}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}$, the cofunctor from Definition \[fsbp\_defn\]. Then the canonical map $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}] \colon {\tilde{F}^{!p}_{{\mathcal{O}}}}(U) {\longrightarrow}{\tilde{F}^{!p}_{{\mathcal{B}}}}(U)$ is a weak equivalence for all $p \geq 0$. Furthermore that map is natural in $U$.
Set ${\mathcal{C}}:= {\widetilde{{\mathcal{B}}}_{k, p}}(M), {\mathcal{D}}:= {\widetilde{{\mathcal{O}}}_{k, p}}(M),$ and $G := {\tilde{F}^p_{{\mathcal{O}}}}$. Since $G$ is locally constant by Example \[loc\_expl\], and since the nerve of $\theta$ is a weak equivalence by Lemma \[pryor1\_lem\], it follows that $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}]$ is a weak equivalence by Lemma \[cisinski\_lem\]. The naturality in $U$ comes directly from the definitions.
\[isop\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then for any $p \geq 0$ the cofunctor ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ (see Definition \[fsbp\_defn\]) is an isotopy cofunctor as well.
Let $U {\hookrightarrow}U'$ be an isotopy equivalence, and let $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U')$ denote the inclusion functor. Consider the cofunctor ${\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U') {\longrightarrow}{\mathcal{M}}$ from Definition \[fsbp\_defn\]. Since ${\tilde{F}^p_{{\mathcal{B}}}}$ is locally constant by Example \[loc\_expl\], and since the nerve of $\theta$ is a weak equivalence by Lemma \[pryor2\_lem\], the desired result follows by Lemma \[cisinski\_lem\].
Grothendieck construction {#gro_const_subsection}
-------------------------
In this subsection we recall the Grothendieck construction, and we give some examples that will be used further. We also recall an important result (see Theorem \[cha\_sch\_thm\]), which regards the homotopy limit of a diagram indexed by the Grothendieck construction.
\[intcf\_defn\] Let ${\mathcal{C}}$ be a small category, and let ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ be a covariant functor from ${\mathcal{C}}$ to the category Cat of small categories. Define $\int_{{\mathcal{C}}} {\mathcal{F}}$ to be the category whose objects are pairs $(c, x)$ where $c \in {\mathcal{C}}$ and $x \in {\mathcal{F}}(c)$. A morphism $(c, x) {\longrightarrow}(c', x')$ consists of a pair $(f, g)$, where $f \colon c {\longrightarrow}c'$ is a morphism of ${\mathcal{C}}$, and $g \colon {\mathcal{F}}(f)(x) {\longrightarrow}x'$ is a morphism of ${\mathcal{F}}(c')$. The construction that sends ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ to $\int_{{\mathcal{C}}} {\mathcal{F}}$ is called the *Grothendieck construction*.
Here are two examples of the Grothendieck construction. The first one will be used in Subsection \[sos\_good\_subsection\], while the second will be used in Section \[poly\_section\].
\[intcf\_expl1\] Let ${\mathcal{D}}_0 {\hookrightarrow}{\mathcal{D}}_1 {\hookrightarrow}{\mathcal{D}}_2 \cdots $ be an increasing inclusion of small categories. Define ${\mathcal{C}}$ to be the category $\{0 {\rightarrow}1 {\rightarrow}2 {\rightarrow}\cdots \}$, and ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ as ${\mathcal{F}}(i) = {\mathcal{D}}_i$. Then one can see that $$\int_{{\mathcal{C}}} {\mathcal{F}}= {\mathcal{C}}\times \left(\bigcup_{i=0}^{\infty} {\mathcal{D}}_i\right).$$
\[intcf\_expl2\] Let $k \geq 0$, and let ${\mathcal{C}}$ be the category defined as $${\mathcal{C}}= \left\{S \subseteq \{0, \cdots, k\}\ \ \text{such that} \ \ S \neq \emptyset \right\}.$$ Given two objects $S, T \in {\mathcal{C}}$, there exists a morphism from $S$ to $T$ if $T \subseteq S$. If $U$ is an open subset of $M$, and $A_0, \cdots, A_k$ are pairwise disjoint closed subsets of $U$, we let $\Omega$ denote the basis (for the topology of $U$) where an element is a subset $B$ diffeomorphic to an open ball such that $B$ intersects at most one $A_i$. In other words, if one introduces the notation $U(S):= U \backslash \cup_{i \in S} A_i$, then $$\Omega := \left\{B \subseteq U| \; \text{$B$ is diffeomorphic to an open ball and $B \subseteq U\left(\{0, \cdots, \hat{i}, \cdots, k \}\right)$} \right\},$$ where the hat means taking out. Certainly $\Omega$ is a good basis (see Definition \[gb\_defn\]). Now, for $V \in {\mathcal{O}}(U)$ we let $\Omega_k(V) \subseteq {\mathcal{O}_k}(V)$ denote the full subposet whose objects are disjoint unions of at most $k$ elements from $\Omega$. Define $${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat} \quad \text{as} \quad {\mathcal{F}}(S) := \Omega_k(U(S)).$$ Clearly, one has ${\mathcal{F}}(S) \subseteq {\mathcal{F}}(T)$ whenever $T \subseteq S$. So ${\mathcal{F}}(S {\rightarrow}T)$ is just the inclusion functor. One can then consider the category $\int_{{\mathcal{C}}} {\mathcal{F}}$ (see Definition \[intcf\_defn\]), which can be described as follows. An object of that category is a pair $(S, V)$ where $\emptyset \neq S \subseteq \{0, \cdots, k\}$, and $V \subseteq U \backslash \cup_{i \in S} A_i$ is the disjoint union of at most $k$ elements from $\Omega$. There exists a morphism $(S, V) {\longrightarrow}(T, W)$ if and only if $T \subseteq S$ and $V \subseteq W$.
[@cha_sch01] \[cha\_sch\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category and let ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ be a covariant functor. Consider a collection $\{G_c \colon \colon {\mathcal{F}}(c) {\longrightarrow}{\mathcal{M}}\}_{c \in {\mathcal{C}}}$ of functors such that for any $f \colon c {\longrightarrow}c'$ in ${\mathcal{C}}$ the following triangle commutes. $$\xymatrix{{\mathcal{F}}(c') \ar[rr]^-{G_{c'}} \ar[d]_-{{\mathcal{F}}(f)} & & {\mathcal{M}}\\
{\mathcal{F}}(c) \ar[rru]_-{G_c} & & }$$ Then the canonical map $$\beta \colon \underset{(c, x) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; G_c(x) {\longrightarrow}\underset{c \in {\mathcal{C}}}{\text{holim}} \; \underset{x \in {\mathcal{F}}(c)}{\text{holim}} \; G_c(x)$$ is a weak equivalence (see Definition \[intcf\_defn\]).
This is the dual of [@cha_sch01 Theorem 26.8].
Special open sets and good cofunctors {#sos_good_subsection}
-------------------------------------
The goal of this subsection is to prove Theorem \[sos\_thm\] and Theorem \[good\_thm\] announced at the beginning of Section \[sos\_good\_section\].
To prove Theorem \[sos\_thm\] we will need Lemma \[fqf\_lem\] below. First we need to introduce some notation. For $U \in {\mathcal{O}(M)}, q \geq 0$ we let ${\widehat{{\mathcal{B}}}_{k, q}}(U)$ denote the poset whose objects are strings $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ of $q$ composable morphisms in ${{\mathcal{B}}_k}(U)$ such that $W_i {\rightarrow}W_{i+1}$ is an isotopy equivalence for all $i$. A morphism from $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ to $W'_0 {\rightarrow}\cdots {\rightarrow}W'_q$ consists of a collection $f = \{f_i \colon W_i {\longrightarrow}W'_i\}_{i=0}^{q}$ of morphisms of ${{\mathcal{B}}_k}(U)$ such that all the obvious squares commute. Now let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor. Define a new cofunctor ${\hat{F}_{{\mathcal{B}}}}^q \colon {\widehat{{\mathcal{B}}}_{k, q}}(U) {\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{fhq}
{\hat{F}_{{\mathcal{B}}}}^q (W_0 {\rightarrow}\cdots {\rightarrow}W_q) := F(W_0).\end{aligned}$$ Also define ${\hat{F}_{{\mathcal{B}}}}^{!q} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{fhsq}
{\hat{F}_{{\mathcal{B}}}}^{!q} (U) := \underset{{\widehat{{\mathcal{B}}}_{k, q}}(U)}{\text{holim}} \; {\hat{F}_{{\mathcal{B}}}}^q. \end{aligned}$$
\[fsp\_bkp\_rmk\]
1. As in Remark \[fsp\_rmk\], the collection ${\hat{F}_{{\mathcal{B}}}}^{!\bullet} (U) = \{{\hat{F}_{{\mathcal{B}}}}^{!q} (U)\}_{q \geq 0}$ is a cosimplicial object in ${\mathcal{M}}$ for all $U \in {\mathcal{O}(M)}$.
2. Recall the poset ${\widetilde{{\mathcal{B}}}_{k, p}}(U)$ and the functor ${\tilde{F}_{{\mathcal{B}}}}^p$ from Definition \[bkp\_defn\] and Definition \[fsbp\_defn\] respectively. Also recall ${\Pi^{\bullet}}(-)$ from Definition \[cr\_defn\]. Then one can easily see that $$\Pi^q {\tilde{F}_{{\mathcal{B}}}}^p = \Pi^p {\hat{F}_{{\mathcal{B}}}}^q$$ since the set of $q$-simplices of the nerve $N({\widetilde{{\mathcal{B}}}_{k, p}}(U))$ is equal to the set of $p$-simplices of the nerve $N({\widehat{{\mathcal{B}}}_{k, q}}(U))$.
\[fqf\_lem\] Let ${\mathcal{M}}$ be a simplicial model category. For $U \in {\mathcal{O}(M)}$, consider the functor $\theta \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\widehat{{\mathcal{B}}}_{k, q}}(U)$ defined as $\theta(V) := V {\rightarrow}\cdots {\rightarrow}V$. Then the canonical map $$\begin{aligned}
\label{fqf_eq}
[\theta; {\hat{F}_{{\mathcal{B}}}}^q] \colon {\hat{F}_{{\mathcal{B}}}}^{!q} (U) {\longrightarrow}F^{!}_{{\mathcal{B}}} (U)\end{aligned}$$ is a weak equivalence. Furthermore this map is natural in $U$.
Let $W_0 {\rightarrow}\cdots {\rightarrow}W_q \in {\widehat{{\mathcal{B}}}_{k, q}}(U)$. The under category $(W_0 {\rightarrow}\cdots {\rightarrow}W_q) \downarrow \theta$ is contractible since it has an initial object, namely $(W_q, f)$ where $f$ is the obvious map from $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ to $W_q {\rightarrow}\cdots {\rightarrow}W_q$. So $\theta$ is homotopy right cofinal, and therefore the map $[\theta; {\hat{F}_{{\mathcal{B}}}}^q]$ is a weak equivalence by Theorem \[htpy\_cofinal\_thm\]. The naturality of that map in $U$ is readily checked.
We are now ready to prove Theorem \[sos\_thm\].
In the following proof we will work with ${\mathcal{B}}'={\mathcal{O}}$. Notice that one can perform exactly the same proof with any good basis ${\mathcal{B}}'$ containing ${\mathcal{B}}$.
Let $U \in {\mathcal{O}(M)}$. Recall ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ from Definition \[fsbp\_defn\]. We will show that the objects $\underset{[p] \in \Delta}{\text{holim}} \; {\tilde{F}_{{\mathcal{B}}}}^{!p} (U)$ and $F^{!}_{{\mathcal{B}}} (U)$ are connected by a zigzag of natural weak equivalences. Consider the following diagram. $$\xymatrix{{\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\cong} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\sim} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{[q] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{B}}}}^p \\
{\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{{\widetilde{{\mathcal{O}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\cong} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\sim} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{[q] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{O}}}}\ar[u]_-{\lambda} }$$
1. In the first row
1. the first map is the homotopy limit of the map (\[phic\_map\]),
2. the second is the homotopy limit of the Bousfield-Kan map from Corollary \[hir\_coro\], and
3. the third is provided by the Fubini Theorem \[fubini\_thm\].
2. The maps in the second row are obtained in the similar way since ${\mathcal{B}}$ is a subposet of ${\mathcal{O}}$.
3. The lefthand vertical map is nothing but the homotopy limit of $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}]$, where $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$ is just the inclusion functor.
4. The three others are the canonial ones induced by the map $\beta^{\bullet}_{{\widetilde{{\mathcal{O}}}_{k, p}}(U)}$ from Remark \[betax\_rmk\].
Certainly the lefthand square commutes by Proposition \[comm\_prop\]. The middle one commutes by the fact that the Bousfield-Kan map ${\text{Tot}}\; Z^{\bullet} {\longrightarrow}\underset{\Delta}{\text{holim}} \; Z^{\bullet}$ is natural in $Z^{\bullet}$. The third square commutes since the map in the Fubini Theorem \[fubini\_thm\] is also natural. So the above diagram is commutative. Therefore, since the first vertical map is a weak equivalence by Proposition \[sosp\_prop\] and Theorem \[fib\_cofib\_thm\], it follows that the last one, $\lambda$, is also a weak equivalence.
Similarly the following diagram (in which the equality comes from Remark \[fsp\_bkp\_rmk\]) is commutative as well. $$\xymatrix{ {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{B}}}}\ar@{=}[r] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^p {\hat{F}_{{\mathcal{B}}}}^q & {\underset{[q] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\hat{F}_{{\mathcal{B}}}}^q \ar[l]_-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{{\widehat{{\mathcal{B}}}_{k, q}}(U)}{\text{holim}}}{\hat{F}_{{\mathcal{B}}}}^q \ar[l]_-{\cong} \\
{\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{O}}}}\ar@{=}[r] \ar[u]_-{\lambda}^-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^p {\hat{F}_{{\mathcal{O}}}}^q \ar[u] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\hat{F}_{{\mathcal{O}}}}^q \ar[l]^-{\sim} \ar[u] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{{\widehat{{\mathcal{O}}}_{k, q}}(U)}{\text{holim}}}\; {\hat{F}_{{\mathcal{O}}}}^q \ar[l]^-{\cong} \ar[u]_-{\varphi}. }$$ So the map $\varphi$ is a weak equivalence.
Now consider the following square. $$\xymatrix{{\underset{[q] \in \Delta}{\text{holim}}}\; {\hat{F}_{{\mathcal{B}}}}^{!q} (U) \ar[rr]^-{\sim} & & {F^{!}_{{\mathcal{B}}}}(U) \\
{\underset{[q] \in \Delta}{\text{holim}}}\; {\hat{F}_{{\mathcal{O}}}}^{!q} (U) \ar[rr]_-{\sim} \ar[u]^-{\varphi}_-{\sim} & & {F^{!}_{{\mathcal{O}}}}(U), \ar[u] }$$ where the top horizontal arrow is the homotopy limit of the map (\[fqf\_eq\]), which is itself a weak equivalence by Lemma \[fqf\_lem\]. In similar fashion the bottom horizontal map is a weak equivalence. The lefthand vertical map is the above map $\varphi$, which is a weak equivalence. So, since the square commutes, it follows that the righthand vertical map is also a weak equivalence. We thus obtain the desired result.
To prove Theorem \[good\_thm\] we will need the following lemma.
\[gd\_lem\] Let $U \in {\mathcal{O}(M)}$, and let ${\overline{{\mathcal{B}}}_k}(U) \subseteq {{\mathcal{B}}_k}(U)$ be the full subposet defined as $$\begin{aligned}
\label{bkb_eq}
{\overline{{\mathcal{B}}}_k}(U) = \{V \in {{\mathcal{B}}_k}(U)| \; \overline{V} \subseteq U\}.\end{aligned}$$ Here $\overline{V}$ stands for the closure of $V$. Let ${\mathcal{M}}$ be a simplicial model category. Consider an isotopy cofunctor $F \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\mathcal{M}}$. Then the canonical map $$\begin{aligned}
\label{good_eq}
[\theta; F] \colon \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{V \in {\overline{{\mathcal{B}}}_k}(U)}{\text{holim}} \; F(V),\end{aligned}$$ induced by the inclusion functor $\theta \colon {\overline{{\mathcal{B}}}_k}(U) {\longrightarrow}{{\mathcal{B}}_k}(U)$, is a weak equivalence.
Let $\overline{{\mathcal{B}}}$ be the following basis for the topology of $U$. $$\overline{{\mathcal{B}}} = \{B \subseteq U| \; \text{$B \in {\mathcal{B}}$ and $\overline{B} \subseteq U$}\}.$$ Certainly $\overline{{\mathcal{B}}}$ is a good basis (see Definition \[gb\_defn\]). One can easily see that each object of ${\overline{{\mathcal{B}}}_k}(U)$ is the disjoint union of at most $k$ elements from $\overline{{\mathcal{B}}}$. So by Theorem \[sos\_thm\], the map $[\theta; F]$ is a weak equivalence.
We begin with part (a) of goodness. Let $U, U' \in {\mathcal{O}(M)}$ such that $U \subseteq U'$. Assume that the inclusion map $U {\hookrightarrow}U'$ is an isotopy equivalence. Then the canonical map $$\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U')}{\text{holim}} \; {\tilde{F}_{{\mathcal{B}}}}^p {\longrightarrow}{\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}_{{\mathcal{B}}}}^p$$ is a weak equivalence by Proposition \[isop\_prop\]. Now, by replacing
1. ${\widetilde{{\mathcal{O}}}_{k, p}}(U)$ by ${\widetilde{{\mathcal{B}}}_{k, p}}(U')$, $${\tilde{F}_{{\mathcal{O}}}}^p \colon {\widetilde{{\mathcal{O}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}\qquad \text{by} \qquad {\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U') {\longrightarrow}{\mathcal{M}},$$ $${\hat{F}_{{\mathcal{O}}}}^q \colon {\widehat{{\mathcal{O}}}_{k, q}}(U) {\longrightarrow}{\mathcal{M}}\qquad \text{by} \qquad {\hat{F}_{{\mathcal{B}}}}^q \colon {\widehat{{\mathcal{B}}}_{k, q}}(U') {\longrightarrow}{\mathcal{M}},$$
2. ${\widehat{{\mathcal{O}}}_{k, q}}(U)$ by ${\widehat{{\mathcal{B}}}_{k, q}}(U')$, ${\hat{F}_{{\mathcal{O}}}}^{!q} (U)$ by ${\hat{F}_{{\mathcal{B}}}}^{!q} (U')$, and ${F^{!}_{{\mathcal{O}}}}(U)$ by ${F^{!}_{{\mathcal{B}}}}(U')$,
in the proof of Theorem \[sos\_thm\], we deduce that the canonical map ${F^{!}_{{\mathcal{B}}}}(U') {\longrightarrow}{F^{!}_{{\mathcal{B}}}}(U)$ is a weak equivalence. This proves part (b) from Definition \[isotopy\_cof\_defn\]. Part (a) from the same definition follows immediately from the fact that $F \colon {{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor by assumption, and from Theorem \[fib\_cofib\_thm\].
Now we show part (b) of goodness. Let $U_0 {\rightarrow}U_1 {\rightarrow}\cdots$ be a string of inclusions of ${\mathcal{O}(M)}$. Consider the following commutative square. $$\xymatrix{ \underset{V \in {{\mathcal{B}}_k}(\cup_i U_i)}{\text{holim}} \; F(V) \ar[rr] \ar[d]_-{\sim} & & \underset{i}{\text{holim}} \; \underset{V \in {{\mathcal{B}}_k}(U_i)}{\text{holim}} \; F(V) \ar[d]^-{\sim} \\
\underset{V \in {\overline{{\mathcal{B}}}_k}(\cup_i U_i)}{\text{holim}} \; F(V) \ar[rr]_-{\sim} & & \underset{i}{\text{holim}} \; \underset{V \in {\overline{{\mathcal{B}}}_k}(U_i)}{\text{holim}} \; F(V). }$$
1. Both vertical maps come from (\[good\_eq\]), and therefore are weak equivalences by Lemma \[gd\_lem\].
2. The bottom horizontal map is a weak equivalence by the following reason. Consider the data from Example \[intcf\_expl1\], and set ${\mathcal{D}}_i := {\overline{{\mathcal{B}}}_k}(U_i)$. Then it is straightforward to see that $$\begin{aligned}
\label{int_eq}
\int_{{\mathcal{C}}} {\mathcal{F}}= {\mathcal{C}}\times \left(\bigcup_i {\mathcal{D}}_i\right) = {\overline{{\mathcal{B}}}_k}(\cup_i U_i). \end{aligned}$$ The first equality is obvious, while the second one comes from the definition of ${\overline{{\mathcal{B}}}_k}(-)$ (see (\[bkb\_eq\])). Furthermore the canonical map $$\underset{(i, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V) {\longrightarrow}\underset{i \in {\mathcal{C}}}{\text{holim}} \; \underset{V \in {\mathcal{D}}_i}{\text{holim}} \; F(V)$$ is a weak equivalence by Theorem \[cha\_sch\_thm\]. But, by (\[int\_eq\]), this latter map is nothing but the map we are interested in.
Hence the top horizontal map is a weak equivalence, and this completes the proof.
Polynomial cofunctors {#poly_section}
=====================
The goal of this section is to prove Theorem \[main\_thm\] announced in the introduction. We will need three preparatory lemmas: Lemma \[cofinal\_lem\], Lemma \[poly\_lem\], and Lemma \[charac\_lem\]. The two latter ones are important themselves.
Let us begin with the following definition.
\[poly\_defn\] A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *polynomial of degree $\leq k$* if for every $U \in {\mathcal{O}(M)}$ and pairwise disjoint closed subsets $A_0, \cdots, A_k$ of $U$, the canonical map $$F(U) {\longrightarrow}\underset{S \neq \emptyset}{\text{holim}} \; F(U \backslash \cup_{i \in S} A_i)$$ is a weak equivalence. Here $S \neq \emptyset$ runs over the power set of $\{0, \cdots, k\}$.
\[cofinal\_lem\] Consider the data from Example \[intcf\_expl2\]. Then the functor $\theta \colon \int_{{\mathcal{C}}} {\mathcal{F}}{\longrightarrow}\Omega_k(U)$, defined as $\theta (S, V) = V$, is homotopy right cofinal (see Definition \[cofinal\_defn\]).
First of all, let us consider the notation (${\mathcal{C}}, U(S), \Omega, \Omega_k(U), {\mathcal{F}}$) introduced in Example \[intcf\_expl2\]. One has the following properties.
1. ${\mathcal{F}}(S \cup T) = {\mathcal{F}}(S) \cap {\mathcal{F}}(T)$ for any $S, T \in {\mathcal{C}}$;
2. for any $X \in \Omega_k(U)$ there exists $j \in \{0, \cdots, k\}$ such that $X \cap A_j = \emptyset$.
The first property follows directly from the definitions. The second comes from the following three facts: (i) By definition, each element of $\Omega$ intersects at most one of the $A_i$’s. (ii) $X$ is the disjoint union of at most $k$ elements from $\Omega$. (iii) The cardinality of the set $\{A_0, \cdots, A_k\}$ is $k+1$, which is greater than the number of components of $X$. This property is nothing but the pigeonhole principle.
Now let $V \in \Omega_k(U)$. We have to prove that the under category (see Definition \[uc\_defn\]) $V \downarrow \theta$ is contractible. It suffices to show that it admits an initial object. Consider the pair $(S, V)$ where $$S = \left\{i \in \{0, \cdots, k\} | \ V \cap A_i = \emptyset \right\}.$$ Certainly $S \neq \emptyset$ by the property (b). So $S$ is an object of ${\mathcal{C}}$. Moreover one can see that $V \in \cap_{i \in S} {\mathcal{F}}(\{i\})$. This amounts to saying that $V \in {\mathcal{F}}(S)$ since $\cap_{i \in S} {\mathcal{F}}(\{i\}) = {\mathcal{F}}(\cup_{i \in S} \{i\})$ by (a). So $(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}$. Hence the pair $((S, V), id_V)$ is an object of $V \downarrow \theta$. We claim that this latter object is an initial object of $V \downarrow \theta$. To prove the claim, let $((T, W), V {\hookrightarrow}W)$ be another object of $V \downarrow \theta$. Since $V \subseteq W$, it follows that $\{i | \ V \cap A_i \neq \emptyset \}$ is a subset of $\{i | \ W \cap A_i \neq \emptyset\}$. This implies that $$\{i | W \cap A_i = \emptyset\} \subseteq \{i | V \cap A_i = \emptyset\} = S.$$ Furthermore, $T$ is a subset of $\{i | W \cap A_i = \emptyset\} $ since $W \in {\mathcal{F}}(T) = \Omega_k(U \backslash (\cup_{i \in T} A_i))$. So $T \subseteq S$, and therefore there is a unique morphism from $((S, V), id_V)$ to $((T, W), V {\hookrightarrow}W)$ in the under category $V \downarrow \theta$. This completes the proof.
\[poly\_lem\] Let ${\mathcal{M}}$ be a simplicial model category.
1. Let ${\mathcal{O}}$ and ${\mathcal{O}_k(M)}$ as in Example \[fso\_expl\]. Let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor $F^{!}_{{\mathcal{O}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ (see Example \[fso\_expl\]) is polynomial of degree $\leq k$.
2. Let ${\mathcal{B}}$ and ${\mathcal{B}}_k(M)$ as in Definition \[fsb\_defn\]. Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor. Then the cofunctor $F^{!}_{{\mathcal{B}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ (see Definition \[fsb\_defn\]) is polynomial of degree $\leq k$.
We begin with the first part. First let us consider again the notation (${\mathcal{C}}, U(S), \Omega, \Omega_k(U), {\mathcal{F}}$) introduced in Example \[intcf\_expl2\]. We will first show that the canonical map $$\Phi_{\Omega} \colon \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V)$$ is a weak equivalence. Next, by using the fact that $\Omega$ is another basis (for the topology of $U$) contained in ${\mathcal{O}}|U$, we will deduce that the canonical map $$\Phi_{{\mathcal{O}}} \colon \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{O}_k}(U(S))}{\text{holim}} \; F(V)$$ is also a weak equivalence.
One can see that the map $\Phi_{\Omega}$ factors through $\underset{(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V)$. That is, there is a commutative triangle $$\xymatrix{ \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) \ar[rr]^-{\Phi_{\Omega}} \ar[rd]_-{\alpha} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V) \\
& \underset{(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V) , \ar[ru]_-{\beta} }$$ where
1. $\alpha$ is nothing but $[\theta; F]$ (see (\[thetax\])), where $\theta \colon {\int_{{\mathcal{C}}} {\mathcal{F}}}{\longrightarrow}\Omega_k(U)$ is the map from Lemma \[cofinal\_lem\].
2. $\beta$ is the canonical map from Theorem \[cha\_sch\_thm\].
Since $\alpha$ is a weak equivalence by Lemma \[cofinal\_lem\] and Theorem \[htpy\_cofinal\_thm\], and since $\beta$ is a weak equivalence by Theorem \[cha\_sch\_thm\], it follows that $\Phi_{\Omega}$ is a weak equivalence as well.
Now consider the following commutative square induced by the inclusion $\Omega_k(U) {\hookrightarrow}{\mathcal{O}_k}(U)$. $$\xymatrix{ \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) \ar[rr]^-{\Phi_{\Omega}}_-{\sim} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V) \\
\underset{V \in {\mathcal{O}}_k(U)}{\text{holim}} \; F(V) \ar[u]^-{\sim} \ar[rr]_-{\Phi_{{\mathcal{O}}}} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{O}_k}(U(S))}{\text{holim}} \; F(V) \ar[u]_-{\sim} }$$ Since the lefthand vertical map is a weak equivalence by Theorem \[sos\_thm\], and since the righthand vertical map is also a weak equivalence by Theorems \[sos\_thm\], \[fib\_cofib\_thm\] (remember that $U(S) := U\backslash \cup_{i \in S}A_i$ and ${\mathcal{F}}(S) := \Omega_k(U(S))$), it follows that $\Phi_{{\mathcal{O}}}$ is a weak equivalence as well. And this proves part (i).
Now we prove part (ii). Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor. We have to show that ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. First, consider the cofunctor $G \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ defined as $G(U) := {F^{!}_{{\mathcal{B}}}}| {\mathcal{O}_k(M)}$, the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{O}_k(M)}$. Certainly $G$ is an isotopy cofunctor since ${F^{!}_{{\mathcal{B}}}}$ is good by Theorem \[good\_thm\]. This implies by the first part that the cofunctor ${G^{!}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is polynomial of degree $\leq k$. Moreover the canonical map ${G^{!}_{{\mathcal{O}}}}{\longrightarrow}(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is a weak equivalence by Theorem \[sos\_thm\]. So $(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is also polynomial of degree $\leq k$. Now, since the canonical map $F {\longrightarrow}G | {{\mathcal{B}}_k}(M)$ is a weak equivalence by Proposition \[fsrik\_prop\], it follows that the induced map ${F^{!}_{{\mathcal{B}}}}{\longrightarrow}(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is a weak equivalence as well. And therefore ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. This proves the lemma.
\[charac\_lem\] Let ${\mathcal{M}}$ be a simplicial model category.
1. Let $F, G \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be good and polynomial cofunctors of degree $\leq k$. Let $\eta \colon F {\longrightarrow}G$ be a natural transformation such that for any $U \in {\mathcal{O}_k(M)}$ the component $\eta[U] \colon F(U) {\longrightarrow}G(U)$ is a weak equivalence. Then for any $U \in {\mathcal{O}(M)}$ the map $\eta[U]$ is a weak equivalence.
2. Let $F, G \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be good and polynomial cofunctors of degree $\leq k$. Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Consider a natural transformation $\eta \colon F {\longrightarrow}G$ such that for any $U \in {\mathcal{B}_k(M)}$ the component $\eta[U] \colon F(U) {\longrightarrow}G(U)$ is a weak equivalence. Then for any $U \in {\mathcal{O}(M)}$ the map $\eta[U]$ is a weak equivalence.
The first part can be proved by following exactly the same steps as those of the proof of Theorem 5.1 from [@wei99]. Now we prove the second part. Let $U \in {\mathcal{O}_k(M)}$. Since ${\mathcal{B}}$ is a basis for the topology of $M$, there exists $V \in {\mathcal{B}_k(M)}$ contained in $U$ and such that the inclusion $V {\hookrightarrow}U$ is an isotopy equivalence. Applying $\eta$ to $V {\hookrightarrow}U$, we get the following commutative square. $$\xymatrix{F(U) \ar[r]^-{\sim} \ar[d]_-{\eta[U]} & F(V) \ar[d]^-{\sim}_-{\eta[V]} \\
G(U) \ar[r]_-{\sim} & G(V).}$$ The top and the bottom maps are weak equivalences since $F$ and $G$ are good by hypothesis. The righthand vertical map is a weak equivalence by assumption. So the lefthand vertical map is also a weak equivalence. Hence $\eta[U]$ is a weak equivalence for every $U \in {\mathcal{O}_k(M)}$. Now the desired result follows from the first part.
We are now ready to prove the main result of the paper.
Assume that $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is good and polynomial of degree $\leq k$. Define $G$ to be the restriction of $F$ to ${\mathcal{B}_k(M)}$. That is, $G:= F|{{\mathcal{B}}_k}(M)$. Since $F$ is good, and then in particular an isotopy cofunctor, it follows that $G$ is an isotopy cofunctor as well. Now we want to show that the canonical map $\eta \colon F {\longrightarrow}G^{!}$ is a weak equivalence. Let $U \in {\mathcal{O}(M)}$. One can rewrite $\eta[U] \colon F(U) {\longrightarrow}G^{!}(U)$ as the composition $$\eta[U] \colon \xymatrix{ F(U) \ar[r]^-{f} & \underset{V \in {\mathcal{O}}(U)}{\text{holim}} \; F(V) \ar[r]^-{g} & G^{!} (U), }$$ where $f$ comes from the fact that $U$ is the terminal object of ${\mathcal{O}}(U)$. The same fact allows us to conclude that $f$ is a weak equivalence. The map $g$ is nothing but $[\theta; F]$, where $\theta \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\mathcal{O}}(U)$ is just the inclusion functor. Now assume $U \in {{\mathcal{B}}_k}(M)$. Then for every $V \in {\mathcal{O}}(U)$ the under category $V \downarrow \theta$ is contractible since it has a terminal object, namely $(U, V {\hookrightarrow}U)$. Therefore, by Theorem \[htpy\_cofinal\_thm\], the map $g$ is a weak equivalence. This implies that $\eta[U]$ is a weak equivalence when $U \in {\mathcal{B}_k(M)}$. So, by Lemma \[charac\_lem\], $\eta[U]$ is also a weak equivalence for any $U \in {\mathcal{O}(M)}$.
Conversely, assume that $G:=F|{{\mathcal{B}}_k}(M)$ is an isotopy cofunctor and that the canonical map $\eta \colon F \stackrel{\sim}{{\longrightarrow}} G^{!}$ be a weak equivalence. By Theorem \[good\_thm\] and Lemma \[poly\_lem\] the cofunctor $G^{!}$ is good and polynomial of degree $\leq k$, which proves the converse. We thus obtained the desired result.
Homogeneous cofunctors {#hc_section}
======================
The goal of this section is to prove Theorem \[main2\_thm\] (announced in the introduction), which roughly says that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. We begin with three definitions. Next we prove Lemma \[homo\_lem\], which is the key lemma here, and which roughly states that homogeneous cofunctors of degree $k$ are determined by their values on open subsets diffeomorphic to the disjoint union of exactly $k$ balls. Note that this lemma is also a useful result in its own right, and its proof is based on the results we obtained in Section \[sos\_good\_section\] and Section \[poly\_section\].
Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be a cofunctor. The *$k$th polynomial approximation* to $F$, denoted $T_kF$, is the cofunctor $T_k F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $$T_k F (U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} F(V).$$
\[hc\_defn\] Let ${\mathcal{M}}$ be a simplicial model category that has a terminal object denoted $0$.
1. A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *homogeneous of degree* $k$ if it satisfies the following three conditions:
1. $F$ is a good cofunctor (see Definition \[good\_defn\]);
2. $F$ is polynomial of degree $\leq k$ (see Definition \[poly\_defn\]);
3. The unique map $T_{k-1} F (U) {\longrightarrow}0$ is a weak equivalence for every $U \in {\mathcal{O}(M)}$.
2. A *linear cofunctor* is a homogeneous cofunctor of degree $1$.
The category of homogeneous cofunctors of degree $k$ and natural transformations will be denoted by ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$.
\[we\_defn\] Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be categories both equipped with a class of maps called weak equivalences.
1. We say that two functors $F, G \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ are *weakly equivalent*, and we denote $F \simeq G$, if they are connected by a zigzag of objectwise weak equivalences.
2. A functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is said to be a *weak equivalence* if it satisfies the following two conditions.
1. $F$ preserves weak equivalences.
2. There is a functor $G \colon {\mathcal{D}}{\longrightarrow}{\mathcal{C}}$ such that $FG$ and $GF$ are both weakly equivalent to the identity. The functor $G$ is also required to preserve weak equivalences.
3. We say that ${\mathcal{C}}$ is weakly equivalent to ${\mathcal{D}}$, and we denote ${\mathcal{C}}\simeq {\mathcal{D}}$, if there exists a zigzag of weak equivalences between ${\mathcal{C}}$ and ${\mathcal{D}}$.
By Definition \[we\_defn\], it follows that if two categories ${\mathcal{C}}$ and ${\mathcal{D}}$ are weakly equivalent, then their localizations with respect to weak equivalences are equivalent in the classical sense. Note that no model structure is required on ${\mathcal{C}}$ and ${\mathcal{D}}$. So our notion of weak equivalences between categories is not comparable, in general, with the well known notion of Quillen equivalence.
As mentioned earlier the following lemma is the key ingredient in proving Theorem \[main2\_thm\].
\[homo\_lem\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Let ${\mathcal{B}^{(k)}}(M) \subseteq {\mathcal{O}(M)}$ denote the subposet whose objects are disjoint unions of exactly $k$ elements from ${\mathcal{B}}$, and whose morphisms are isotopy equivalences. Let ${\mathcal{M}}$ be a simplicial model category. Assume that ${\mathcal{M}}$ has a zero object $0$ (that is, an object which is both terminal an initial).
1. Then the category ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ of homogeneous cofunctors of degree $k$ (see Definition \[hc\_defn\]) is weakly equivalent (in the sense of Definition \[we\_defn\]) to the category ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ of isotopy cofunctors ${\mathcal{B}^{(k)}}(M) {\longrightarrow}{\mathcal{M}}$ (see Definition \[isotopy\_cof\_defn\]). That is, $${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\simeq {\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}).$$
2. For $A \in {\mathcal{M}}$ we have the weak equivalence $${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_A({\mathcal{B}^{(k)}}(M); {\mathcal{M}}),$$ where ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}})$ is the category from Theorem \[main2\_thm\] and ${\mathcal{F}}_A({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ denotes the category of isotopy cofunctors $F \colon {\mathcal{B}^{(k)}}(M) {\longrightarrow}{\mathcal{M}}$ such that $F(U) \simeq A$ for every $U \in {\mathcal{B}^{(k)}}(M)$.
We will prove the first part; the proof of the second part is similar. The idea of the proof is to define a new category and show that it is weakly equivalent to both ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ and ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$. To define that category, let us first recall the notation ${\mathcal{B}_k(M)}$ from Definition \[fsb\_defn\]. Define ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$ to be the category whose objects are isotopy cofunctors $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ such that the restriction to ${\mathcal{B}}_{k-1} (M)$ is weakly equivalent to the constant functor at $0$. That is, $$\begin{aligned}
\label{wd_cond}
\text{for all $U \in {\mathcal{B}}_{k-1}(M)$,} \quad F(U) \simeq 0. \end{aligned}$$ Now consider the following diagram $$\begin{aligned}
\label{psii_phii}
\xymatrix{{\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}) \ar@<1ex>[r]^-{\psi_1} & {\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}}) \ar@<1ex>[l]^-{\phi_1} \ar@<1ex>[r]^-{\psi_2} & {{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\ar@<1ex>[l]^-{\phi_2} }\end{aligned}$$ where the maps are defined as follows.
1. $\phi_1$ is the restriction functor. That is, $\phi_1(F) = F|{\mathcal{B}^{(k)}}(M)$.
2. $\phi_2$ is also the restriction functor. To see that it is well defined, let $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be an object of ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$. We have to check that $F$ satisfies condition (\[wd\_cond\]). So let $U \in {\mathcal{B}}_{k-1}(M)$. Recalling the notation $[-; -]$ from Proposition \[induced\_holim\_prop\], we have the following commutative diagram $$\xymatrix{\underset{V \in {\mathcal{O}}_{k-1}(U)}{\text{holim}} \; F(V) \ar[rr]^-{\sim} \ar[d]_-{[\theta; F]}^-{\sim} & & 0 \\
\underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; F(V) \ar[rru]^-{\sim} & & F(U). \ar[ll]^-{\sim} \ar[u] }$$ Here $\theta \colon {\mathcal{B}}_{k-1}(U) {\hookrightarrow}{\mathcal{O}}_{k-1}(U)$ is the inclusion functor. The bottom horizontal map is a weak equivalence since $U$ is the terminal object of ${\mathcal{B}}_{k-1}(U)$ (see Proposition \[fsrik\_prop\]). The top one is a weak equivalence since $F$ is homogeneous of degree $k$. Since the lefthand vertical map is also a weak equivalence (by Theorem \[sos\_thm\]), it follows that $F(U)$ is weakly equivalent to $0$.
3. $\psi_1$ is defined as $$\psi_1(F)(U) := \left\{ \begin{array}{cc}
F(U) & \text{if $U \in {\mathcal{B}^{(k)}}(M)$ } \\
0 & \text{otherwise,}
\end{array} \right.$$ Certainly $\psi_1(F)$ satisfies (\[wd\_cond\]) and is an isotopy cofunctor. This latter assertion comes from the fact that if $U \subseteq U'$ is an isotopy equivalence, then $U$ and $U'$ definitely have the same number of connected components. On morphisms $\psi_1$ is defined in the obvious way.
4. $\psi_2$ is defined as $\psi_2(F) := {F^{!}_{{\mathcal{B}}}}$ (see Definition \[fsb\_defn\]). On morphisms $\psi_2$ is defined by the fact that the homotopy right Kan extension is functorial. By Theorem \[good\_thm\] and Lemma \[poly\_lem\], it is clear that $\psi_2(F)$ is good and polynomial of degree $\leq k$. To see that $\psi_2(F)$ satisfies condition (c) from Definition \[hc\_defn\], let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an object of ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$. Consider the following commutative diagram $$\xymatrix{\underset{V \in {\mathcal{O}}_{k-1}(U)}{\text{holim}} \; {F^{!}_{{\mathcal{B}}}}(V) \ar[r]^-{[\theta; {F^{!}_{{\mathcal{B}}}}]}_-{\sim} \ar[d] & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; {F^{!}_{{\mathcal{B}}}}(V) \ar@{=}[r] & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; \underset{W \in {{\mathcal{B}}_k}(V)}{\text{holim}} \; F(W) \ar[lld]^-{\sim} \\
0 & & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; F(V), \ar[ll]^-{\sim} \ar[u]_-{\sim}}$$ where the righthand vertical map is induced by the canonical map $F(V) {\longrightarrow}\underset{W \in {{\mathcal{B}}_k}(V)}{\text{holim}} \; F(W)$. Since $V$ belongs to ${\mathcal{B}}_{k-1}(U)$ it follows that $V$ is the terminal object of ${{\mathcal{B}}_k}(V)$, and therefore this latter map is a weak equivalence (by Proposition \[fsrik\_prop\]). The bottom horizontal map is a weak equivalence since $F$ belongs to ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$, and then satisfies (\[wd\_cond\]). Regarding the map $[\theta; {F^{!}_{{\mathcal{B}}}}]$, it is a weak equivalence by Theorem \[sos\_thm\]. All this implies that the lefthand vertical map is a weak equivalence as well. So ${F^{!}_{{\mathcal{B}}}}$ satisfies condition (c).
Certainly $\phi_1, \psi_1$ and $\phi_2$ preserve weak equivalences. The functor $\psi_2$ preserves weak equivalences as well by Theorem \[fib\_cofib\_thm\] and condition (a) from Definition \[isotopy\_cof\_defn\]. Moreover, it is clear that $\phi_1\psi_1 =id$ and $\psi_1 \phi_1 \simeq id$. So the category ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ is weakly equivalent to the category ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$. Furthermore, by using Propsosition \[fsrik\_prop\], one can easily come to $\phi_2\psi_2 \simeq id$ and $\psi_2\phi_2 \simeq id$. So the categories ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$ and ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ are also weakly equivalent. This proves the lemma.
We are now ready to prove Theorem \[main2\_thm\].
We will prove the first part; the proof of the second part is similar. Recall the notation $F_k(M)$, which is that of the space of unordered configuration of $k$ points in $M$. The proof of part (i) follows from the following three weak equivalences: (\[we1\]), (\[we2\]), and (\[we3\]). The first one $$\begin{aligned}
\label{we1}
{{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\simeq {\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}),\end{aligned}$$ is nothing but Lemma \[homo\_lem\] -(i). The second $$\begin{aligned}
\label{we2}
{\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}) \cong {\mathcal{F}}({\mathcal{B}}'^{(1)} (F_k(M)); {\mathcal{M}}),\end{aligned}$$ is actually an isomorphism where ${\mathcal{B}}'$ is the basis for the topology of $F_k(M)$ whose elements are products of exactly $k$ elements from ${\mathcal{B}}$. This isomorphism comes from the fact that ${\mathcal{B}^{(k)}}(M) \cong {\mathcal{B}}'^{(1)} (F_k(M))$. The last weak equivalence $$\begin{aligned}
\label{we3}
{\mathcal{F}}({\mathcal{B}}'^{(1)} (F_k(M)); {\mathcal{M}}) \simeq {\mathcal{F}}_1 ({\mathcal{O}}(F_k(M)); {\mathcal{M}}),\end{aligned}$$ is again Lemma \[homo\_lem\] -(i).
Isotopy cofunctors in general model categories {#iso_cof_section}
==============================================
This section is independent of previous ones, and its goal is to prove Theorem \[iso\_cof\_thm\] (announced in the introduction), which says that the cofunctor $F^{!}={F^{!}_{{\mathcal{O}}}}$ from Example \[fso\_expl\] is an isotopy cofunctor provided that $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor (here ${\mathcal{M}}$ is a general model category). This result is proved in Theorem \[good\_thm\] when ${\mathcal{M}}$ is a simplicial model category. To prove Theorem \[good\_thm\] we used several results/properties (about homotopy limits in simplicial model categories) including Theorem \[fubini\_thm\], Proposition \[comm\_prop\]. This latter result involves the notion of totalization of a cosimplicial object, which does not make sense in a general model category. So the method we used before do not work here anymore. In this section we present a completely different approach, but rather lengthy, that uses only two properties of homotopy limits (see Theorem \[fib\_cofib\_thmg\] and Theorem \[htpy\_cofinal\_thmg\]). That approach is inspired by our work in [@paul_don17]. For the plan of this section, we refer the reader to the table of contents and the outline given at the introduction. For a faster run through the section, the reader could, after reading the Introduction, jump directly to the beginning of Section \[iso\_cof\_subsection\] to get a better idea of the proof of Theorem \[iso\_cof\_thm\].
Homotopy limits in general model categories {#holim_subsectiong}
-------------------------------------------
This subsection recalls some useful properties of homotopy limits in general model categories. We also recall two results (Proposition \[induced\_holim\_propg\] and Proposition \[fsrik\_propg\]) that will be used in next subsections.
Homotopy limits and colimits in general model categories are constructed in [@hir03; @dhks04] by W. Dwyer, P. Hirschhorn, D. Kan, and J. Smith. They use the notion of *frames* that we now recall briefly. Let ${\mathcal{M}}$ be a model category, and let $X$ be an object of ${\mathcal{M}}$. A *cosimplicial frame* on $X$ is a cofibrant replacement (in the Reedy model category of cosimplicial objects in ${\mathcal{M}}$) of the constant cosimplicial object at $X$ that satisfies certain properties. A *simplicial frame* on $X$ is the dual notion. For a more precise definition we refer the reader to [@hir03 Definition 16.6.1]. A *framing* on ${\mathcal{M}}$ is a functorial cosimplicial and simplicial frame on every object of ${\mathcal{M}}$. A *framed model category* is a model category endowed with a framing (see also [@hir03 Definition 16.6.21]). A typical example of a framed model category is any simplicial model category as we considered in previous sections.
In [@hir03 Theorem 16.6.9] it is proved that there exists a framing on any model category. It is also proved that two any framings are weakly equivalent [@hir03 Theorem 16.6.10]. Throughout this section, ${\mathcal{M}}$ is a model category endowed with a fixed framing.
Using the notion of framing, one can define the homotopy limit and colimit of a diagram in ${\mathcal{M}}$. We won’t give that definition here since it is not important for us (the reader who is interested in that definition can find it in [@hir03 Definition 19.1.2 and Definition 19.1.5]). All we need are some properties of that homotopy limit and colimit (see Theorem \[fib\_cofib\_thmg\] and Theorem \[htpy\_cofinal\_thmg\] below).
[@hir03 Theorem 19.4.2] \[fib\_cofib\_thmg\] Let ${\mathcal{M}}$ be a model category, and let ${\mathcal{C}}$ be a small category. Let $\eta \colon F {\longrightarrow}G$ be a map of ${\mathcal{C}}$-diagrams in ${\mathcal{M}}$.
1. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence of cofibrant objects, then the induced map of homotopy colimits $\text{hocolim} \; F {\longrightarrow}\text{hocolim} \; G$ is a weak equivalence of cofibrant objects of ${\mathcal{M}}$.
2. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence of fibrant objects, then the induced map of homotopy limits $\text{holim} \; F {\longrightarrow}\text{holim} \; G$ is a weak equivalence of fibrant objects of ${\mathcal{M}}$.
[@hir03 Theorem 19.6.7] \[htpy\_cofinal\_thmg\] Let ${\mathcal{M}}$ be a model category. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is homotopy left cofinal (respectively homotopy right cofinal) (see Definition \[cofinal\_defn\]), then for every objectwise fibrant covariant (respectively contravariant) functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, the natural map $[\theta; F]$ from Proposition \[induced\_holim\_propg\] is a weak equivalence.
We will also need the following two propositions. The first is a generalization of Proposition \[induced\_holim\_prop\], while the second is a generalization of Proposition \[fsrik\_prop\].
[@hir03 Proposition 19.1.8] \[induced\_holim\_propg\] Let ${\mathcal{M}}$ be a model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between two small categories. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram, then there is a canonical map $$\begin{aligned}
\label{thetaxg}
[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F.\end{aligned}$$ (see (\[theta\_starx\])) Furthermore, this map is natural in both variables $\theta$ and $F$ as in Proposition \[induced\_holim\_prop\].
\[fsrik\_propg\] Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Let ${\mathcal{M}}$ be a model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor.
1. There is a natural transformation $\eta$ from $F$ to the restriction ${F^{!}_{{\mathcal{B}}}}| {\mathcal{B}_k(M)}$ (see Definition \[fsb\_defn\]), which is an objectwise weak equivalence.
2. If $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), then so is the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{B}_k(M)}$.
This is very similar to the proof of Proposition \[fsrik\_prop\].
The category ${{\mathcal{D}}(V)}$ {#dv_subsection}
---------------------------------
Consider the following data:
1. $U, U' \in {\mathcal{O}(M)}$ such that $U \subseteq U'$;
2. $V \in {\mathcal{O}_k}(U)$;
3. $L \colon U \times I {\longrightarrow}U', (x, t) \mapsto L_t(x):= L(x, t)$ is an isotopy from $U$ to $U'$ (see Definition \[iso\_eq\_defn\]).
The aim of this subsection is to define an important category ${{\mathcal{D}}(V)}$ (see Definition \[dv\_defn\]) out of these data. The definition of ${{\mathcal{D}}(V)}$ is rather technical. Roughly speaking, an object of ${{\mathcal{D}}(V)}$ is a zigzag $x$ of isotopy equivalences between $W$ and $L_1(W)$, where $W$ is a object of ${\mathcal{O}_k}(V)$. Morphisms of ${{\mathcal{D}}(V)}$ are inclusions. There are two types of morphisms from $x$ to $y$ depending on the fact that $x$ and $y$ have the same length or not. If $x$ and $y$ have the same length, a morphism from $x$ to $y$ is just an inclusion. Otherwise a morphism is still an inclusion, but more subtle. We also prove Proposition \[dv\_contractible\_prop\], which says that ${{\mathcal{D}}(V)}$ is contractible.
Let us begin with the following notation, and some technical definition.
\[ssie\_not\] Given two objects $W, T \in {\mathcal{O}(M)}$, we use the notation $W {\subseteq_{ie}}T$ to mean that $W$ is a subset of $T$ and the inclusion map $W {\hookrightarrow}T$ is an isotopy equivalence.
In [@paul_don17 Section 3.2] we introduced the concept of *admissible family* $x=\{a_0, \cdots, a_{n+1}\}$ with respect to $L$ and a compact subset $K \subseteq U$. If one has different compact for each interval $[a_i, a_{i+1}]$, the family $x$ is said to be *piecewise admissible*. More precisely, we have the following definition.
\[pw\_adm\_defn\] Let $W \in {\mathcal{O}_k}(U)$, and let $[a, b] \subseteq [0, 1]$. Let $x=\{a_0, \cdots, a_{n+1}\} \subseteq [0, 1]$ be a family such that $a_0 = a, a_{n+1} = b,$ and $a_i \leq a_{i+1}$ for all $i$. Let $K = \{K_0, \cdots, K_n\}$ be a family of nonempty compact subsets $K_i \subseteq L_{a_i}(W)$ such that $\pi_0(K_i {\hookrightarrow}L_{a_i}(W))$ is surjective. The family $x$ is said to be *piecewise admissible* with respect to $\{K, L\colon W \times I {\longrightarrow}U'\}$ (or just *piecewise admissible*) if for every $i$ there exists an object $W_{i(i+1)}$ of ${\mathcal{O}_k(M)}$ such that for all $s \in [a_i, a_{i+1}]$, $$\begin{aligned}
L_s(L_{a_i}^{-1}(K_i)) \subseteq W_{i(i+1)} {\subseteq_{ie}}L_s(W),\end{aligned}$$ and $$\begin{aligned}
\label{vii_eqn}
\overline{W_{i(i+1)}} \subseteq L_s(W),\end{aligned}$$ where $L_{a_i} \colon W {\longrightarrow}L_{a_i}(W)$ is the canonical homeomorphism induced by $L$, and $\overline{W_{i(i+1)}}$ stands for the closure of $W_{i(i+1)}$.
The following proposition, which will be used in the proof of Proposition \[dv\_contractible\_prop\], can be deduced easily from [@paul_don17 Proposition 3.10 ].
\[existence\_adm\_prop\] Let $[a, b] \subseteq I,$ and let $t \in [a, b]$. Let $W \in {\mathcal{O}_k}(U)$, and let $K \subseteq L_t(W)$ be a nonempty compact subset such that $\pi_0(K {\hookrightarrow}L_a(W))$ is surjective.
1. If $t = a$ (respectively $t=b$), there exists $t' > t$ (respectively $t'' < t$) such that the family $\{t, t'\}$ (respectively $\{t'', t\}$) is admissible with respect to $\{K, L\}$.
2. If $t \in (a, b)$, there exists $\epsilon_t >0$ such that the family $\{t-\epsilon_t, t+\epsilon_t\}$ is admissible with respect to $\{K, L\}$.
\[mcale\_defn\] Define ${\mathcal{E}}$ to be the category whose objects are finite subsets $A= \{a_0, \cdots, a_{n+1}\}$ of the interval $[0, 1]$ such that $a_0 = 0, a_{n+1} = 1$ and $a_i \leq a_{i+1}$ for all $i$. Morphisms of ${\mathcal{E}}$ are inclusions.
\[ia\_defn\] Let $A = \{a_0, \cdots, a_{n+1}\}$ be an object of ${\mathcal{E}}$. Define ${\mathcal{I}}_A$ to be the poset whose objects are $$\{a_0\}, \{a_1\} \cdots, \{a_n\}, \{a_0, a_1\}, \{a_1, a_2\}, \cdots, \{a_{n-1}, a_n\},$$ and whose morphisms are inclusions $\{a_i\} {\longrightarrow}\{a_i, a_{i+1}\}$ and $\{a_{i+1}\} {\longrightarrow}\{a_i, a_{i+1}\}, 0 \leq i \leq n$.
The category ${\mathcal{I}}_A$ looks like a zigzag starting at $\{a_0\} = \{0\}$ and ending at $\{a_{n+1}\} =\{1\}$. For instance, if $n=2$, then $${\mathcal{I}}_A = \left\{\xymatrix{\{a_0\} \ar[r] & \{a_0, a_1\} & \{a_1\} \ar[l] \ar[r] & \{a_1, a_2\} & \{a_2\} \ar[l] } \right\}.$$
\[theta\_ab\_prop\] The construction that sends $A$ to ${\mathcal{I}}_A$ is a contravariant functor ${\mathcal{E}}{\longrightarrow}\text{Cat}$ from ${\mathcal{E}}$ to the category Cat of small categories.
Given $A, B \in {\mathcal{E}}$ such that $A \subseteq B$ with $B = \{b_0, \cdots, b_{m+1}\}$, we need to define a morphism $$\begin{aligned}
\label{theta_ab}
\theta_{AB} \colon {\mathcal{I}}_B {\longrightarrow}{\mathcal{I}}_A.\end{aligned}$$ Let us begin with an example. Take $A = \{a_0, a_1, a_2\}$ and $B = \{b_0, b_1, b_2, b_3\}$ such that $b_2 = a_1$ as shown Figure \[ab\_fig\].
The idea of the definition of $\theta_{AB}$ is as follows. First consider the elements of $A \cap B = \{b_0, b_2, b_3\}$, and define $\theta_{AB}(\{b_0\}) = \{a_0\}, \theta_{AB}(\{b_2\}) = \{a_1\}$, and $\theta_{AB}(\{b_3\}) = \{a_2\}$. Next consider $B \backslash A = \{b_1\}$. Since $D =[a_0, a_1]$ is the smallest closed interval containing $b_1$ such that $\text{Inf} D \in A, \text{Sup} D \in A$, and $D \cap A = \{a_0, a_1\}$, we have $\theta_{AB}(\{b_1\}) := \{a_0, a_1\}$. A similar observation gives $
\theta_{AB}(\{b_0, b_1\}) := \{a_0, a_1\}, \theta_{AB}(\{b_1, b_2\}):= \{a_0, a_1\}, \text{ and } \theta_{AB}(\{b_2, b_3\}) = \{a_1, a_2\}.
$ The following diagram summarizes the definition of $\theta_{AB}$. $$\xymatrix{\{a_0\} \ar[rr] & & \{a_0, a_1\} & & \{a_1\} \ar[ll] \ar[r] & \{a_1, a_2\} & \{a_2\} \ar[l] \\
\{b_0\} \ar[r] \ar[u] & \{b_0, b_1\} \ar[ru] & \{b_1\} \ar[u] \ar[l] \ar[r] & \{b_1, b_2\} \ar[lu] & \{b_2\} \ar[l] \ar[r] \ar[u] & \{b_2, b_3\} \ar[u] & \{b_3\}. \ar[l] \ar[u]}$$
Now we give a precise definition of $\theta_{AB}$. For $b \in B$, define $$c(b) := \text{max}\{x \in A | \ x \leq b\} \quad \text{and} \quad d(b) := \text{min}\{x \in A| \ x \geq b\}.$$ Now define $\theta_{AB}$ as $$\theta_{AB}(\{b\}) = \left\{ \begin{array}{ccc}
\{b\} & \text{if} & b \in A \\
\{c(b), d(b)\} & \text{if} & b \notin A,
\end{array} \right.$$ and $$\theta_{AB}(\{b_i, b_{i+1}\}) = \{c(b_i), d(b_{i+1})\}.$$ On morphisms of ${\mathcal{I}}_B$, $\theta_{AB}$ is defined in the most obvious way. Regarding the composition, if $A, B, C \in {\mathcal{E}}$ such that $A \subseteq B \subseteq C$, then one obviously has $$\begin{aligned}
\label{theta_abc}
\theta_{AC} = \theta_{AB}\theta_{BC}, \end{aligned}$$ which completes the proof.
We are now ready to define ${{\mathcal{D}}(V)}$.
\[dv\_defn\] Recall the posets ${\mathcal{E}}$ and ${\mathcal{I}}_A$ from Definition \[mcale\_defn\] and Definition \[ia\_defn\] respectively. Also recall the isotopy $L$ from the beginning of this subsection. The category ${{\mathcal{D}}(V)}$ is defined as follows.
1. An object is a triple $(W, A, {\mathcal{X}_A})$ (or just a pair $(W, {\mathcal{X}_A}$)) where $W \in {\mathcal{O}_k}(V)$, $A = \{a_0, \cdots, a_{n+1}\} \in {\mathcal{E}}$, and ${\mathcal{X}_A}\colon {\mathcal{I}}_A {\longrightarrow}{\mathcal{O}_k(M)}$ is a contravariant functor that satisfies the following three conditions:
1. ${\mathcal{X}_A}(\{a\}) = L_a(W)$ for all $a \in A$.
2. For every $i \in \{0, \cdots, n\}$, for every $s \in [a_i, a_{i+1}]$, $${\mathcal{X}_A}(\{a_i, a_{i+1}\}) {\subseteq_{ie}}L_s(W).$$ (See Notation \[ssie\_not\] for the meaning of ${\subseteq_{ie}}$.)
3. For every $i \in \{0, \cdots, n\}$, for every $s \in [a_i, a_{i+1}]$, $$\begin{aligned}
\label{xai_eqn}
\overline{{\mathcal{X}_A}(\{a_i, a_{i+1}\})} \subseteq L_s(W).
\end{aligned}$$
2. A morphism from $(W, A, {\mathcal{X}_A})$ to $(T, B, {\mathcal{Y}_B})$ consists of a triple $(f, g, \Lambda_{AB})$ (or just $\Lambda_{AB}$) where $f \colon W {\hookrightarrow}T$ and $g \colon A {\hookrightarrow}B$ are both the inclusion maps, and $\Lambda_{AB} \colon {\mathcal{X}_A}\theta_{AB} {\longrightarrow}{\mathcal{Y}_B}$ is a natural transformation.
In other words, an object of ${{\mathcal{D}}(V)}$ is a zigzag of isotopy equivalences between $L_0(W) = W$ and $L_1(W)$, where $W \in {\mathcal{O}_k}(V)$. For instance, when $A = \{a_0, a_1, a_2\}$, an object looks like (\[xzig\]). $$\begin{aligned}
\label{xzig}
(W,{\mathcal{X}_A}) = \left\{ \xymatrix{W=X_0 & X_{01} \ar[l]_-{\simeq} \ar[r]^-{\simeq} & X_1 & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} & X_2 = L_1(W)} \right\}.\end{aligned}$$ There are two kind of morphisms from $(W,{\mathcal{X}_A})$ to $(T, {\mathcal{Y}_B})$ depending on the fact that $A =B$ or $A$ is a proper subset of $B$. These morphisms are illustrated by (\[mor1\]) and (\[mor2\]). $$\begin{aligned}
\label{mor1}
\xymatrix{X_0 \ar[d] & X_{01} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_1 \ar[d] & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_2 \ar[d] \\
Y_0 & Y_{01} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_1 & Y_{12} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_2. }\end{aligned}$$
$$\begin{aligned}
\label{mor2}
\xymatrix{X_0 \ar[d] & & X_{01} \ar[ll]_-{\simeq} \ar[rr]^-{\simeq} \ar[d] \ar[ld] \ar[rd] & & X_1 \ar[d] & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_2 \ar[d] \\
Y_0 & Y_{01} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_1 & Y_{12} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_2 & Y_{23} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_3.}\end{aligned}$$
\[associated\_rmk\] To any piecewise admissible family $A$ (see Definition \[pw\_adm\_defn\]), one can associate a canonical object $(W, {\mathcal{X}_A})$ of ${{\mathcal{D}}(V)}$ by letting ${\mathcal{X}_A}(\{a_i, a_{i+1}\}) := W_{i(i+1)}.$
\[dv\_contractible\_prop\] The category ${{\mathcal{D}}(V)}$ is contractible.
It suffices to show that ${{\mathcal{D}}(V)}$ is *filtered*, that is, it satisfies the following two conditions:
1. For every pair of objects $(W,{\mathcal{X}_A})$ and $(T, {\mathcal{Y}_B})$ there are morphisms to a common object $(W, {\mathcal{X}_A}) {\longrightarrow}(S, {\mathcal{Z}_C})$ and $(T, {\mathcal{Y}_B}) {\longrightarrow}(S, {\mathcal{Z}_C})$;
2. For every pair of parallel morphisms $\Lambda_{AB}, \Lambda'_{AB} \colon (W, {\mathcal{X}_A}) {\longrightarrow}(T, {\mathcal{Y}_B})$, there is some morphism $\Lambda_{BC} \colon (T, {\mathcal{Y}_B}) {\longrightarrow}(S, {\mathcal{Z}_C})$ such that $\Lambda_{BC}\Lambda_{AB} = \Lambda_{BC} \Lambda'_{AB}$.
Since ${{\mathcal{D}}(V)}$ is a poset by definition, it clearly satisfies (2). To check (1), let $(W, {\mathcal{X}_A}), (T, {\mathcal{Y}_B}) \in {{\mathcal{D}}(V)}$. Set $D = A \cup B$. Certainly $D$ is a finite subset, denoted $\{d_0, \cdots, d_{p+1}\}$, of $I$ such that $d_0 = 0, d_{p+1} =1$, and $d_i \leq d_{i+1}$ for all $i$. One can write the intervals $[d_i, d_{i+1}], 0 \leq i \leq p,$ as $$[d_i, d_{i+1}] = \big[a_{r(i)}, a_{r(i)+1}\big] \cap \big[b_{s(i)}, b_{s(i)+1}\big],$$ where $$a_{r(i)} := \text{max}\left\{x \in A| \ x \leq d_i \right\} \quad \text{and} \quad b_{s(i)} := \text{max}\left\{y \in B| \ y \leq d_i\right\}.$$ Of course, $a_{r(i)+1}$ (respectively $b_{s(i)+1}$) is the successor of $a_{r(i)}$ in $A$ (respectively the successor of $b_{s(i)}$ in $B$). Note that the interior of $[d_i, d_{i+1}]$ does not intersect either $A$ or $B$.
Take $S = V$. The idea of the construction of ${\mathcal{Z}_C}$ is to subdivide each $[d_i, d_{i+1}]$ into small intervals $[a, b]$ such that there exists $Z_{ab} \in {\mathcal{O}_k(M)}$ that is contained in $L_a(V) \cap L_b(V)$ and that contains both ${\mathcal{X}_A}\left(\{a_{r(i)}, a_{r(i)+1}\}\right)$ and ${\mathcal{Y}_B}\left(\{b_{s(i)}, b_{s(i)+1}\}\right)$. So let $i \in \{0, \cdots, p\}$, and let $t \in [d_i, d_{i+1}]$. Thanks to (\[xai\_eqn\]) one can consider the compact subset ${\mathcal{K}}_i \subseteq L_t(V)$ defined as $${\mathcal{K}}_i = \overline{{\mathcal{X}_A}\left(\left\{a_{r(i)}, a_{r(i)+1} \right\} \right)} \bigcup \overline{{\mathcal{Y}_B}\left(\left\{b_{s(i)}, b_{s(i)+1} \right\} \right)}.$$ If $t \in (d_i, d_{i+1})$ then by Proposition \[existence\_adm\_prop\] there exist $\epsilon_t > 0$ and $Z_{i(i+1)} \in {\mathcal{O}_k(M)}$ such that for all $u \in [t-\epsilon_t, t+\epsilon_t]$, $$L_u(L_t^{-1}({\mathcal{K}}_i)) \subseteq Z_{i(i+1)} {\subseteq_{ie}}L_u(V) \quad \text{and} \quad \overline{Z_{i(i+1)}} \subseteq L_u(V).$$ Clearly one has $${\mathcal{X}_A}\left(\{a_{r(i)}, a_{r(i)+1}\}\right) \subseteq L_{t-\epsilon_t}(V) \cap Z_{i(i+1)} \cap L_{t+\epsilon} (V)$$ and $${\mathcal{Y}_B}\left(\{b_{s(i)}, b_{s(i)+1}\}\right) \subseteq L_{t-\epsilon_t}(V) \cap Z_{i(i+1)} \cap L_{t+\epsilon} (V).$$ If $t= d_i$ (respectively $t=d_{i+1}$) there is an admissible family $\{d_i, t'\}$ (respectively $\{t'', d_{i+1}\}$) again by Proposition \[existence\_adm\_prop\]. Letting $t$ vary in $[d_i, d_{i+1}]$ one obtains an open cover, $[d_i, t') \cup \{(t-\epsilon_t, t+\epsilon_t)\}_t \cup (t'', d_{i+1}])$, of $[d_i, d_{i+1}]$. Now, applying the compactness we get an ordered finite subset $C^i = \{c^i_0, \cdots, c^i_{n_i}\}$ of $[d_i, d_{i+1}]$ such that $c^i_0 = d_i$, $c^i_{n_i} = d_{i+1}$, and for every $j$ the interval $[c^i_j, c^i_{j+1}]$ is contained in one of the open subsets from the cover. This implies that $C:= \cup_{i=0}^p C^i$ is piecewise admissible and contains both $A$ and $B$. Moreover, it is clear that the associated object $(V, {\mathcal{Z}_C})$ of ${{\mathcal{D}}(V)}$ (as in Remark \[associated\_rmk\]) has the desired property. This ends the proof.
\[theta\_zo\_lem\] The functors $$\theta_0 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(V) \quad \text{and} \quad \theta_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(L_1(V))$$ defined as $\theta_0(W, {\mathcal{X}_A}) = W$ and $\theta_1(W, {\mathcal{X}_A}) = L_1(W)$ are both homotopy right cofinal (see Definition \[cofinal\_defn\]).
For every $W \in {\mathcal{O}_k}(V)$ the under category $W \downarrow \theta_0$ is contractible. This works exactly as the proof of Proposition \[dv\_contractible\_prop\]. Similarly, one can show that $\theta_1$ is homotopy right cofinal.
\[dv\_functor\_prop\] The construction ${\mathcal{D}}\colon {\mathcal{O}_k}(U) {\longrightarrow}\text{Cat}$ that sends $V$ to ${{\mathcal{D}}(V)}$ is a covariant functor.
It is very easy to establish. For a morphism $V {\hookrightarrow}V'$ of ${\mathcal{O}_k}(U)$, we define $\theta \colon {{\mathcal{D}}(V)}{\longrightarrow}{{\mathcal{D}}(V')}$ as $\theta(W, {\mathcal{X}_A}) = (W, {\mathcal{X}_A})$. Certainly this defines a functor from ${{\mathcal{D}}(V)}$ to ${{\mathcal{D}}(V')}$.
The functors $H, P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ {#hp_subsection}
----------------------------------------------------------------------------------
The goal of this subsection is to define three important functors, $H, P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, and two natural weak equivalences, $\eta_0 \colon H {\longrightarrow}P_0$ and $\eta_1 \colon H {\longrightarrow}P_1$. We will use them in the proof of Theorem \[iso\_cof\_thm\], which will be done at Subsection \[iso\_cof\_subsection\]. In this subsection ${\mathcal{M}}$ is a model category, $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), and $F^{!} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is the cofunctor defined by (\[fsrik\_defn\]). We continue to use the same data as those provided at the beginning of Subsection \[dv\_subsection\].
### The functor $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$
Before we define $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, we need to first recall a certain model category of diagrams, and next define $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ (for the categories ${{\mathcal{D}}(V)}$ and ${\mathcal{E}}$, see Definition \[dv\_defn\] and Definition \[mcale\_defn\] respectively), which is functorial in each argument.
For $E \in {\mathcal{E}}$, consider the category ${\mathcal{M}}^{{\mathcal{I}}_E}$ of ${\mathcal{I}}_E$-diagrams in ${\mathcal{M}}$ (recall the poset ${\mathcal{I}}_E$ from Definition \[ia\_defn\]). In the literature there exist many model structures on ${\mathcal{M}}^{{\mathcal{I}}_E}$. But for our purposes we endow it with the one described by Dwyer and Spalinski in [@dwyer_spa95 Section 10]. First recall that this model structure is only defined for diagrams indexed by *very small categories* (see the paragraph just after 10.13 from [@dwyer_spa95]), which is the case for ${\mathcal{I}}_E$. Next recall that this model structure states that weak equivalences and cofibrations are both objectwise. A map ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ is a fibration if certain explicit morphisms in ${\mathcal{M}}$ associated with ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ are fibrations. (See for example (10.9), (10.10), and Proposition 10.11 from [@dwyer_spa95].) One of the advantages of this model structure is the fact that any diagram admits an explicit fibrant replacement as shown the following illustration.
\[fib\_repl\_expl\] Consider the following objectwise fibrant diagram in ${\mathcal{M}}$. $${\mathcal{X}}= \left\{ \xymatrix{X_0 \ar[r]^-{f_0} & X_{01} & X_1 \ar[l]_-{f_1} \ar[r]^-{f_2} & X_{12} & X_2 \ar[l]_-{f_3}} \right\}$$ Then its fibrant replacement, $R{\mathcal{X}}$, is the second row of the following commutative diagram $$\xymatrix{X_0 \ar[r]^-{f_0} \ar@{>->}[d]^-{\sim}_-{g_0} & X_{01} \ar@{>->}[d]^-{\sim}_-{id} & X_1 \ar[l]_-{f_1} \ar[r]^-{f_2} \ar@{>->}[d]^-{\sim}_-{g_1} & X_{12} \ar@{>->}[d]^-{\sim}_-{id} & X_2 \ar[l]_-{f_3} \ar@{>->}[d]^-{\sim}_-{g_2} \\
{\widetilde{X}}_{0} \ar@{->>}[r] & X_{01} & {\widetilde{X}}_1 \ar@{->>}[l] \ar@{->>}[r] & X_{12} & {\widetilde{X}}_2. \ar@{->>}[l] }$$ To get $R{\mathcal{X}}$, first we take a fibrant replacement ${\widetilde{X}}_{i(i+1)}, 0 \leq i \leq n$ (here $n=1$), of $X_{i(i+1)}$ in ${\mathcal{M}}$. Since ${\mathcal{X}}$ is objectwise fibrant, we then take ${\widetilde{X}}_{i(i+1)} = X_{i(i+1)}$. Next the functorial factorization of the composition $idf_0$ (respectively $idf_3$) provides ${\widetilde{X}}_0$ (respectively ${\widetilde{X}}_{n+1} = {\widetilde{X}}_2$). Lastly, ${\widetilde{X}}_1$ comes from the functorial factorization $$\xymatrix{X_1 \ar[rr]^-{(idf_1, idf_2)} \ar@{>->}[rd]^-{\sim}_-{g_1} & & X_{01} \times X_{12} \\
& \widetilde{X}_1 \ar@{->>}[ru] }$$
\[trx\_rmk\] Let $\theta \colon {\mathcal{I}}{\longrightarrow}{\mathcal{J}}$ be a functor between small categories, and let ${\mathcal{X}}\colon {\mathcal{J}}{\longrightarrow}{\mathcal{M}}$ be an ${\mathcal{J}}$-diagram in ${\mathcal{M}}$. Then $\theta^*(R{\mathcal{X}})$ is not equal to $R \theta^*({\mathcal{X}})$ in general, but there is a natural map $\theta^*(R{\mathcal{X}}) {\longrightarrow}R \theta^*({\mathcal{X}})$. This map comes directly from the way we construct our fibrant replacements.
Now we define $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$. First recall the covariant functor $\theta_{AB} \colon {\mathcal{I}}_B {\longrightarrow}{\mathcal{I}}_A$ defined in the course of the proof of Proposition \[theta\_ab\_prop\]. For $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$ such that $A \subseteq E$, one can consider the composition $$\begin{aligned}
\label{ieae_comp}
\xymatrix{{\mathcal{I}}_E \ar[rr]^-{\theta_{AE}} & & {\mathcal{I}}_A \ar[rr]^-{{\mathcal{X}_A}} & & {\mathcal{O}_k(M)}\ar[rr]^-{F^{!}} & & {\mathcal{M}}}, \end{aligned}$$ which is nothing but an ${\mathcal{I}}_E$-diagram in ${\mathcal{M}}$. Define $\Psi((W,{\mathcal{X}_A}), E) \in {\mathcal{M}}$ as $$\begin{aligned}
\label{psi_defn}
\Psi((W,{\mathcal{X}_A}), E) = \left\{ \begin{array}{ccc}
\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta_{AE}^*({\mathcal{X}_A}) & \text{if} & A \subseteq E \\
\emptyset & \text{if} & \text{$A$ is not contained in $E$},
\end{array} \right.\end{aligned}$$ where $\emptyset$ stands for the initial object of ${\mathcal{M}}$.
The construction $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ that sends $((W,{\mathcal{X}_A}), E)$ to $\Psi((W,{\mathcal{X}_A}), E)$ is contravariant in the first variable and covariant in the second one.
Let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. We have to prove two things.
1. Functoriality in the first variable. Let $(T,{\mathcal{Y}_B}) \in {{\mathcal{D}}(V)}$ such that $A \subseteq B$, and let $\Lambda_{AB}$ be a morphism in ${{\mathcal{D}}(V)}$ from $(W,{\mathcal{X}_A})$ to $(T,{\mathcal{Y}_B})$. Then, by Definition \[dv\_defn\], $\Lambda_{AB} \colon {\mathcal{X}_A}\theta_{AB} {\longrightarrow}{\mathcal{Y}_B}$ is a natural transformation. If $B \subseteq E$, then one has the composition $F^{!}\Lambda_{AB} \theta_{BE} \colon F^{!}{\mathcal{X}_A}\theta_{AB}\theta_{BE} {\longleftarrow}F^{!}{\mathcal{Y}_B}\theta_{BE}$ (remember that $F^{!}$ is contravariant, and that $\theta_{BE}$ is covariant), which is the same as $$\begin{aligned}
\label{fab}
F^{!}\Lambda_{AB} \theta_{BE} \colon F^{!}\theta_{AE}^*({\mathcal{X}_A}) {\longleftarrow}F^{!}\theta_{BE}^*({\mathcal{Y}_B})\end{aligned}$$ since $\theta_{AB}\theta_{BE} = \theta_{AE}$ by (\[theta\_abc\]). This induces a morphism $$\Psi(\Lambda_{AB}, id) := \text{lim}(RF^{!}\Lambda_{AB}\theta_{BE}) \colon \Psi((W,{\mathcal{X}_A}), E) {\longleftarrow}\Psi((T,{\mathcal{Y}_B}), E).$$ If $E$ does not contain $B$, then $\Psi((T,{\mathcal{Y}_B}), E)$ is the initial object by definition, and therefore $\Psi(\Lambda_{AB}, id)$ is the unique morphism from $\emptyset$ to $\Psi((W,{\mathcal{X}_A}), E)$.
2. Functoriality in the second variable. Let $E' \in {\mathcal{E}}$ such that $E \subseteq E'$. If $A \subseteq E$, then we have $$\theta^*_{EE'}\left(F^{!}\theta^*_{AE}({\mathcal{X}_A}) \right) = F^{!}\theta^*_{AE'}({\mathcal{X}_A})$$ by (\[theta\_abc\]) and (\[theta\_starx\]). The map $$\Psi(id, E {\hookrightarrow}E') \colon \Psi((W,{\mathcal{X}_A}), E) {\longrightarrow}\Psi((W,{\mathcal{X}_A}), E')$$ is then defined to be the composition $$\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta_{AE}^* ({\mathcal{X}_A}) {\longrightarrow}\underset{{\mathcal{I}}_{E'}}{\text{lim}} \; \theta^*_{EE'} \left(RF^{!} \theta_{AE}^* ({\mathcal{X}_A})\right) {\longrightarrow}\underset{{\mathcal{I}}_{E'}}{\text{lim}} \; R\theta^*_{EE'} \left(F^{!} \theta_{AE}^* ({\mathcal{X}_A})\right).$$ Here the first arrow is the canonical map induced by $\theta_{EE'} \colon {\mathcal{I}}_{E'} {\longrightarrow}{\mathcal{I}}_E$, and the second is the natural map that comes directly from the way fibrant replacements of ${\mathcal{I}}_{E'}$-diagram are constructed (see Example \[fib\_repl\_expl\] and Remark \[trx\_rmk\]). As before, the case where $A$ is not contained in $E$ is obvious.
This proves the proposition.
Before we define $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, we need to equip the category ${\mathcal{M}}^{{\mathcal{E}}}$ of ${\mathcal{E}}$-diagrams in ${\mathcal{M}}$ with a nice model structure. Thanks to the fact that the category ${\mathcal{E}}$ is a *direct category* (see [@hovey99 Definition 5.1.1]), and therefore a *Reedy category* (see [@hovey99 Definition 5.2.1]), we can endow ${\mathcal{M}}^{{\mathcal{E}}}$ with a Reedy model structure that we now recall. For $E \in {\mathcal{E}}$, we define the *latching space* functor $L_E \colon {\mathcal{M}}^{{\mathcal{E}}} {\longrightarrow}{\mathcal{M}}$ as follows. Let ${\mathcal{E}}_E$ be the category of non-identity maps in ${\mathcal{E}}$ with codomain $E$, and define $L_E$ to be the composite $$L_E \colon \xymatrix{{\mathcal{M}}^{{\mathcal{E}}} \ar[r] & {\mathcal{M}}^{{\mathcal{E}}_E} \ar[rr]^-{\text{colim}} & & {\mathcal{M}}}$$ where the first arrow is restriction. Clearly there is a natural transformation $L_E{\mathcal{X}}{\longrightarrow}{\mathcal{X}}(E)$.
[@hovey99 Theorem 5.1.3] \[struc\_me\_thm\] There exists a model structure on the category ${\mathcal{M}}^{{\mathcal{E}}}$ of ${\mathcal{E}}$-diagrams in ${\mathcal{M}}$ such that weak equivalences and fibrations are objectwise. Furthermore, a map ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ is a (trivial) cofibration if and only if the induced map ${\mathcal{X}}(E) \coprod_{L_E{\mathcal{X}}} L_E{\mathcal{Y}}{\longrightarrow}{\mathcal{Y}}(E)$ is a (trivial) cofibration for all $E$.
Note that any object ${\mathcal{X}}$ of ${\mathcal{M}}^{{\mathcal{E}}}$ has an explicit cofibrant replacement $Q{\mathcal{X}}\colon {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ obtained by induction as follows. (Recall that by definition $\{0, 1 \}$ is the initial object of ${\mathcal{E}}$). First take the cofibrant replacement $Q{\mathcal{X}}(\{0, 1\})$ of ${\mathcal{X}}(\{0, 1\})$. Next, for any other object $E \in {\mathcal{E}}$, $Q{\mathcal{X}}(E)$ comes from the functorial factorization of the obvious map $$\underset{E' \subset E}{\text{colim}} \; Q{\mathcal{X}}(E') {\longrightarrow}{\mathcal{X}}(E),$$ where $E' \in {\mathcal{E}}$ runs over the set of proper subsets of $E$. As an example, the cofibrant replacement of $\Psi((W,{\mathcal{X}_A}), -)$ is an ${\mathcal{E}}$-diagram on the form $$\begin{aligned}
\label{qpsi_shape}
Q\Psi((W,{\mathcal{X}_A}), -) = \xymatrix{ & & \bullet \cdots \ar@{.}[d] \\
\emptyset \ar[r] & Q\Psi((W,{\mathcal{X}_A}), A) \ar@{.>}[ru] \ar@{.>}[r] \ar@{.>}[rd] & \bullet \cdots \ar@{.}[d]\\
& & \bullet \cdots } \end{aligned}$$
\[cofibrant\_rmk\] By construction, every object of the diagram $Q{\mathcal{X}}$ is cofibrant in ${\mathcal{M}}$.
\[hoco\_co\_prop\] The natural map $$\underset{{\mathcal{E}}}{\text{hocolim}} \; Q\Psi((W, {\mathcal{X}_A}), -) {\longrightarrow}\underset{{\mathcal{E}}}{\text{colim}} \; Q\Psi((W, {\mathcal{X}_A}), -)$$ is a weak equivalence.
This follows from [@hir03 Theorem 19.9.1].
We come to the definition of $H$.
Recall $\Psi$ from (\[psi\_defn\]), and define the functor $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{functor_h}
H(W, {\mathcal{X}_A}) = \underset{E \in {\mathcal{E}}}{\text{colim}} \; Q\Psi((W, {\mathcal{X}_A}), E). \end{aligned}$$
Let ${\mathcal{E}}_A \subseteq {\mathcal{E}}$ denote the full subcategory whose objects are $E$ containing $A$, and let ${\widetilde{\Psi}}((W,{\mathcal{X}_A}), -)$ denote the restriction of $\Psi((W,{\mathcal{X}_A}), -)$ to ${\mathcal{E}}_A$. Then one has $$\begin{aligned}
\label{hxa_formula}
H(W,{\mathcal{X}_A}) = \underset{ {\mathcal{E}}}{\text{colim}} \; Q\Psi((W,{\mathcal{X}_A}), -) \cong \underset{ {\mathcal{E}}_A}{\text{colim}} \; Q{\widetilde{\Psi}}((W,{\mathcal{X}_A}), -). \end{aligned}$$ The isomorphism $\cong$ follows from the fact that the diagram (\[qpsi\_shape\]) contains the initial object, $\emptyset$, of ${\mathcal{M}}$.
Now we define another map (the map $h$ below) which will be used in the next subsection. Recalling the data provided at the beginning of Subsection \[dv\_subsection\], and using definitions, we can easily see that for every $E \in {\mathcal{E}}_A$, for every $x \in {\mathcal{I}}_E$, one has ${\mathcal{X}_A}\theta_{AE}(x) \subseteq U'$. Applying the contravariant functor $F^{!}$ to this latter inclusion, we get maps $F^{!}(U') {\longrightarrow}F^{!} {\mathcal{X}_A}\theta_{AE}(x)$. This induces a natural transformation $F^{!}(U') {\longrightarrow}{\widetilde{\Psi}}({\mathcal{X}_A}, -)$ between two ${\mathcal{E}}_A$-diagrams in ${\mathcal{M}}$, the first one being the constant diagram (recall that $RF^{!}(U') = F^{!}(U')$ by the assumption that $F$ is objectwise fibrant and by Theorem \[fib\_cofib\_thm\]). Now taking the cofibrant replacement of this latter map, passing to the colimit, and using (\[hxa\_formula\]) we have a map $$\begin{aligned}
\label{map_h}
h \colon QF^{!}(U') {\longrightarrow}H(W,{\mathcal{X}_A}), \end{aligned}$$ which is natural in $(W,{\mathcal{X}_A})$.
### The functors $P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$
To define $P_0$ and $P_1$, we will first define $$\Phi_0 \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}\quad \text{and} \quad \Phi_1 \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}.$$ To do this, we need to introduce some notation. If $E= \{a_0, \cdots, a_{n+1}\}$ is an object of ${\mathcal{E}}$ and ${\mathcal{X}}\colon {\mathcal{I}}_E {\longrightarrow}{\mathcal{M}}$ is a functor, we define two objects $\phi_0 {\mathcal{X}}$ and $\phi_1 {\mathcal{X}}$ of ${\mathcal{M}}$ as $$\phi_0 {\mathcal{X}}:= {\mathcal{X}}(\{a_0\}) \quad \text{and} \quad \phi_1 {\mathcal{X}}:= {\mathcal{X}}(\{a_{n+1}\}).$$ In other words, $\phi_0 {\mathcal{X}}$ is the first object of the zigzag ${\mathcal{X}}$, while $\phi_1 {\mathcal{X}}$ is the last one.
Let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. If $A \subseteq E$, then one can consider the composition $F^{!}{\mathcal{X}_A}\theta_{AE} \colon {\mathcal{I}}_E {\longrightarrow}{\mathcal{M}}$ (from (\[ieae\_comp\])), which is an object of ${\mathcal{M}}^{{\mathcal{I}}_E}$. Let $RF^{!}{\mathcal{X}_A}\theta_{AE}$ denote its fibrant replacement with respect to the Dwyer-Spalinski model structure we described before (see Example \[fib\_repl\_expl\] for an illustration of what we call fibrant replacement). Define $\Phi_0((W,{\mathcal{X}_A}), E)$ as $$\begin{aligned}
\label{phiz_defn}
\Phi_0((W,{\mathcal{X}_A}), E) = \left\{ \begin{array}{ccc}
\phi_0R F^{!} {\mathcal{X}_A}\theta_{AE} & \text{if} & A \subseteq E \\
\emptyset & \text{if} & \text{$A$ is not contained in $E$}.
\end{array} \right.\end{aligned}$$
Replacing $\phi_0$ by $\phi_1$ in (\[phiz\_defn\]), we have the definition of $\Phi_1((W,{\mathcal{X}_A}), E)$. The following remark about $\Phi_0$ and $\Phi_1$ is important.
\[phixa\_rmk\] By inspection, for every $((W,{\mathcal{X}_A}),E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$, one has $$\Phi_0((W,{\mathcal{X}_A}), E) = \Phi_0((W,{\mathcal{X}_A}), A) \quad \text{and} \quad \Phi_1((W,{\mathcal{X}_A}), E) = \Phi_1((W,{\mathcal{X}_A}), A),$$ provided that $A \subseteq E$. This easily comes from three things: the definition of ${{\mathcal{D}}(V)}$, that of $\theta_{AE}$, and the way fibrant replacements of ${\mathcal{I}}_E$-diagrams in ${\mathcal{M}}$ are constructed (see Example \[fib\_repl\_expl\]).
The construction $\Phi_i \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}, i=0, 1$, that sends $((W,{\mathcal{X}_A}), E)$ to $\Phi_i((W,{\mathcal{X}_A}), E)$ is contravariant in the first variable and covariant in the second one.
For the functoriality in the first variable, let $\Lambda_{AB}$ be a morphism of ${{\mathcal{D}}(V)}$ from $(W,{\mathcal{X}_A})$ to $(T,{\mathcal{Y}_B})$, and consider the map $F^{!}\Lambda_{AB}\theta_{BE}$ from (\[fab\]). Its fibrant replacement gives $$\Phi_0(\Lambda_{AB}, id) \colon \Phi_0((W,{\mathcal{X}_A}), E) {\longleftarrow}\Phi_0((T,{\mathcal{Y}_B}), E).$$ The functoriality in the second variable is obvious by Remark \[phixa\_rmk\]. In fact, if $i \colon E {\hookrightarrow}E'$ is a morphism of ${\mathcal{E}}$ then $\Phi_0((W,{\mathcal{X}_A}), i) = id$ when $A \subseteq E$. A similar proof can be performed with $\Phi_1$ in place of $\Phi_0$.
Recall $\Phi_0$ and $\Phi_1$ from (\[phiz\_defn\]), and define $P_0 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ and $P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{functor_po}
P_0(W,{\mathcal{X}_A}) = \underset{{\mathcal{E}}}{\text{colim}} \; Q\Phi_0((W,{\mathcal{X}_A}), -) \quad \text{and} \quad P_1(W,{\mathcal{X}_A}) = \underset{{\mathcal{E}}}{\text{colim}} \; Q\Phi_1((W,{\mathcal{X}_A}), -),\end{aligned}$$ where $Q\Phi_i((W, {\mathcal{X}_A}), -), i \in \{0, 1\}$, is the cofibrant replacement of the ${\mathcal{E}}$-diagram $\Phi_i((W, {\mathcal{X}_A}), -)$ with respect to the model structure given by Theorem \[struc\_me\_thm\].
Now we define two important maps ($p_0$ and $p_1$ below) that will be also used in the next subsection. First, recalling the definition of ${{\mathcal{D}}(V)}$ (from Definition \[dv\_defn\]) and that of $\theta_{AB}$ (from (\[theta\_ab\])), one can see that the functor $F^{!}{\mathcal{X}_A}\theta_{AE}$ from (\[ieae\_comp\]) is nothing but a zigzag in ${\mathcal{M}}$ starting at $F^{!}(W)$ and ending at $F^{!}(L_1(W))$. If ${\widetilde{\Phi}}_i({\mathcal{X}_A}, -), i \in \{0, 1 \}$ denotes the restriction of $\Phi_i({\mathcal{X}_A}, -)$ to ${\mathcal{E}}_A$, by the definition of the fibrant replacement, (\[phiz\_defn\]) immediately implies the existence of two natural weak equivalences: $$F^{!}(W) \stackrel{\sim}{{\longrightarrow}} {\widetilde{\Phi}}_0((W,{\mathcal{X}_A}), -) \quad \text{and} \quad F^{!} (L_1(W)) \stackrel{\sim}{{\longrightarrow}} {\widetilde{\Phi}}_1((W,{\mathcal{X}_A}), -).$$ Of course the functors involves are viewed as ${\mathcal{E}}_A$-diagrams. Taking the cofibrant replacement with respect to the model structure described in Theorem \[struc\_me\_thm\] of those maps, we get $QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} Q{\widetilde{\Phi}}_0((W,{\mathcal{X}_A}), -)$ and $QF^{!}(L_1(W)) \stackrel{\sim}{{\longrightarrow}} Q{\widetilde{\Phi}}_1((W,{\mathcal{X}_A}), -)$. Passing to the colimit, and using the observation (\[hxa\_formula\]) we did for $H(W,{\mathcal{X}_A})$, we have weak equivalences $$\begin{aligned}
\label{qfv_colim}
p_0 \colon QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} P_0(W,{\mathcal{X}_A}) \quad \text{and} \quad p_1 \colon QF^{!}(L_1(W)) \stackrel{\sim}{{\longrightarrow}} P_1(W,{\mathcal{X}_A}).
\end{aligned}$$ Notice that these maps are both natural in $(W,{\mathcal{X}_A})$. Also notice that by Remark \[phixa\_rmk\] the diagram ${\widetilde{\Phi}}_i((W,{\mathcal{X}_A}), -)$ is in fact the constant diagram whose value is ${\widetilde{\Phi}}_i((W,{\mathcal{X}_A}), A)$.
### The maps $\eta_i \colon H {\longrightarrow}P_i, i=0, 1$
The aim here is to show that the definitions of $H$ (\[functor\_h\]), $P_0,$ and $P_1$ (\[functor\_po\]) imply the existence of natural weak equivalences $$\begin{aligned}
\label{etai}
\eta_0 \colon \xymatrix{H \ar[r]^-{\sim} & P_0} \quad \text{and} \quad \eta_1 \colon \xymatrix{H \ar[r]^-{\sim} & P_1}. \end{aligned}$$ We will show the existence of $\eta_0$; the existence of $\eta_1$ is similar. The idea is to define for every $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$ a natural map $$\alpha[(W,{\mathcal{X}_A}), E] \colon \Psi((W,{\mathcal{X}_A}), E) {\longrightarrow}\Phi_0((W,{\mathcal{X}_A}), E),$$ and show that it is a weak equivalence. Applying the cofibrant replacement functor, and then the colimit functor to $\alpha[(W,{\mathcal{X}_A}), -]$, and using Remark \[cofibrant\_rmk\], Theorem \[fib\_cofib\_thmg\], and Proposition \[hoco\_co\_prop\] we will then deduce that $\eta_0$ is a weak equivalence. So let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. If $A$ is not contained in $E$, then $\Psi((W,{\mathcal{X}_A}), E) = \emptyset = \Phi_0((W, {\mathcal{X}_A}), E)$ by definition, and therefore $\alpha[(W,{\mathcal{X}_A}), E]$ is the obvious map. Assume $A \subseteq E$. Then $\Psi((W,{\mathcal{X}_A}), E) = \underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!}\theta^*_{AE}({\mathcal{X}_A})$ by definition. Define $\alpha[(W,{\mathcal{X}_A}), E]$ to be obvious map $$\begin{aligned}
\label{lim_phi}
\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta^*_{AE}({\mathcal{X}_A}) {\longrightarrow}\phi_0RF^{!} \theta^*_{AE}({\mathcal{X}_A}) = \Phi_0((W,{\mathcal{X}_A}), E).\end{aligned}$$ This map is so obvious because $\phi_0RF^{!} \theta^*_{AE}({\mathcal{X}_A})$ is nothing but one piece from the diagram $RF^{!} \theta^*_{AE}({\mathcal{X}_A})$. It is straightforward to check the naturality of $\alpha[(W,{\mathcal{X}_A}), E]$ in both variables. One can also see that (\[lim\_phi\]) is a weak equivalence essentially by the following reason. First, since ${\mathcal{X}_A}$ is a zigzag of isotopy equivalences by the condition (b) from Definition \[dv\_defn\], and since $F^{!} | {\mathcal{O}_k(M)}$ is an isotopy cofunctor by Proposition \[fsrik\_propg\], it follows that every morphism of the diagram $F^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence. This implies that every morphism of $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence as well. Moreover $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is fibrant. So every morphism of the diagram $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence which is also a fibration. Thanks to the shape of this diagram, one can compute its limit by taking successive pullbacks. An illustration of this is given by (\[pb\_diag\]). $$\begin{aligned}
\label{pb_diag}
\xymatrix{{\widetilde{X}}_0 \ar@{->>}[r]^-{\sim} & X_{01} & {\widetilde{X}}_1 \ar@{->>}[l]_-{\sim} \ar@{->>}[r]^-{\sim} & X_{12} & {\widetilde{X}}_2 \ar@{->>}[l]_-{\sim} \\
& Y_1 \ar[lu] \ar[u] \ar[ru] & & Y_2 \ar[lu] \ar[u] \ar[ru] & \\
& & Z \ar[lu] \ar[ru] & & }\end{aligned}$$ Now applying the fact that the pullback of a fibration is again a fibration, and the fact that the pullback of a weak equivalence along a fibration is again a weak equivalence, we deduce that the map from $\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta^*_{AE}({\mathcal{X}_A})$ to each piece of the diagram is a weak equivalence.
Proof of the main result of the section {#iso_cof_subsection}
---------------------------------------
The goal here is to prove Theorem \[iso\_cof\_thm\], which is the main result of this section.
Let $U {\hookrightarrow}U'$ be an isotopy equivalence of ${\mathcal{O}(M)}$, and let $L \colon U \times I {\longrightarrow}U', (x, t) \mapsto L_t(x)$, be an isotopy from $U$ to $U'$. Our aim is to show that the canonical map $F^{!}(U') {\longrightarrow}F^{!}(U)$ is a weak equivalence. The idea is to first consider the commutative diagram (\[big\_diag\]), which will be defined below (for $V \in {\mathcal{O}_k}(U)$). Next we will show that the map $$\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (f_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; B_1(V)$$ is a weak equivalence. By the two-out-of-three axioms, we will deduce successively that the maps $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; ({\widetilde{f}}_1)$, $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (g)$, $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; ({\widetilde{f}}_0),$ and $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (f_0)$ are weak equivalences. Using the fact that this latter morphism is a weak equivalence, we will deduce the theorem.
$$\begin{aligned}
\label{big_diag}
\xymatrix{ & & {\widetilde{B}}_0(V) \ar[rrd]^-{k_0}_-{\sim} \ar[d]^-{\sim} & & \\
QF^{!}(U') \ar[rru]^-{{\widetilde{f}}_0} \ar[rrd]^{g} \ar[d]_-{\sim} & & B_0(V) & & A_0(V) \\
F^{!}(U') \ar[rru]_{f_0} \ar[rrd]^{f_1} & & G(V) \ar[rru]^-{\sim}_-{l_0} \ar[rrd]^-{l_1}_-{\sim} & & \\
QF^{!}(U') \ar[rru]_{g} \ar[rrd]_-{{\widetilde{f}}_1} \ar[u]^-{\sim} & & B_1(V) & & A_1(V) \\
& & {\widetilde{B}}_1(V) \ar[u]_-{\sim} \ar[rru]^-{\sim}_-{k_1} & & }\end{aligned}$$
Now we explain the construction of the diagram (\[big\_diag\]).
1. Q(-) is the cofibrant replacement functor in ${\mathcal{M}}$.
2. The objects $B_i(V), {\widetilde{B}}_i(V), i \in \{0, 1\}$, are defined as $$B_i(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; F^{!}(L_i(W)) \quad \text{and} \quad {\widetilde{B}}_i(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; QF^{!}(L_i(W)).$$ (Recall that $L_0 \colon U {\longrightarrow}U'$ is the inclusion functor; so $L_0(W) = W$.) Clearly these objects are functorial in $V$. Indeed, if $V {\hookrightarrow}V'$ is a morphism of ${\mathcal{O}_k}(U)$ then we have the inclusion functor $\theta \colon {{\mathcal{D}}(V)}{\longrightarrow}{{\mathcal{D}}(V')}$ defined in the course of the proof of Proposition \[dv\_functor\_prop\]. This latter functor and Proposition \[induced\_holim\_propg\] allow us to get the desired functoriality.
3. G(V) is defined as $G(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; H(W, {\mathcal{X}_A})$, where $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ is the functor from (\[functor\_h\]). As before the construction that sends $V$ to $G(V)$ is a contravariant functor.
4. The maps $f_i, {\widetilde{f}}_i, i \in \{0, 1\}$, are defined as $f_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (h_i)$ and ${\widetilde{f}}_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (Qh_i)$ where $h_i \colon F^{!}(U') {\longrightarrow}F^{!}(L_i(W))$ is the obvious map obtained by applying $F^{!}$ to the inclusion $L_i(W) \subseteq U'$.
5. The map $g$ is defined as $g = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (h)$, where $h \colon QF^{!}(U') {\longrightarrow}H(W, {\mathcal{X}_A})$ is the map from (\[map\_h\]).
6. The objects $A_i(V), i \in \{0, 1\}$, are defined as $A_i(V) := \underset{(W,{\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; P_i(W, {\mathcal{X}_A}),$ where $P_i \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ comes from (\[functor\_po\]). Again as before $A_i(V)$ is functorial in $V$.
7. The maps $k_i, i \in \{0, 1\}$, are defined as $k_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (p_i)$, where $p_i \colon QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} P_i(W, {\mathcal{X}_A})$ comes from (\[qfv\_colim\]). Since $p_i$ is a weak equivalence, and since by assumption $F$ is objectwise fibrant, it follows that $k_i$ is a weak equivalence as well by Theorem \[fib\_cofib\_thmg\] [^3].
8. Lastly, the maps $l_i, i \in \{0, 1\}$, are defined as $l_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (\eta_i)$, where $\eta_i \colon H \stackrel{\sim}{{\longrightarrow}} P_i$ is the natural transformation from (\[etai\]). As before, $l_i$ is a weak equivalence.
Using definitions, one can see that every map from the diagram (\[big\_diag\]) is natural in $V$. One can also check that the squares containing $f_i, {\widetilde{f}}_i, k_i$ and $l_i, i \in \{0, 1\}$, are both commutative. Now applying the homotopy limit functor (when $V$ runs over ${\mathcal{O}_k}(U)$) to each morphism of (\[big\_diag\]), we get a new diagram, denoted $\mathbb{D}$, in which the map $\text{holim}(f_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; B_1(V)$ is a weak equivalence because of the following. First consider the following commutative diagram constructed as follows. $$\begin{aligned}
\label{small_diag}
\xymatrix{\underset{W \in {\mathcal{O}_k}(L_1(V))}{\text{holim}}\; F^{!}(W) \ar[rr]^-{\sim} & & B_1(V) \\
F^{!}(L_1(V)) \ar[u]^-{\sim} & & F^{!}(U'). \ar[u]_-{f_1} \ar[ll]^-{h_1} \ar[llu]_-{q} }\end{aligned}$$
1. The top map is nothing but $[\theta_1; F^{!}]$ (see Proposition \[induced\_holim\_propg\] for the notation $[-;-]$), where $\theta_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(L_1(V))$ is defined as $\theta_1(W, {\mathcal{X}_A}) = L_1(W)$, and $F^{!} \colon {\mathcal{O}_k}(L_1(V)) {\longrightarrow}{\mathcal{M}}$ is just the restriction of $F^{!}$ to ${\mathcal{O}_k}(L_1(V))$. By Corollary \[theta\_zo\_lem\] the functor $\theta_1$ is homotopy right cofinal, and therefore the map $[\theta_1; F^{!}]$ is a weak equivalence by Theorem \[htpy\_cofinal\_thmg\].
2. The maps $f_1$ and $h_1$ have been defined before, while $q$ is induced by the canonical map $F^{!}(U') {\longrightarrow}F^{!}(W)$.
3. The lefthand vertical map is the map $\text{holim} (\eta)$, where $ \eta[W] \colon F(W) \stackrel{\sim}{{\longrightarrow}} F^{!}(W)$ is the map from Proposition \[fsrik\_propg\].
Applying the homotopy limit functor (when $V$ runs over ${\mathcal{O}_k}(U)$ of course) to each morphism of (\[small\_diag\]), we get a new commutative diagram, denoted $\mathbb{S}$, in which the map $\text{holim} (h_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} F^{!}(L_1(V))$ is a weak equivalence because of the following reason. Consider the functor $\theta \colon {\mathcal{O}_k}(U) {\longrightarrow}{\mathcal{O}_k}(U')$ defined as $\theta(V) = L_1(V)$. Also consider $F^{!} \colon {\mathcal{O}_k}(U') {\longrightarrow}{\mathcal{M}}$. Clearly $\theta$ is an isomorphism since $L_1 \colon U {\longrightarrow}U'$ is a homeomorphism. So for any $W \in {\mathcal{O}_k}(U')$, the pair $(\theta^{-1}(W), id)$ is the initial object of the under category $W \downarrow \theta$. This shows that $\theta$ is homotopy right cofinal, and therefore the map $[\theta; F^{!}] \colon \underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F^{!}(V) {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; (\theta^{*}F^{!})(V)$ is a weak equivalence by Theorem \[htpy\_cofinal\_thmg\]. By inspection, the map $\text{holim}(h_1)$ is nothing but the composition $$\xymatrix{\underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F(V) \ar[rr]^-{\text{holim}(\eta)}_-{\sim} & & \underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F^{!}(V) \ar[rr]^-{[\theta; F^{!}]}_-{\sim} & & \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; (\theta^{*}F^{!})(V),}$$ where $\eta[V] \colon F(V) \stackrel{\sim}{{\longrightarrow}} F^{!}(V)$ is again the map from Proposition \[fsrik\_propg\]. Now, applying the two-out-of-three axiom to the diagram $\mathbb{S}$ we deduce that the map $\text{holim}(f_1)$ is a weak equivalence.
We come back to the diagram $\mathbb{D}$. As we said before, the two-out-of-three axiom shows successively that the maps $\text{holim}({\widetilde{f}}_1)$, $\text{holim}(g), \text{holim} ({\widetilde{f}}_0), $ and $\text{holim} (f_0)$ are weak equivalences. Now, replacing $1 $ by $0$ in the diagram (\[small\_diag\]), one can see that the map $\text{holim}(h_0) \colon F^{!}(U') {\longrightarrow}F^{!}(U)$ is a weak equivalence by the two-out-of-three axiom. But this is what we had to show.
[99]{} A.K. Bousfield and D. M. Kan, *Homotopy limits, completion and localizations*, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304. W. Chacholski and J. Scherer, *Homotopy theory of diagrams*, Mem. Amer. Math. Soc. 155 (2002), no. 736, x+90 pp. D.C. Cisinski, *Locally constant functors*, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 593–614. W. Dwyer, P. Hirschhorn, D. Kan, and J. Smith, *Homotopy limit functors on model categories and homotopical categories*, Mathematical Surveys and Monographs, 113, American Mathematical Society, Providence, RI, 2004. viii+181 pp. W. Dwyer, J. Spalinski, *Homotopy theories and model categories*, Handbook of algebraic topology, 73–126, North-Holland, Amsterdam, 1995. T. Goodwillie, Calculus III, *Taylor series*, Geom. Topol. 7 (2003) 645–711. P. Hirschhorn, *Models categories and their localizations*, Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. P. Hirschhorn, *Notes on homotopy colimits and homotopy limits*, Work in progress available online at P. Hirschhorn’s homepage: $http://www-math.mit.edu/~psh/\#hocolim$. M. Hovey, *Model categories*, Mathematical Surveys and Monographs, vol. 63, *American Mathematical Society, Providence, RI*, 1999. B. A. Munson, *Introduction to the manifold calculus of Goodwillie-Weiss*, Morfismos, Vol. 14, No. 1, 2010, pp. 1–50. D. Pryor, *Special open sets in manifold calculus*, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 89–103. P. A. Songhafouo Tsopméné, D. Stanley, *Very good homogeneous functors in manifold calculus*, Preprint. 2017, arXiv:1705.0120. P. A. Songhafouo Tsopméné, D. Stanley, *Classification of homogeneous functors in manifold calculus*, Preprint. 2018, arXiv:1807.06120. M. Weiss, *Embedding from the point of view of immersion theory I*, Geom. Topol. 3 (1999), 67–101.
*E-mail address: [email protected]*
*E-mail address: [email protected]*
[^1]: In this paper the word cofunctor means contravariant functor
[^2]: One of the axioms of the definition of a *simplicial model category* ${\mathcal{M}}$ [@hir03 Definition 9.1.6] says that for an object $Y \in {\mathcal{M}}$ and every simplicial set $K$, there exists an object $Y^K$ of ${\mathcal{M}}$
[^3]: If $P_i(W, {\mathcal{X}_A})$ is not fibrant, one can always substitute it by its fibrant replacement
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[Using the Feynman-Kac and Cameron-Martin-Girsanov formulas, we obtain a generalized integral fluctuation theorem (GIFT) for discrete jump processes by constructing a time-invariable inner product. The existing discrete IFTs can be derived as its specific cases. A connection between our approach and the conventional time-reversal method is also established. Different from the latter approach that were extensively employed in existing literature, our approach can naturally bring out the definition of a time-reversal for a Markovian stochastic system. Intriguingly, we find the robust GIFT usually does not result into a detailed fluctuation theorem. ]{}'
author:
- Fei Liu
- 'Yu-Pin Luo'
- 'Ming-Chang Huang'
- 'Zhong-can Ou-Yang'
title: A generalized integral fluctuation theorem for general jump processes
---
Introduction
============
One of important progresses in nonequilibrium statistic physics in the past two decades is the discovery of a various of fluctuation theorems. They are thought of to be a nonperturbative extension of the fluctuation-dissipation theorems in near-equilibrium region to far-from equilibrium region. According to their mathematical expressions, these theorems are loosely divided into two types. One is called the integral fluctuation theorems (IFT) [@JarzynskiPRE97; @JarzynskiPRL97; @Crooks99; @Crooks00; @Maes; @SeifertPRL05; @HatanoSasa; @Speck; @Schmiedl; @QianJPC; @Harris], and the other is called the detailed fluctuation theorems (DFT) [@Evans; @Gallavotti; @Kurchan; @Lebowitz; @Crooks00]. The former follows a unified expression $$\begin{aligned}
\langle \exp[-{\cal A}]\rangle=1, \label{IFT}\end{aligned}$$ where $\cal{A}$ is a functional of a stochastic trajectory of a concerned stochastic system, and angular brackets denote an average over the ensemble of the trajectories that start from an given initial distribution. For instance, $\cal{A}$ may be the dissipated work along a trajectory and eq. (\[IFT\]) is the celebrated Jarzynski equality (JE) [@JarzynskiPRE97; @JarzynskiPRL97].
Due to the insightful work of Hummer and Szabo [@Hummer01], we now know that these IFTs have an intimate connection with the famous Feynman-Kac formula (FK) [@Feynman; @Kac] in the stochastic theory of diffusion processes [@Stroock]. Recently, several works including us reinvestigated this issue from mathematic generalization and rigors [@QianJPC; @Ge; @Chetrite; @Ao; @LiuF]. One of findings is that the application of the FK formula in proving the IFTs is based on a construction of a time-reversed process of a diffusion process [@Chetrite; @LiuF]. Because the definition of a time-reversal has some certain arbitrariness [@Chetrite], we have obtained a generalized IFT (GIFT) by constructing a time-invariable integral and employing the FK and Cameron-Martin-Girsanov (CMG) formulas [@Cameron; @Girsanov] simultaneously, and the several IFTs [@JarzynskiPRE97; @SeifertPRL05; @HatanoSasa; @QianJPC] were specific cases of the GIFT [@LiuF]. We should emphasize that all of the works were concerning with continuous diffusion processes described by Fokker-Planck (FK) equation.
In addition to continuous case, there are still another kind of stochastic jump processes described by Markovian discrete master equations. In many practical physical systems, a description of discrete jump process is more satisfactory than a description using continuous diffusion process, e.g., the systems only involving few individual objects [@Gardiner]. One may naturally think of that there exists a GIFT in discrete version, and the discrete IFTs in literature [@SeifertPRL05; @Harris; @SeifertJPA04] are specific cases of it. At a first sight, this effort seems trivial since a continuous diffusion process can be always discretized to a discrete jump process. However, In addition that one hardly ensures that the “discrete" GIFT achieved in this way is really exact, we know that a jump process is not always equivalent to a discretization of a certain continuous process [@Gardiner]. Additionally, to our knowledge, fewer works have formally studied the IFTs for general jump processes employing the FK and CMG formulas, though several authors have mentioned this possibility [@QianJP; @GeJMP] earlier. Therefore, in our opinion a rigorous derivation of an exact GIFT for discrete jump processes is essential and meaningful. In this work we present this effort. Because we focus on the general Markovian jump processes, fewer physics are mentioned here. The detailed discussions about the specific IFTs in previous literature [@Harris] should make it up.
Generalized integral fluctuations for jump processes
====================================================
We start with a Markovian jumping process described by a discrete master equation $$\begin{aligned}
\frac{dp_n(t)}{dt}=\left[{\textbf H}(t){\textbf p}(t)\right]_n,
\label{forwardeq}\end{aligned}$$ where the $N$-dimension column vector ${\textbf
p}(t)=(p_1,\cdots,p_N)^{\rm T}$ is the probabilities of the system at individual states at time $t$ (the state index $n$ may be a vector), the matrix element of the time-dependent (or time-independent) rate ${\textbf H}_{mn}>0$ ($m\neq n$) and ${\textbf H}_{nn}=-\sum_{m\neq n}{\textbf H}_{mn}$. Given a normalized positive column vector ${\textbf
f}(t)=(f_1,\cdots,f_N)^{\rm T}$ and a $N\times N$ matrix ${\textbf
A}$ that satisfies conditions $f_n{\textbf H_{mn}+\textbf
A}_{mn}>0$ ($m\neq n$) and ${\textbf A}_{nn}=-\sum_{m\neq
n}{\textbf A}_{mn}$, we state that an inner product ${\textbf
f}^{\rm T}(t'){\textbf v}(t')$ is time-invariable if the column vector $\textbf v(t')=(v_1,\cdots,v_N)^{\rm T}$ satisfies $$\begin{aligned}
\frac{dv_n(t')}{dt'}=-\left[{\textbf H}^{\rm T}{\textbf
v}\right]_n-f_n^{-1}\left[\partial_{t'}{ \textbf f}-{\textbf H}{
\textbf f}\right]_nv_n+f_n^{-1} \left[\left({ \textbf A} \textbf
1\right)_nv_n-\left({\textbf A}^{\rm T} {\bf\text
v}\right)_n\right], \label{backwardeq}\end{aligned}$$ where the final condition of $v_n(t)$ is $q_n$ ($t'$$<$$t$), and the column vector ${\textbf 1}=(1,\cdots,1)^{\rm T}$. This is easily proved by noting a time differential $d_{t'}\left[{\textbf
f}^{\rm T}(t'){\textbf v}(t')\right]= d_{t'}({\textbf f}^{\rm
T}){\textbf v}+{\textbf f}^{\rm T}d_{t'}({\textbf v})$ and the transpose property of a matrix. Employing the Feynman-Kac and Cameron-Martin-Girsanov formulas for jump processes (a simple derivation about the latter see the Appendix I), eq. (\[backwardeq\]) has a stochastic representation given by $$\begin{aligned}
v_n(t')=E^{n,t'}\left[e^{-{\cal J}[{\textbf x},\textbf f,\textbf
A]} q_{s(t)}\right] \label{stochrep}\end{aligned}$$ and $$\begin{aligned}
{\cal J}[{\textbf x},\textbf f,\textbf A]=\int_{t'}^tf_{\textbf
x(\tau)}^{-1}\left[-\partial_{\tau}\textbf f+\textbf H\textbf
f+\textbf A\textbf 1\right]_{{\textbf
x}(\tau)}d\tau-\int_{t'}^tf_{{\textbf x}(\tau)}^{-1}\textbf
A_{{\textbf x}(\tau){\textbf x}(\tau)}d\tau
-\sum_{i=1}^k\ln\left[1+\frac{{\textbf A}_{{\textbf
x}(t_i^+){\textbf x}(t_{i}^-)}(t_i)}{f_{{\textbf
x}(t_i^-)}(t_i)\textbf H_{{\textbf x}(t_i^+){\textbf
x}(t_{i}^-)}(t_i)} \right],\label{functional}\end{aligned}$$ the expectation $E^{n,t'}$ is over all trajectories $\textbf x$ generated from eq.(\[forwardeq\]) with fixed initial state $n$ at time $t'$, $\textbf x(t')$ is the discrete state at time $t'$, $\textbf x(t_i^{-})$ and $\textbf x(t_i^{+})$ represent the states just before and after a jump occurs at time $t_i$, respectively, and we assumed the jumps occur $k$ times for a process. The readers are reminded that the first and last two terms of the functional are the consequences of the FK and GCM formulas, respectively. We see that the last term is significantly different from that in the continuous processes \[eq. (11) in ref. [@LiuF]\]. Combining the stochastic representation and the time-invariable quantity and choosing $t'=0$, we obtain the exact discrete GIFT for a jump process, $$\begin{aligned}
\sum_{m=1}^N f_m(0) E^{m,0}\left\{e^{-J[{\textbf x},\textbf
f,\textbf A]} q_{{\textbf x}(t)}]\right\}={\textbf f}^{\rm T}(t)
{\textbf q} \label{GIFT}\end{aligned}$$ Particulary, the right hand side of the equation become $1$ if $\textbf q=1$.
Relationship between the GIFT and existing IFTs for jump processes {#secIII}
==================================================================
The abstract eq. (\[GIFT\]) includes several discrete IFTs in literature. First we investigate the case in which the discrete system has a transient steady-state solution ${\textbf H}(t) {\textbf p}^{\rm ss}(t)=0$. Choosing the matrix $\textbf A$=0 and the vector $\textbf f(t)={\textbf p}^{\rm
ss}(t)$, eq. (\[functional\]) is immediately simplified into $$\begin{aligned}
{\cal J}=-\int_0^t\partial_\tau p^{\rm ss}_{\textbf
x(\tau)}(\tau)d\tau \label{JarzynskiHantanFunctional}\end{aligned}$$ If one further thinks of ${\textbf p}^{\rm ss}$ satisfying a time-dependent detailed balance condition ${\textbf
H}_{mn}(t)p^{ss}_n(t)={\textbf H}_{nm}(t)p^{ss}_m(t)$, the above functional may be analogous to the dissipated work and eq. (\[GIFT\]) is the discrete version of the JE [@JarzynskiPRE97; @JarzynskiPRL97]. On the other hand, if ${\textbf p}^{\rm ss}$ is a transient nonequilibrium steady-state without detailed balance, eq. (\[JarzynskiHantanFunctional\]) could be rewritten to $$\begin{aligned}
{\cal J}=\ln\frac{p^{\rm ss}_{\textbf x(0)}(0)}{p^{\rm
ss}_{\textbf x(t)}(t)}+\sum_{i=1}^k\ln\frac{p^{\rm ss}_{\textbf
x(t_i^+)}(t_i)}{p^{\rm ss}_{\textbf x(t_i^-)}(t_i)}.
\label{HantanFunctional}\end{aligned}$$ where we used the following relationship $$\begin{aligned}
{d_t}\ln p^{\rm ss}_{\textbf x(t)}(t)=\partial_t\ln p^{\rm
ss}_{\textbf x(t)}(t)+\sum_{i=1}^{k}\delta(t-t_i)
\ln\left[p^{\rm ss}_{\textbf x(t_i^+)}(t_i)/p^{\rm
ss}_{\textbf x(t_i^-)}(t_i)\right].
\label{differentialrelation}\end{aligned}$$ Then we may interpret the first term in eq. (\[HantanFunctional\]) as the entropy change of system and the second term as the “excess" heat of the driven jump process. Under this circumstance eq. (\[GIFT\]) is the discrete version of the Hatano-Sasa equality [@HatanoSasa].
The last case is about nonvanishing $\textbf A(t)$. Choosing the matrix element ${\textbf A}_{mn}(t)={\textbf
H}_{nm}(t)f_m(t)-{\textbf H}_{mn}(t)f_n(t)$ ($m\neq n$), or the flux $J_{mn}(t)$ between states $m$ and $n$ for a distribution $\textbf f(t)$. Obviously, the condition of $f_n\textbf
H_{mn}+\textbf A_{mn}$$>0$. Substituting this matrix into eq. (\[functional\]), we obtain $$\begin{aligned}
{\cal J}=-\int_0^t \partial_{\tau}\ln p_{{\textbf
x}(\tau)}(\tau)d\tau+\sum_{i=1}^{k}\ln\frac{{\textbf H}_{{\textbf
x}(t_i^-)\textbf x(t_i^+)}(t_i)p_{\textbf x(t_i^+)}(t_i)}{{\textbf
H}_{\textbf x(t_i^+)\textbf x(t_i^-)}(t_i)p_{\textbf
x(t_i^-)}(t_i)} \label{orgtotentropy}\end{aligned}$$ To achieve obvious physical meaning of the above expression, we employ eq. (\[differentialrelation\]) again and have $$\begin{aligned}
{\cal
J}=\ln\frac{f_{\textbf x(0)}(0)}{f_{\textbf
x(t)}(t)}+\sum_{i=1}^k\ln\frac{{\textbf H}_{\textbf
x(t_i^+)\textbf x(t_i^-)}(t_i)}{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}(t_i)}. \label{totalentropy}\end{aligned}$$ Hence, if $\textbf f(t)$ is the distribution of the system itself satisfying the evolution eq. (\[forwardeq\]), the first term in the equation is just the entropy change of the system and the second term is interpreted as entropy change of environment [@SeifertPRL05; @Harris]. In other words, the GIFT with eq. (\[totalentropy\]) is about the total entropy change of a stochastic jump process.
The GIFT and time reversal for jump processes
=============================================
Like the case of continuous diffusion processes, we can connect the time-invariable inner product to be a jump process that is regarded to be a time-reversal of the original jump process [@LiuF]. Multiplying $f_n(t')$ and rearranging on both sides of eq. (\[backwardeq\]), we have $$\begin{aligned}
\frac{d}{dt'}\left[f_n(t')v_n(t')\right]=-\sum_{m=1}^N
f_m^{-1}\left[\textbf H_{mn}f_n+{\textbf
A}_{mn}\right]f_mv_m+f_nv_n\sum_{m=1}^Nf_n^{-1}\left[\textbf
H_{nm}f_m+{\textbf A}_{nm}\right] \label{orgtimereversal}\end{aligned}$$ Then we define a new function $q_{\bar n}(s)=f_n(t')v_n(t')$, where $s=t-t'$ and $\bar n$ represents an index whose components are the same or the minus of the components of the index $n$ depending on whether they are even or odd under time reversal ($t\to -t$). We also define a new rate matrix $\overline{\textbf
H}(s)$ whose elements are $$\begin{aligned}
\overline{\textbf H}_{{\bar n}{\bar
m}}(s)=f_m^{-1}(t')\left[\textbf H_{mn}(t')f_n(t')+\textbf
A_{mn}(t')\right] \label{ratetimereversal}\end{aligned}$$ for $m\neq n$, and $\overline{\textbf H}_{mm}(s)=-\sum_{n\neq
m}\overline{\textbf H}_{nm}(s)$, respectively. Then eq. (\[orgtimereversal\]) is rewritten as $$\begin{aligned}
\frac{dq_{\bar n}(s)}{ds}=[\overline{\textbf H}(s)\textbf
q(s)]_{\bar n}. \label{timereversal}\end{aligned}$$ Because of the variable $s=t-t'$, we interpret $\overline{\textbf
H}(t)$ to be a time-reversal of the original $\textbf H(t)$. Equation (\[timereversal\]) directly presents the reason of the time-invariable inner product $\textbf f^{\rm T}(t')\textbf v(t')$ that equals $\textbf 1^{\rm T}\textbf q(s)$; the latter is a constant due to probability conservation.
The generalized time-reversal (\[ratetimereversal\]) includes several types of time-reversal in literature [@HatanoSasa; @Chernyak; @Harris]. For convenience, we only consider even components only in the state-index $n$. First, if the matrix $\textbf A=0$ and $\textbf f(t')=\textbf p^{\rm
ss}(t')$ satisfying the detailed balance condition, the time-reversed rate matrix $\overline{\textbf H}(t')={\textbf
H}(s)$ simply. The process determined by this rate matrix was termed backward process [@Harris] (or a reversed protocol in Ref. [@Chernyak]). In contrast, if $\textbf p^{\rm ss}(t')$ is transient nonequilibrium steady-state, a process determined by $\overline{\textbf H}_{mn}(t')=f_n(s){\textbf H}_{nm}(s)/f_m(s)$ was termed an adjoint process [@Harris] (or the current reversal in Ref. [@Chetrite]). Intriguingly, if we choose $\textbf A_{mn}(s)$ to be the flux $J_{mn}(t)$ between the states $m$ and $n$ for a distribution $\textbf f(s)$, we reobtain $\overline{\textbf H}(t')={\textbf H}(s)$ that is the same with case of the detailed balance condition. Considering that these choices of $\textbf f$ and $\textbf A$ here are corresponding to those in Sec. \[secIII\], respectively, we see that the JE and the IFT of the total entropy have the same physical origin. It is expected in physics since a realization of a reversed protocol is usually possible and does not depend on whether the system satisfies detailed balance condition. We should point out that one may construct infinite time-reversals, because $\textbf f$ and $\textbf A$ are almost completely arbitrary, e.g., $\textbf
A_{mn}(s)=\alpha J_{mn}(s)$ and $0\leq\alpha\leq 1$. Before ending this section, we give two comments about the relationship $q_{\bar
n}(s)=f_n(t')v_n(t')$. First, for a time-independent $\textbf H$, if $f_n$ is the equilibrium solution of the rate matrix, eq. (\[backwardeq\]) with zero $\textbf A$ is just the backward master equation [@Gardiner]. Second, employing the relationship repeatedly, we may obtain the detailed DFTs for the specific vectors $\textbf f(t')$ and matrixes $\textbf A(t')$ in Sec. \[secIII\] (the details see the Appendix II).
Conclusion
==========
In this work we derived a GIFT for general jump processes. The existing IFTs for discrete master equations are its special cases. We see that, in form the GIFT for the jump cases is apparently distinct from that for the continuous diffusions that we obtained earlier [@LiuF]. Additionally, we also find this robust GIFT usually does not result into a detailed fluctuation theorem. Compared to other approaches, the major advantage of the current and previous works is that the time-reversal can come out automatically during the constructions of the time-invariable integral or the inner product, which should be direct and obvious, at least from point of view of us. Of course, A limit of our two works is that we did not show some applications of the two GIFTs in concrete physical systems. We hope that this point would be remedied in near future.\
[This work was supported in part by Tsinghua Basic Research Foundation and by the National Science Foundation of China under Grant No. 10704045 and No. 10547002.]{}
Appendix I: The Cameron-Martin-Girsanov formula for jump processes {#appendix-i-the-cameron-martin-girsanov-formula-for-jump-processes .unnumbered}
==================================================================
Compared to the CMG formula for continuous diffusion processes, little literature discussed the CMG formula for discrete jump processes. For the convenience of the readers, we give a simple derivation of the formula here. Given a master equation with rate matrix ${\textbf H}$. The probability observing a trajectory ${\textbf x}(\cdot)$ which starts state $n_1$ at time $t_0=0$, jumps at time $t_1$ to state $n_2$,$\cdots$, finally jumps at time $t_k$ to $n_{k+1}$ and stay till time $t_{k+1}=t$ is $$\begin{aligned}
{\rm prob}[{\textbf
x}(\cdot)]=&&\prod_{i=1}^{k}\exp\left[\int_{t_{i-1}}^{t_i}{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^-)}(\tau)d\tau\right]{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^+)}\times
\exp\left[\int_{t_{k}}^{t}{\textbf
H}_{n_{k+1}n_{k+1}}(\tau)d\tau\right],\end{aligned}$$ where $\textbf x(t_i^{-})=n_i$ and $\textbf x(t_i^{+})=n_{i+1}$ ($i=1,\cdots,k$). Assuming that there is another master equation with a different rate matrix ${\textbf H'}={\textbf H}+{\textbf
A}$, where the matrix elements of ${\textbf A}$ may be negative. Then the ration of the probabilities observing the same trajectory in these two equations is simply $$\begin{aligned}
\frac{{\rm prob}'[{\textbf x}(\cdot)]}{{\rm prob}[{\textbf
x}(\cdot)]}=e^{-Q[{\textbf x}(\cdot)]}. \label{CGMformula}\end{aligned}$$ where $$\begin{aligned}
Q[{\textbf x}(\cdot)]=-\int_{0}^{t}{\textbf A}_{\textbf
x(\tau)\textbf
x({\tau})}(\tau)d\tau-\sum_{i=1}^{k}\ln\left(1+\frac{ {\textbf
A}_{\textbf x(t_i^-)\textbf x(t_i^+)}}{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}}\right)\end{aligned}$$ Obviously, eq. (\[CGMformula\]) results into a IFT $$\begin{aligned}
\langle e^{-Q[{\textbf x}(\cdot)]}\rangle =1,\end{aligned}$$ where the average is over an ensemble of trajectories generated from the stochastic system with the rate matrix $\textbf H$ and with any initial distribution. Choosing a specific $$\begin{aligned}
{\textbf A}_{mn}(t)=p^{\rm ss}_n(t)^{-1}\left[{\textbf
H}_{nm}(t)p^{\rm ss}_m(t)-{\textbf H}_{mn}(t)p^{\rm
ss}_n(t)\right]{\text (m\neq n)}\end{aligned}$$ and ${\textbf A}_{nn}=-\sum_{m\neq n}{\textbf A}_{mn}(t)=0$, we obtain the IFT of the house-keeping heat [@Speck; @Harris] in discrete version, where $$\begin{aligned}
Q_{\rm hk}[{\textbf x}(\cdot)]=\sum_{i=1}^{k}\frac{{\textbf
H}_{\textbf x(t_i^-)\textbf x(t_i^+)}(t_i)p^{\rm ss}_{\textbf
x(t_i^+)}(t_i)}{{\textbf H}_{\textbf x(t_i^+)\textbf x(t_i^-)}
(t_i)p^{\rm ss}_{\textbf x(t_i^-)}(t_i)}\end{aligned}$$ Intriguingly, replacing $p_m^{\rm ss}$ above by the real probability distribution $p_m(t)$ of the system ${\textbf H}$, one obtains an IFT with $$\begin{aligned}
Q[{\textbf x}(\cdot)]=\int_0^t \partial_{\tau}\ln p_{{\textbf
x}(\tau)}(\tau)d\tau+\sum_{i=1}^{k}\frac{{\textbf H}_{\textbf
x(t_i^-)\textbf x(t_i^+)}(t_i)p_{\textbf x(t_i^+)}(t_i)}{{\textbf
H}_{\textbf x(t_i^+)\textbf x(t_i^-)}(t_i)p_{\textbf
x(t_i^-)}(t_i)} \label{psudoentropy}\end{aligned}$$ We notice that this functional is the almost same with that of the IFT of total entropy eq. (\[orgtotentropy\]) except that the first term becomes minus here. In addition, the average of $\langle Q[{\textbf s}(\cdot)]\rangle$ is the same with the average of the total entropy since the first terms of eqs. (\[totalentropy\]) and (\[orgtotentropy\]) vanish. We are not very clear whether eq. (\[psudoentropy\]) has new physical interpretation.
Appendix II: the detailed fluctuation theorem {#appendix-ii-the-detailed-fluctuation-theorem .unnumbered}
=============================================
Given the transition probability of eq. (\[timereversal\]) to be $q_{\bar n}(s'|m,s)$ (0$<$$s$$<$$s'$$<$$t$), the previous relationship implies $$\begin{aligned}
q_{\bar n}(s'|m,s)f_{\bar
m}(t-s)=f_{n}(t-s')E^{n,t-s'}\left[e^{-{{\cal
J}(t-s',t-s)}}\delta_{{\textbf x}(t-s),\bar m}\right]
\label{Gdetailedbalance}\end{aligned}$$ if one notices the initial condition $q_{\bar
n}(s|m,s)=\delta_{\bar n,m}$, where we use ${\cal J}(t-s',t-s)$ to denote the functional eq. (\[functional\]) with the lower and upper limits $t-s'$ and $t-s$, respectively, and $\delta$ is the Kronecker’s. Now we consider a mean of a $(k+1)$-point function over the time-reversed system (\[timereversal\]), $$\begin{aligned}
\langle{F}[{\bar {\textbf x}}(s_k),\cdots,{\bar {\textbf
x}}(s_0)]\rangle_{\rm TR}=\sum_{n_0,\cdots,n_k}
q_{n_k}(s_k|n_{k-1},s_{k-1})\cdots
q_{n_1}(s_1|n_0,s_0)q_{n_0}(s_0){F}(\bar n_k,\cdots,\bar n_0)\end{aligned}$$ where $0$=$s_0$$<$$s_1$$<$$\cdots$$<$$s_k$$=$$t$ and $\textbf
q(s_0)$ is the initial distribution. If we choose a specific $\textbf q_{n_0}(s_0)= f_{\bar n_0}(t-s_0)$ and employ eq. (\[Gdetailedbalance\]) repeatedly, the right hand side of the above equation becomes $$\begin{aligned}
\langle{e^{-{\cal J}[{\textbf x},\textbf f,\textbf A]}F}[{{\textbf
x}}(t_0),\cdots,{{\textbf x}}(t_k)]\rangle=\sum_{\bar n_k}f_{\bar
n_k}(t-s_k)E^{\bar n_k,t-s_k}\left\{ e^{-{\cal
J}(0,t)}{F}[{{\textbf x}}(t-s_k),\cdots,{{\textbf
x}}(t-s_0)]\right\}.\end{aligned}$$ Here we define $t_i=t-s_{k-i}$ and $0$$=$$t_0$$<$$t_1$$<$$\cdots$$<$$t_k$$=$$t$. On the basis of the above discussion, if $k\to\infty$ the function $F$ becomes a functional ${\cal F}$ over the space of all trajectories $\textbf
x$, and we get an identity $$\begin{aligned}
\langle{\bar{\cal F}}\rangle_{\rm TR}= \langle e^{-{{\cal
J}[{\textbf x},\textbf f,\textbf A]}}{\cal F}\rangle,
\label{GCrook}\end{aligned}$$ where ${\cal {\bar F}}(\textbf x)={\cal F}({\bar {\textbf x}})$ and $\bar {\textbf x}$ is simply the time-reversed trajectory of $\textbf x$. This is a generalization of Crooks’ relation [@Crooks00]. Obviously, choosing ${\cal F}$ constant, one obtains the GIFT (\[GIFT\]). An important following question is whether the GIFT results into a DFT. For the specific matrixes $\textbf A(t)$ and vectors $\textbf f(t)$ in sec. \[secIII\], we indeed obtain several DFTs $$\begin{aligned}
P_{\rm TR}(-J)=P(J)e^{-J}, \label{DFT}\end{aligned}$$ by choosing ${\cal F}(\textbf x)=\delta({\cal J}[{\textbf
x},\textbf f,\textbf A]-J)$, where $P(J)$ is the probability distribution for the quantity $\cal J$ achieved from the jump process (\[forwardeq\]) and $P_{\rm TR}(J)$ is the corresponding distribution from the time-revered system (\[timereversal\]). For any a pair of $\textbf A$ and $\textbf f$, eq. (\[DFT\]) usually does not hold.
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| {
"pile_set_name": "ArXiv"
} |
[**A Multiple Integral Explicit Evaluation Inspired by The Multi-WZ Method**]{}
Akalu Tefera[^1]
[**Abstract**]{}
We give an identity which is conjectured and proved by using an implementation \[3\] in multi-WZ \[5\].
[**0. Introduction**]{}
There are relatively few known non-trivial evaluations of n-dimensional integrals, with [*arbitrary $n$*]{}. Celebrated examples are the Selberg and the Metha-Dyson integrals, as well as the Macdonald constant term ex-conjectures for the various infinite families of root systems. They are all very important. See \[1\] for a superb exposition of the various known proofs and of numerous intriguing applications.
At present, the (continuous version of the) WZ method\[5\] is capable of mechanically proving these identities only for a fixed $n$. In principle for [*any*]{} fixed $n$ (even, say, $n=100000$), but in practice only for $n \leq 5$. However, by interfacing a human to the computer-generated output, the human may discern a pattern, and generalize the computer-generated proofs for $n=1,2,3,4$ to an arbitrary $n$.
Using this strategy, Wilf and Zeilberger\[5\] gave a WZ-style proof of Selberg’s integral evaluation. But just giving yet another proof of an already known identity, especially one that already had (at least) three beautiful proofs (Selberg’s, Aomoto’s, and Anderson’s, see \[1\]), is not very exciting.
In this article we present a [*new*]{} multi-integral evaluation, that was [*first*]{} found using the author’s implementation of the continuous multi-WZ method\[3\]. Both the conjecturing part, and the proving part, were done by a close human-machine collaboration. Our proof hence may be termed [*computer-assisted*]{} but not yet [*computer-generated*]{}.
Now that the result is known and proved, it may be of interest to have a non-WZ proof, possibly by performing an appropriate change of variables, converting the multi-integral to a double integral. My advisor, Doron Zeilberger, is offering \$100 for such a proof, provided it does not exceed the length of the present proof.
[**1. Notation**]{}
---------------------------------------------------------------------- -----------------------------------
${\bf x} = (x_{1}, \ldots, x_{k})$, $(y)_{m} = \prod_{i=0}^{m-1}(y +
i)$,
$d{\bf x} = dx_{1} \cdots dx_{k}$, $e_1({\bf x}) = \sum_{i=1}^{k}
x_{i}$,
$\hat {{\bf x}}_{i} = (x_{1}, \ldots, \hat {x}_{i}, \ldots, x_{k})$, $e_2({\bf
x}) = \sum_{i<j} x_{i} x_{j}$,
$\Delta_n F(n, {\bf x}) = F(n + 1, {\bf x}) - F(n, {\bf x})$
---------------------------------------------------------------------- -----------------------------------
[**2. The Integral Evaluation**]{}
[**Theorem** ]{} $$\int_{[0, +\infty)^{k}} (e_{2}({\bf
x}))^{m} (e_{1}({\bf x}))^{n} e^{-e_{1}({\bf x})} d{\bf x} = \frac {m! (2 m
+ n
+ k - 1)! {(k/2)}_{m} } {(2 m + k - 1)!} \left(\frac {2 (k - 1)}{k}
\right)^{m}T_{k}(m)$$ for all $k$ in ${{\rm I\!N}}$, and for all $m$, $n$ in ${\mathbf{Z}}_{\geq 0}$, where, $$T_{k}(m) - T_{k}(m - 1) = \frac {(k (k - 2))^{m} {((k - 1)/2)}_{m}}{(k - 1)^{2
m}{(k/2)}_{m}} T_{k - 1}(m)$$ for all $k \geq 2$, $T_{1} (m) = 0$, for all $m$ in ${\mathbf{Z}}_{\geq 0}$, and $ T_{k} (0) = 1 $ for all $ k \geq 2$.
[**3. Proof of the Integral Evaluation**]{}
If $k = 1 $, then trivially, both sides of the integral equate to zero. Let $k > 1$ and $A_{k}(m, n)$ be the left side of the integral divided by $$\frac {m! (2 m + n + k - 1)! {(k/2)}_{m} } {(2 m + k - 1)!} \left(\frac
{2
(k - 1)}{k} \right)^{m}.$$ We want to show $A_{k}(m, n) = T_{k}(m)$, for all $m$, $n$ in ${\mathbf{Z}}_{\geq 0}$. Let $$F_{k}(m, n; {\bf x}) := \frac {(2 m + k - 1)!} {m! (2 m + n +
k - 1)! {(k/2)}_{m} } \left(\frac {k} {2 (k - 1)} \right)^{m} (e_{2}({\bf
x}))^{m}
(e_{1}({\bf x}))^{n} e^{-e_{1}({\bf x})}$$ We construct[^2] $$R(u; v_{1}, \ldots, v_{k-1}) := \frac {u}{2 m + n + k},$$ with the motive that $${\mbox {(WZ 1)}} \qquad \Delta_{n} F_{k}(m, n; {\bf x}) = -
\sum_{i=1}^{k}
D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i}) F_{k}(m, n; {\bf x})].$$ Now, we verify (WZ 1), $$\begin{aligned}
\lefteqn{ \frac {F_{k} (m, n + 1; {\bf x}) - F_{k}(m, n; {\bf x}) +
\sum_{i=1}^{k}
D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i}) F_{k}(m, n; {\bf x})]
}{F_{k}(m, n; {\bf x})}}\\
& = & \frac {F_{k} (m, n + 1; {\bf x})}{F_{k}(m, n;{\bf x})} - 1 +
\sum_{i=1}^{k}
D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i})] + R(x_{i}; \hat{{\bf
x}}_{i})D_{x_{i}}
\left[log \left(F_{k}(m, n; {\bf x})\right) \right]\\
& = & \frac {e_{1}({\bf x})}{2 m + n + k} - 1 + \frac {k}{2 m + n + k} + \\
& & \sum_{i=1}^{k} \left(\frac {n}{e_{1}({\bf x})}\frac {x_{i}}{2 m + n + k}
+
\frac{m e_{1}(\hat {{\bf x}}_{i})}{e_{2}({\bf x})}
\frac {x_{i}}{2 m + n + k} - \frac {x_{i}}{2 m + n + k} \right)\\
& = & \frac {e_{1}({\bf x})}{2 m + n + k} - 1 + \frac {k}{2 m + n + k} +
\frac
{n}{2 m + n + k} + \frac{2 m }{2 m + n + k} -\frac {e_{1}({\bf x})}{2 m + n
+
k}\\
& = & 0.\end{aligned}$$
Hence, by integrating both sides of (WZ 1) w.r.t $x_{1}, \ldots, x_{k}$ over $[0, \infty)^{k}$, we get $$A_{k}(m, n + 1) - A_{k}(m, n) \equiv 0.$$
To complete the proof we show $A_{k}(m, 0) = T_{k}(m)$.
To this end, set $A_{k}(m) := A_{k}(m, 0)$ and $F_{k}(m; {\bf x}) :=
F_{k}(m,
0; {\bf x})$. Now, we construct[^3], $$R(u; v_{1}, \ldots, v_{k-1}) := \frac {((k - 1) (m + 1) + e_{1}(v_{1},
\ldots, v_{k-1})) u + e_{2}(v_{1}, \ldots, v_{k-1})}{(k - 1)(m + 1)(2 m +
k)}$$ with the motive that $${\mbox {(WZ 2)}} \qquad F_{k}(m + 1; {\bf x}) - F_{k}(m; {\bf x}) = -
\sum_{i=1}^{k} D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i}) F_{k}(m; {\bf x})].$$ Verification of (WZ 2): $$\begin{aligned}
\lefteqn{ \frac {F_{k}(m + 1; {\bf x}) - F_{k}(m; {\bf x}) + \sum_{i=1}^{k}
D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i}) F_{k}(m; {\bf x})]
}{F_{k}(m; {\bf x})}}\\
& = & \frac {F_{k} (m + 1; {\bf x})}{F_{k}(m; {\bf x})} - 1 +
\sum_{i=1}^{k}
D_{x_{i}}[R(x_{i}; \hat{{\bf x}}_{i})] + \sum_{i=1}^{k}
R(x_{i}; \hat{{\bf x}}_{i})D_{x_{i}} \left[log \left(F_{k}(m; {\bf
x})\right)
\right]\\
& = & \frac{k e_{2}({\bf x})}{(m + 1) (k - 1) (2 m + k)} - 1 +
\sum_{i=1}^{k}
\frac {(k - 1) (m + 1) + e_{1}(\hat{{\bf x}}_{i})}{(m + 1) (k - 1) (2 m +
k)} +
\\
& &\sum_{i=1}^{k} \frac {(k - 1) (m + 1) x_{i} + e_{2}({\bf x}))}{(m + 1) (k
-
1) (2 m + k)} \left (\frac {m e_{1}({\hat {\bf
x}}_{i})}{e_{2}({\bf x})} - 1 \right)\\
& = & \frac{k e_{2}({\bf x})}{(m + 1) (k - 1) (2 m + k)} - 1 + \frac{k}{2 m
+
k} + \frac {e_{1}({\bf x})}{(m + 1)(2 m + k)} + \frac {2 m}{2 m + k} - \frac
{e_{1}({\bf x})}{2 m + k} + \\
& & \frac {m e_{1}({\bf x})}{(m + 1)(2 m + k)} -\frac {k e_{2}({\bf x})}{(m
+
1)(k - 1)(2 m + k)}\\
& = & 0.\end{aligned}$$
Hence, by integrating both sides of (WZ 2) w.r.t. $x_{1}, \ldots, x_{k}$ over $[0, \infty)^{k}$, we obtain,
$$A_{k}(m + 1) - A_{k}(m) = \frac {(k (k - 2))^{m + 1} {((k - 1)/2)}_{m +
1}}{(k - 1)^{2 (m + 1)}{(k/2)}_{m + 1}} A_{k - 1}(m + 1),$$
and noting that $A_{k}(0) = 1$, $A_{1} (m) = 0$, it follows that $A_{k}(m) =
T_{k} (m)$, for all $m$ in ${\mathbf{Z}}_{\geq 0}$. Consequently, $A_{k}(m,
n) = T_{k} (m)$ for all $m$, $n$ in ${\mathbf{Z}}_{\geq 0}$. $ \Box$
By unfolding the recurrence equation for $T_{k}(m)$, we obtain the following identity.
[**Corollary**]{} $$\begin{aligned}
\lefteqn{ \int_{[0, +\infty)^{k}} (e_{2}({\bf
x}))^{m} (e_{1}({\bf x}))^{n} e^{-e_{1}({\bf x})} d{\bf x} =\frac {m! (2 m
+ n + k - 1)! {(k/2)}_{m} } {(2 m + k - 1)!} \left(\frac {2 (k - 1)}{k}
\right)^{m}}\\
& & \left( 1 + \sum_{r = 1}^{k - 2}\sum_{1 \leq s_{r} \leq \cdots \leq s_{1}
\leq m} \prod_{i = 1}^r \frac {((k - i)^2 - 1)^{s_{i}} ((k
- i)/2)_{s_{i}}}{(k - i)^{2 s_{i}} ((k - i + 1)/2)_{s_{i}}} \right) \end{aligned}$$
[**1.**]{} From the computational point of view, the recurrence form of the integral is [*nicer*]{} than its indefinite summation form (the above corollary), for the former requires $O(m k)$ calculations, whereas the latter requires $O(m^{k})$ calculations. However, in both forms the evaluation of the right side of the integral is much faster (for specific $m$, $n$, and $k$) than the direct evalution of the left side of our intergal. Hence both forms are indeed complete [*answers*]{} in the sense of Wilf\[4\].
[**2.**]{} The present paper is an example of what Doron Zeilberger\[6\] calls [*WZ Theory, Chapter 1 1/2*]{}. Even though, at present, our proof, for general $n$, was human-generated, it looks almost computer-generated. It seems that by using John Stembridge’s\[2\] Maple package for symmetric functions, SF, or an extension of it, it should be possible to write a new version of [SMint]{} that should work for [*symbolic*]{}, i.e. arbitrary, $n$, thereby fulfilling the hope raised in \[6\].
[**Acknowledgement:**]{} I thank Doron Zeilberger, my Ph.D. thesis advisor, for very helpful suggestions and valuable support.
[**References**]{}
\[1\] G. Andrews, R. Askey, and R. Roy, [ *Special Functions*]{}, Cambridge University\
Press, 1998.
\[2\] J.R. Stembridge, [*A Maple package for symmetric functions*]{}, J.\
Symbolic Comput., [**20**]{}(1995), 755-768.
\[3\] A. Tefera, [**SMint**]{}[*(A Maple Package for Multiple Integrals)*]{},\
.
\[4\] H.S. Wilf,[*What is an answer?*]{}, Amer. Math. Monthly, [**89**]{} (1982), 289-292.
\[5\] H.S. Wilf and D. Zeilberger, [*An Algorithmic proof theory for hypergeometric\
(ordinary and “q”) multisum/integral identities*]{}, Invent. Math., [**108**]{} (1992),\
575-633.
\[6\] D. Zeilberger, [*WZ Theory, Chapter II*]{}, The Personal Journal of S.B. Ekhad\
and D. Zeilberger, [http://www.math.temple.edu/$\sim$zeilberg/pj.html]{}.
[^1]: Department of Mathematics, Temple University, Philadelphia, PA 19104.\
[email protected]
[^2]: for specific $k$, the rational function $R$ is obtained by using [**SMint**]{} \[3\] and the output is available from [ *http://www.math.temple.edu/$\sim$akalu/maplepack/rational1.output*]{}
[^3]: for specific $k$, the rational function $R$ is obtained by using [**SMint**]{} \[3\] and the output is available from [ *http://www.math.temple.edu/$\sim$akalu/maplepack/rational2.output*]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is shown how cosmological perturbation theory arises from a fully quantized perturbative theory of quantum gravity. Central for the derivation is a non-perturbative concept of gauge-invariant local observables by means of which perturbative invariant expressions of arbitrary order are generated. In particular, in the linearised theory, first order gauge-invariant observables familiar from cosmological perturbation theory are recovered. Explicit expressions of second order quantities are presented as well.'
author:
- 'Romeo Brunetti$^{1,a}$, Klaus Fredenhagen$^{2,b}$, Thomas-Paul Hack$^{3,c}$, Nicola Pinamonti$^{4,d}$ and Katarzyna Rejzner$^{5,e}$'
title: Cosmological perturbation theory and quantum gravity
---
$^1$Dipartimento di Matematica, Università di Trento,\
$^2$II Institute für Theoretische Physik, Universität Hamburg,\
$^3$Institute für Theoretische Physik, Universität Leipzig,\
$^4$Dipartimento di Matematica, Università di Genova, and INFN, Sezione di Genova,\
$^5$Department of Mathematics, University of York.
Email: $^[email protected], $^[email protected], $^[email protected], $^[email protected], $^[email protected]
Introduction
============
The fluctuations of the cosmological microwave background provide a deep insight into the early history of the universe. The most successful theoretical explanation is inflationary cosmology where a scalar field (the inflaton) is coupled to the gravitational field. Usually, the theory is considered in linear order around a highly symmetric background, typically the spatially flat Friedmann-Lema\^ itre-Robertson-Walker spacetime.
Extending the theory to higher orders is accompanied by severe obstacles. Already in a classical analysis the definition of gauge-invariant observables turns out to be rather complicated; moreover, one is immediately confronted with the problem of constructing a theory of quantum gravity. Previous treatments of higher-order cosmological perturbation theory include [@Bartolo:2006fj; @Bruni:1996im; @Langlois:2010vx; @Maldacena:2002vr; @Malik:2008im; @Nakamura:2004rm; @Noh:2004bc; @Hwang:2012aa]; many further references on the subject can be found e.g. in [@Langlois:2010vx].
In a recent paper [@BFR] three of us reanalysed the field theoretical construction of quantum gravity from the view point of locally covariant quantum field theory. This analysis was based on the methods of perturbative Algebraic Quantum Field Theory (pAQFT), see [@FR-Advances; @in; @AQFT] and references therein, and on an adapted version of the Batalin-Vilkovisky formalism for the treatment of local gauge symmetries [@Hollands; @FR-BVQ]. The result was that a consistent theory (in the sense of an expansion into a formal power series) exists and is independent of the background. Due to non-renormalisability, however, in each order of perturbation theory new dimensionful coupling constants occur, which have to be fixed by experiments; hence the theory should be interpreted as an effective theory that is valid at scales where these new constants are irrelevant. One might hope that non-perturbative effects improve the situation in the sense of Weinberg’s concept of asymptotic safety, since there are encouraging results supporting this perspective; see for example [@Reuter; @Reuter2]. Furthermore, it is difficult to observe any effects of quantum gravity, so it seems reasonable to start from the hypothesis that at presently accessible scales the influence of these higher order contributions is small.
One of the main questions addressed by [@BFR] in the construction of the theory was the existence of local observables. It was answered, in a way familiar from classical general relativity, by using physical scalar fields, e.g. curvature scalars, as coordinates, and by expressing other fields as functions of these coordinates. Since quantization in the framework of pAQFT relies on a field theoretical version of deformation quantization of classical theories (first introduced in [@DF]), the classical construction can be transferred to the quantum realm.
The procedure works as follows. One selects $4$ scalar fields $X_{\Gamma}^a, a=1,\ldots 4$, which are functionals of the field configuration $\Gamma$ which includes the spacetime metric $g$, the inflaton field $\phi$ and possibly other fields. The fields $X_{\Gamma}^a$ are supposed to transform under diffeomorphisms $\chi$ as $$\label{equivariance}
X_{\chi^*\Gamma}^a=X_{\Gamma}^a\circ\chi\ ,$$ where $\chi^*$ denotes the pullback (of sections of direct sums of tensor products of the cotangent bundle) via $\chi$. We choose a background $\Gamma_0$ such that the map $$X_{\Gamma_0}:x\mapsto (X_{\Gamma_0}^1,\ldots,X_{\Gamma_0}^4)$$ is injective. In order to achieve injectivity on cosmological backgrounds $\Gamma_0$, we shall be forced to include the coordinates $x$ in the construction of $X_\Gamma$ in a way which is compatible with . We then consider $\Gamma$ sufficiently near $\Gamma_0$ and set $$\alpha_{\Gamma}=X_{\Gamma}^{-1}\circ X_{\Gamma_0}\,.$$ We observe that $\alpha_{\Gamma}$ transforms under diffeomorphisms – which leave the background $\Gamma_0$, that is by definition fixed, invariant – as $$\alpha_{\chi^*\Gamma}=\chi^{-1}\circ\alpha_{\Gamma}\,.$$ Let now $A_{\Gamma}$ be any other scalar field which is a local functional of $\Gamma$ and transforms under diffeomorphisms as in . Then the field $$\mathcal {A}_{\Gamma}:=A_{\Gamma}\circ\alpha_{\Gamma}$$ is invariant under diffeomorphisms and may be considered as a local observable. Note that invariance is obtained by shifting the argument of the field in a way which depends on the configuration.
The physical interpretation of this construction is as follows: the fields $X^a_\Gamma$ are configuration-dependent coordinates such that $[A_{\Gamma}\circ X_{\Gamma}^{-1}](Y)$ corresponds to the value of the quantity $A_{\Gamma}$ provided that the quantity $X_{\Gamma}$ has the value $X_{\Gamma}=Y$. Thus $A_\Gamma \circ X_\Gamma^{-1}$ is a partial or relational observable [@Rovelli:2001bz; @Dittrich:2005kc; @Thiemann:2004wk], and by considering $\mathcal {A}_{\Gamma} = A_\Gamma \circ X_\Gamma^{-1} \circ X_{\Gamma_0}$ we can interpret this observable as a field on the background spacetime.
Clearly, to make things precise, one also has to characterise the region in the configuration space where all the maps are well defined and restrict oneself to configurations $\Gamma$ in the appropriate neighbourhood of the background $\Gamma_0$, see [@BFR; @Igor] for details.
Fortunately, in formal deformation quantization as well as in perturbation theory, only the Taylor expansion of observables around some background configuration enters, hence it is sufficient to establish the injectivity of $X_{\Gamma_0}$ in order for the expansion of $\mathcal{A}_{\Gamma_0+\delta\Gamma}$ around $\Gamma_0$ to be well-defined. As an example we compute this expansion up to the first order. We obtain $$\mathcal{A}_{\Gamma_0+\delta\Gamma}=A_{\Gamma_0}+\left\langle\frac{\delta A_{\Gamma}}{\delta \Gamma}(\Gamma_0),\delta\Gamma\right\rangle+\frac{\partial A_{\Gamma_0}}{\partial x^{\mu}}\left\langle\frac{\delta\alpha_{\Gamma}^{\mu}}{\delta\Gamma}(\Gamma_0),\delta\Gamma\right\rangle + O(\delta \Gamma^2)\ .$$ The third term on the right hand side is necessary in order to get gauge-invariant fields (up to first order). We calculate $$\frac{\delta\alpha_{\Gamma}^{\mu}}{\delta\Gamma}(\Gamma_0)=-\left(\left(\frac{\partial X_{\Gamma_0}}{\partial x}\right)^{-1}\right)^{\mu}_ a\frac{\delta X_{\Gamma}^a}{\delta\Gamma}(\Gamma_0)\ .$$
In this work we apply this general idea to inflationary cosmology. In contrast to other systematic or covariant attempts to define gauge-invariant quantities in higher-order cosmological perturbation theory, see for example [@Langlois:2010vx; @Malik:2008im; @Nakamura:2014kza; @Hwang:2012aa], our construction works off-shell, is based on a clear and simple concept which is applicable to general backgrounds such that cosmological perturbation theory may be viewed as a particular application of perturbative quantum gravity [@BFR]. Moreover, we construct non-perturbative gauge-invariant quantities whose perturbative expansion to arbitrary orders may be computed algorithmically without the need for additional input at each order.
This paper is organised as follows: In the second section we recall a few basic facts about perturbation theory of the Einstein-Klein-Gordon system on cosmological backgrounds. In the third section we describe the general method to obtain gauge invariant observables at all orders on generic backgrounds. We furthermore discuss how to treat the case of a FLRW background where the large symmetry prevents us from using coordinates constructed from the dynamical fields alone. The fourth section contains the analysis of two gauge invariant observables at second order. The steps necessary for the construction of a full all-order quantum theory are briefly sketched in Section 5. Finally a number of conclusions are drawn in the last section.
Perturbations of the Einstein-Klein-Gordon system on a FLRW spacetime {#sec:perturbationsintro}
=====================================================================
We consider the Einstein-Klein-Gordon system, namely a minimally coupled scalar field $\tilde\phi$ with potential $V(\tilde\phi)$ propagating on a Lorentzian spacetime $(M,\tilde g)$ with field equations $$\label{eq:EKG}
R_{ab}-\frac{1}{2}R \tilde g_{ab} = T_{ab} ,\qquad - \Box \tilde\phi + V^{(1)}(\tilde\phi) = 0,$$ where $T_{ab}$ is the stress tensor of $\tilde\phi$, $R_{ab}$ the Ricci tensor and $R$ the Ricci scalar. We discuss perturbations of this system around a background. A linearised theory is obtained starting from a one-parameter family of solutions $\lambda \mapsto \Gamma_\lambda := (\tilde g_{\lambda},\tilde\phi_{\lambda})$ and considering $$\delta \Gamma := (\gamma,\varphi) := \left. \frac{d}{d\lambda}(\tilde g_{\lambda},\tilde\phi_{\lambda}) \right|_{\lambda = 0},$$ hence $\Gamma_0:=( g,\phi):=(\tilde g_0,\tilde\phi_0)$ is the background configuration while $\delta \Gamma = (\gamma,\varphi)$ is the linearised perturbation.
The background solution we choose consists of a flat Friedmann-Lema\^ itre-Robertson-Walker (FLRW) spacetime $(M,g)$ together with a scalar field $\phi$ which is constant in space. We recall that a flat FLRW spacetime is conformally flat and that $$\label{eq:FRW}
M = I \times \mathbb{R}^3, \qquad g= a^2(\tau)(-d\tau\otimes d\tau+\sum_i d{x}^i\otimes d{x}^i ),$$ where $I\subset \mathbb{R}$ is an open interval, the scale factor $a(\tau)$ is a function of the conformal time $\tau$ and where $x^i$ are three-dimensional Cartesian (comoving) coordinates. The background equations of motion of the system are best displayed in terms of the auxiliary function $$\mathcal{H} := \frac{a'}{a} ,$$ where $a'$ indicates the derivative with respect to the conformal time. $\mathcal{H}$ is related to the Hubble parameter $H=\mathcal{H}a^{-1}$ and to the Ricci scalar $R = 6(\mathcal{H}'+ \mathcal{H}^2) a^{-2}$. The background equations of motion are $$\begin{gathered}
\mathcal{H}^2= (\phi')^2+2a^2 V(\phi), \qquad
2(\mathcal{H}'+ 2\mathcal{H}^2) = - (\phi')^2+2a^2 V(\phi),\\
\phi'' + 2\mathcal{H} \phi'+ a^2V^{(1)}(\phi) = 0.\end{gathered}$$
A generic perturbation $\gamma$ of the FLRW metric $g$ can be decomposed in the following way $$\label{eq:perturbation}
\gamma =
a(\tau)^2\begin{pmatrix}
-2A && (-\partial_i B + V_i)^t \\
-\partial_i B + V_i && 2(\partial_i\partial_j E+\delta_{ij}D+\partial_{(i}W_{j)}+T_{ij})
\end{pmatrix}$$ where $A,B,D,E$ are scalars, $V, W$ are three dimensional vectors and $T$ is a tensor on 3-dimensional Euclidean space. The decomposition is unique if all these perturbations vanish at infinity and if $${T_i}^i = 0 , \qquad \partial_i {T^i}_j = 0 , \qquad \partial_i {V^i} = 0 , \qquad \partial_i {W^i} = 0$$ (see e.g. Proposition 3.1 in [@Hack]).
Under an infinitesimal first order gauge transformation the linear perturbations transform in the following way $$\gamma_{ab} \mapsto \gamma_{ab} + \mathcal{L}_{\xi} g_{ab} = \gamma_{ab} + 2\nabla_{(a} \xi_{b)} , \qquad
\varphi \mapsto \varphi + \mathcal{L}_{\xi} \phi = \varphi + \xi(\phi).$$ In particular $$\begin{gathered}
A\mapsto A+(\partial_\tau+\mathcal{H})r, \qquad
B\mapsto B+r-s', \qquad
D\mapsto D+\mathcal{H}r, \qquad
E\mapsto E+s, \\
\varphi\mapsto \varphi+\phi'r, \qquad
V_i\mapsto V_i + v_i', \qquad
W_i\mapsto W_i + v_i, \qquad
T_{ij}\mapsto T_{ij}, \qquad \end{gathered}$$ where the generator $\xi$ of one-parameter gauge transformations is also decomposed as $$\label{eq:perturbation_diffeo}
\xi^0 = r,\qquad \xi^i = \partial_i s + v_i, \qquad \partial_i v^i = 0.$$ Notice that the gauge transformations do not mix scalar, vector or tensor perturbations at linear order.
Furthermore, we observe that tensor perturbations are gauge-invariant and that gauge-invariant vector perturbations can be obtained considering $X_i := W'_i-V_i$. Regarding the scalar perturbations we see that the following fields are gauge-invariant $$\label{eq:Bardeen}
\Phi:= A-(\partial_t+\mathcal{H})(B+E'), \qquad
\Psi:= D- \mathcal{H}(B+E'), \qquad \chi := \varphi - \phi' (B+E').$$ The first two of them are called Bardeen potentials.
Let us recall the form of the linearised equations of motions satisfied by the gauge-invariant perturbations. The first observation is that the equations of motion respect the decomposition in scalar, vector and tensor perturbations. In particular, for the vector and tensor perturbations, it holds that $$\label{eq:linear-equations}
\Delta X_i =0, \qquad (\partial_t + 2\mathcal{H})X_i =0, \qquad
\frac{1}{a^2} (\partial_t^2 +2 \mathcal{H} \partial_t - \Delta ) T_{ij} =0.$$ For the scalar part the equations of motion are better displayed in terms of the Mukhanov-Sasaki variable $$\label{eq:muk-sas}
\mu := \chi - \frac{\phi'}{\mathcal{H}} \Psi = \varphi - \frac{\phi'}{\mathcal{H}} D.$$ The equation of motion for this variable is decoupled also from the other scalars of the theory, in fact $$\left( -\Box +\frac{R}{6} - \frac{z''}{z a^2} \right) \mu = 0 , \qquad z := \frac{a \phi'}{\mathcal{H}}.$$ The other scalar perturbations can be obtained in terms of $\mu$. In particular the Bardeen potential $\Phi$ is the unique solution of $$\label{eq:onshellfirstorder}
\Delta \Phi = \frac{\phi^\prime}{2}\left(\mu^\prime + \left(\frac{\mathcal{H}^\prime}{\mathcal{H}}-\frac{\phi^{\prime\prime}}{\phi^\prime}\right)\mu\right)
$$ while the other scalar perturbations are given by $$\label{eq:onshellfirstorder2}
\Psi = -\Phi , \qquad
\chi = \frac{2}{\phi'} (\partial_\tau + \mathcal{H}) \Phi.$$
We briefly discuss the situation beyond linear order. According to [@Sonego:1997np], infinitesimal diffeomorphisms may be approximated by so-called knight diffeomorphisms, which are of the form $\exp \mathcal{L}_{\xi}$ with $\xi = \lambda \xi_1 + \frac12 \lambda^2 \xi_2 + O(\lambda^3)$. Analogously we may expand a configuration $\Gamma$ as $ \Gamma = \Gamma_0+\delta \Gamma=\Gamma_0 + \lambda \delta\Gamma_1 + \frac12 \lambda^2 \delta\Gamma_2 + O(\lambda^3)$, and determine the transformation behaviour of separate orders by considering $\exp \mathcal{L}_{\xi} \, \Gamma$ at fixed order in $\lambda$, see for example [@Bartolo:2006fj; @Bruni:1996im; @Malik:2008im; @Nakamura:2004rm; @Noh:2004bc]. Assuming that $\xi$ and $\delta \Gamma$ vanish at spatial infinity, each order $\xi_i$ and $\delta \Gamma_i$ may be uniquely decomposed as in and . The transformation behaviour of the components of the latter decomposition becomes more complicated than at linear order, since higher-order gauge transformations mix scalar, vector and tensor quantities in a non-local fashion, as do the higher-order equations of motion. We shall not be concerned with the explicit form of higher-order gauge transformations in this work, as our constructions do not rely on these details and the quantities we consider are manifestly all-order gauge-invariant from the outset.
For the remainder of this work we shall use the following notation motivated by the fact that the space of configurations is an affine space. We decompose a general configuration $\Gamma$ as $\Gamma := (\tilde g, \tilde \phi) :=\Gamma_0 + \delta \Gamma $, where $\tilde g := g + \gamma$, $\tilde \phi := \phi + \varphi$ and $\delta \Gamma := (\gamma,\varphi)$ effectively subsumes linear and higher orders of the perturbation of the background $\Gamma_0 := (g,\phi)$. This applies analogously to the components of the decomposition of $\gamma$.
For later use we recall a useful observation regarding Bardeen potentials. The linear Bardeen potentials $\Phi$, $\Psi$ and the gauge-invariant scalar field perturbation $\chi$ in have the advantage that they coincide with $A$, $D$, and $\varphi$ respectively in the so-called longitudinal or conformal gauge where the components $B$ and $E$ of the metric perturbation $\gamma$ vanish. This gauge and the definition of the gauge-invariant quantities $\Phi$, $\Psi$ and $\chi$ may be extended to higher orders, such that also at higher orders $\Phi=A$, $\Psi = D$, $\chi=\varphi$ if $B=E=0$, see for example [@Malik:2008im].
All-order gauge-invariant observables on FLRW backgrounds {#sec_covcoords}
=========================================================
In this section we provide details on the general construction of all-order gauge-invariant quantities on general and FLRW backgrounds before discussing examples in the next section.
In perturbative Algebraic Quantum Field Theory (pAQFT) – the conceptual framework underlying perturbative quantum gravity in [@BFR] – observables of a field theory are described as functionals of smooth field configurations $\Gamma=(\tilde g,\tilde\phi)$. For the purpose of cosmological perturbation theory, we need the additional restriction that configurations vanish at spatial infinity. In order to be able to operate on the functionals, some regularity is required: the functional derivatives to all orders should exist as distributions of compact support.
Moreover, we restrict our attention to local functionals, i.e. those functionals whose $n-$th order functional derivatives are supported on the diagonal of $M^n$ for every $n$. Examples of objects of this form are $$\label{eq:field}
A_{\Gamma}(f):= \int_M A_{\Gamma} f$$ where $A_{\Gamma}$ is a smooth scalar function which is a polynomial in the derivatives of the field configuration $\Gamma=(\tilde g,\tilde\phi)$ (i.e. $A_{\Gamma}(x)=F(j_x(\Gamma))$ with $F$ a smooth function on the appropriate jet bundle) and where $f$ is a smooth compactly supported test density. However, later on in this work we are forced to consider also functionals which violate this locality condition as well as the condition of compact support. The diffeomorphisms $\chi$ of the spacetime act on configurations via pullback $\Gamma\mapsto\chi^*\Gamma$, and candidates for gauge-invariant fields are equivariant in the sense that $$\label{eq_equivariant}
A_{\chi^*\Gamma}=A_{\Gamma}\circ\chi\ .$$ Thus in order to exhibit gauge-invariant functionals one has to consider test densities $f_{\Gamma}$ which depend on the field configuration $\Gamma$ such that $$f_{\chi^*\Gamma}=\chi_*f_{\Gamma}\,,$$ where $\chi_*$ is the pushforward of test densities via $\chi$.
As described in the Introduction, in the general case we solve the problem by choosing four scalar fields $X_{\Gamma}^a$ which constitute a coordinate system $X_{\Gamma}$ for a given background $\Gamma_0$, and define the $\Gamma$-dependent diffeomorphism $$\alpha_{\Gamma}=X_{\Gamma}^{-1}\circ X_{\Gamma_0}\ .$$ For arbitrary test densities $f$, we may now consider the $\Gamma$-dependent test densities $f_{\Gamma}=\alpha_{\Gamma}{}_*f$ in order to obtain gauge-invariant observables $A_\Gamma(f_\Gamma)$ by means of . Equivalently, we may directly consider the gauge-invariant field $$\label{eq:ref3}
\mathcal{A}_{\Gamma}=A_{\Gamma}\circ\alpha_{\Gamma}\ .$$
Scalars that can be used as coordinates on generic backgrounds $\Gamma_0$ are e.g. traces of powers of the Ricci operator ${\mathbf{R}}$ $$\label{eq:ricci-scalars}
X_\Gamma^a:=\text{Tr} ({\mathbf{R}}^{a}), \qquad a\in\{1,2,3,4\}$$ (the operator which maps one forms to one forms and whose components are given in terms of the Ricci tensor ${R_{a}}^b$). When other (matter) fields are present in the considered model, also these can serve as coordinates, e.g., in the case of a Einstein-Klein-Gordon system, the scalar field $\tilde \phi$.
In view of renormalisation it is advisable to use coordinates $X_\Gamma$ which are local functionals of the configuration $\Gamma$. As we shall discuss in the following, this does not seem to be possible in cosmological perturbation theory on account of the symmetries of FLRW backgrounds $\Gamma_0$.
Perturbative expansion up to second order {#sec:generalexpansion2}
-----------------------------------------
To illustrate the general procedure we compute the second order expansion of the gauge-invariant field $\mathcal{A}_\Gamma$ which was to first order described in the Introduction.
We observe that we have to calculate the functional derivatives of the diffeomorphisms ${\alpha_{\Gamma}}$ with respect to $\Gamma$. We use the notation $$\left\langle\frac{\delta^n}{\delta \Gamma^n}X_{\Gamma}(\Gamma_0),\delta\Gamma^{\otimes n}\right\rangle=:X_n\ ,\quad \left\langle\frac{\delta^n}{\delta\Gamma^n}\alpha_{\Gamma}(\Gamma_0),\delta\Gamma^{\otimes n}\right\rangle=:x_n$$ and find up to second order $$x_0^{\mu}(x)=x^{\mu}\,,\quad x_1^{\mu}=-J^{\mu}_aX^a_1\,,$$ where $J$ is the inverse of the Jacobian of $X_{\Gamma_0}$, and $$x_2^{\mu}=-J^{\mu}_aX_2^a-J^{\mu}_aJ^{\nu}_bJ^{\rho}_c\frac{\partial^2 X_0^a}{\partial x^{\nu}\partial x^{\rho}}X_1^bX_1^c+2J^{\mu}_aJ^{\nu}_b\frac{\partial X_1^a}{\partial x^\nu}X_1^{b}\ .$$ We use an analogous notation for the Taylor expansions of the fields $A_{\Gamma}$ and $\mathcal{A}_{\Gamma}$ and find $$\label{eq_gaugeinvexp1}
\mathcal{A}_0=A_0\,,\quad \mathcal{A}_1=A_1+\frac{\partial A_0}{\partial x^{\mu}}x_1^{\mu}\,,$$ and $$\label{eq_gaugeinvexp2}
\mathcal{A}_2=A_2+2\frac{\partial A_1}{\partial x^{\mu}}x_1^{\mu}+\frac{\partial A_0}{\partial x^{\mu}}x_2^{\mu}+\frac{\partial^2 A_0}{\partial x^{\mu}\partial x^{\nu}}x_1^{\mu}x_1^{\nu}\ .$$
Non-degenerate covariant coordinates on FLRW backgrounds {#sec:coordinates}
--------------------------------------------------------
In order to obtain these expansions we need a $4$-tuple of equivariant fields which define a non-degenerate coordinate system on the background $\Gamma_0$. This is possible in the generic case, e.g. by using the ansatz , but creates problems, if the background metric possesses non-trivial symmetries. This applies to the case of FLRW backgrounds $\Gamma_0$ where only time functions can be constructed out of the background metric $g$ and the background scalar field $\phi$. In the following we present a construction of non-degenerate coordinates which solves the above-mentioned problem at the expense of being non-local, albeit in a controlled way. Note that introducing additional external fields as reference coordinates like in the Brown-Kuchař model [@Brown:1994py] is not useful in the context of cosmological perturbation theory because these fields would appear in the final gauge-invariant expressions and thus an interpretation of these in terms of only the fundamental dynamical fields is difficult. The construction we present in the following does involve the comoving spatial coordinates $x^i$ of the FLRW spacetime as an external input. However the explicit dependence on $x^i$ disappears from the final expressions because these depend on $X_{\Gamma_0}$ only via its Jacobian.
The simplest choice of the time coordinate is provided by the inflaton field itself, so we set $$\label{eq_coord0}
X_\Gamma^0 = \tilde \phi = \phi + \varphi\,.$$ The construction of the spatial coordinates $X^i_\Gamma$ needs a bit of preparation. To this end, we consider the unit time-like vector $$\label{eq_nphi}
n_\phi = \frac{\tilde g^{-1}(d\tilde \phi,\cdot)}{\sqrt{|\tilde g^{-1}(d\tilde \phi,d\tilde \phi)}|} = \frac{1}{a}(1-A)\partial_\tau + \frac{1}{a}\left(\partial^i B -\frac{\partial^i \varphi}{\phi^\prime}\right) \partial_i + O(\delta \Gamma^2)$$ and the tensor $$\label{eq_hphi}
h_\phi = \tilde g + \tilde g(n_\phi,\cdot) \otimes \tilde g(n_\phi,\cdot)\,,$$ where $\partial^i := \partial_i := \partial/\partial x^i$ and $x^i$ for $i\in\{1,2,3\}$ are comoving spatial coordinates on the FLRW spacetime $(M,g)$. $n_\phi$ is a unit normal on the hypersurfaces of constant $\tilde \phi$ and $h_\phi$ is the induced metric on these hypersurfaces.
Let $\Delta_\phi$ denote the Laplacian for $h_\phi$ and $G_\phi$ its inverse, which we choose by imposing the boundary condition that the background value of $G_\phi$ is the Coulomb potential $G_\Delta$ with suitable factors of the scale factor $a$. We define and compute $$\Delta_\phi := \Delta_0 + \delta\Delta\,,\qquad \Delta_0 := \frac{\Delta}{a^2}\,,\qquad \Delta := \sum^3_{i=1} \partial^2_i$$ $$\delta\Delta = -\lambda\left( \frac{ 2( D + \Delta E)\Delta
-(\partial^i ( D - \Delta E))\partial_i}{a^2}+\frac{(\Delta \varphi ) \partial_\tau+(\partial^i \varphi)(2\partial_\tau + {{\mathcal H}}) \partial_i}{a^2 \phi^\prime}\right)+O(\delta \Gamma^2)$$ $$G_\phi := G_0 + \delta G\,,\qquad G_0 := a^2 G_\Delta\,, \qquad G_\Delta \circ \Delta = \1\quad \text{on functions that vanish at spatial infinity}\,,$$ $$\delta G = \sum^\infty_{n=1} (-1)^n G_0 \circ (\delta \Delta \circ G_0)^{\circ n} = - G_0 \circ \delta \Delta \circ G_0 + O(\delta\Gamma^2)\,.$$ Using these objects, we obtain $$\label{eq:Ycoords}
Y_\Gamma^i := \left(1-G_\phi \circ \Delta_\phi \right)x^i = x^i + \partial_i ( E+G_\Delta\mathfrak{R})+O(\delta \Gamma ^2)\,,\qquad \mathfrak{R} := \frac{{{\mathcal H}}}{\phi^\prime}\mu\,.$$ We observe that $Y_\Gamma^i$ are harmonic coordinates for $\Delta_\phi $ that we have constructed by means of $x^i$, i.e. harmonic coordinates for $\Delta_0$. The construction of $Y_\Gamma^i$ makes sense for all configurations $\Gamma$ which vanish at spatial infinity, but not in general. The restriction to this set of configurations from the outset is natural in the context of cosmological perturbation theory – recall that the decomposition is unique only in this case – and does not create problems for the pAQFT framework. For consistency, we have to restrict the class of infinitesimal diffeomorphisms we consider in the same manner. In fact, a straightforward computation reveals that the functionals $Y_\Gamma^i$ are equivariant with respect to all diffeomorphisms $\chi$ that vanish at spatial infinity $$\chi^* Y_\Gamma^i = Y_{\chi^*\Gamma}^i + (1-G_{\chi^* \phi}\circ \Delta_{\chi^* \phi})(\chi^* x^i - x^i ) = Y_{\chi^*\Gamma}^i\,,$$ but not with respect to arbitrary diffeomorphisms. Here $\Delta_{\chi^* \phi}$ denotes the Laplacian constructed analogous to $\Delta_{\phi}$ but with $\chi^* \tilde \phi$ instead of $\tilde \phi$ and $G_{\chi^* \phi}$ denotes its inverse with the discussed boundary condition. Consequently, the observables constructed by means of the equivariant coordinates and via are gauge-invariant with respect to diffeomorphisms which vanish at spatial infinity. As anticipated, the coordinates $Y_\Gamma^i$ are non-local, but the non-locality of $G_\phi $ is relatively harmless since its wave front set is that of the $\delta$-function, and renormalisation of expressions involving such objects is well under control, cf. Section \[sec:quantization\].
The coordinates are not entirely well-suited for practical computations because of the fact that the rescaled Mukhanov-Sasaki variable $\mathfrak{R}$ appears convoluted with the Coulomb potential. In order to remedy this we use a different family of spatial hypersurfaces and a corresponding modification of the spatial Laplacian and its inverse. To this end we consider a number of additional quantities related to the slicing induced by the time-function $\tilde \phi$: the lapse function $N_\phi$, the extrinsic curvature $K_{\phi,ab}$, and the spatial Ricci scalar $R^{(3)}_\phi $ which are defined and computed respectively as $$\begin{gathered}
\label{eq_lapsenongi}
N_\phi:= |\tilde{g}^{-1}_\lambda(d\tilde\phi,d\tilde \phi)|^{-1/2} = \frac{a}{\phi^\prime}\left(1 - \frac{\varphi'}{\phi'} + A\right) + O(\delta\Gamma^2)\,,\\
K_{\phi,ab} := {h_{\phi,a}}^c\nabla_{c} n_{\phi,b}\,,\qquad K_\phi := {K_{\phi,a}}^a = \frac{3{{\mathcal H}}}{a} + O(\delta \Gamma)\,,\label{eq_traceK}\\
R^{(3)}_\phi := K_{\phi,ab}K_\phi ^{ba}-K_\phi ^2+2\left(R_{ab}-\frac12 R \tilde g_{ab}\right)n_\phi^an_\phi^b = \frac{4}{a^2} \Delta {{\mathfrak R}}+ O(\delta\Gamma^2)\,, \label{eq_spatialCurv}\end{gathered}$$ where $n_\phi$ and $h_\phi$ are defined respectively in and . Using these quantities, we define a new time function $$\mathfrak{t}:=\tilde \phi - \frac{3 N_\phi}{4 K_\phi } G_\phi R^{(3)}_\phi = \phi + \frac{\phi^\prime}{{{\mathcal H}}} D+O(\delta\Gamma ^2)\,,$$ If we define the spatial metric $h_\mathfrak{t}$, the Laplacian $\Delta_\mathfrak{t}$ and its inverse $G_\mathfrak{t}$ in analogy to $h_\phi$, $\Delta_\phi $ and $G_\phi $ by replacing $\tilde \phi$ with $\mathfrak{t}$ we obtain $$\label{eq:Xcoords}
X_\Gamma^i := \left(1-G_\mathfrak{t} \circ \Delta_\mathfrak{t}\right)x^i = x^i + \partial_i E+O(\delta \Gamma ^2)\,,$$ and the spatial coordinates $X_\Gamma^i$ share the qualitative properties of the initially defined $Y_\Gamma^i$.
Examples of gauge-invariant observables at second order {#sec_obs}
=======================================================
In the previous sections we have developed a principle to construct gauge-invariant perturbative observables from non-gauge-invariant ones. In the following we demonstrate this principle at the example of two observables which are relevant in Cosmology. To this end we use the covariant coordinates and .
Despite the mild non-locality inherent in the covariant spatial coordinates $\eqref{eq:Xcoords}$, we are interested in observables $A_\Gamma$ which are local functionals of the configuration $\Gamma$. The non-locality of $\mathcal{A}_\Gamma = A_\Gamma \circ \alpha_\Gamma$ implied by the non-locality of $X^i_\Gamma$ in appears only because we consider the local functional $A_\Gamma$ relative to the non-local functional $X_\Gamma$. Since the background $\Gamma_0$ depends only on time the same applies to the background value of any local functional $A_\Gamma$. Consequently, at first order only the field $X^0_\Gamma$ chosen as time coordinate enters the formula for gauge-invariant fields. At second order also the fields used as spatial coordinates $X^i_\Gamma$ enter the expression.
The inverse $J$ of the Jacobi matrix of the coordinate transform $X_{\Gamma_0}$ on the background is $$J=\left(\begin{array}{cccc}
\frac{1}{\phi'}&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}\right).$$ The field dependent shifts from Section \[sec:generalexpansion2\] with respect to these coordinates up to second order are $$x_1^0=-\frac{\varphi}{\phi^{\prime}}\,,\qquad \ x_1^i=-\partial_i E\,,$$ and $$x_2^{0}=-\frac{\phi''{\varphi}^2}{(\phi')^3}+\frac{2}{\phi'}\left(\frac{\varphi'\varphi}{\phi'}+(\partial_i \varphi)\partial^i E\right),$$ $$x_2^{i}=\frac{2\varphi}{\phi'}\partial_i E^\prime +2(\partial^i \partial^j E)\partial_j E - (X^i_\Gamma - x^i - \partial_i E)\,.$$ Thus, for a field $A_\Gamma$ whose value on the background depends only on time the contributions up to second order for the gauge-invariant modification $\mathcal{A}_\Gamma = A_\Gamma \circ \alpha_\Gamma$ are $$\mathcal{A}_0=A_0\,,\qquad \mathcal{A}_1=A_1-\frac{A_0'\varphi}{\phi'}\,,$$ $$\mathcal{A}_2=A_2-\frac{2A_1'\varphi}{\phi'}-2(\partial_iA_1)\partial^i E+A_0'\left(-\frac{\phi''{\varphi}^2}{(\phi')^3}+\frac{2}{\phi'}\left(\frac{\varphi'\varphi}{\phi'}+(\partial_i \varphi)\partial^i E\right)\right)+\frac{A_0''\varphi^2}{(\phi')^2}\,.$$ If we were to use the fields $Y^i_\Gamma$ as spatial coordinates rather than the fields $X^i_\Gamma$ , then the corresponding expression for $\mathcal{A}_1$ would remain unchanged whereas $\mathcal{A}_2$ would change by replacing all occurrences of $\partial_i E$ by $\partial_i E + G_\Delta \partial_i \mathfrak{R}$. This demonstrates the dependence of the gauge-invariant constructions on the chosen covariant coordinate system.
The lapse function {#sec_obs_lapse}
------------------
The Sachs-Wolfe effect is one of the main building blocks of the current understanding of the Cosmic Microwave Background (CMB). A rough estimate of this effect can be obtained using the Tolman idea, see e.g. [@Mukhanov:2003xr]. Given a spacetime with a (conformal) timelike Killing field $\kappa$ and a state in equilibrium relative to the $\kappa$-flow with absolute temperature $T$, an observer with four-velocity $u\propto \kappa$ measures the temperature $\widetilde T=T/N$ with $N$ denoting the lapse function $N=\sqrt{|g(\kappa,\kappa)|}$.
In the context of Cosmology we use the Klein-Gordon field $\tilde\phi$ as a time coordinate and consider the vector $$\kappa_\phi := N_\phi n_\phi = \frac{1}{\phi'}\partial_\tau + O(\delta \Gamma)$$ with $N_\phi$, $n_\phi$ defined in and respectively as an approximate conformal Killing vector – in the sense that $\mathcal{L}_{\kappa_\phi} \tilde g - 2 {{\mathcal H}}/\phi' \tilde g = O(\phi'', \delta \Gamma)$. The corresponding lapse function is $N_\phi = a/\phi^\prime + O(\delta \Gamma)$. Its background value is not vanishing and thus it is not automatically gauge-invariant at linear order.
As described in Section \[sec\_covcoords\], we may obtain a non-perturbatively gauge-invariant version of the lapse function by setting and computing $$\begin{aligned}
{{\mathcal N}}_\phi:= & N_\phi \circ \alpha_\Gamma = \frac{a}{\phi^\prime}\left(1-\left((\partial_\tau + \mathcal{H})\frac{\varphi}{\phi^\prime} - A\right)\right)+O(\delta\Gamma^2)\label{eq_lapsegi}\\
=&\frac{a}{\phi^\prime}\left(1-\left((\partial_\tau + \mathcal{H})\frac{\chi}{\phi^\prime} - \Phi\right)\right)+O(\delta\Gamma^2)\,,\notag\end{aligned}$$ where $\Phi$ and $\chi$ are the gauge-invariant fields reviewed in Section \[sec:perturbationsintro\]. Using the on-shell identities , and the definition of the Mukhanov-Sasaki field $\mu$ we can rewrite the linear term as $$\begin{aligned}
{{\mathcal N}}_{\phi,1}=& \lambda\frac{a}{\phi^\prime}\left((\partial_\tau + \mathcal{H})\frac{\chi}{\phi^\prime} - \Phi\right)=\frac{a}{(\phi^\prime)^2}\left(\mu^\prime + \left(\frac{\mathcal{H}^\prime}{\mathcal{H}}-\frac{\phi^{\prime\prime}}{\phi^\prime}\right)\mu\right)\\
=&\frac{2a}{(\phi^\prime)^3} \Delta \Phi = -\frac{2a}{(\phi^\prime)^3} \Delta \Psi\,.\end{aligned}$$ Using the quantities introduces in Section \[sec:coordinates\], we may extract the Bardeen potential on-shell from $N_{\phi}$ as $$\left[\frac{1}{2 N_{\phi}^3} G^2_\phi \Delta_\phi N_{\phi}\right]\circ \alpha_\Gamma = \Phi + O(\delta \Gamma^2)\,.$$ In fact, one could use the above equation as a covariant, gauge-invariant, all-order (and on shell) definition of $\Phi$; however, we shall refrain from doing so.
In order to display second order expressions in a readable form we omit terms containing the metric perturbation components $V_i$, $W_j$ and $T_{ij}$ and use once more the Bardeen potentials $\Phi$, $\Psi$ and the gauge-invariant scalar field perturbation $\chi$. We stress that the particular expressions of these fields at linear and higher order are not needed for the actual computations but just for a compact display of the result. Using this, we arrive at the following second order form of the gauge-invariant lapse function $$\begin{aligned}
{{\mathcal N}}_{\phi,2} &= \frac{a}{\phi'}\left(-\Phi^2 -2 \left(\frac{\Phi \chi}{\phi'}\right)' -2 \mathcal{H}\frac{\Phi \chi}{\phi'} +2 \left(\left(\frac{\chi}{\phi'}\right)'\right)^2+\left( \frac{\phi''}{\phi'}+2\mathcal{H} \right)\left(\frac{\chi^2}{\phi'^2}\right)'
+
\right.
\\
&\qquad \left.+\left(\mathcal{H}^2+\mathcal{H}'+\frac{\phi'''}{\phi'}+\mathcal{H}\frac{\phi''}{\phi'}- \frac{\phi''^2}{\phi'^2}\right)\frac{\chi^2}{\phi'^2} + \sum^3_{i=1}\left(\partial_i\left(\frac{\chi}{\phi'}\right)\right)^2
+2\frac{\chi}{\phi'}\left(\frac{\chi}{\phi'}\right)''
\right),\end{aligned}$$ where, as before, we use the notation that e.g. $\Phi = \lambda \Phi_1 + \frac12 \lambda^2 \Phi_2+O(\lambda^3)$ and omit the second order terms linear in $\Phi$, $\chi$ displayed already in .
The spatial curvature
---------------------
A further observable of interest is the scalar curvature of the spatial metric induced by a particular slicing because for a large class of slicings this quantity vanishes in the background and thus is automatically gauge-invariant at linear order. Moreover, for the slicing defined by the inflation field it is related to the Mukhanov-Sasaki field $\mu$ which has a very simple dynamical equation.
We have already discussed the spatial curvature relative to the slicing induced by $\tilde \phi$. It may be computed as $$R^{(3)}_\phi = \frac{4}{a^2} \Delta {{\mathfrak R}}+ O(\delta\Gamma^2)\,,\qquad \mathfrak{R} = \frac{{{\mathcal H}}}{\phi^\prime}\mu =\frac{{{\mathcal H}}}{\phi^\prime}\varphi - D \,.$$ In the literature, the quantity ${{\mathfrak R}}$ is usually called the *comoving curvature perturbation*. This is due to the fact that the $\tilde \phi$-slicing may be equivalently characterised by the condition that $$T(\tilde \phi)_{ab} n_\phi^a = -\tilde g_{ab} n_\phi^a T(\tilde \phi)_{cd} n_\phi^c n_\phi^d\,,$$ i.e. that the energy flux of $\tilde \phi$ is parallel to $n_\phi$, where $T(\tilde \phi)_{ab}$ is the stress tensor of $\tilde \phi$.
An alternative slicing considered in the literature is the one defined by the energy density $\tilde\rho$ of $\tilde \phi$ $$\tilde\rho := T(\tilde \phi)_{ab} n_\phi^a n_\phi^b = \rho + \varrho\,,$$ $$\rho := \frac{(\phi')^2}{2 a^2}\,,\qquad \varrho := V^{(1)}(\phi)\varphi + \frac{\phi'(\varphi' - \phi' A)}{a^2}+O(\delta \Gamma^2)\,.$$ The spatial curvature $R^{(3)}_\rho$ with respect to this slicing, defined in analogy to $R^{(3)}_\phi$, reads $$R^{(3)}_\rho = \frac{4}{a^2}\Delta \zeta + O(\delta \Gamma)\,,\qquad \zeta := \frac{{{\mathcal H}}}{\rho'}\varrho-D \,,$$ where $\zeta$ is called *uniform density perturbation* because $\tilde \rho$ is by definition constant on the hypersurfaces in the slicing relative to $\tilde \rho$. The global sign in the definition of $\zeta$ is conventional.
As anticipated, the background contributions of $R^{(3)}_\phi$ and $R^{(3)}_\rho$ vanish and thus $${{\mathcal R}}^{(3)}_\phi := R^{(3)}_\phi \circ \alpha_\Gamma = R^{(3)}_\phi + O(\delta\Gamma^2)\,,\qquad {{\mathcal R}}^{(3)}_\rho := R^{(3)}_\rho \circ \alpha_\Gamma = R^{(3)}_\rho + O(\delta\Gamma^2)\,,$$ cf. , . In order to display the second order contribution to ${{\mathcal R}}^{(3)}_\phi$, we make the simplifications discussed for the lapse function in Section \[sec\_obs\_lapse\]. Proceeding like this, we find $$\begin{aligned}
\label{eq_spatialgi}
{{\mathcal R}}^{(3)}_{\phi,2}=&\frac{8}{a^2}\left(\Delta\left(2{{\mathfrak R}}^2 - \frac{\chi}{\phi'}(\partial_\tau + 2{{\mathcal H}}){{\mathfrak R}}+ \frac12\left({{\mathcal H}}'+2{{\mathcal H}}^2-\frac{{{\mathcal H}}\phi''}{\phi'}\right)\left(\frac{\chi}{\phi'}\right)^2\right)\right.\\
&\qquad\qquad \left.- \frac{5(\partial_i {{\mathfrak R}}) \partial^i {{\mathfrak R}}}{2}\right).\notag\end{aligned}$$ We omit the result for ${{\mathcal R}}^{(3)}_{\rho,2}$ computed with the coordinate system $X_\Gamma$ defined in and , because it is rather long due to the “mismatch” between the time coordinate $\tilde \phi$ used in $X^0_\Gamma$ and the time coordinate $\tilde \rho$ used in the definition of $R^{(3)}_\rho$. Clearly, using $\tilde \rho$ as a time coordinate in both aspects we would obtain a second order expression ${{\mathcal R}}^{(3)}_{\rho,2}$ which is of the form up to the replacements $$\label{eq_replacements}
{{\mathfrak R}}\mapsto \zeta\,,\qquad \phi\mapsto \rho\,,\qquad \chi\mapsto \pi:= V^{(1)}(\phi)\chi + \frac{\phi'(\chi'-\phi'\Phi)}{a^2}\,,$$ where $\pi$ is gauge-invariant with $\pi = \varrho + O(\delta \Gamma^2)$ in the longitudinal gauge.
On shell and at first order, $\mu$, and thus ${{\mathfrak R}}$, are preferred observables because they have canonical equal-time Poisson brackets and thus in the quantized theory they commute at spacelike separations, in contrast to $\Psi$, $\Phi$ and $\chi$ [@Eltzner; @Hack]. Moreover, again on shell and at first order, one may compute $$\zeta = {{\mathfrak R}}-\frac{2 \Delta \Phi}{3 (\phi')^2} = {{\mathfrak R}}- \frac{{{\mathfrak R}}'}{3 {{\mathcal H}}}\,.$$ Consequently, $\zeta$ shares the causality properties of $\mu$ and ${{\mathfrak R}}$.
Apart from the phenomenological relevance of an all-order definition of ${{\mathfrak R}}$, $\mu$ and $\zeta$, it is interesting on conceptual grounds to investigate whether the causality property of these fields persists at higher orders. To this end, we need a fully covariant and gauge-invariant all-order definition of ${{\mathfrak R}}$, $\mu$ and $\zeta$. Such a definition may be given by means of covariant quantities introduced in Section \[sec:coordinates\]: $$\begin{aligned}
\label{eq_higherorderR}
\left[\frac{1}{4} G_\phi R^{(3)}_\phi\right]\circ \alpha_\Gamma &= \frac{a^2}{4} G_\Delta \mathcal{R}^{(3)}_\phi- a^2 G_\Delta \delta\Delta {{\mathfrak R}}+ O(\delta\Gamma^3)\notag\\
&={{\mathcal H}}\frac{\chi}{\phi'}-\Psi + {{\mathfrak R}}^2 - 2 {{\mathcal H}}\frac{\chi}{\phi'}{{\mathfrak R}}+ \frac12\left({{\mathcal H}}'+2{{\mathcal H}}^2-\frac{{{\mathcal H}}\phi''}{\phi'}\right)\left(\frac{\chi}{\phi'}\right)^2 + \\
\notag&\qquad + G_\Delta\left(\frac{(\partial_i {{\mathfrak R}}) \partial^i {{\mathfrak R}}}{2}\right) + O(\delta\Gamma^3)\,,\end{aligned}$$ $$\label{eq_higherordermu}
\left[\frac{3 N_\phi}{4 K_\phi } G_\phi R^{(3)}_\phi\right]\circ \alpha_\Gamma = \mu + O(\delta\Gamma^2)\,,\qquad \left[\frac{1}{4} G_\phi R^{(3)}_\rho\right]\circ \alpha_\Gamma = \zeta + O(\delta\Gamma^2)\,.$$ In we wrote the $O(\delta \Gamma)$ term as ${{\mathcal H}}\chi/\phi' - \Psi$ instead of ${{\mathfrak R}}$ because the fields $\chi$, $\Psi$ are defined in such a way that they are invariant also with respect to second order gauge transformations (cf. the end of Section \[sec:perturbationsintro\]), whereas ${{\mathfrak R}}={{\mathcal H}}\varphi/\phi' - D$ is only gauge-invariant up to the first order.
In analogy to our discussion of $R^{(3)}_\rho$, using $\tilde \rho$ rather than $\tilde \phi$ both as the time coordinate $X^0_\Gamma$ and as the time function defining a foliation of spacetime, we obtain a higher order definition of $\zeta$ which is of the form up to the replacements in (whereby a second order generalisation of $\pi$, which can be constructed in analogy to the second order Bardeen potentials, is needed).
In the literature, several possible second order gauge-invariant corrections to ${{\mathfrak R}}$ are considered. One often encounters constructions where in a gauge with $\varphi=0$ (or $D=0$), the second order corrections to ${{\mathfrak R}}$ vanish – at least in situations where spatial derivatives can be neglected in comparison to temporal ones, see e.g. [@Maldacena:2002vr; @Malik:2008im; @Prokopec:2012ug; @Vernizzi:2004nc]. In fact ${{\mathfrak R}}$ is often defined by the condition ${{\mathfrak R}}= -D$ in a gauge where $\varphi=0$. A quick analysis reveals that this is not the case in our construction . In [@Vernizzi:2004nc] it is argued that expressions for ${{\mathfrak R}}$ valid up to second order that are not of this form, e.g. the one in [@Acquaviva:2002ud], are potentially physically ill-behaved because they are not conserved on “super-Hubble scales”. Here, conservation of a function $f(\tau,\vec x)$ on “super-Hubble scales” means that the Fourier transform $\hat f(\tau,\vec k)$ of $f$ with respect to $\vec x$ satisfies $\partial_\tau \hat f(\tau,\vec k) = O(|\vec k|/{{\mathcal H}})$. This property, whose relevance is explained e.g. in [@Maldacena:2002vr; @Vernizzi:2004nc], usually holds only on-shell. It would be interesting to check whether our result for ${{\mathfrak R}}$ as given in (and the analogous result for $\zeta$) is conserved in this sense; however, this is beyond the scope of the present work.
Quantization {#sec:quantization}
============
In the previous sections we have prepared the ground for an all-order perturbative quantization of the Einstein-Klein-Gordon system on FLRW backgrounds, i.e. for a conceptually clear higher-order generalisation of quantized cosmological perturbation theory. In this section we would like to sketch the steps necessary for a full construction of the quantum theory. A detailed account will be given in a future work [@longpaper].
BRST quantization
-----------------
It is known that a direct quantization of non-linear gauge-invariant observables in a theory with local gauge symmetries is difficult. The standard way out is to perform a gauge fixing in the sense of the BRST method, or more generally, the BV formalism, as treated in [@Hollands; @Fredenhagen:2011an; @FR-BVQ]. There one adds a Fermionic vector field $c^{\mu}$ (the ghost field), which describes the infinitesimal gauge transformations, auxiliary scalar fields $b_{\mu}, \bar{c}_{\mu}$, where $b_{\mu}$ (the Nakanishi-Lautrup field) is Bosonic and $\bar{c}_{\mu}$ (antighost) is Fermionic, $\mu=0,\ldots,3$. Infinitesimal coordinate transformations are described by the BRST operator $s$, which acts on scalar local functionals $A$ of the metric, the inflaton and the $b$ fields by $$s(A)(x)=c^{\mu}(x)\partial_{\mu}A(x)\ ,$$ on the components of the ghost field by $$s(c^{\mu})(x)=c^{\nu}(x)\partial_{\nu}c^{\mu}(x)\ ,$$ on antighosts by $$s(\bar{c}_{\mu})(x)=ib_{\mu}(x)-c^{\nu}(x)\partial_{\nu}\bar{c}_{\mu}(x)$$ and satisfies on products the graded Leibniz rule so that $s^2=0$. One can characterise the classical observables as functionals in the kernel of $s$ modulo those in the image of $s$ (i.e. classical observables belong to the $0$-th cohomology group of $s$).
The field equations for the extended system are the usual field equation for $\tilde\phi$ as well as $$R_{\mu\nu}=T(\tilde\phi)_{\mu\nu}-\frac12 T(\tilde\phi)\tilde g_{\mu\nu}+s(i\partial_{(\mu}\bar{c}_{\nu)})$$ $$\square_{\tilde g} c^{\mu}=0$$ $$\square_{\tilde g} \bar{c}_{\mu}=0$$ $$|\mathrm{det}\tilde g|^{-\frac12}\partial_{\mu}|\mathrm{det}\tilde g|^{\frac12}\tilde g^{\mu\nu}=\kappa^{\mu\nu}b_{\mu}\,.$$ Here $\kappa$ is a non-degenerate fixed tensor.
The quantization of the extended system now proceeds largely analogous to the pure gravity treatment in [@BFR]. The main idea is to use deformation quantization to deform the algebra of functionals as well as the BRST operator $s$. Elements of the cohomology of the quantized (i.e. deformed) BRST operator $s$ are then interpreted as quantized versions of the gauge-invariant fields discussed in the previous sections.
Renormalisation
---------------
A conceptual and technical difference to the pure gravity case treated in [@BFR] arises because of the fact that we have introduced a mild non-locality via the non-local spatial coordinates $X^i_\Gamma$ . In [@BFR] renormalisation was treated in the Epstein-Glaser framework which is initially only suitable for local functionals. As we have to deal with non-local expressions, we need to extend this framework from local quantities to non-local ones. Recall that $$X_{\Gamma}^i=(1-G_\phi \Delta_\phi )x^i = \sum_{k=0}^\infty (-G_0\,\delta \Delta)^k x^i\,,$$ where $\Delta_\phi =\Delta_0+\delta \Delta $ is the Laplacian relative to the $\tilde \phi$-slicing and $G_\phi =\sum_{k=0}^\infty (-G_0\,\delta \Delta)^kG_0$ is its Green’s function for suitable boundary conditions, cf. Section \[sec:coordinates\].
Our gauge-invariant observables can be expanded as Taylor series in $X_\Gamma^a$, so in order to discuss the renormalisation of non-local contributions it is sufficient to discuss the kind of singularities that arise from considering the time-ordered products involving $X_\Gamma^i$. The general strategy is similar to the standard setting. We start with non-renormalised expressions where the $n$-fold time-ordered product involving $X_\Gamma^i$ and local functionals $F_1$,…, $F_{n-1}$ is given by $$\mathcal{T}_n(X_\Gamma^i,F_1,\dots,F_{n-1}):= m\circ e^{\hbar\sum_{0\leq k<l\leq n-1}D_{\mathrm{F}}^{kl}}(X_\Gamma^i\otimes F_1\otimes\dots\otimes F_{n-1})\,.$$ where $m$ denotes pointwise multiplication and $D_{\mathrm{F}}^{kl}\doteq \langle\Delta_{S_0}^{\mathrm{F}},\frac{\delta^2}{\delta\Gamma_k\delta\Gamma_l}\rangle$ with $\Delta_{S_0}^{\mathrm{F}}$ denoting the Feynman propagator of the full linearised theory. For simplicity, we suppress all indices. This expression is then expanded into graphs. The non-locality is expressed by the fact that our graphs have now two kinds of vertices and two kinds of propagators. Namely, there are the “usual” Feynman propagators of the theory (for simplicity all denoted by
(-0.07,0) rectangle +(1.2,-0.02); (0,0.1) – (1,0.1);
), but also the “internal” propagators $G_0$ corresponding to lines
(-0.07,0) rectangle +(1.2,-0.05); (0,0.13) – (1,0.13); (0,0.08) – (1,0.08);
.
As for the vertices, there are the external vertices
(-0.07,0) rectangle +(0.1,-0.1); circle (1.5pt);
arising from local functionals $F_{1},\dots,F_{n-1}$ and from the vertex corresponding to the explicit spacetime dependence of $X^i_\Gamma$, but also the internal vertices
(-0.07,0) rectangle +(0.1,-0.1); circle (1.5pt);
obtained from the $\delta \Delta$ operators. An example contribution would be
(36.57,47.12) rectangle +(66.49,25.85); (39.40,50.07) circle (0.84mm); (59.92,49.95) circle (0.84mm); (80.07,50.20) circle (0.84mm); (100.22,50.07) circle (0.84mm); (40.24,50.28) – (59.14,50.23); (39.94,49.62) – (59.18,49.63); (60.77,50.31) – (79.30,50.31); (60.87,49.69) – (79.36,49.71); (80.88,50.28) – (99.45,50.24); (80.58,49.62) – (99.52,49.64); (39.73,70.23) circle (0.74mm); (39.63,69.62) – (39.63,50.74); (39.83,70.23) – (59.33,50.64); (39.83,70.15) .. controls (47.90,71.01) and (60.00,57.62) .. (59.83,50.93) .. controls (59.83,50.89) and (59.37,50.64) .. (59.40,50.67); (70.01,70.00) circle (0.74mm); (70.01,69.78) – (60.44,50.58); (69.90,69.89) – (79.76,51.04);
To see that such graphs can be renormalised, consider the simplest divergent case, namely
(36.57,47.12) rectangle +(66.49,25.85); (39.40,50.07) circle (0.84mm); (59.92,49.95) circle (0.84mm); (80.07,50.20) circle (0.84mm); (40.24,50.28) – (59.14,50.23); (39.94,49.62) – (59.18,49.63); (60.77,50.31) – (79.30,50.31); (60.87,49.69) – (79.36,49.71); (39.73,70.23) circle (0.74mm); (39.63,69.62) – (39.63,50.74); (39.83,70.23) – (59.33,50.64);
The kernel of $G_0$ considered as a distribution on $M^2$ is of the form $$G_0(x,y) = c(\tau_x) \delta(\tau_x,\tau_y) \frac{1}{|\vec {x}-\vec{y}|}$$ with a smooth function $c$. The wave front set of $G_0(x,y)$ is the one of $\delta(x,y)$ and its scaling degree is 2. The vertex operators $\delta \Delta$ are differential operators of at most second order. By direct inspection we thus see that the only singularity of the loop in the above example is at the total diagonal and by power counting we find that the degree of divergence of this loop is at most 2, so that the appropriately renormalised expression is unique up to at most two derivatives of $\delta$ distributions of the three loop vertices. In general the degree of divergence of a loop containing “internal” propagators may be higher or lower than in the above example depending on the number of Feynman propagators appearing in the loop; the same applies to the renormalisation freedom of general loops.
These arguments indicate that the new types of graphs do not create new problems in the UV regime. We briefly sketch why we do not expect additional IR problems. We have already pointed out that our setup is only meaningful if we restrict the admissible classical configurations to those which vanish at spatial infinity. By consistency we need the same behaviour for the correlation functions of the quantized theory, in particular for the Feynman propagators of the linearised model. Provided quantum states (or more general Hadamard parametrices) with this property exist – this is not obvious and needs to be proven – we expect that the integrals corresponding to the “internal” vertices will converge.
The remaining problem is to deal with the combinatorics of such graphs and ensure that the renormalisation can be performed systematically order by order. This can be done by a slight generalisation of the standard framework and will be discussed in detail in our forthcoming paper [@longpaper]. In the same publication we will also prove the validity of Ward identities analogous to the ones proven by Hollands for the Yang-Mills theory [@Hollands].
Conclusions
===========
We described how cosmological perturbation theory may be derived from a full theory of perturbative quantum gravity. This demonstrates that perturbative quantum gravity can already be tested by present observations. Moreover, on a more practical side, our definition of gauge-invariant observables provides a conceptually simple way of extending the observables which are relevant for the interpretation of cosmological observations to arbitrary high orders.
However, even in linear order, our discussion clarifies the choice of good observables, as we have indicated at the example of the lapse function $N_\phi$ with respect to the spatial hypersurfaces of constant inflaton field. Initially $N_\phi$ is not gauge-invariant, but our construction yields a gauge-invariant version which at linear order and on shell may be expressed in terms of the Bardeen potential $\Phi$ that is related to the temperature fluctuations of the CMB via the Sachs-Wolfe effect.
We computed examples of gauge-invariant observables beyond linear order and found a second-order expression for the comoving curvature perturbation which seems to differ from constructions in other works. As in the literature there is some debate about whether some constructions are physically well-behaved, see. e.g. [@Vernizzi:2004nc], it would be interesting to investigate the physical properties of our result, even though it is clear from the outset that it has a transparent geometric interpretation.
Finally we have sketched the details of the quantization of the Einstein-Klein-Gordon system on cosmological backgrounds beyond linear order. We believe that the strategy outlined here leads to a full renormalised all-order theory of cosmological perturbations by means of which higher order corrections to standard results in cosmology may be computed.
[**Acknowledgements**]{}\
K.F., N.P. and K.R. would like to thank the Erwin Schrödinger Institute in Vienna, where part of the research reported here was carried out, for the kind hospitality.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The $(2 + 1)$-dimensional Maxwell-Chern-Simons gauge model consisting of two complex scalar fields interacting through a common Abelian gauge field is considered. It is shown that the model has a solution that describes a soliton system consisting of vortex and Q-ball constituents. This two-dimensional soliton system possesses a quantized magnetic flux and a quantized electric charge. Moreover, the soliton system has a nonzero angular momentum. Properties of this vortex-Q-ball system are investigated by analytical and numerical methods. It is found that the system combines properties of a vortex and a Q-ball.'
address: 'Tomsk Polytechnic University, 634050 Tomsk, Russia'
author:
- 'A.Yu. Loginov'
- 'V.V. Gauzshtein'
bibliography:
- 'article.bib'
title: |
A two-dimensional soliton system in the Maxwell-Chern-Simons\
gauge model
---
vortex ,flux quantization ,Q-ball ,Noether charge ,Chern-Simons term
Introduction {#seq:I}
============
Topological solitons of $(2+1)$-dimensional field models play an important role in various areas of field theory, physics of condensed state, cosmology, and hydrodynamics. Among them, we should first mention vortices of the effective theory of superconductivity [@abr], vortices of the $(2+1)$-dimensional Abelian Higgs model [@nielsen], and lumps of the $(2+1)$-dimensional nonlinear $O(3)$ sigma model [@belpol].
In contrast to the $(3 + 1)$-dimensional case, electrically charged solitons do not exist in the $(2 + 1)$-dimensional Maxwell electrodynamics for a fairly straightforward reason: the electric field goes like $1/r$, so the electric field’s energy diverges logarithmically. In $(2+1)$ dimensions, however, the dynamics of gauge field may be governed not only by the Maxwell term but also by the Chern-Simons term [@JT; @schonfeld; @DJT]. In the presence of the Chern-Simons term, a gauge field becomes topologically massive, thus making possible the existence of two-dimensional electrically charged solitons. These solitons exist both in the Maxwell-Chern-Simons models [@paul; @khare_rao_227; @khare_255] and in the Chern-Simons models [@hong; @jw1; @jw2; @bazeia_1991; @ghosh] and can be both topological and nontopological. Topological solitons are electrically charged vortices, whereas nontopological ones are two-dimensional electrically charged spinning (possessing an angular momentum) Q-balls. The numerical research of such a two-dimensional Q-ball has been performed in [@deshaies_2006]. The three-dimensional counterparts of these Q-balls have been described in [@volkov_2002; @radu]. More recently, the influence of the Chern-Simons term on electrically charged and spinning solitons of several $(2 + 1)$-dimensional Abelian gauge models has been studied in [@navarro_2017].
In this Letter a two-dimensional soliton system in the Maxwell-Chern-Simons gauge model is considered. As well as in the Maxwell gauge model [@loginov_plb_777], the soliton system consists of a vortex and a Q-ball interacting through a common Abelian gauge field. This vortex-Q-ball system possesses a radial electric field, carries a quantized magnetic flux, and has a nonzero angular momentum, but in contrast to [@loginov_plb_777], it also has a quantized electric charge. It is shown that the vortex-Q-ball system combines properties of topological and nontopological solitons.
Lagrangian and field equations of the model {#seq:II}
===========================================
The Lagrangian density of the model is $$\begin{aligned}
\mathcal{L} & = & -\frac{1}{4}F_{\mu \nu }F^{\mu \nu } + \frac{\mu }{4}\epsilon
^{\rho \sigma \tau}F_{\rho \sigma}A_{\tau} \nonumber \\
&& + \left( D_{\mu }\phi \right) ^{\ast }D^{\mu }\phi - V\left( \left\vert \phi
\right\vert \right) \nonumber \\
&& + \left( D_{\mu }\chi \right) ^{\ast }D^{\mu }\chi - U\left( \left\vert \chi
\right\vert \right), \label{1}\end{aligned}$$ where the complex scalar fields $\phi$ and $\chi$ are minimally coupled to the Abelian gauge field $A_{\mu}$ through the covariant derivatives: $$D_{\mu }\phi =\partial_{\mu }\phi + ieA_{\mu }\phi,\quad
D_{\mu }\chi =\partial_{\mu }\chi + iqA_{\mu }\chi. \label{2}$$ The self-interaction potentials used in this paper are the same as those used in [@loginov_plb_777]: $$\begin{aligned}
V\left( \left\vert \phi \right\vert \right) &=\frac{\lambda }{2}\left(
\phi ^{\ast }\phi -v^{2}\right) ^{2}\!, \nonumber \\
U\left( \left\vert \chi \right\vert \right) &=m^{2}\chi ^{\ast }\chi
-g\left( \chi^{\ast }\chi \right)^{2}+
h \left( \chi^{\ast }\chi \right)^{3}\!. \label{3}\end{aligned}$$ In Eq. (\[3\]), $\lambda$, $g$, and $h$ are the positive self-interaction constants, $m$ is the mass of the scalar $\chi$-particle, and $v$ is the vacuum average of the complex scalar field $\phi$. We suppose that the parameters $m$, $g$, and $h$ satisfy the condition $$\frac{g^{2}}{4m^{2}}<h<\frac{g^{2}}{3m^{2}}. \label{4}$$ In this case, the potential $U\left(\left\vert\chi\right\vert\right)$ has the two minima: the global minimum at $\chi = 0$ and a local one at some nonzero $\left\vert \chi \right\vert$.
The model’s action $S = \int\mathcal{L}d^{3}x$ is invariant under the local gauge transformations: $$\begin{aligned}
\phi \left( x\right) &\rightarrow &\phi ^{\prime }\left( x\right) = \exp
\left(-ie\Lambda \left( x\right) \right) \phi\left( x\right) , \nonumber \\
\chi \left( x\right) &\rightarrow &\chi ^{\prime }\left( x\right) = \exp
\left(-iq\Lambda \left( x\right) \right) \chi\left( x\right) , \nonumber \\
A_{\mu }\left( x\right) &\rightarrow &A_{\mu }^{\prime }\left( x\right)
=A_{\mu }\left( x\right) +\partial _{\mu }\Lambda \left( x\right) \label{5}\end{aligned}$$ if the local gauge parameter $\Lambda \left( x\right)$ decreases rapidly at infinity. Because of neutrality of the Abelian gauge field $A_{\mu}$, the Lagrangian density (\[1\]) is also invariant under the two independent global gauge transformations: $$\begin{aligned}
\phi \left( x\right) &\rightarrow &\phi ^{\prime }\left( x\right) = \exp
\left(-i\alpha \right) \phi \left( x\right) , \nonumber \\
\chi \left( x\right) &\rightarrow &\chi ^{\prime }\left( x\right) = \exp
\left(-i\beta \right) \chi \left( x\right). \label{6}\end{aligned}$$ This invariance leads to the two Noether currents: $$\begin{aligned}
j_{\phi }^{\mu } &=& i\left[ \phi ^{\ast }D^{\mu }\phi -\left( D^{\mu }\phi
\right) ^{\ast }\phi \right], \nonumber \\
j_{\chi }^{\mu } &=& i\left[ \chi ^{\ast }D^{\mu }\chi -\left( D^{\mu }\chi
\right) ^{\ast }\chi \right]. \label{7}\end{aligned}$$ Under the discrete transformations $C$, $P$, and $T$, the Chern-Simons term $\mathcal{L}_{C\!S} = \mu \epsilon^{\rho\sigma\tau }F_{\rho \sigma}A_{\tau }/4$ behaves as follows: $$\mathcal{L}_{\mathrm{CS}}^{\left( C\right) }=\mathcal{L}_{\mathrm{CS}},\;
\mathcal{L}_{\mathrm{CS}}^{\left( P\right) }=-\mathcal{L}_{\mathrm{CS}},\;
\mathcal{L}_{\mathrm{CS}}^{\left( T\right) }=-\mathcal{L}_{\mathrm{CS}}.
\label{7a}$$ It follows from Eq. (\[7a\]) that the Chern-Simons term breaks the $P$, $CP$, and $T$-invariance of the model’s Lagrangian.
The field equations of the model have the form:
&\_F\^+..\^ F\^=j\^, \[8a\]\
& D\_D\^+( \^- v\^[2]{}) = 0, \[8b\]\
& D\_D\^+( m\^[2]{}-2g( \^) +3h( \^)\^[2]{}) = 0, \[8c\]
where the dual field strength $\left.\! \hspace{-0.05cm} \right.^{\ast } \!
\hspace{-0.05cm}F^{\nu} = \epsilon^{\nu \alpha \beta } F_{\alpha \beta}/2$, and the electromagnetic current $j^{\nu}$ is expressed in terms of the Noether currents: $$j^{\nu } = e j_{\phi }^{\nu } + q j_{\chi }^{\nu }. \label{9}$$ Integrating the left and right hand sides of Eq. (\[8a\]) with the index $\nu = 0$ over the spatial plane, we obtain an important relation between the electric charge and the magnetic flux: $$Q = e Q_{\phi} + q Q_{\chi} = - \mu \Phi, \label{9b}$$ where $Q_{\phi}=\int j_{\phi}^{0}d^{2}x$ and $Q_{\chi}=\int j_{\chi}^{0}d^{2}x$ are the conserved Noether charges.
The symmetric energy-momentum tensor of the model and the corresponding expression for the energy density are written as: $$\begin{aligned}
T_{\mu \nu } =&-F_{\mu \lambda }F_{\nu }^{\;\lambda }+\frac{1}{4}g_{\mu
\nu }F_{\lambda \rho }F^{\lambda \rho } \nonumber \\
&+\left( D_{\mu }\phi \right) ^{\ast }D_{\nu }\phi +\left( D_{\nu }\phi
\right) ^{\ast }D_{\mu }\phi \nonumber \\
&-g_{\mu \nu }\left( \left( D_{\mu }\phi \right) ^{\ast }D^{\mu }\phi
-V\left( \left\vert \phi \right\vert \right) \right) \nonumber \\
&+\left( D_{\mu }\chi \right) ^{\ast }D_{\nu }\chi +\left( D_{\nu }\chi
\right) ^{\ast }D_{\mu }\chi \nonumber \\
&-g_{\mu \nu }\left( \left( D_{\mu }\chi \right) ^{\ast }D^{\mu }\chi
-U\left( \left\vert \chi \right\vert \right) \right), \label{10}\end{aligned}$$ $$\begin{aligned}
T_{00} = & \frac{1}{2}E_{i}E_{i}+\frac{1}{2}B^{2} \label{11} \\
& +\left( D_{0}\phi \right) ^{\ast }D_{0}\phi + \left( D_{i}\phi \right)
^{\ast }D_{i}\phi +V\left( \left\vert \phi \right\vert \right) \nonumber \\
& +\left( D_{0}\chi \right) ^{\ast }D_{0}\chi +\left( D_{i}\chi \right)
^{\ast }D_{i}\chi +U\left( \left\vert \chi \right\vert \right), \nonumber\end{aligned}$$ where $E_{i} = F_{0i}$ are the components of electric field strength and $B = -F_{12}$ is the magnetic field strength.
In this paper we adopt the following gauge condition: $\partial _{0}\phi = 0$. Using the analogy of Q-ball, we find a soliton solution of model (\[1\]) that minimizes the energy $E = \int T_{00}d^{2}x = H = \int \mathcal{H} d^{2}x$ ($\mathcal{H}$ is the density of the Hamiltonian $H$) at a fixed value of the Noether charge $Q_{\chi} = \int j_{\chi }^{0}d^{2}x$. In this case, the soliton solution is an unconditional extremum of the functional $$F=\int \mathcal{H}d^{2}x-\omega \int j_{\chi }^{0}d^{2}x =
H - \omega Q_{\chi}, \label{14}$$ where $\omega$ is the Lagrange multiplier. The Noether charge $Q_{\chi}$ is written in terms of the canonically conjugated fields $\chi$, $\chi^{\ast}$, $\pi_{\chi } = \partial \mathcal{L}/\partial
\left(\partial_{0} \chi \right) = \left(D_{0} \chi \right)^{\ast}$, and $\pi_{
\chi ^{\ast }}= \partial \mathcal{L}/\partial \left( \partial _{0} \chi^{\ast}
\right) = D_{0} \chi$ as follows: $$Q_{\chi } = - i \int \left( \chi \pi _{\chi }-\chi ^{\ast }\pi _{\chi ^{\ast
}}\right) d^{2}x. \label{15}$$ From Eq. (\[15\]), we obtain the variation of the Noether charge $Q_{\chi}$ in terms of the canonically conjugate fields: $$\delta Q_{\chi } = - i \int \left( \chi \delta \pi_{\chi } + \pi_{\chi } \delta
\chi - \text{c.c.}\right) d^{2}x. \label{16}$$ At the same time, the first variation of the functional $F$ vanishes for the soliton solution: $$\delta F = \delta H - \omega \delta Q_{\chi } = 0. \label{17}$$ From Eqs. (\[16\]) and (\[17\]), we obtain the following Hamilton field equations: $$\partial _{0}\chi = \frac{\delta H}{\delta \pi _{\chi }} = - i \omega \chi
,\quad\partial _{0}\chi ^{\ast }=\frac{\delta H}{\delta \pi_{\chi^{\ast }}}
= i \omega \chi^{\ast}, \label{18}$$ while time derivatives of the other model’s fields are equal to zero. We see that the scalar field $\chi$ has the time dependence of Q-ball type: $$\chi \left( x\right) = \chi \left( \mathbf{x}\right)
\exp \left( - i \omega t \right), \label{19}$$ whereas the other model’s fields do not depend on time for the adopted gauge condition $\partial_{0} \phi = 0$. Extremum condition (\[17\]) leads to the important relation for the soliton solution: $$\frac{dE}{dQ_{\chi }} = \omega, \label{20}$$ where it is understood that the Lagrange multiplier $\omega$ is some function of the Noether charge $Q_{\chi}$.
The ansatz and some properties of the solution {#seq:III}
==============================================
To solve field equations (\[8a\]) – (\[8c\]), we apply the following ansatz: $$\begin{aligned}
A^{\mu }\left( x\right) &= \left( \frac{A_{0}\left( r\right) }{e r},\frac{1}
{e r} \epsilon_{ij} n_{j} A\left( r\right) \right) , \nonumber \\
\phi \left( x\right) & = v\exp \left( i N \theta \right) F\left( r\right),
\nonumber \\
\chi \left( x\right) & = \sigma \left( r\right) \exp \left(-i\omega t\right),
\label{21}\end{aligned}$$ where $\epsilon_{ij}$ are the components of the two-dimensional antisymmetric tensor $\left( \epsilon_{12} = 1 \right)$ and $n_{j}$ are those of the two-dimensional radial unit vector $\mathbf{n} = \left(\cos\left(\theta\right)
\!,\,\sin\left(\theta\right)\right)$. We suppose that the function $\sigma\left( r \right)$ is real, so ansatz (\[21\]) completely fixes the model’s gauge.
It can be shown that ansatz (\[21\]) is consistent with field equations (\[8a\]) – (\[8c\]), so we get the system of second order nonlinear differential equations for the ansatz functions: $$\begin{aligned}
&& A_{0}^{\prime \prime }(r)-\frac{A_{0}^{\prime }(r)}{r}+\frac{A_{0}(r)}
{r^{2}} - \mu A^{\prime }(r) \nonumber \\
&& - 2 \left(e^{2}v^{2}F\left( r\right) ^{2} + q^{2}\sigma \left( r\right)
^{2}\right) \nonumber \\
&& \times A_{0}(r) + 2 e q \omega r \sigma \left( r \right)^{2} = 0,
\label{22}\end{aligned}$$ $$\begin{aligned}
&& A^{\prime \prime }(r)-\frac{A^{\prime }(r)}{r}-\mu A_{0}^{\prime }(r)
\nonumber \\
&& -2e^{2}v^{2}\left( N+A(r)\right) F\left( r\right) ^{2}
\nonumber \\
&& -2q^{2}\sigma \left(r\right)^{2}A\left( r\right)+\mu\frac{A_{0}(r)}{r}=0,
\label{23}\end{aligned}$$ $$\begin{aligned}
&& F^{\prime \prime }(r)+\frac{F^{\prime }(r)}{r} -
\frac{F(r)}{r^{2}} \nonumber \\
&& \times \left((N + A(r))^{2}-A_{0}(r)^{2}\right) \nonumber \\
&& +\lambda v^{2}\left( 1-F(r)^{2}\right) F(r) = 0, \label{24}\end{aligned}$$ $$\begin{aligned}
&& \sigma ^{\prime \prime }(r)+\frac{\sigma ^{\prime }(r)}{r}+
\sigma \left( r\right) \nonumber \\
&& \times \left( \left(\omega - \frac{q}{e}\frac{A_{0}
\left( r\right)}{r}\right)^{2}-\frac{q^{2}}{e^{2}}
\frac{A\left( r\right) ^{2}}{r^{2}}\right) \label{25} \\
&& -\left( m^{2}-2g\sigma \left( r\right) ^{2}+3h\sigma
\left( r\right)^{4}\right) \sigma \left( r\right) = 0. \nonumber\end{aligned}$$ The expression for the energy density $\mathcal{E}=T_{00}$ can also be written in terms of the ansatz functions:
=& + ( ( ) \^) \^[2]{}+v\^[2]{}F\^\^[2]{}\
& + v\^[2]{}F\^[2]{}\
&+v\^[4]{}( F\^[2]{}-1) \^[2]{}+\^\^[2]{}\
&+( -q) \^[2]{}\^[2]{}+\^[2]{}\
&+m\^[2]{}\^[2]{}-g\^[4]{}+h\^[6]{}. \[26\]
The regularity of the soliton solution at $r = 0$ and the finiteness of the soliton’s energy $E = 2\pi \int\nolimits_{0}^{\infty}\mathcal{E}\left(r\right)r
dr$ lead us to the following boundary conditions for the ansatz functions: $$\begin{aligned}
A_{0}(0) &= 0, \quad A_{0}(r) \underset{r\rightarrow \infty }{\longrightarrow}0,
\nonumber \\
A(0) &= 0, \quad \, A(r) \underset{r\rightarrow \infty }{\longrightarrow} - N,
\nonumber \\
F(0) &= 0, \quad \, F(r) \underset{r\rightarrow \infty }{\longrightarrow }1,
\nonumber \\
\sigma^{\prime }(0) &= 0, \quad \, \sigma (r) \underset{r\rightarrow \infty}
{\longrightarrow } 0. \label{27}\end{aligned}$$ From the boundary conditions for $A(r)$ it follows that the magnetic flux of the vortex-Q-ball system is quantized: $$\Phi = 2 \pi \int_{0}^{\infty }B\left(r\right) rdr=\frac{2\pi }{e}N, \label{28}$$ where $B(r)=-A^{\prime }(r)/(er)$ is the magnetic field strength. Moreover, from Eqs. (\[9b\]) and (\[28\]) it follows that the total electric charge of the vortex-Q-ball system is also quantized: $$Q = -\frac{2 \pi}{e} \mu N. \label{28h}$$
In terms of the ansatz functions, the $C$-transformation is written as $$\begin{gathered}
\omega \rightarrow - \omega ,\;\ N\rightarrow - N,\;A_{0}\rightarrow - A_{0},\;
\nonumber \\
A \rightarrow -A,\;\ F \rightarrow F,\;\sigma \rightarrow \sigma. \label{28aa}\end{gathered}$$ It is easily shown that Eqs. (\[22\]) – (\[25\]) are invariant under transformation (\[28aa\]) as well as energy density (\[26\]). According to Eq. (\[7a\]), $C$-transformation (\[28aa\]) is the only discrete one that leaves Eqs. (\[22\]) – (\[25\]) invariant. All other discrete transformations ($P$, $CP$, and $T$) do not leave Eqs. (\[22\]) – (\[25\]) invariant. We see that transformation (\[28aa\]) changes the sign of $\omega$, but at the same time, it also changes the sign of the soliton’s winding number. From this it follows that the energy of a soliton with a given fixed $N$ is not invariant under the change of sign of the phase frequency: $E \left(-\omega
\right) \neq E\left(\omega \right)$. Similarly, it can be shown that $Q_{\phi ,\chi}\left(-\omega\right) \neq -
Q_{\phi ,\chi }\left(\omega \right)$, so the absolute values of the Noether charges are also not invariant.
From Eqs. (\[22\]) – (\[25\]) and boundary conditions (\[27\]) it follows that at $r = 0$, the power expansion of the soliton solution has the form: $$\begin{aligned}
A_{0}\left( r\right)& =a_{1}r+\frac{a_{3}}{3!}r^{3}+O\left( r^{5}\right),
\nonumber \\
A\left( r\right)& =\frac{b_{2}}{2!}r^{2}+\frac{b_{4}}{4!}r^{4}+
O\left(r^{6}\right),
\nonumber \\
F\left( r\right)& =\frac{c_{\left|N\right|}}{\left|N\right|!}r^{\left|N\right|}
\!+\!\frac{c_{\left|N\right|+2}}{\left( \left|N\right|\!+\!2\right)!}
r^{\left|N\right|+2}\!+\!O\!\left(\!r^{\left|N\right|+4}\!\right)\!,
\nonumber \\
\sigma \left( r\right)& =d_{0}+\frac{d_{2}}{2!}r^{2}+O\left( r^{4}\right).
\label{28a}\end{aligned}$$ In Eq. (\[28a\]), the expressions of the next-to-leading coefficients $a_{3}$, $b_{4}$, $c_{\left| N \right| + 2}$, and $d_{2}$ are $$\begin{aligned}
a_{3} = & \, 3 q d_{0}^{2}\left( a_{1}q - e\omega \right)+\frac{3}{2}\mu b_{2},
\nonumber \\
b_{4} = & \, 3 \left( q^{2}b_{2}d_{0}^{2}+2 N e^{2}v^{2}c_{\left|N\right|}^{2}
\delta_{1,\left|N\right|}\right) + \mu a_{3}, \nonumber \\
c_{\left|N\right|+2} = & -\frac{c_{\left|N\right|}}{4}
\left(\left|N\right|+2\right)\left( a_{1}^{2}+\left|N\right|\left|b_{2}\right|
+\lambda v^{2}\right), \nonumber \\
d_{2} = & \frac{d_{0}}{2}\left[ d_{0}^{2}\left( 3d_{0}^{2}h-2g\right) \right.
\nonumber \\
&\left. +e^{-2}\left( qa_{1}+e\left( m-\omega \right) \right) \right.
\nonumber \\
&\left. \times \left( -qa_{1}+e\left( m+\omega \right) \right) \right],
\label{28b}\end{aligned}$$ where $\delta_{1,\left|N\right|}$ is the Kronecker symbol. Linearization of Eqs. (\[22\]) – (\[25\]) at large $r$ and use of corresponding boundary conditions (\[27\]) lead us to the asymptotic form of the solution as $r \rightarrow \infty$: $$\begin{aligned}
A_{0}\left( r\right) &\sim a_{\infty }\sqrt{m_{A}r}\exp \left(
-m_{A}r\right), \nonumber \\
A\left( r\right) &\sim - N + a_{\infty }\sqrt{m_{A}r}\exp \left(
-m_{A}r\right), \nonumber \\
F\left( r\right) &\sim 1 + c_{\infty}\frac{\exp \left(-m_{\phi }r\right)}
{\sqrt{m_{\phi }r}}, \nonumber \\
\sigma \left( r\right)& \sim d_{\infty }\frac{\exp
\left(-\sqrt{m^{2}-\omega^{2}} r \right) }
{\sqrt{m r}}, \label{28c}\end{aligned}$$ where $$m_{A}=\sqrt{2e^{2}v^{2}+\frac{\mu ^{2}}{4}} - \frac{\mu }{2} \label{28d}$$ is the mass of the gauge boson and $m_{\phi} = \sqrt{2 \lambda} v$ is the mass of the scalar $\phi$-particle.
For symmetric energy-momentum tensor (\[10\]), the angular momentum tensor has the form $$J^{\lambda \mu \nu }=x^{\mu }T^{\lambda \nu }-x^{\nu }T^{\lambda \mu }.
\label{29}$$ Use of Eqs. (\[10\]), (\[21\]), and (\[29\]) leads us to the angular momentum’s density expressed in terms of the ansatz functions: $$\begin{aligned}
\mathcal{J} &=\frac{1}{2}\epsilon _{ij}J^{0ij}=- r B E_{r}+2\frac{q}{e}A\left(
\omega - q\frac{A_{0}}{er}\right) \sigma ^{2} \nonumber \\
&-2\frac{A_{0}\left( N+A\right) }{r}v^{2}F^{2}. \label{30}\end{aligned}$$ In Eq. (\[30\]), $E_{r}(r)=-\left( A_{0}\left(r\right)/\left( er\right)\right
)^{\prime}$ is the radial electric field strength. Integrating the term $-rBE_{r}=-e^{-2}A^{\prime }\left(A_{0}/r\right)^{\prime}$ by parts, taking into account boundary conditions (\[27\]), and using Eq. (\[22\]) to eliminate $A_{0}^{\prime \prime}$, we obtain the expression for the soliton’s angular momentum $J = 2 \pi \int_{0}^{\infty} \mathcal{J}
\left(r\right)rdr$: $$J = -4 \pi N v^{2} \int_{0}^{\infty }A_{0}\left( r\right)F^{2}(r)dr +
\pi \frac{\mu}{e^{2}}N^{2}. \label{31}$$ At the same time, Eqs. (\[7\]) and (\[21\]) lead us to the following expression of the Noether charge $Q_{\phi}$: $$Q_{\phi }=-4\pi v^{2}\int_{0}^{\infty}A_{0}\left(r\right)F^{2}(r)dr. \label{32}$$ From Eqs. (\[9b\]), (\[28h\]), (\[31\]), and (\[32\]) it follows that for the vortex-Q-ball system the important relation holds between the angular momentum $J$ and the Noether charges $Q_{\phi}$ and $Q_{\chi}$:
& J = N Q\_+N\^[2]{} = -NQ\_ - N\^[2]{}. \[33\]
We see that the angular momentum depends linearly on the Noether charges of the scalar fields.
![\[fig1\] The numerical solution $m^{-1/2}A_{0}(r)/(er)$ (dotted), $A(r)$ (dashed), $F(r)$ (solid), and $m^{-1/2}\sigma(r)$ (dash-dotted) for the vortex-Q-ball system. The model’s parameters are $e = q = 0.3\,m^{1/2}$, $\mu =
0.5\, m$, $\lambda = 0.335\, m$, $v = 1.221\, m^{1/2}$, $g=1.0\, m$, $h =
0.26$, and $N = 1$. The phase frequency $\omega = 0.38\,m$.](Fig1.eps){width="7.8cm"}
Any solution of field equations (\[8a\]) – (\[8c\]) is an extremum of the action $S=\int \mathcal{L}d^{2}xdt$. It is readily seen, however, that the Lagrangian density (\[1\]) does not depend on time if the field configurations are those of ansatz (\[21\]). It follows that any solution of system (\[22\]) – (\[25\]) is an extremum of the Lagrangian $L = \int \mathcal{L} d^{2}x$. Let $A_{0}(r)$, $A(r)$, $F(r)$, and $\sigma(r)$ be a solution of system (\[22\]) – (\[25\]) satisfying boundary conditions (\[27\]). After the scale transformation of the solution’s argument $r \rightarrow\lambda
r$, the Lagrangian $L$ becomes a function of the scale parameter $\lambda$. The function $L\left(\lambda\right)$ has an extremum at $\lambda = 1$, so its derivative with respect to $\lambda$ vanishes at this point: $\left. dL/d\lambda
\right\vert_{\lambda = 1} = 0$. From this equation it follows that the virial relation holds for the vortex-Q-ball system: $$2 \left( E^{\left( E\right) }-E^{\left( H\right) }+E^{\left( P\right)
}\right) +L^{\left(C\!S\right) }-\omega Q_{\chi } = 0, \label{34}$$ where $$E^{\left( E\right) }=\frac{1}{2}\int E_{i}E_{i} d^{2}x =
\pi \int_{0}^{\infty}\left( \left( \hspace{-0.03in}
\frac{A_{0}}{er}\hspace{-0.03in}\right)^{\prime }\right)^{2}r dr \label{35}$$ is the electric field’s energy, $$E^{\left( B\right) }=\frac{1}{2}\int B^{2}d^{2}x =
\pi \int_{0}^{\infty} \frac{A^{\prime}{}^{2}}{e^{2}r} dr \label{36}$$ is the magnetic field’s energy, $$E^{\left( P\right) }=2\pi\int_{0}^{\infty}
\left[V\left( \left\vert \phi \right\vert \right)
+U\left( \left\vert \chi \right\vert \right) \right] r dr \label{37}$$ is the potential part of the soliton’s energy, and $$L^{\left(C\!S\right) }=\frac{\mu }{4}\int \epsilon ^{\rho \sigma \tau}
F_{\rho \sigma}A_{\tau}d^{2}x \label{37a}$$ is the Chern-Simons part of the model’s Lagrangian.
Numerical results {#seq:IV}
=================
![\[fig2\] The numerical solution $m^{-1/2}A_{0}(r)/(er)$ (dotted), $A(r)$ (dashed), $F(r)$ (solid), and $m^{-1/2}\sigma(r)$ (dash-dotted) for the system of vortex and Q-ball that do not interact with each other. The model’s parameters are the same as in Fig. 1.](Fig2.eps){width="7.8cm"}
Now we present some numerical results concerning the vortex-Q-ball system. For numerical calculations, we use the natural units $c = 1$, $\hbar = 1$. In addition, the mass $m$ of scalar $\chi$-particle is used as the energy unit. After that, the model is completely determined by the seven parameters: $e$, $q$, $\mu$, $\lambda$, $v$, $g$, and $h$. We chose the following values of these parameters: $e = q = 0.3\,m^{1/2}$, $\mu=0.5\,m$, $\lambda=0.335\,m$, $v=1.221\,m^{1/2}$, $g=1.0\,m$, and $h=0.26$, where the parameters’ dimensions correspond to the $(2+1)$-dimensional case. The correctness of the numerical solution were checked by use of Eqs. (\[9b\]), (\[20\]), (\[33\]), and (\[34\]).
In Fig. 1, we can see the dimensionless zero component $m^{-1/2}A_{0}(r)/(er)$ of the gauge potential along with the dimensionless ansatz functions $A(r)$, $F(r)$, and $m^{-1/2}\sigma(r)$. The vortex part of the soliton system is in the topological sector with $N=1$, the phase frequency $\omega$ is equal to $0.38\,m$. Figure 2 presents the numerical solution for the case $q = 0$, whereas the other model’s parameters remain the same as in Fig. 1. This corresponds to superimposed gauged vortex and non-gauged Q-ball that do not interact with each other. From Figs. 1 and 2, we can conclude that the gauge interaction between the vortex and Q-ball components leads to significant changes in the shapes of the ansatz functions $A_{0}(r)$, $A(r)$, and $\sigma(r)$, while the shape of $F(r)$ does not change significantly.
![\[fig3\] The dimensionless versions of the electric field strength $\tilde{E}_{r}(r) = m^{-3/2}E_{r}(r)$ (solid), the magnetic field strength $\tilde{B}(r) = m^{-3/2}B(r)$ (dashed), the scaled energy density $0.5\,
\tilde{\mathcal{E}}(r) = 0.5\,m^{-3}\mathcal{E}(r)$ (dash-dotted), the electric charge density $\tilde{j}_{0}(r) = m^{-5/2}j_{0}(r)$ (dash-dot-dotted), and the scaled angular momentum’s density $0.5\,\tilde{\mathcal{J}}(r) = 0.5\,m^{-2}
\mathcal{J}(r)$ (dotted), corresponding to the solution in Fig. 1.](Fig3.eps){width="7.8cm"}
Figure 3 shows the dimensionless versions of the electric field strength $\tilde{E}_{r}(r)=m^{-3/2}E_{r}(r)$, the magnetic field strength $\tilde{B}(r)=
m^{-3/2}B(r)$, the scaled energy density $0.5\,\tilde{\mathcal{E}}(r) = 0.5\,
m^{-3}\mathcal{E}(r)$, the electric charge density $\tilde{j}_{0}(r)=m^{-5/2}
j_{0}(r)$, and the scaled angular momentum’s density $0.5\,\tilde{\mathcal{J}}
(r) = 0.5\,m^{-2} \mathcal{J}(r)$ that correspond to the soliton solution in Fig. 1. We see that just as in [@loginov_plb_777], the vortex-Q-ball system consists of three parts: the central transition region, the inner region, and the external transition region. We also see that the densities of the energy and the angular momentum are approximately constant in the inner region, while the electric and magnetic field strengths are close to zero there.
In Fig. 4, we can see the dimensionless soliton energy $\tilde{E} = m^{-1}E$ as a function of the dimensionless phase frequency $\tilde{\omega} = m^{-1}
\omega$. The function $\tilde{E}\left( \tilde{\omega} \right)$ is presented in the range from the minimum values of $\left\vert \tilde{\omega} \right\vert$ that we have reached by numerical methods to its maximum value of $1$. The most important feature of Fig. 4 is that the soliton’s energy is not invariant under the change of sign of the phase frequency: $\tilde{E}\left(
-\tilde{\omega}\right) \neq \tilde{E}\left( \tilde{\omega} \right)$. This fact is a direct consequence of the $T$-invariance breaking, which is caused by the Chern-Simons term in the Lagrangian (\[1\]). From Fig. 4 it follows that the soliton’s energy $E$ tends to infinity as $\left \vert \tilde{ \omega} \right \vert$ tends to its minimum values (thin-wall regime). In the thin-wall regime, the spatial size of the soliton’s inner region increases indefinitely, so the main contribution to the soliton’s energy and angular momentum comes from this region.
![\[fig4\] The dimensionless soliton energy $\tilde{E} = m^{-1}E$ as a function of the dimensionless phase frequency $\tilde{\omega} = m^{-1}
\omega$. The model’s parameters are the same as in Fig. 1.](Fig4.eps){width="7.8cm"}
As $\tilde{ \omega} \rightarrow 1$, the vortex-Q-ball system goes into the thick-wall regime. As well as in the thin-wall regime, the soliton’s Noether charge $Q_{\chi}$ and energy $E$ tend to infinity in the thick-wall regime. It was found numerically that $Q_{\chi}(\tilde{\omega})$ and $\tilde{E}(\tilde{
\omega})$ have the following behaviour as $\tilde{ \omega} \rightarrow 1$: $$\begin{aligned}
& & Q_{\chi } \underset{\tilde{\omega} \rightarrow 1}{\longrightarrow} B +
A \left(1-\tilde{\omega} \right)^{-\frac{1}{2}}, \nonumber \\
& & \tilde{E} \underset{\tilde{\omega} \rightarrow 1}{\longrightarrow} C +
A\left(2-\tilde{\omega}\right) \left( 1-\tilde{\omega} \right)^{-\frac{1}{2}},
\label{38}\end{aligned}$$ where $A$, $B$, and $C$ are positive constants. From Eq. (\[38\]) it follows that the behaviour of $Q_{\chi}\left(\tilde{
\omega}\right)$ and $\tilde{E}\left(\tilde{\omega}\right)$ in the thick-wall regime is in agreement with Eq. (\[20\]). Such behaviour of $Q_{\chi}(\tilde{\omega})$ and $\tilde{E}(\tilde{\omega})$ in a neighborhood of the maximum value $\tilde{\omega} = 1$ is very different from that of the two-dimensional non-gauged Q-ball [@lee]. It is also quite different from the behaviour of the vortex-Q-ball system [@loginov_plb_777] in the Maxwell gauge model. However, the behaviour of $Q_{\chi}(\tilde{\omega})$ and $\tilde{E}(\tilde{
\omega})$ in a neighborhood of $\tilde{\omega} = 1$ is similar to that of the usual three-dimensional Q-ball [@lee].
In Fig. 5, we can see the dependences of the dimensionless soliton energy $\tilde{E}$ and the absolute value of Noether charge $Q_{\chi}$ (which is negative for $\tilde{\omega} < 0$) on the absolute value of $\tilde{\omega}$ in a neighborhood of $\left\vert \tilde{\omega} \right\vert = 1$. From Fig. 5 it follows that the behaviour of the vortex-Q-ball system in the neighborhoods of $\tilde{\omega} = -1$ and $\tilde{\omega} = 1$ is completely different. Indeed, its behaviour near $\tilde{\omega} = 1$ corresponds to thick-wall regime (\[38\]). At the same time, its behaviour near $\tilde{\omega} = -1$ is rather unusual. Firstly, there is no thick-wall regime here. Secondly, the Q-ball component of the vortex-Q-ball system disappears at $\tilde{\omega} = -0.93$. For $\tilde{\omega} \in \left( -1, -0.93 \right)$, we have only the single Maxwell-Chern-Simons vortex without any Q-ball component.
![\[fig5\] The dimensionless soliton energy $\tilde{E} = m^{-1}E$ (solid for $\tilde{\omega} > 0$ and dash-dotted for $\tilde{\omega} < 0$) and the absolute value of Noether charge $Q_{\chi}$ (dashed for $\tilde{\omega} >0$ and dotted for $\tilde{\omega} < 0$) as functions of the absolute value of dimensionless phase frequency $\left \vert \tilde{\omega} \right \vert$ in a neighborhood of $\left\vert \tilde{\omega} \right\vert = 1$.](Fig5.eps){width="7.8cm"}
Figure 6 shows the dimensionless energy $\tilde{E}$ as a function of the absolute value of Noether charge $Q_{\chi}$ for the both signs of $\tilde{\omega}$. It also shows the similar dependence $\tilde{E}\left(\left\vert Q_{\chi} \right
\vert\right)$ for the two-dimensional non-gauged Q-ball with the same parameters $m$, $g$, and $h$ as for the vortex-Q-ball system. In addition, the straight line $\tilde{E} = \left\vert Q_{\chi} \right\vert$ is also shown in Fig. 6. We can see that the curve $\tilde{E}( \left\vert Q_{\chi} \right\vert )$ corresponding to the two-dimensional Q-ball is tangent to the straight line $\tilde{E} = \left\vert Q_{\chi} \right\vert$ at some nonzero $\left\vert Q_{
\chi} \right\vert$ as it should be [@lee]. In contrast to this, the vortex-Q-ball system is described by the two curves, which correspond to the both signs of the phase frequency $\tilde{\omega}$. The curve $\tilde{E}(Q_{\chi})$ corresponding to the positive $\tilde{\omega}$ is similar to that of three-dimensional Q-ball. In particular, it has the cusp and consists of two branches. As $Q_{\chi} \rightarrow \infty$, the lower branch goes into the thin-wall regime, while the upper one goes into the thick-wall regime. At the same time, the curve $\tilde{E}(-Q_{\chi})$ corresponding to the negative $\tilde{\omega}$ has no cusp and consists of only one branch. The curve starts at $Q_{\chi} = 0$ and goes into the thin-wall regime as $Q_{\chi} \rightarrow -\infty$. From Fig. 6, we can conclude that in the thin-wall regime the Q-ball component of the the vortex-Q-ball system is stable to the decay into the massive scalar $\chi$-particles.
![\[fig6\] The vortex-Q-ball system’s dimensionless energy $\tilde{E
}$ as a function of the absolute value of Noether charge $Q_{\chi}$ for $\tilde{\omega} > 0$ (solid) and for $\tilde{\omega} < 0$ (dash-dotted), and that for the two-dimensional non-gauged Q-ball (dash-dot-dotted) with the same parameters $m$, $g$, and $h$ as for the vortex-Q-ball system. The dashed line is the straight line $\tilde{E} = \left\vert Q_{\chi} \right\vert$.](Fig6.eps){width="7.8cm"}
Conclusions {#seq:V}
===========
In the present paper, we have researched the soliton system consisting of a vortex and a Q-ball that interact with each other through a common Abelian gauge field. Like a vortex, this two-dimensional soliton system has quantized magnetic flux (\[28\]). Due to the Chern-Simons term in the Lagrangian, the quantized magnetic flux leads to quantized electric charge (\[28h\]) of the soliton system and, as a consequence, to a nonzero radial electric field. As a result, the soliton system possesses nonzero angular momentum (\[33\]) that depends linearly on the Noether charges of the scalar fields. Owing to the Chern-Simons term, the energy of the vortex-Q-ball system is not invariant under the sign reversal of the phase frequency $\omega$. This in turn leads to the significant change of the dependence $E(Q_{\chi})$ in comparison with the vortex-Q-ball system [@loginov_plb_777] and with the two-dimensional non-gauged Q-ball [@lee]. The vortex-Q-ball system combines properties of both nontopological (Eq. (\[20\])) and topological (boundary condition (\[27\]) for $A(r)$ and, as a consequence, magnetic flux quantization (\[28\])) solitons.
Finally, let us point out a possible application of the results obtained in [@loginov_plb_777] and in the present paper. A vortex-Q-ball string may arise when a cosmic string passes through a charged scalar condensate. Such a condensate could exist in the early universe; electrically charged boson stars [@jetzer_227], if they exist, also consist of such a condensate. A part of the condensate may be carried away by the passing cosmic vortex string, with the result that the vortex-Q-ball string arises. In this case, the gauge interaction between vortex and Q-ball components of the vortex-Q-ball string leads to significant changes of their properties.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research is carried out at Tomsk Polytechnic University within the framework of Tomsk Polytechnic University Competitiveness Enhancement Program grant.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Planck Collaboration: A. Abergel'
- 'P. A. R. Ade'
- 'N. Aghanim'
- 'M. Arnaud'
- 'M. Ashdown'
- 'J. Aumont'
- 'C. Baccigalupi'
- 'A. Balbi'
- 'A. J. Banday'
- 'R. B. Barreiro'
- 'J. G. Bartlett'
- 'E. Battaner'
- 'K. Benabed'
- 'A. Benoît'
- 'J.-P. Bernard'
- 'M. Bersanelli'
- 'R. Bhatia'
- 'J. J. Bock'
- 'A. Bonaldi'
- 'J. R. Bond'
- 'J. Borrill'
- 'F. R. Bouchet'
- 'F. Boulanger'
- 'M. Bucher'
- 'C. Burigana'
- 'P. Cabella'
- 'J.-F. Cardoso'
- 'A. Catalano'
- 'L. Cayón'
- 'A. Challinor'
- 'A. Chamballu'
- 'L.-Y Chiang'
- 'C. Chiang'
- 'P. R. Christensen'
- 'S. Colombi'
- 'F. Couchot'
- 'A. Coulais'
- 'B. P. Crill'
- 'F. Cuttaia'
- 'T. M. Dame'
- 'L. Danese'
- 'R. D. Davies'
- 'R. J. Davis'
- 'P. de Bernardis'
- 'G. de Gasperis'
- 'A. de Rosa'
- 'G. de Zotti'
- 'J. Delabrouille'
- 'J.-M. Delouis'
- 'F.-X. Désert'
- 'C. Dickinson'
- 'S. Donzelli'
- 'O. Doré'
- 'U. Dörl'
- 'M. Douspis'
- 'X. Dupac'
- 'G. Efstathiou'
- 'T. A. En[ß]{}lin'
- 'F. Finelli'
- 'O. Forni'
- 'M. Frailis'
- 'E. Franceschi'
- 'S. Galeotta'
- 'K. Ganga'
- 'M. Giard'
- 'G. Giardino'
- 'Y. Giraud-Héraud'
- 'J. González-Nuevo'
- 'K. M. Górski'
- 'S. Gratton'
- 'A. Gregorio'
- 'I. A. Grenier'
- 'A. Gruppuso'
- 'F. K. Hansen'
- 'D. Harrison'
- 'S. Henrot-Versillé'
- 'D. Herranz'
- 'S. R. Hildebrandt'
- 'E. Hivon'
- 'M. Hobson'
- 'W. A. Holmes'
- 'W. Hovest'
- 'R. J. Hoyland'
- 'K. M. Huffenberger'
- 'T. R. Jaffe'
- 'A. H. Jaffe'
- 'W. C. Jones'
- 'M. Juvela'
- 'E. Keihänen'
- 'R. Keskitalo'
- 'T. S. Kisner'
- 'R. Kneissl'
- 'L. Knox'
- 'H. Kurki-Suonio'
- 'G. Lagache'
- 'A. Lähteenmäki'
- 'J.-M. Lamarre'
- 'A. Lasenby'
- 'R. J. Laureijs'
- 'C. R. Lawrence'
- 'S. Leach'
- 'R. Leonardi'
- 'C. Leroy'
- 'P. B. Lilje'
- 'M. Linden-V[ø]{}rnle'
- 'M. López-Caniego'
- 'P. M. Lubin'
- 'J. F. Macías-Pérez'
- 'C. J. MacTavish'
- 'B. Maffei'
- 'N. Mandolesi'
- 'R. Mann'
- 'M. Maris'
- 'D. J. Marshall[^1]'
- 'E. Martínez-González'
- 'S. Masi'
- 'S. Matarrese'
- 'F. Matthai'
- 'P. Mazzotta'
- 'P. McGehee'
- 'P. R. Meinhold'
- 'A. Melchiorri'
- 'L. Mendes'
- 'A. Mennella'
- 'M.-A. Miville-Deschênes'
- 'A. Moneti'
- 'L. Montier'
- 'G. Morgante'
- 'D. Mortlock'
- 'D. Munshi'
- 'A. Murphy'
- 'P. Naselsky'
- 'P. Natoli'
- 'C. B. Netterfield'
- 'H. U. N[ø]{}rgaard-Nielsen'
- 'F. Noviello'
- 'D. Novikov'
- 'I. Novikov'
- 'S. Osborne'
- 'F. Pajot'
- 'R. Paladini'
- 'F. Pasian'
- 'G. Patanchon'
- 'O. Perdereau'
- 'L. Perotto'
- 'F. Perrotta'
- 'F. Piacentini'
- 'M. Piat'
- 'S. Plaszczynski'
- 'E. Pointecouteau'
- 'G. Polenta'
- 'N. Ponthieu'
- 'T. Poutanen'
- 'G. Prézeau'
- 'S. Prunet'
- 'J.-L. Puget'
- 'J. P. Rachen'
- 'W. T. Reach'
- 'R. Rebolo'
- 'W. Reich'
- 'C. Renault'
- 'S. Ricciardi'
- 'T. Riller'
- 'I. Ristorcelli'
- 'G. Rocha'
- 'C. Rosset'
- 'J. A. Rubiño-Martín'
- 'B. Rusholme'
- 'M. Sandri'
- 'D. Santos'
- 'G. Savini'
- 'D. Scott'
- 'M. D. Seiffert'
- 'P. Shellard'
- 'G. F. Smoot'
- 'J.-L. Starck'
- 'F. Stivoli'
- 'V. Stolyarov'
- 'R. Stompor'
- 'R. Sudiwala'
- 'J.-F. Sygnet'
- 'J. A. Tauber'
- 'L. Terenzi'
- 'L. Toffolatti'
- 'M. Tomasi'
- 'J.-P. Torre'
- 'M. Tristram'
- 'J. Tuovinen'
- 'G. Umana'
- 'L. Valenziano'
- 'J. Varis'
- 'P. Vielva'
- 'F. Villa'
- 'N. Vittorio'
- 'L. A. Wade'
- 'B. D. Wandelt'
- 'A. Wilkinson'
- 'N. Ysard'
- 'D. Yvon'
- 'A. Zacchei'
- 'A. Zonca'
---
[^1]: Corresponding author: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
address: |
INFN, sezione di Ferrara, Via Paradiso, 12 - 44100 Ferrara, Italy\
and\
ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia
author:
- 'A. D. DOLGOV'
title: PARTICLE PRODUCTION IN COSMOLOGY AND IMAGINARY TIME METHOD
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Personal Recollections. \[s:psrn\]
==================================
It is difficult to write about a close friend who died so unexpectedly and so early. Time runs fast and it seems that it was just yesterday that we saw each other and talked a lot on every possible subject. I really miss these discussions now. Last time that I saw Misha Marinov was spring 1999. I was visiting Weizmann Institute of Science in frameworks of Landau-Weizmann Program and used this opportunity to come to Haifa where Misha lived and worked in Technion. It was a nice sunny and fresh morning when we arrived with my wife Inna to Haifa railway station, where Misha met us and drove along a beautiful road up the Carmel Mountain to his home where Lilia, his wife, waited us with a delicious lunch. Misha was in high spirits, the four of us being old friends, were very glad to see each other, but slightly complained about, as he said, small pain in his spine. None of us knew at that moment that it was a first sign of fast and terrible disease.
Our friendship with Misha began, I think, in 1965 when we both were graduate students. We happened to be in the same plane on the way to Yerevan to the First International Nor-Ambert School on Particle Physics. Together with another young physicist from ITEP, Misha Terentev, we shared a small hut and enjoyed first in our lives international conference and charming and hospitable town of Yerevan.
Misha was two years older than me but I had a feeling that his knowledge of physics, especially of mathematical physics, was at a professor level and benefited a lot from our communications. Later we both worked in ITEP theory group and it was always instructive and interesting to talk with him not only about physics but practically any subject, especially history where Misha had unusually deep and wide knowledge would it be ancient or modern.
Our friendship turned into friendship between families when in 1970 we started to live in the same apartment building near ITEP and the distance between our apartments was only 1-2 minute walk up or down the stairs. In 1979 Misha quit his position in ITEP and applied for permission to emigrate with the aim to live in Israel. Immediately life became much harder for him. It was difficult to find a job that could give enough money to support his family of four. Special rules existed at that time in the Soviet Union to prevent people from working without strict state control. In summer seasons (plus a part of spring and autumn) Misha worked as a construction worker, building small private houses (dachas) in the country. In winter he did some work for the official “Center of Translations” translating scientific papers or books from English into Russian or vice versa. However this kind of job was allowed only if one has another permanent place of work at one or other state enterprise, which Misha had not. At some stage they requested from him a certificate that he had such a job and since nothing can be presented, he was fired. So my wife, Inna, formally took this job and fetch for him papers to translate from the Center. Misha translated them, Inna presented the translated papers to the Center, received the money and brought it to Misha or his wife, Lilia. That’s how it worked.
I have to confess that I also participated in a similar deception activity, a few papers and books translated into Russian under my name were in fact translated by Misha. In particular, the review paper by S. Coleman “The magnetic monopole fifty years later”[@coleman82] was translated for Uspekhi Fiz. Nauk, vol. 144, by Marinov but under my name. Moreover, the editors wanted to have a short review on the activity related to magnetic monopoles to the moment when the paper was translated, i.e. two years after the original one had been written. Again Misha wrote the paper and I only signed it (honestly I also read it and liked it very much). So he got the money and I got the fame. Now I have to restore justice and to change the reference[@dolgov84] into[@marinov84].
Only in 1987 Marinovs received permission to emigrate and left for Israel. As we all thought that time, emigration meant leaving for good with practically zero chances to see or contact each other again. However things were changing fast and freedom to travel abroad, unbelievable at Soviet times, came to our country. In 1990 Inna was able to go to Israel and for a whole month enjoyed friendly atmosphere of Marinov’s home. She and Lilia even now recall with mutual pleasure how nice was that time. After a couple of years Misha’s life in Israel was successfully arranged. He got a professor position in Technion and was happy to be there. I remember how proudly he showed me the campus, labs, students during my first visit to Haifa. He enthusiastically returned to research that was interrupted for 6-7 years. As I can judge by what and how I learned from him, he was a very good teacher and did teaching with vigor and love. On the other hand, he kept warm feelings toward ITEP, and often going in the morning to his office in Technion he used to say, addressing Inna and Lilia, “Buy girls, I am going to ITEP”.
There are several fields where Misha made very important contributions, despite a long break in his scientific activity. But I am not going to describe them all, since, I think, this will be described in the Introduction to this volume. I will mention only two which have some relation to me. Misha’s results on application of path integral methods to complicated quantum systems are internationally renowned and I am proud that I recommended Maurice Jacob to publish Marinov’s review on the subject in Physics Reports[@marinov79]. This was the last paper written by Misha, while he was still in ITEP. Another subject where M. Marinov made a very essential contribution together with V. Popov was electron-positron production by an external electromagnetic field[@marinov72]. The method developed in these works was applied to the non-perturbative calculations of cosmological particle production by scalar (inflaton) field in our paper with D. Kirilova[@dolgov90], which is discussed below.
Particle Production in Cosmology; Brief Historical Review \[s:hstr\]
====================================================================
There are two different cases of quantum particle production by external classical fields that are cosmologically interesting. The first is the production by time-dependent background metric or, in other words, by gravitational field and the second is the transformation of classical (oscillating) inflaton field into elementary particles and the corresponding universe (re)heating. Particle production by gravity might be essential in the very early universe near cosmological singularity when the strength of gravitational field was close to the Planck value. Creation of particles by isotropic Friedman-Robertson-Walker (FRW) metric was pioneered by Parker[@parker68] and further developed in a series of papers[@parker69; @grib69; @chernikov72]. Particle production by gravity in anisotropic cosmologies was considered in refs.[@zeldovich70]. As argued in these papers, particle production in anisotropic case creates anisotropic distribution of matter and back reaction of the created matter on the metric could lead to isotropization of the latter. Thus, in principle, the observed FRW cosmology might originate from a rather general initial state. More references to the subsequent works and detailed discussion can be found in the books[@books].
There is an important difference between particle production in isotropic and anisotropic cosmologies. Isotropic FRW metric is known to be conformally flat, i.e. after a suitable coordinate transformation it can be reduced to the form: ds\^2 = g\_ dx\^dx\^= a\^2 (r,) ( d\^2 - dr\^2 ) \[ds2\] From this expression follows, in particular, that FRW metric cannot create massless particles if the latter are described by conformally invariant theory[@parker68]. If the particle mass, $m$, is non-vanishing but the interactions are conformally invariant, their production rate is suppressed as a power of the ratio $(m/m_{Pl})$. (Of course, non-vanishing masses break conformal invariance.) These statements can be easily checked in perturbation theory. The coupling of gravity to matter fields is given by $\left(g^{\mu\nu}-\eta_{\mu\nu}\right) T^{\mu\nu}$, where $\eta^{\mu\nu}$ is the Minkowsky metric tensor and $ T^{\mu\nu}$ is the energy-momentum tensor of matter. If the metric tensor is given by expression (\[ds2\]), the coupling to matter is proportional to the trace of the energy-momentum tensor that vanishes in conformally invariant theory.
A well known example of the theory which is conformally invariant at classical level (i.e. without quantum corrections) is electrodynamics with massless charged fermions, or any other (possibly non-abelian) gauge theory describing interacting massless gauge bosons and fermions. However quantum trace anomaly[@chanowitz73] breaks conformal invariance and gives rise to a non-zero trace of $T_{\mu\nu}$. In $SU(N)$ gauge theory with $N_f$ number of fermions the trace of the energy-momentum tensor of matter is equal to: T\^\_= ( [11 N 3]{} - [2N\_f 3]{} ) G\_ G\^ \[tmumu\] where $G_{\mu\nu}$ is the gauge field strength tensor. This anomaly could strongly enhance generation of electromagnetic field (or any other gauge fields) in the early universe[@dolgov80].
Another simple and important theory of a free massless scalar field $\phi$ is not conformally invariant even at the classical level if $\phi$ is minimally coupled to gravity (that is through covariant derivatives only). The energy-momentum tensor of such field is given by the expression: T\_ () = (1/2)\_ \_- (1/4) g\_ \_ \^\[tmunuphi\] and its trace $T^\mu_\mu = -(1/2)\partial_\alpha \phi\,\partial^\alpha \phi$ is generally non-vanishing. Conformal invariance can be restored if one adds to the free Lagrangian the nonminimal coupling to gravity in the form $R \phi^2 /12$ (see e.g. refs. [@books]). However it would be better not to restore it because generation of primordial density perturbations at inflation[@guth82], which serve as seeds for large scale structure formation, is possible only for non-conformal fields.
Another realistic example of conformally non-invariant theory with massless fields is gravity itself. It was shown that gravitational waves are not conformally invariant in the standard General Relativity[@grishchuk75]. This explains efficient production of gravitational waves during inflationary stage[@starobinsky79].
A renewed interest to gravitational particle production arose in connection with a possible explanation of the observed ultra-high energy cosmic rays by heavy particle decays[@berezinsky97]. There are two competing mechanisms of creation of such particles in cosmology: by background metric and by inflaton field. The former was considered in refs. [@kuzmin98] (for a review see[@kuzmin99]), while particle production by inflaton will be discussed below.
In the earlier papers[@dolgov82] the universe (re)heating at the final stage of inflation through particle production by the oscillating inflaton field was treated in a simplified perturbation theory approximation. First non-perturbative treatment was performed in two papers[@dolgov90; @traschen90]. In what follows we concentrate on the approach of ref.[@dolgov90] where the imaginary time method was used. In both papers[@dolgov90; @traschen90] a possibility of parametric resonance enhancement of particle production rate, noticed long ago[@narozhnyi73], was mentioned. However, it was argued in the first of them that the resonance was not effective because the produced particles were quickly removed from the resonance band by the cosmological expansion and elastic scattering on the background. A more careful analysis of the subsequent paper[@traschen90] showed that under certain condition expansion might be irrelevant and did not destroy the resonance. In this case a strong amplification of the production probability and much faster process of post-inflationary (re)heating could be expected. The issue of the parametric resonance (re)heating attracted great attention after the paper[@kofman94], and now the number of published papers on the subject is measured by a few hundreds. However, a review of this activity is outside the scope of the present paper and below we will confine ourselves to the problem of fermion production by a time dependent scalar field where parametric resonance is not effective.
Concerning production of fermions, there is a contradiction in the literature between the paper[@dolgov90], where non-perturbative production of fermions was pioneered, and the subsequent ones. While in the paper[@dolgov90] was stated that fermion production is always the strongest in perturbation theory regime, and in the opposite, quasiclassical limit the production is noticeably weaker, in subsequent works was argued that in non-perturbative regime fermion production was strongly enhanced so that it could even compete with resonant boson production. Calculations in ref. [@dolgov90] have been performed by imaginary time method, while other works either used numerical calculations or some approximate analytical estimates. I will argue in what follows that there is practically no difference between the results of all calculations, earlier and later ones, but the difference is in the interpretation of the results and fermion production by the inflaton is always weak, weaker than that found in perturbation theory.
Particle Production in Perturbation Theory \[s:pert\]
=====================================================
Let us start from consideration of production in the case when perturbation theory is applicable and calculations are straightforward and simple. In this section we will neglect the universe expansion and assume that the external scalar field periodically changes with time according to: (t) = \_0 t \[phioft\] Here $\phi_0$ is the amplitude of the field, it can be slowly varying function of time, and the frequency of oscillations $\omega$ coincides with the mass of $\phi$ if the latter lives in the harmonic potential $U(\phi) = m^2_\phi \phi^2/2$.
We assume that $\phi$ is coupled to fermions through the Yukawa interaction: \_ = |( i[ /]{}+m\_0 ) + g |\[lpsi\] Perturbation theory would be valid if the coupling constant is small, $g\ll 1$, which is well fulfilled for the inflaton field, and if the fermion mass is smaller then the mass of the inflaton, $m_\phi = \omega$. The last condition may not be true even if $m_0 < m_\phi$ because the interaction with $\phi$ introduces effective time-dependent mass m\_1 (t) = g \_0 t \[m1oft\] and for a large amplitude $\phi_0$ the latter may be large in comparison with $\omega$ for most of the oscillation period, except for a small part, when $\cos \omega t$ is close to zero. In this case perturbation theory is invalid.
It is practically evident, even without calculations, that in perturbative case the rate of particle production is equal to the width of the decay of the scalar boson $\phi$ into a pair of fermions: n\_/n\_= \_= g\^2 /8\[dotn/n\] where $n_{\psi ,\phi}$ are the number densities of $\psi$ and $\phi$ particles per unit volume respectively and we assumed for simplicity that the fermion mass $m_0 =0$ (it is straightforward to lift this restriction).
Still to make comparison with subsequent non-perturbative calculations we will sketch below the derivation of this result. According to general rules of quantum field theory the amplitude of production of a pair of particles with momenta $\vec p_1$ and $\vec p_2$ by an external time-dependent field $\phi(t)$ in first order in perturbation theory is given by A(p\_1,p\_2) = gd\^4 x (t) p\_1,p\_2 | |(x) (x) | vac \[apt\] where the state $\langle \vec p_1,\vec p_2 |$ is produced by action on vacuum of the creation operators in the standard second-quantized decomposition of Dirac operators $\psi $ and $\bar \psi$: (x) = \_s \[u\_k\^s b\_k\^s e\^[-ik x]{} + v\_k\^s d\_k\^[s ]{} e\^[ik x]{}\] \[psiofx\] where $b_k^s$ and $d_k^{s{\dagger}}$ are respectively annihilation and creation operators for particles and antiparticles with momentum $k$ and spin $s$.
After the usual anti-commutation algebra we will arrive to the integral d\^3 k d\^3 k’ (k - p\_1) (k’- p\_2) e\^[i(E+E’)t - i(k+ k’) x ]{} \[intdp\] The integral can be trivially taken and substituted into the integral over $d^3x dt$ (\[apt\]). Integration over $d^3 x$ gives $\delta (\vec p_1 +\vec p_2)$ and we are left with the Fourier transform: A(p\_1,p\_2) \~g\^2 \^[(3)]{} (p\_1 +p\_2) dt (t) e\^[i(E\_1+E\_2)t]{} \[adelta\] (for details and more rigorous consideration in terms of Bogolyubov coefficients see e.g. appendix A in ref. [@dolgov96]).
The probability of particle production is proportional to $ |A(\vec p_1,\vec p_2) |^2$ and contains the square of momentum delta-function. The latter is treated in the standard way, \^2 = 2V (p\_1 +p\_2) \[delta2\] where $V$ is the total space volume. The origin of the volume factor is evident: since the external field is space-point independent, so is the probability of production per unit volume and the total probability is proportional to the total volume.
Similar situation is realized for the time dependence in the case of periodic external fields, if one neglects back reaction of the produced particles on the field evolution and on the probability of production. The former can be taken into account by a (slow) decrease of the field amplitude $\phi_0(t)$, while the latter is determined by the statistics of the produced particles: the probability of boson production is proportional to the phase space density of already produced bosons, $(1+f_k)$, while the probability of fermion production is inhibited by the factor $(1-f_k)$. This back reaction effect is absent for Boltzmann statistics, which we will mostly assume in what follows. Thus, for a periodic external field one would expect that the probability of production is proportional to the total time interval, during which the external field was operating. In the idealistic case of $\phi \sim \exp (i\omega t)$, its Fourier transform gives $\delta (2E -\omega)$ and the square of the latter is, as above, $t_{tot} \delta (2E -\omega)$. The second factor ensures energy conservation and is infinitely large for $E=\omega/2$. It means that the phase space density of the produced particles becomes very large after period of time when the energy conservation is approximately established. One can check that this time is much shorter than $1/\Gamma$ (where $\Gamma$ is the perturbative decay rate) but still the time of transition of energy from the inflaton field to the produced fermions is given by $1/\Gamma$. This fact is commonly agreed upon in the case of perturbative production. The statements in the literature that in non-perturbative regime fermion production could be very strong is possibly related to this trivial rise of the occupation numbers and does not mean that fermion production can compete with production of bosons (see below).
In the case when external field operates during a finite period of time, starting e.g. from $t=0$, or if one is interested in the number of produced particles at the running moment $t$, the integral in expression (\[adelta\]) should be taken in the limits $(0,t)$ and for the particular case of $\phi = \phi_0 \cos \omega t$ one obtains: I(t; E, ) && \_0\^t dt e\^[2iEt]{} t\
&=&e\^[i(E-/2) t]{}\[int0t\]
For $E$ close to $\omega/2$ the first term dominates and the number of produced fermions rises as $t^2$ till $t \sim 1/|2E-\omega|$. At larger times it oscillates. The same phenomena was found in non-perturbative calculations. Indeed, the phase space number density of the produced particles (we use this term interchangeably with the “occupation number”) is given by f\_p = g\^2 \_0\^2 I (t; E,) \^2 \[fpptbl\] As we have argued above, usually one has $ \mid I (t; E,\omega) \mid^2 = 2\pi t\,\delta (2E-\omega)$. In this case the number density of the produced particles as a function of time is given by: n(t) = f\_p = [g\^2 8]{}\_0\^2t = n\_t \[noftbl\] where $n_\phi = \phi_0^2 \omega$ is the number density of $\phi$-bosons and $\Gamma$, given by eq. (\[dotn/n\]), is their decay width.
A detailed explanation of the discussed phenomena can be found in textbooks on quantum mechanics in the section where perturbation theory for time dependent potential is presented, see e.g.[@landau].
Returning to the occupation number (\[fpptbl\]) we see that for $(\omega- 2E)t <1$ it evolves as $f_p \approx g^2\phi_0^2 t^2$ and reaches unity for $t=t_1 = 1/g\phi_0$. This is much earlier than $t_d = 1/\Gamma$ which is the characteristic decay time of $\phi (t)$: t\_d /t\_1 = (8/g)(\_0 /) \[tdt1\] Formally taken this ratio may reach the value $10^8-10^9$. This is an explanation of statement that fermions could be very quickly produced by inflaton. On the other hand, though some fermionic bands (approximately satisfying energy conservation) might be quickly populated, the total transfer of energy from the inflaton to the produced particles is determined by the total decay rate and is much slower. Roughly speaking $f_p = 1$ corresponds to production of only one pair of fermions and, of course, the energy of this pair is negligible in comparison with the total energy accumulated in the classical field $\phi (t)$.
Perturbation theory is not applicable if the effective mass of fermions $m_{eff}= (m_0 + g\phi_0)$ is larger than the frequency of the oscillations of the scalar field. For example, the probability of pair production by two-quanta process, when the energy of each produced fermion would be equal to $\omega$, is related to one-quantum process, when $E=\omega/2$, as $W_2/W_1 \sim (g\phi_0 /\omega)^2$. It is still possible that $\phi_0/g\omega \gg 1$, while $g\phi_0/\omega <1$, so that perturbation theory is reliable and the relation $t_d/t_1 \gg 1$ still holds. However in many practically interesting cases $g\phi_0/\omega >1$ and in this range of parameters the result obtained above can serve only for the purpose of illustration and for more precise statements we have to go beyond perturbation theory. This will be done in the following section by the imaginary time method[@nikishov69; @popov71; @marinov72]. (For recent applications of this method and a more complete list of references see[@ringwald01].) Qualitatively clear that non-perturbative effects could only diminish the rate of particle production because the non-perturbative calculations take into account non-vanishing and large value of the effective mass of the produced particles and this leads to a smaller rate of the production in comparison with the case when the interaction is taken in the form $g\phi\bar\psi\psi$ but its contribution into fermion effective mass is neglected. As we see below, the suppression of the production rate in nonperturbative regime[@dolgov90] in comparison with perturbation theory is given by the factor $(\omega/g\phi)^{1/2}$ in qualitative agreement with these simple arguments.
Effects of quantum statistics were neglected above, and thus the results obtained are valid only if $f_p <1$. The corresponding corrections can be approximately introduced by multiplication of the r.h.s. of eq. (\[fpptbl\]) by the factor $(1\pm f_p)$ and correspondingly $f_p^{(f,b)} = g^2 \phi_0^2|I|^2 / (1 \pm g^2 \phi_0^2 |I|^2)$, where the signs $''\pm''$ refer for fermions and bosons respectively. One sees that the production of fermions effectively stops (as one should expect) when $f_p^{(f)} \sim 1$, while production of bosons tends to infinity. Presumably a more accurate treatment would not allow bosons to reach infinitely large density in a finite time but the message is clear, the production of bosons becomes explosive in perturbation theory with characteristic time of the order of $t_1 = 1/(g\phi_0)$ and all the energy of the inflaton would go into that of the produced bosons during approximately this time. There are several effects that can weaken this conclusion. One is a possible inapplicability of perturbation theory for a large $g\phi_0/\omega$. This effect qualitatively acts in the same way as in fermionic case discussed above. Still, even if the $g\phi_0/\omega>>1$ the effect of explosive production of bosons would survive due to parametric resonance in equation of motions for the produced modes[@dolgov90; @traschen90; @kofman94]. Another two effects that could diminish the production are the cosmological red-shift of momenta of the produced particles and their scattering on other particles in the background. Both would push the produced particles away from the resonance band and could significantly slow down the production in the case of narrow resonance[@dolgov90], while in the case of wide resonance the effect survives[@traschen90; @kofman94].
On the other hand, both red-shift and scattering of the produced fermions back react on their production in exactly opposite (to bosons) way. These phenomena “cleans” the occupied zone and allows for production of more fermions.
Quasiclassical Limit; Imaginary Time Method. \[s:imt\]
======================================================
Small Mass Case. \[ss:small\]
-----------------------------
Usually non-perturbative calculations are not simple but in the case that we are considering there is a fortunate circumstance that in the anti-perturbative limit quasiclassical approximation works pretty well. The latter can be efficiently treated by the imaginary time method[@nikishov69; @popov71; @marinov72]. Below we will essentially repeat the paper[@dolgov90] correcting some typos and algebraic errors, though the basic results of the paper remain intact.
The coupling of $\phi(t)$ to the produced particles is equivalent to prescription of the time dependent mass to the latter, $m(t) = m_0 + g\phi(t)$. The classical Lagrange function for a relativistic particle with such a mass has the form L\_[cl]{} = - m(t) ( 1 - V\^2)\^[1/2]{} \[lcl\] where $\vec V$ is the particle velocity. The corresponding Hamiltonian is = \^[1/2]{} (t) \[ham\] The quantization of this system can be achieved by the path integral method. The Green’s function of the quantum particle has the form (see e.g. [@marinov79]): G(x\_f,t\_f; x\_i, t\_i) = Dp Dx . \[gxtint\] The functional integral in this case can be easily taken, giving: G(x\_f,t\_f; x\_i, t\_i) = \[gxt\]
According to the general rules of quantum mechanics the amplitude of the transition from the state given by the initial wave function $\Psi_i$ into that given by $\Psi_f$ is equal to A(p\_1,p\_2) = d\^3 x\_i d\^3 x\_f \^\*\_f (x\_f) G(x\_f,t\_f; x\_i, t\_i) \_i (x\_i). \[ap1p2\] where for $\Psi_{i,f}$ plane waves are usually substituted.
If we want to obtain the amplitude of creation of a pair of particles the contour of integration over time should be shifted into complex $t$-plane in such a way that it goes around the branching point of the energy $\Omega$ in the direction of changing the sign of energy from negative to positive one. This corresponds to transition from the lower continuum of the Dirac sea to the upper one, i.e. to pair creation. Thus we find: A(p\_1,p\_2) = ( 2)\^3 ( p\_1 + p\_2 ) , \[apair\] where the contour $C(t_i,t_f)$ starts at $t=t_i$ and ends at $t=t_f$ and turns around the branching point of $\Omega$ in the way specified above.
The position of the branching points $t_b = t' +it'' $ can be found from the equation: p\^2 + ( m\_0 + g \_0 t )\^2 =0. \[brpoint\] Correspondingly m\_0 + g\_0 ( ’” -i ’ ” ) = i p, \[tt\] where $\tau = \omega t$. In what follows we assume that $m_0 = 0$ and it will grossly simplify technical details. In this limit $\tau' = \pi/2 + n\pi$ and $\sinh \tau'' = \pm (p/g\phi_0)$.
The integral along the cut $\tau = \tau' +i\eta$ is real and, according to our prescription, negative. It gives exponential suppression factor for the production probability, $W \sim \exp (-2Q)$, with Q =(2/) \_0\^[”]{} d( p\^2 - g\^2\_0\^2\^2 )\^[1/2]{}. \[Q\] This integral can be expressed through complete elliptic functions[@grad]: Q = [2 ]{} \[qell\] where m\_1 = g\_0,[and]{} = p/. \[beta\] For small $\beta$ these functions can be expanded as $K(\beta) \approx (\pi/2) (1+\beta^2 /4)$ and $E(\beta) \approx (\pi/2) (1-\beta^2 /4)$, so that $Q\approx (\pi/2) (p^2/ \omega \, m_1 ) $.
The total production amplitude is equal to the sum of expressions (\[apair\]) with all the contours encircling the proper branch points between $t_i$ and $t_f$. Since the integrals along imaginary direction $id\eta$ are all real and have the same value for all branch points, their contribution to the amplitude gives the common factor $\exp (-Q)$. The integrals over real time axis corresponding to different contours $C$ around neighboring branch points differ by the phase factor $A_{n+2}/A_n =\exp (2i\alpha)$, because the energy changes sign after the integration contour turns around branch points. The absence of the contribution from the nearest cut is related to the particle statistics and is discussed e.g. in ref.[@popov71; @marinov72]. The phase $\alpha$ is given by: = \_0\^[2]{} dt = [4 ]{} E() \[alpha\] All this is true if the free fermion mass is vanishing, $m_0 =0$, otherwise equations become significantly more complicated. In the limit of small $\beta$ we find[@grad]: E()\[E\] while for $\beta$ close to 1 the necessary expressions are presented after eq. (\[beta\]) with the interchange $\beta^2 \leftrightarrow (1-\beta^2)$.
Summing over all branch points we obtain: A(p\_1,p\_2) = ( 2)\^3 ( p\_1 + p\_2 ) [(N ) -1 ( ) -1]{} \[atot\] where $N$ is the total number of branch points included in the amplitude; it is approximately equal to the total time in units $1/\omega$ during which the particles are produced, $N= {\rm Integer}[ (t_f-t_i)/\omega]$. The last factor reminds that coming from the integration over time in perturbation theory discussed in sec. \[s:pert\] and in fact its physical nature is the same. For very large $N$, formally for $N\rightarrow\infty$ it tends to \_j ( -j ) \[delofal\] These delta-functions impose energy conservation for the production of pair of particles by $j$ quanta of the field $\phi$. Note that in contrast to the lowest order perturbation theory, when only a single quanta production is taken into account, the expression (\[atot\]) includes production of a pair by many quanta of the field $\phi$. For example, in the limit of high momenta of the produced particles these delta-functions are reduced to $\delta (2p -j\omega)$, the same as in perturbation theory for j-quanta production.
Treating again, as in sec \[s:pert\], the square of delta-function as a product of the single delta-function and $\delta (0) = \pi N$ with $N$ expressed through the total time $t$, during which the particles have been produced, as the integer part of $t\omega $, we find the following expression for the rate of production per unit time and unit volume[@dolgov90]: n =\_j (-2Q) ( - j) \[dotn\] In the limit of $m_1 \gg \omega$ one obtains: Q && [2]{}[p\^2 m\_1]{}\
&& [4m\_1 ]{}, \[Qalpha\] and hence n = [12]{}\_[j\_m]{} \[dotnpj\] Here summation starts from the minimum integer value $j_m \geq (4m_1/\pi\omega)$ and $p_j$ is determined from the equation $\alpha = j\pi$, i.e. p\_j\^2 (m\_1 /2)(j-4m\_1/) /\[(4m\_1/p\_j) +1\] \[pj2\] A rough estimate gives $\dot n \sim \omega^{5/2} m_1^{3/2}$. Correspondingly the characteristic rate of the inflaton decay in the quasiclassical approximation is given by \_[q]{} = n/n\_=n /(\^2\_0) \~( /m\_1 )\^[1/2]{} \[gammaq\] where $\Gamma$ is the decay rate in perturbation theory (\[dotn/n\]). One sees that in the quasiclassical limit the decay rate is suppressed in comparison with the formal result of perturbation theory as a square root of the ratio of the oscillation frequency to the amplitude of the scalar field. This suppression can be understood as follows[@dolgov90]. Most of the time the instant value of the field $\phi(t)$ and the effective mass of the fermions, $m_{eff}=g\phi_0\,\cos \omega t$ are large in comparison with the oscillation frequency. As is well known (see also sec. \[ss:large\] below) the probability of particle production in this case is exponentially suppressed. However, when $\cos \omega t$ is very close to zero the effective mass of the produced particles would be smaller than $\omega$ and they are essentially produced at this short time moments. This results in a much milder suppression of the production, not exponential but only as $(\omega/g\phi_0)^{1/2}$.
For the case of finite and not too big $N$ we will see that, according to the calculations of reference[@dolgov90] presented above, the occupation number $f_p$ would reach unity in a much shorter time than $1/\Gamma_q$. This result was rediscovered later in the papers[@baacke98; @green99] by numerical calculations and reconfirmed by analytical methods in ref. [@peloso00]. However, as it has been already argued, this does not mean that non-perturbative production of fermions is strong, it is always weaker than the perturbative one.
The calculations presented above do not include the effects of quantum statistics, so strictly speaking, they are valid for “boltzons”. Thus, they present an upper bound for the production of fermions. In the fermionic case, the production would stop when the occupation number, $f_p$, approaches unity, while production of “boltzons” would go unabated. However, if the particles from the occupied Fermi band are quickly removed by scattering or red-shift (as we discussed above) the production of fermions would go essentially with the same rate as production of “boltzons”.
For a finite number of oscillations $N$ the occupation number of the produced particles is equal to (see eq. (\[atot\])): f\_p (N) = (-2Q)( [(N ) -1 ( ) -1]{} )\^2 \[fpnonpt\] The last factor is rather similar to that in eq. (\[int0t\]). This is an oscillating function of $N$. For $\alpha= \pi (1-\epsilon)$ with a small $\epsilon$ it rises roughly as $N^2$ during $N= 1/(2\epsilon)$ oscillations. The occupation number increases with time discontinuously as a series of discrete jumps as time $t/\omega$ reaches integer values. During this stage $f_p$ may quickly rise with the speed much faster then the rate $\Gamma_q$ (\[gammaq\]) in complete analogy with the perturbative case considered in sec. \[s:pert\]. However, as we have already stressed, this does not mean that the production of fermions goes faster than in perturbation theory.
After this period of increase, $f_p$ starts to go down and approaches zero at $N_0\approx 1/\epsilon$. This oscillating behavior of the number of produced particles was noticed long ago in the problem of $e^+e^-$-pair creation by periodic electric field (for the list of references see e.g. the book by Grib et al in ref. [@books]). Thus it looks as though particles are produced by the field and after a while they all are absorbed back. This behavior is difficult to digest. Note that it is absent if time is very large, tending to infinity, as is discussed above. In this case the energy conservation is strictly imposed by the delta-function, $\alpha = \pi n$ (where $n$ is an integer), or in other words $\epsilon =0$ and $N_0 \rightarrow \infty$.
Possibly this mysterious phenomenon of re-absorption of the produced particles is related to the fact that during finite time the external field $\phi(t)$ does not disappear and the particle vacuum is not well defined over this time dependent background. To resolve the ambiguity one may calculate the transition of energy from the time-varying field $\phi (t)$ into other quantum states which are not necessarily determined in terms of particles. Energy density, in contrast to the particle number density, can be unambiguously defined in terms of local fields operators and does not suffer from any ambiguity related to the non-local character of the latter. The energy density of the quantum field $\psi$, defined as the expectation value of the time-time component of its energy-momentum operator, may also exhibit the oscillating behavior described above but the correct interpretation is possibly not production of $\psi$-particles but some excitation (“classical”?) of the (fermion) field $\psi$ coupled to $\phi(t)$.
Large Mass Case. \[ss:large\]
-----------------------------
Let us now consider the case when the fermion mass $m_0$ is large in comparison with the oscillation frequency $\omega$ and with the amplitude of the oscillations, $m_0 \gg g\phi_0$, so that the total effective fermion mass, $m_{tot} = m_0 + g\phi_0 \cos \omega t$ never vanishes and always large. The calculations for this case have been only done in ref. [@dolgov90] and we will reproduce them here. To be more precise we will reproduce only imaginary time part, while in ref. [@dolgov90] the method of Bogolyubov coefficients was used as well.
Following this paper we will consider production of bosons. It will be technically simpler allowing to make all calculations analytically, but qualitatively the same results should be valid also for fermions, because for a large $m_0$ the production is weak and the occupation numbers remain small. We assume that the effective mass has the form m\^2(t) = m\_0\^2 + g\^2 \_0\^2 \^2 t \[m2oft\] This case is realized if the interaction of the inflaton field with the produced particles ($\chi$-bosons) has the form $g^2|\chi^2| \phi^2$. The probability of production can be found from the expressions of the previous subsection by the substitution $p^2 \rightarrow p^2 +m_0^2$. In particular, the exponential damping factor is given, instead of (\[qell\]), by: Q’ = [2 ]{} \[qell’\] where (’)\^2 1- u\^2 = 1 - [m\_1\^2 m\_0\^2 +m\_1\^2 + p\^2]{} \[beta’2\] and the complete elliptic integrals in the case of small $k$ are expanded as[@grad]: K(’) && + [u\^24]{} ( -1)\
E(’) && 1 + [u\^22]{} ( -[12]{} ) \[KE\] The phase difference over the period of oscillations is now given by: ’ &=& [4 ]{} E()\
&& [2 ]{} \[alpha’\]
We can repeat the same calculations as in the previous subsection to find the occupation number and the number density of the produced particles. The production probability is now exponentially suppressed, as $\exp \{{-2 \sqrt{m_0^2+m_1^2}\ln [16(m_0^2+m_1^2)/m_1^2] /\omega}\}$. For a sufficiently large ratio $m_0/\omega$ the production would be very weak, all occupation numbers would be small in comparison with unity and bosons and fermions would be equally poorly produced.
Back Reaction and Cosmological Expansion Effects. \[s:evol\]
============================================================
.
Now we briefly comment on applicability of the results discussed above to realistic case of universe (re)heating after inflation. We have neglected universe expansion and damping of the field $\phi$ due to energy transfer to the produced particles. The effect of expansion can be easily taken into account in conformal coordinates where the metric takes the form (\[ds2\]) with space point independent cosmological scale factor $a(\tau)$. Under transformation to conformal coordinates and simultaneous redefinition of the gravitational, scalar, and fermionic fields respectively as $g_{\mu\nu} \rightarrow a^2 g_{\mu\nu}$, $\phi \rightarrow \phi/a$, and $\psi \rightarrow \psi /a^{3/2}$, the mode equation for the scalar field takes the form: \_k” +(k\^2 + m\^2 a\^2 - a”/a ) \_k = 0, \[phi”\] where the derivatives are taken with respect to conformal time and $k$ is comoving momentum. The presence of the term $a''/a$ demonstrates breaking of conformal invariance even for massless scalar field, as has been already mentioned in sec. \[s:hstr\]. All masses enter equation of motion in the combination $ma$, so mass terms explicitly break conformal invariance. The interactions of the types $g\phi \bar \psi \psi$, $\lambda \phi^4$ and $f\phi^2 \chi^* \chi$ are invariant with respect to the transformation of the fields specified above (note that the presence of the $\sqrt{det[g_\mu\nu]}$ in the action integral gives the necessary factor $a^4$ to ensure this invariance).
The expressions for the scale factors through conformal time in three most interesting cosmologies are the following: a&\~& e\^[Ht]{} = - [1/ H]{}, [De Sitteruniverse, inflation]{},\
a&\~& t\^[1/2]{} \~, [radiationdominance]{},\
a&\~& t\^[2/3]{} \~\^2, [matterdominance]{}. \[expregm\] In particular, in the radiation dominated universe with conformally invariant interactions, scalar field is conformally invariant but this is not true for other expansion regimes. Correspondingly, particles production by massless scalar field with the self-potential $\lambda \phi^4$ can be reduced to the flat space case discussed in the previous section. The difference between the potentials of $\phi$ in these two cases, $\omega^2 \phi^2$ and $\lambda \phi^4$, is not essential and the obtained above results can be easily translated to the $\lambda \phi^4/4$ potential. Indeed, the equation of motion of spatially homogeneous field $\phi$ in flat space-time (in conformal coordinates) has the form: ” +\^3 = 0 \[ddotphi\] This equation is solved in terms of Jacobi elliptic functions[@grad]: () &=& \_0 [cn]{} ( \_0; 2 )\
&=& [2 ]{} \_[n=1]{} [1 + ]{} \[cnoft\] where $\kappa = \Gamma^2 (1/4) /4\sqrt{\pi}$. The expansion is well approximated by the first term and particle production rate can be estimated using results of the previous section. Significant deviations from those results can be expected only in the case of heavy particle production when higher frequency terms in expansion (\[cnoft\]) may compete with the exponentially suppressed contribution coming from lower terms (see eq. (\[qell’\])).
It should be repeated, however, that these results are true only for radiation dominated regime of expansion. For other cosmologies the term $a''/a$ in eq. (\[phi”\]) is non-vanishing and must be taken into account.
Another effect, in addition to expansion, that results in a decrease of the amplitude of the field $\phi(t)$, is back reaction of the particle production. Energy that is transferred to the produced particles is taken from the field $\phi$ so the energy density of the latter should become smaller. For harmonic oscillations (in the case of the potential $\omega^2 \phi^2$) only the amplitude of the field diminishes, while frequency remains the same. For quartic potential both the frequency and the amplitude of oscillations go down, as one can see from eq. (\[cnoft\]) with $\phi_0 (t)$.
In the case of quickly oscillating field the effect can be easily estimated in adiabatic approximation. One has to solve the equation for energy balance in expanding background: = -3H ( + P ) \[dotrho\] where $\rho$ and $P$ are respectively energy and pressure densities of the field $\phi$ and the produced particles. For the former the solution of the standard equation of motion without interactions should be substituted with the effect of production included in a slow decrease of the amplitude $\phi_0$.
More accurate consideration demands using equation of motion modified by the production process. Usually this is described by the introduction into equation of motion, in addition to Hubble friction, the “production friction term”: + 3H + U’() = -. \[dotphi\] where $U(\phi)$ is the potential of $\phi$ and derivative is taken with respect to $\phi$. This anzats gives reasonable results only for harmonic potential but in all other cases this approximation is not satisfactory. A better approximation has been derived in refs. [@dolgov95; @dolgov99]. One starts with exact quantum operator equation of motion for the field $\phi$ and some other fields $\chi$ that are coupled to $\phi$. The production of the latter by oscillations of $\phi$ results in a damping term in the equation of motion for $\phi$. As an example let us consider a simple case of scalar $\chi$ with trilinear coupling $f\phi \chi^2$. The corresponding equations of motion have the form (expansion neglected for simplicity): -+ V\^[’]{}() &=& f \^2 , \[eqvp\]\
\^2 +m\^2\_ &=& 2 f . \[eqchi\] The next step is to make quantum averaging of these equations in the presence of classical field $\phi_c (t)$ (in what follows we omit sub-c and neglect the mass of $\chi$). This can be easily done in one-loop approximation (some subtleties related to renormalization of mass and coupling constants are discussed in ref. [@dolgov99]) and one comes to the equation that contains only the field $\phi$ and accounts for the backreaction from the production of the quanta of $\chi$: + V’() = \_0\^[t-t\_[in]{}]{} (t-) , \[scalres\] where $t_{in}$ is an initial time, when the particle production was switched on (it is assumed that $t>t_{in}$). The term in the r.h.s. that describes the influence of the particle production is non-local in time as one should have expected because the impact of the produced particle on the evolution of $\phi$ depends upon all the previous history. To use this equation for realistic calculations one has to make proper renormalization procedure. It is described in detail in ref. [@dolgov99]. The coupling to fermions as well as quartic coupling $\lambda' \phi^2 \chi^2$ are also considered in that paper. Similar one-loop approach was used in ref. [@baacke98] but no self-contained equation for $\phi$ was derived there.
Both effects of cosmological expansion and of damping of $\phi$ due to particle production can be easily incorporated into imaginary time method. This is especially simple in the case of fast oscillations and slow decrease of the amplitude of $\phi$. In this case the results obtained above practically do not change. One should only substitute there $\phi_0 (t)$ and to determine the law of the evolution of the latter from the energy balance equation (\[dotrho\]) or, more accurately, from eq. (\[scalres\]).
One more phenomenon deserves a comment here. As we have already mentioned, production of bosons may be strongly amplified due to the presence of the earlier produced bosons in the same final state. In classical language this effect is described by the parametric resonance in the equation of motion of the produced particles, while in quantum language it is the so called stimulated emission well known in laser physics. When the amplitude of the driving field $\phi$ drops below a certain value, the resonance would not be excited and the rest of $\phi$ would decay slowly. If the mass of $\phi$ is non-zero, this field would behave as non-relativistic matter and its cosmological energy density would drop as $1/a^3$. On the other hand, the produced particles are mostly relativistic with energy density decreasing as $1/a^4$. Thus for a sufficiently slow decay rate of $\phi$ the latter may dominate the cosmological energy density once again, when previously produced particles are red-shifted away. This would result in a low second reheating temperature, much lower than in parametric resonance scenario. On the other hand, the phenomenon of stimulated emission persists in perturbation theory even with a very small amplitude of $\phi$. Possibly even in this limit the production is not very fast as well, because the width of the band is quite narrow and the produced bosons are quickly pushed away from the band due to cosmological red-shift and collisions. More detailed consideration is desirable here.
Conclusion. \[s:concl\]
=======================
It is demonstrated that imaginary time method very well describes particle production by scalar field. It is very simple technically and permits to obtain physically transparent results. The calculations here were done for a particular case of periodic or quasiperiodic oscillations of the field but, as shows the experience with production of $e^+e^-$-pairs by electric field (for a review see e.g. third paper in ref.[@marinov72]), the method also works well in the opposite case of short pulse fields. The method is applicable in the quasiclassical limit. In the opposite case perturbation theory is applicable and hence one can obtain simple and accurate (semi)analytical estimates practically in all parameter range.
The results of calculations in the quasiclassical limit are in a good agreement with subsequent numerical ones[@baacke98; @green99]. An important difference between the latter papers and the initial one[@dolgov90] lays in the interpretation of the results. According to all these papers the occupation numbers of the produced particles quickly approaches unity but, in contrast to refs.[@baacke98; @green99], it is argued in the paper[@dolgov90] that the total production rate is nevertheless suppressed in comparison to perturbation theory and the production of fermions by the inflaton with Yukawa coupling to fermions is always weak. This conclusion is verified above. As is shown in this paper, the occupation numbers may quickly reach unity both in perturbation theory and in non-perturbative case. Still even the production rate of particles obeying Boltzmann statistics is very weak to ensure fast (pre,re)heating. In the case of fermion production the rate is evidently much weaker because the production must stop when the occupation number reaches unity and to continue the process the produced fermions should be eliminated from the band. As is argued in sec. \[ss:small\], the non-perturbative effects can only diminish the production rate.
The bosonic case is opposite: more bosons are in the final state, the faster is production. Thus even in perturbation regime the boson production can be strongly amplified because their occupation number may reach unity in much shorter time than $1/\Gamma$ and the energy may be transferred from the inflaton to the produced bosons much faster than is given by the original perturbative estimates[@dolgov82], where the effect of stimulated emission was not taken into account. Of course to realize this regime the band should not be destroyed by expansion and scattering, as argued in ref. [@dolgov90].
To summarize, we have shown that perturbation theory gives a good estimate of production of light fermions and bosons if Fermi exclusion principle or stimulated emission respectively are taken into account. The formally calculated production rate in perturbation theory is always larger than the non-perturbative one, at least in the simple cases that we have considered. So the results of perturbation theory may be used as upper bounds for production rates. Moreover, perturbation theory helps to understand physical meaning of the obtained results and to interpret them correctly.
In many realistic cases (e.g. for large $g\phi_0$ or $m_0$) perturbation theory is not applicable and to calculate the real production rate (not just an upper bound) one has to make more involved non-perturbative calculations. In quasiclassical (anti-perturbative) limit imaginary time method permits to obtain accurate and simple results and to avoid complicated numerical procedure
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to S. Hansen and A. Vainshtein for critical comments on the manuscript.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We find a counterpart of the classical fact that the regular representation $\mathfrak R(G)$ of a simple complex group $G$ is spanned by the matrix elements of all irreducible representations of $G$. Namely, the algebra of functions on the big cell $G_0 \subset G$ of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category $\mathcal O$ of representations of the Lie algebra $\g$ of $G$, and has the structure of a $\ggbar$-module.
We extend both regular representations to the affine group $\hat G$, and show that the loop form of the Bruhat decomposition of $\hat G$ yields modified versions of $\mathfrak R(\hat G)$. They involve pairings of positive and negative level modules, with the total value of the central charge required for the existence of non-trivial semi-infinite cohomology. In this paper we consider in detail the case $G=SL(2,\C)$, the corresponding finite-dimensional and affine Lie algebras, and the closely related to them Virasoro algebra.
Using the Fock space realization, we show that both types of modified regular representations for the affine and Virasoro algebras become vertex operator algebras, whereas the ordinary regular representations have instead the structure of conformal field theories. We identify the inherited algebra structure on the semi-infinite cohomology when the central charge is generic. We conjecture that for the integral values of the central charge the semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.
address:
- |
Department of Mathematics, Yale University\
New Haven, CT 06520, USA
- |
Max-Planck-Institut für Mathematik\
D-53111 Bonn, Germany
author:
- 'Igor B. Frenkel'
- Konstantin Styrkas
title: |
Modified regular representations\
of affine and Virasoro algebras, VOA structure\
and semi-infinite cohomology.
---
Introduction.
=============
The study of the regular representation of a simple complex Lie group $G$ is at the foundation of representation theory of $G$. Realized as the space of regular functions on $G$, the regular representation $\mathfrak R(G)$ carries two compatible structures of a $G$-bimodule and of a commutative associative algebra. An algebro-geometric version of the Peter-Weyl theorem establishes the decomposition of $\mathfrak R(G)$ into a direct sum of subspaces, spanned by matrix elements of all irreducible finite-dimensional representations $V_\lambda$ of $G$, indexed by integral dominant highest weights $\lambda \in \mathbf P^+$. In other words, we have an isomorphism of $G$-bimodules $$\label{eq:Peter-Weyl classical}
\mathfrak R(G) = \bigoplus_{\lambda \in \mathbf P^+} V_\lambda \o V_\lambda^*.$$ where $V_\lambda^*$ is the dual representation of $G$. The multiplication in $\mathfrak R(G)$ can be described in representation-theoretic terms as a pairing of intertwining operators for the left and right $\g$-actions with appropriate structural coefficients. Thus the algebra structure on $\mathfrak R(G)$ encodes the information about the tensor category of finite-dimensional $\g$-modules.
The representations of $G$ can also be viewed as modules over the simple complex Lie algebra $\g$ associated with $G$. In the case of the Lie algebra $\g$ it is natural to consider a larger collection of modules - namely, the Bernstein-Gelfand-Gelfand category $\mathcal O$. Infinite-dimensional $\g$-modules from the category $\mathcal O$ are not integrable, and therefore their matrix elements cannot be regarded as functions on $G$. However, one can interpret them as functions on the open dense subset $G^o \subset G$, given by the Gauss decomposition $$\label{eq:Gauss decomposition}
G^o = N_- \cdot T \cdot N_+,$$ where $N_\pm$ is the upper and lower triangular unipotent subgroup of $G$, and $T$ is the diagonal maximal abelian subgroup. The space $\mathfrak R(G^o)$ of regular functions on $G^o$ does not have the structure of a representation of $G$. Nevertheless, the left and right infinitesimal actions of the Lie algebra $\g$ on this space are well-defined, and can be expressed in terms of explicit differential operators in the parameters of the Gauss decomposition . The enlarged regular representation $\mathfrak R(G^o)$ decomposes into the direct sum of bimodules spanned by the matrix coefficients of all “big” projective $\g$-modules $P_\lambda$, indexed by strictly anti-dominant highest weights $\lambda \in - \mathbf P^{++} = - (\mathbf P^+ + \rho)$, where $\rho$ is the half-sum of all positive roots of $\g$. Thus we obtain an isomorphism of $\g$-bimodules $$\label{eq:Peter-Weyl twisted}
\mathfrak R(G^o) \cong \bigoplus_{\lambda \in -\mathbf P^{++}} \( P_\lambda \o P_\lambda^* \) \biggr/ I_\lambda,$$ where $P_\lambda^*$ is the module dual to $P_\lambda$, and $I_\lambda$ is the sub-bimodule of the matrix coefficients which vanish identically on the universal enveloping algebra $\mathcal U(\g)$.
It is important to note that the dual modules $P_\lambda^*$ do not belong to the category $\mathcal O$, but to its “mirror image”, in which all highest weight modules are replaces by lowest weight modules. In order to stay in the category $\mathcal O$ we replace the open subset $G^o$ coming from the Gauss decomposition by the maximal cell in the Bruhat decomposition $$\label{eq:Bruhat decomposition}
G_0 = N_+ \cdot \mathbf{w_0} \cdot T \cdot N_+,$$ where $\mathbf {w_0}$ is the longest element of the Weyl group $W$. Then we obtain a version of the isomorphism , $$\label{eq:Peter-Weyl projective}
\mathfrak R(G_0) \cong \bigoplus_{\lambda \in -\mathbf P^{++}} \( P_\lambda \o P_\lambda^\star \) \biggr/ I_\lambda,$$ where the ’twisted’ duals $P_\lambda^\star$ differs from $P_\lambda^*$ by the automorphism $\omega$ of $\g$, which is induced by $\mathbf w_0$ and interchanges the positive and negative roots.
The theorems of Peter-Weyl type and the Gauss decomposition can be extended to the central extension of the loop group $\hat G$ associated to $G$, and to the corresponding affine Lie algebra $\ghat$ and its universal enveloping algebra $\mathcal U(\ghat)$. In this infinite-dimensional case the space $\mathfrak R(\hat G)$ of regular functions on $\hat G$ is decomposed into the direct sum of subspaces $\mathfrak R_k(\hat G)$, corresponding to the value $k \in \mathbb Z$ of the central charge. Using the version of the Gauss decomposition known as the Birkhoff decomposition, one can show (see [@PS]) that for any $k \in \mathbb Z$ there is an isomorphism $$\label{eq:Peter-Weyl affine}
\mathfrak R_k(\hat G) \cong \bigoplus_{\lambda \in \mathbf P^k_+} \hat V_{\lambda,k} \o \hat V_{\lambda,k}^*,$$ where $\lambda$ runs over the truncated alcove $\mathbf P^+_k \subset \mathbf P^+$, depending on $k$, and $\hat V_{\lambda,k}$ are the corresponding irreducible modules. Similarly, one can obtain decompositions of $\mathfrak R_k(\hat G^o)$ analogous to , where $\hat G^o$ is the maximal cell in the Birkhoff decomposition. Viewing the decomposition in terms of the Lie algebra $\ghat$ allows to extend it for all values of $k$, with $\mathbf P^+_k = \mathbf P^+$ for $k \notin \mathbb Q$.
To transform the dual module $\hat V_{\lambda,k}^*$ into a module from the category $\mathcal O$ for $\ghat$, one might apply again an automorphism of $\ghat$ which interchanges the positive and negative affine roots. However, it no longer belongs to the affine Weyl group, and the Bruhat decomposition for $\hat G$ does not have a maximal cell. To overcome this obstacle we consider instead an intermediate between the Birkhoff and the affine Bruhat decompositions - the loop version of the finite-dimensional Bruhat decomposition, and the corresponding big cell $$\label{eq:loop Bruhat decomposition}
\hat G_0 = LN_+ \cdot \mathbf{w_0} \cdot \widehat{LT} \cdot LN_+,$$ where $LN_\pm$ denote the loop groups with values in $\n_\pm$, and $\widehat{LT}$ is the central extension of the loop group with values in $T$. The decomposition is especially useful for explicit realizations of the left and right regular $\ghat$-actions in terms of differential operators. However, we are still in “semi-infinite” distance from the category $\mathcal O$, and need to further apply a well-known procedure of “changing the vacuum”, which has originated from the free field realizations of the Wakimoto modules and the irreducible representations $\hat V_{\lambda,k}$ (see [@FeFr; @BF]). As a result of this procedure we obtain a modified affine version of the extended regular representation , $$\label{eq:Peter-Weyl affine twisted}
\mathfrak R'_k(\hat G_0) \cong \bigoplus_{\lambda \in -\mathbf P^{++}}
\( \hat P_{\lambda,k-h^\vee} \o \hat P_{\lambda,-k-h^\vee}^\star \) \biggr/ \hat I_{\lambda,k},$$ where $\hat P_{\lambda,k-h^\vee}$ and $\hat P^\star_{\lambda,-k-h^\vee}$ are the projective $\ghat$-modules and their ’twisted’ duals, $\hat I_{\lambda,k}$ are appropriate sub-bimodules, and we assume $k \notin\mathbb Q$. The levels are shifted by the dual Coxeter number $h^\vee$, so that the diagonal $\ghat$-action has the level $-2h^\vee$.
Like the Wakimoto modules, the bimodule $\mathfrak R'_k(\hat G_0)$ is realized as a certain Fock space, with two commuting $\ghat$-actions described explicitly. This realization is similar to the standard realization of the Wakimoto modules, but the actions of $\ghat$ contain a crucial new ingredient - the vertex operators, directly related to the screening operators used to construct intertwining operators for the affine Lie algebra. We also establish that for $k \notin \mathbb Q$ the structure of the socle filtration of the non-semisimple bimodule $\mathfrak R'_k(\hat G_0)$ is the same as in the finite-dimensional case. In particular, $\mathfrak R'_k(\hat G_0)$ contains the distinguished sub-bimodule $$\label{eq:positive Peter-Weyl}
\mathfrak R'_k(\hat G) \cong \bigoplus_{\lambda \in \mathbf P^{+}} \hat V_{\lambda,k-h^\vee} \o \hat V_{\lambda,-k-h^\vee}^\star.$$ The shifts of the central charge by the dual Coxeter number no longer allow the interpretation of the bimodules in and as spaces of matrix elements of $\ghat$-modules. Nevertheless, the structures of these bimodules are completely analogous to those of bimodules $\mathfrak R(G_0)$ and $\mathfrak R(G)$!
The vacuum module $\hat V_{0,k}$ of the affine Lie algebra $\ghat$ carries an extremely rich additional structure of a vertex operator algebra (VOA); other $\ghat$-modules $\hat V_{\lambda,k}$ become its representations (see [@FZ]). We show in this paper that the bimodules $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ also admit a vertex operator structure, compatible with $\ghat$-actions. In the proof we use the explicit Fock space realization of these bimodules. As in the finite-dimensional case, the VOA structure of the modified regular representations $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ encodes the information about fusion rules of the corresponding tensor categories of $\ghat$-modules. Thus besides the vacuum modules there is a class of vertex operator algebras associated to affine Lie algebras with fixed central charges. It is also related to the algebras of chiral differential operators over the simple algebraic group $G$ recently studied in [@GMS] and [@AG]. A study of this relation might help to understand the geometric nature of the modified regular representations.
On the other hand, the original regular representation $\mathfrak R_k(\hat G)$ does not seem to have a VOA structure. Instead it has the structure of a two-dimensional conformal field theory, which is an object of a different nature despite having local properties similar to those of a VOA. The bimodule $\mathfrak R_k(\hat G_0)$ has a structure of a generalized (non-semisimple!) conformal field theory.
It is well-known that the representation theory of $\ghat$ is closely related to the representation theory of the corresponding $\mathcal W$-algebra via the quantum Drinfeld-Sokolov reduction. In particular, one expects to have an analogue of the Peter-Weyl theorem for the $\mathcal W$-algebras. In this paper we consider in detail the simplest case of $\g = \sl(2,\C)$, when the corresponding $\mathcal W$-algebra is the infinite-dimensional Virasoro algebra. We give explicit realizations of the Virasoro bimodules, analogous to and , and equip them with compatible VOA structures. The structures of the non-semisimple modified regular representations are quite parallel in all cases; we fully describe their socle filtrations. Generalizations of our constructions to higher rank Lie algebras are straightforward, but their $\mathcal W$-algebra versions require more technicalities; full details will be presented in a subsequent paper.
A remarkable feature of all the modified bimodules that appear in the decompositions of Peter-Weyl type is that the central charge of the diagonal subalgebra is always equal to the special values that appear in the semi-infinite cohomology theory [@Fe; @FGZ] - namely, $-2h^\vee$ for the affine Lie algebras and 26 for Virasoro. Moreover, thanks to a general result of [@LZ], the corresponding semi-infinite cohomology spaces inherit a VOA structure from the modified regular representations. In our case, they degenerate into commutative associative superalgebras, and for generic central charge we establish isomorphisms between cohomology groups with coefficients in the corresponding modified regular representations of the affine and Virasoro algebras and their finite-dimensional counterparts. In particular, we show that the 0th semi-infinite cohomologies of the affine and Virasoro algebras are isomorphic to the Grothendieck ring of finite-dimensional representations of $G$. We conjecture that for integral $k$ they lead to the Verlinde algebra and its projective counterpart.
This paper is organized as follows. In Section 1 we consider the Bruhat decomposition and the Peter-Weyl theorems in the finite-dimensional case with $G=SL(2,\C)$. We give a Fock space realization of the algebra $\mathfrak R(G_0)$, and obtain explicit formulas for the $\g$-actions and decomposition theorems, which will later be used as prototypes of the infinite-dimensional case. In the last subsection we compute the Lie algebra cohomology with coefficients in $\mathfrak R(G)$ and $\mathfrak R(G_0)$. In Section 2 we study the affine case, and use the loop version of the finite-dimensional Bruhat decomposition to obtain the modified Peter-Weyl theorems for the spaces $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$. The Fock space realization of these spaces equips them with VOA structures compatible with the regular $\ghat$-actions. The semi-infinite cohomology of $\ghat$ with coefficients in the modified regular representations $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ for generic central charge is shown to be isomorphic to its finite-dimensional counterpart. In Section 3 we construct the analogues of the modified regular representations of the Virasoro algebra using the quantum Drinfeld-Sokolov reduction. We compute the corresponding semi-infinite cohomology groups using methods developed in string theory, and prove that they are isomorphic to their affine counterparts. Finally, in Section 4 we describe another class of vertex operator algebras obtained by the pairing of $\slhat$ and Virasoro modules. We also discuss generalizations of our results to Lie algebras of other types, and to the integral values of the central charge. We conclude with conjectures on relations of the semi-infinite cohomology of $\mathfrak R'_k(\hat G)$ and $\mathfrak R'_k(\hat G_0)$ for $k \in \Z_{>0}$ with the Verlinde algebra, its projective counterpart and twisted equivariant K-theory.
We wish to thank G. Zuckerman for sharing his expertise on semi-infinite cohomology, and S. Arkhipov, F. Malikov for valuable comments. I.B.F. is supported in part by NSF grant DMS-0070551.
Regular representation of $\sl(2,\C)$ on the big cell.
======================================================
Regular representations of $\sl(2,\C)$.
---------------------------------------
Let $G = SL(2,\C)$. We define the left and right regular actions of $G$ on the space $\mathfrak R(G)$ of regular functions on $G$ by $$\label{eq:group regular actions}
(\pi_l(g)\psi)(h) = \psi(g^{-1} h), \qquad
(\pi_r(g)\psi)(h) = \psi(h \, g), \qquad
g,h \in G.$$ The multiplication in $\mathfrak R(G)$ intertwines both left and right regular actions.
Let $T, N_+$ denote the diagonal and unipotent upper-triangular subgroups of $G$. The group $W = \operatorname{Norm}(T)/T$ is called the Weyl group. The Bruhat decomposition $G = N_+ \cdot W \cdot T \cdot N_+$ implies that every $g \in G$ can be factored as $g = n \cdot w \cdot t \cdot n'$ for some $n,n' \in N_+, t \in T, w \in W$.
We denote by $G_0$ the big cell of the Bruhat decomposition, corresponding to the longest Weyl group element ${\mathbf w_0}$. Explicitly, $G_0$ is the dense open subset of $G$, consisting of $g \in G,$ $$\label{eq:explicit Bruhat decomposition}
g =
\begin{pmatrix} 1 && x \\ 0 && 1 \end{pmatrix}
\begin{pmatrix} 0 && -1 \\ 1 && 0 \end{pmatrix}
\begin{pmatrix} \zeta && 0 \\ 0 && \zeta^{-1} \end{pmatrix}
\begin{pmatrix} 1 && y \\ 0 && 1 \end{pmatrix}$$ for some $x,y \in \C$ and $\zeta \in \C^\times$. The variables $x,y,\zeta$ can be viewed as coordinates on $G_0$, and thus the algebra $\mathfrak R(G_0)$ of regular functions on $G_0$ is identified with the space $\C[x,y,\zeta^{\pm1}]$.
Let $\g = \sl(2,\C)$ be the Lie algebra of $G$, with the standard basis $$\mathbf e = \begin{pmatrix} 0 && 1 \\ 0 && 0 \end{pmatrix}, \qquad
\mathbf h = \begin{pmatrix} 1 && 0 \\ 0 && -1 \end{pmatrix}, \qquad
\mathbf f = \begin{pmatrix} 0 && 0 \\ 1 && 0 \end{pmatrix},$$ satisfying the commutation relations $[\mathbf h, \mathbf e] = 2 \mathbf e, \
[\mathbf h, \mathbf f] = -2 \mathbf f, \
[\mathbf e, \mathbf f] = \mathbf h.$ The nilpotent subalgebras $\n_{\pm}$ and the Cartan subalgebra $\h$ of $\g$ are defined by $\n_+ = \C \mathbf e, \, \h = \C \mathbf h, \, \n_- = \C \mathbf f.$ The element $\mathbf {w_0} \in W$ determines a Lie algebra involution $\omega$ of $\g$, such that $\omega(\n_\pm) = \n_\mp$ and $\omega(\h) = \h$, defined by $$\label{eq:Cartan automorphism}
\omega(\mathbf e) = -\mathbf f, \qquad
\omega(\mathbf h) = -\mathbf h, \qquad
\omega(\mathbf f) = -\mathbf e.$$
The infinitesimal regular actions of $\g$ on $\mathfrak R(G)$, corresponding to , are given by $$\label{eq:algebra regular actions}
(\pi_l(x) \psi)(g) = \frac d{dt} \psi(e^{-t \, x} g) \biggr|_{t=0}, \qquad
(\pi_r(x) \psi)(g) = \frac d{dt} \psi(g \, e^{t \, x} ) \biggr|_{t=0}, \qquad x \in \g, \, g \in G.$$
These formulas also define left and right infinitesimal actions of $\g$ on the space $\mathfrak R(G_0)$. (These actions cannot be lifted to the group $G$, because $G_0$ is not invariant under left and right shifts). Elementary calculations yield the following explicit description of the regular $\g$-actions (cf. [@FP]).
\[thm:classical action\] The regular $\g$-actions on $\mathfrak R(G_0)$ are given by $$\begin{split} \label {eq:classical left action}
\pi_l(\mathbf e) & = -\pd x,\\
\pi_l(\mathbf h) & = \zeta \pd \zeta - 2 x \pd x,\\
\pi_l(\mathbf f) & = - x \zeta \pd \zeta + x^2 \pd x + \zeta^{-2} \pd y.
\end{split}$$ $$\begin{split} \label {eq:classical right action}
\pi_r(\mathbf e) & = \pd y ,\\
\pi_r(\mathbf h) & = \zeta \pd \zeta - 2 y \pd y,\\
\pi_r(\mathbf f) & = y \zeta \pd \zeta - y^2 \pd y - \zeta^{-2} \pd x.
\end{split}$$
Bosonic realizations
--------------------
We now reformulate the constructions of the previous section in terms of Fock modules for certain Heisenberg algebras. These realizations admit generalizations to the affine and Virasoro cases, where the geometric approach to the regular representations becomes more subtle.
The operators $\beta = x, \ \gamma = -\pd x$ acting on polynomials in $y$ give a representation of the Heisenberg algebra with generators $\beta,\gamma$ and relation $[\beta,\gamma] = 1$. The polynomial space $\C[y]$ is then identified with its irreducible representation $F(\beta,\gamma)$, generated by a vector $\1$ satisfying $\gamma \, \1 = 0$.
The operators $\bar\beta = -y, \ \bar\gamma = \pd y$ generate a second Heisenberg algebra, acting irreducibly in the space $F(\bar\beta,\bar\gamma) \cong \C[x]$. Here and everywhere else in this paper the ’bar’ notation is used to denote the second copies of algebras and their generators; it does not denote the complex conjugation.
We identify $\h^* \cong \C$ so that $\mathbf P \cong \Z$. Whenever possible, we use the more invariant notation in order to avoid possible numeric coincidences. The operators $\1_\lambda =\zeta^\lambda$ and $a = \zeta \pd \zeta$ gives rise to the semi-direct product $\C[a] \ltimes \C[\mathbf P]$, with $\C[a]$ acting on $\C[\mathbf P]$ by derivations: $a \1_\lambda = \lambda \1_\lambda$.
Thus, we get a realization of the algebra $\mathfrak R(G_0)$ of regular functions on $G_0$, with the $\ggbar$-action described by the abstract versions of the formulas , .
\[thm:classical bimodule action\] The space $\mathbb F= F(\beta,\gamma) \otimes F(\bar\beta,\bar\gamma) \otimes \C[\mathbf P]$ gives a realization of the algebra $\mathfrak R(G_0)$. In particular,
1. The space $\mathbb F$ has a $\ggbar$-module structure, given by $$\label{eq:classical boson left}
\begin{split}
\mathbf e & = \gamma, \\
\mathbf h & = 2 \, \beta \gamma + a, \\
\mathbf f & = -\beta^2 \gamma - \beta \, a + \bar \gamma \, \1_{-2},
\end{split}$$ $$\label{eq:classical boson right}
\begin{split}
\bar {\mathbf e} & = \bar\gamma, \\
\bar {\mathbf h} & = 2 \, \bar\beta \bar\gamma + a, \\
\bar {\mathbf f} & = -\bar\beta^2 \bar\gamma - \bar\beta \, a + \gamma \, \1_{-2}.
\end{split}$$
2. The space $\mathbb F$ has a compatible commutative algebra structure (i.e. the multiplication in $\mathbb F$ intertwines the $\ggbar$-action).
By specializing the action to the subspace $\ker \bar\gamma \subset \mathbb F$, we get the following well-known realizations of $\g$-action in the spaces $F_\lambda = F(\beta,\gamma) \o \C\1_\lambda$: $$\begin{split} \label{eq:classical boson}
\mathbf e & = \gamma, \\
\mathbf h & = 2\beta \gamma + \lambda,\\
\mathbf f & = -\beta^2 \gamma - \lambda \, \beta.
\end{split}$$
Simultaneous rescaling of the extra terms in ,, involving the shift $\1_{-2}$, by any multiple $\epsilon$ would preserve all the $\ggbar$ commutation relations. For $\epsilon = 0$ such $\ggbar$-action degenerates into the product of two standard $\g$-actions . However, the multiplication in this naïve bimodule loses much of its rich structure, and no longer encodes the information about the fusion rules in the tensor category of finite-dimensional $\g$-modules.
$\ggbar$-module structure of the modified regular representation
----------------------------------------------------------------
In this subsection we describe the socle filtration of the $\ggbar$-module $\mathbb F$.
For any $\lambda \in \h^*$, we denote by $V_\lambda$ the irreducible $\g$-module, generated by a highest weight vector $v_\lambda$ satisfying $\mathbf e \, v_\lambda = 0$ and $\mathbf h \, v_\lambda = \lambda \, v_\lambda$.
Recall that a $\g$-module $V$ is said to have a weight space decomposition, if $$V = \bigoplus_{\mu\in\h^*} V[\mu], \qquad V[\mu] = \left\{v \in V \, \bigr| \, \mathbf h \, v = \mu \, v \right\}.$$ The restricted dual space $V' = \bigoplus_{\mu\in\h^*} V[\mu]'$ can be equipped with a $\g$-action, defined by $$\<g \, v', v\> = - \< v', \, \omega(g) \, v\>, \qquad g \in \g, \, v \in V, \, v' \in V',$$ where $\omega$ is as in . We denote the resulting dual module $V^\star$.
We have an involution $\lambda \mapsto \lambda^\star$ of $\h^*$, determined by the condition $(V_\lambda)^\star \cong V_{\lambda^\star}$. This involution can also be defined by $\lambda^\star = - \mathbf{w_0} (\lambda)$, where $\mathbf {w_0}$ is the longest Weyl group element.
For $\g = \sl(2,\C)$, we have $\lambda^\star = \lambda$. However, we keep the notation $\lambda^\star$, to indicate how our constructions generalize to Lie algebras of higher rank, where the involution is nontrivial.
\[thm:classical bimodule structure\] There exists a filtration $$\label{eq:classical filtration}
0 \subset \mathbb F^{(0)} \subset \mathbb F^{(1)} \subset \mathbb F^{(2)} = \mathbb F$$ of $\ggbar$-submodules of $\mathbb F$, such that $$\begin{aligned}
\mathbb F^{(2)}/\mathbb F^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} V_{-\lambda-2} \o V^\star_{-\lambda-2} \label{eq:classical socle F2},\\
\mathbb F^{(1)}/\mathbb F^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( V_\lambda \o V^\star_{-\lambda-2} \oplus V_{-\lambda-2} \o V^\star_\lambda \) \label{eq:classical socle F1},\\
\mathbb F^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} V_\lambda \o V^\star_\lambda \label{eq:classical socle F0}.\end{aligned}$$
We introduce a filtration of $\ggbar$-submodules of $\mathbb F$ $$\label{eq:classical Fock filtration}
\dots \subset \mathbb F_{\le -2} \subset \mathbb F_{\le -1} \subset \mathbb F_{\le 0} \subset \mathbb F_{\le 1} \subset \mathbb F_{\le 2} \subset \dots,$$ satisfying $\bigcap_{\lambda\in\mathbf P} \ \mathbb F_{\le \lambda} = 0$ and $\bigcup_{\lambda\in\mathbf P} \ \mathbb F_{\le \lambda} = \mathbb F,$ where $$\mathbb F_{\le \lambda} = F(\beta,\gamma) \o F(\bar\beta,\bar\gamma) \o \bigoplus_{\mu \le \lambda} \C\1_\mu, \qquad \lambda \in \mathbf P.$$ It is clear that $\mathbb F_{\le \lambda} / \mathbb F_{<\lambda} \cong F_\lambda \o F_{\lambda^\star}$; moreover, for $\lambda <0$ we have $F_\lambda \cong V_\lambda$, and for $\lambda \ge 0$ there is a short exact sequence $0 \to V_\lambda\to F_\lambda \to V_{-\lambda-2}\to 0$. The linking principle for $\g$-modules implies that the successive quotients $\mathbb F_{\le \lambda} / \mathbb F_{<\lambda}$ and $\mathbb F_{\le \mu} / \mathbb F_{<\mu}$ of this filtration may be non-trivially linked only if $\mu = -\lambda-2$.
Thus we see that the $\ggbar$-module $\mathbb F$ splits into the direct sum of blocks $$\label{eq:classical double blocks}
\mathbb F = \mathbb F(-1) \oplus \bigoplus_{\lambda \in\mathbf P^+} \mathbb F(\lambda),$$ where $\mathbb F(-1) \cong V_{-1} \o V^\star_{-1}$, and $\mathbb F(\lambda)\cong
\(V_{-\lambda-2} \o V^\star_{-\lambda-2}\) + \(F_\lambda \o F_{\lambda^\star}\)$ for $\lambda \in \mathbf P^+$; another way to obtain the decomposition is by using the Casimir operator.
It remains to describe the structure of $\mathbb F(\lambda)$ for each $\lambda \in \mathbf P^+$. By construction, $\mathbb F(\lambda)$ can be included in a short exact sequence $0 \to V_{-\lambda-2} \o V^\star_{-\lambda-2} \to \mathbb F(\lambda) \to F_\lambda \o F_{\lambda^\star} \to 0.$ We conclude that there exists a filtration $0 \subset \mathbb F(\lambda)^{(0)} \subset \mathbb F(\lambda)^{(1)} \subset \mathbb F(\lambda)^{(2)} = \mathbb F(\lambda),$ such that $$\begin{aligned}
\mathbb F(\lambda)^{(2)}/\mathbb F(\lambda)^{(1)} &\cong
V_{-\lambda-2} \o V^\star_{-\lambda-2},\\
\mathbb F(\lambda)^{(1)}/\mathbb F(\lambda)^{(0)} &\cong
\( V_\lambda \o V^\star_{-\lambda-2} \) \oplus \( V_{-\lambda-2} \o V^\star_\lambda \) ,\\
\mathbb F(\lambda)^{(0)} &\cong
\( V_{-\lambda-2} \o V^\star_{-\lambda-2} \) + \( V_\lambda \o V^\star_\lambda \).\end{aligned}$$ In fact, the linking principle implies that the sum in $\mathbb F(\lambda)^{(0)}$ is direct: $$\mathbb F(\lambda)^{(0)} \cong
\( V_{-\lambda-2} \o V^\star_{-\lambda-2} \) \oplus \( V_\lambda \o V^\star_\lambda \).$$ Finally, we construct the filtration by setting $$\mathbb F^{(0)} = \mathbb F(-1) \oplus \bigoplus_{\lambda \in \mathbf P^+} \mathbb F(\lambda)^{(0)},
\qquad
\mathbb F^{(1)} = \mathbb F(-1) \oplus \bigoplus_{\lambda \in \mathbf P^+} \mathbb F(\lambda)^{(1)},$$ which obviously satisfies the required conditions , , .
For a Lie algebra $\g$ of higher rank, we will get a similar filtration of length $2\, l(\mathbf{w_0}) + 1$, and in addition to the regular blocks, corresponding to $\lambda \in \mathbf P^+$, and the most degenerate block $\mathbb F(-1)$, there will be all intermediate types.
The natural inclusion of algebras $\mathfrak R(G) \subset \mathfrak R(G_0)$ can be seen in the Fock space realizations.
\[thm:classical positive subalgebra\] There exists a subspace $\mathbf F \subset \mathbb F$ satisfying the following properties.
1. $\mathbf F$ is a subalgebra of $\mathbb F$, and is generated by the elements from the submodule $V_1 \o V_1^\star$, corresponding to the matrix elements of the canonical representation of $G$.
2. $\mathbf F$ is a $\ggbar$-submodule of $\mathbb F$, and is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$. We have $$\label{eq:classical Peter-Weyl}
\mathbf F = \bigoplus_{\lambda\in\mathbf P^+} \mathbf F(\lambda) \cong \bigoplus_{\lambda\in \mathbf P^+} V_\lambda \o V^\star_\lambda.$$
3. The space $\mathbf F$ is a realization of the algebra $\mathfrak R(G)$.
In the polynomial realization, the generators of $\mathbf F$ from $V_1 \o V_1^\star$ are identified with functions $$\psi_{11} = \zeta, \qquad \psi_{12} = x \zeta, \qquad \psi_{21} = y \zeta, \qquad \psi_{22} = x y \zeta + \zeta^{-1},$$ which satisfy the relation $\psi_{11} \psi_{22} - \psi_{12} \psi_{21} = 1$. This establishes a very direct connection with the space of regular functions on the group $G = SL(2,\C)$.
The generalized Peter-Weyl theorem
----------------------------------
In this section we interpret the space $\mathfrak R(G_0)$ of regular functions on $G_0$ and its Fock space realization $\mathbb F$ as the algebra of matrix elements of all modules from the category $\mathcal O$.
Recall that the Bernstein-Gelfand-Gelfand category $\mathcal O$ consists of all finitely generated, locally $\n_+$-nilpotent $\g$-modules. In particular, $V_\lambda\in\mathcal O$ for any $\lambda$. If $V \in \mathcal O$, then $V^\star \in \mathcal O$.
For any $\g$-module $V$ we define $\mathbb M(V)$ to be the subspace of $\mathcal U(\g)'$, spanned by functionals $$\label{eq:matrix element}
\phi_{v,v'}(x) = \<v', x \, v\>, \qquad v \in V, \, v' \in V', \, x \in \mathcal U(\g),$$ where $\<\cdot,\cdot\>$ stands for the natural pairing between $V$ and $V'.$ The functionals are called matrix elements of the representation $V.$
\[thm:matrix elements\] Introduce a $\ggbar$-module structure on the restricted dual $\mathcal U(\g)'$ by $$\label{eq:dual enveloping bimodule}
(\pi_l(g) \phi) (x) = \phi(x g), \qquad\quad (\pi_r(g) \phi) (x) = - \phi(\omega(g) x)$$ for any $\phi \in \mathcal U(\g)', \, g\in \g, \, x \in \mathcal U(\g)$. Then
1. For any $\g$-module $V$, the space $\mathbb M(V)$ is a $\ggbar$-submodule of $\mathcal U(\g)'$.
2. For any $\varphi \in \mathcal U(\g)',$ there exists a $\g$-module $V$, such that $\varphi \in \mathbb M(V).$ Moreover, if $\varphi$ is $\n_+ \oplus \n_+$-nilpotent, then $V$ can be chosen from the category $\mathcal O.$
To show that $\mathbb M(V)$ is invariant under the left action of $\g$, we compute $$(\pi_l(g) \phi_{v,v'}) (x) = \phi_{v,v'}(x g) = \<v', x g \, v\> = \phi_{g v,v'}(x),$$ for any $x \in \mathcal U(\g),\, g \in \g,\, v \in V, \, v' \in V'.$ This shows that $y \phi_{v,v'}\in \mathbb M(V).$ The invariance under the right action follows from the computation $$(\pi_r(g) \phi_{v,v'}) (x) = - \phi_{v,v'}(\omega(g) x) = - \<v', \omega(g) x \, v\> = \<y v', x \, v\> = \phi_{v, y v'}(x).$$
For the second part, assume $\varphi \in \mathcal U(\g)'$. Denote by $V$ the subspace of $\mathcal U(\g)'$, generated from $\varphi$ by the left action of $\g$. Let $\varphi'$ be the restriction to $V$ of the unit $1 \in \mathcal U(\g) = \mathcal U(\g)''$. Equivalently, $\varphi'$ is the linear functional on $V$, determined by $\<\varphi', \psi\> = \psi(1),$ for any $\psi \in V \subset \mathcal U(\g)'$. We claim that $\varphi = \phi_{\varphi,\varphi'} \in \mathbb M(V)$. Indeed, for any $x \in \mathcal U(\g)$ we have $$\phi_{\varphi,\varphi'} (x) = \<\varphi', x \varphi\> = (x\varphi)(1) = \varphi(x).$$
Finally, if $\varphi$ is left-$\n_+$-nilpotent, then $V$ is locally $\n_+$-nilpotent. Since $V$ is generated by a single element $\varphi,$ it belongs to category $\mathcal O.$ The right-$\n_+$-nilpotency condition guarantees that $\varphi'$ belongs to the [*restricted*]{} dual space $V'$.
The elements of the universal enveloping algebra $\mathcal U(\g)$ may be regarded as the differential operators, acting on $\mathfrak R(G)$. This gives an interpretation of the regular functions on $G$ (or even on $G_0$) as linear functionals on $\mathcal U(\g)$, and thus to identifications of the spaces $\mathfrak R(G)$ and $\mathfrak R(G_0)$ with certain subspaces of $\mathcal U(\g)'$. In the explicit realizations $\mathbf F$ and $\mathbb F$ this correspondence is constructed using the algebraic analogue of the “co-unit” element of the Hopf algebra $\mathfrak R(G)$ - the linear functional $\<\cdot\>: \mathbb F \to \C$, defined by $$\<\beta^m \bar\beta^n \1_\lambda\> = \delta_{m,0} \delta _{n,0}.$$
The linear map $\vartheta: \mathbb F \to \mathcal U(\g)'$, defined by $v \mapsto \vartheta_v$, $$\label{eq:map iota}
\vartheta_v(x) = \<\pi_l(x) v\>, \qquad v \in \mathbb F, \ x \in \mathcal U(\g).$$ is an injective $\ggbar$-homomorphism.
In terms of the polynomial realization, $\<\cdot\>$ corresponds to evaluating a function $\psi(x,y,\zeta) \in \mathfrak R(G_0)$ at the element $\mathbf{w_0}$: $\<\psi\> = \psi(0,0,1)$. This implies that for any $v \in \mathbb F$ $$\label{eq:classical contravariant functional}
\<\mathbf e v\> = -\<\bar{\mathbf f} v\>, \qquad
\<\mathbf h v\> = -\<\bar{\mathbf h} v\>, \qquad
\<\mathbf f v\> = -\<\bar{\mathbf e} v\>.$$ Therefore, for any $g \in \g$ and $x \in \mathcal U(\g)$ we have $$\vartheta_{g v}(x) = \<x \, g v\> = \vartheta_v(x g) = (\pi_l(g)\vartheta_v)(x),$$ $$\vartheta_{\bar g v}(x) = \<x \, \bar g v\> = \<\bar g \, x v\> = - \<\omega(g) x v\> =
- \vartheta_v(\omega(g) x) = (\pi_r(g)\vartheta_v)(x).$$ We conclude that the map $\vartheta$ is a $\ggbar$-homomorphism. To prove that it is injective, we need to show that for any nonzero $v \in \mathbb F$ there exists an element $x \in \ggbar$ such that $\<x v\> \ne 0.$
Since $\mathbb F$ is locally $\n_+$-nilpotent, we can pick $k\ge0$ such that $\mathbf e^k v \ne 0,$ but $\mathbf e^{k+1} v = 0.$ Replacing $v$ by $\mathbf e^k v,$ we see that it suffices consider the case of $v \ne 0$ such that $\mathbf e v = 0.$ Similarly, we may assume that $\bar{\mathbf e} v = 0.$
A vector $v$ satisfying $\mathbf e v = 0 = \bar{\mathbf e} v$ must have the form $v = \sum_{\lambda\in\mathbf P} c_\lambda \1_\lambda$ with only finitely many $c_\lambda \ne 0.$ Using the formula for the Vandermonde determinant and the fact that $$\<\mathbf h^m v\> = \sum_{\lambda\in\mathbf P} c_\lambda \lambda^m, \qquad m \ge 0,$$ we conclude that $\<\mathbf h^k v\> = 0$ for all $k\ge0$ if and only if all $c_\lambda$ vanish. Thus, $\theta_v = 0$ is equivalent to $v = 0$, which means that $\vartheta$ is an injection.
The following statement is an algebraic version of the classical Peter-Weyl theorem.
The space $\mathfrak R(G)$ of regular functions on $G$ is spanned by the matrix elements of finite-dimensional irreducible $\g$-modules, $$\mathfrak R(G) \cong \bigoplus_{\lambda\in \mathbf P^+} \mathbb M(V_\lambda).$$ The decomposition of $\mathfrak R(G)$ as a $\ggbar$-module is given by $$\mathfrak R(G) \cong \bigoplus_{\lambda\in \mathbf P^+} V_\lambda \o V^\star_\lambda.$$
The subspace of $\mathcal U(\g)'$, corresponding to $\mathfrak R(G)$, is invariantly characterized as the restricted Hopf dual $\mathcal U(\g)'_{Hopf} \subset \mathcal U(\g)'$, defined by $$\mathcal U(\g)'_{Hopf} = \{ \phi\in \mathcal U(\g)' \bigr|
\exists \text{ two-sided ideal } J\subset \mathcal U(\g) \text { such that } \phi(J)=0 \text{ and }\operatorname{codim} J < \infty \}.$$
The extended space $\mathfrak R(G_0)$ corresponds to a larger subalgebra of $\mathcal U(\g)'$, spanned by the matrix elements of all modules in the category $\mathcal O$.
Recall that the category $\mathcal O$ has enough projectives; we denote by $P_\lambda$ the indecomposable projective cover of the irreducible module $V_\lambda$. It is known that every indecomposable module in the category $\mathcal O$ with integral weights is isomorphic to a subfactor of the projective module, corresponding to some anti-dominant integral weight $\lambda$. In particular, this means that it suffices to consider the matrix elements of the big projective modules $\{P_\lambda\}_{\lambda<0}$.
The following result can be regarded as a non-semisimple generalization of the Peter-Weyl theorem.
\[thm:generalized Peter-Weyl classical\] The space $\mathfrak R(G_0)$ of regular functions on $G_0$ is spanned by the matrix elements of all big projective modules in the category $\mathcal O$, $$\mathfrak R(G_0) \cong \bigoplus_{\lambda\in \mathbf P^+} \mathbb M(P_\lambda).$$ As a $\ggbar$-module, $\mathfrak R(G_0)$ is given by $$\mathfrak R(G_0) \cong \bigoplus_{\lambda\in-\mathbf P^{++}} \( P_{\lambda} \o P^\star_{\lambda} \) \biggr/ I_\lambda$$ where $I_\lambda$’s are the $\ggbar$-submodules of $P_{\lambda} \o P^\star_{\lambda}$, corresponding to identically vanishing matrix elements.
We use the realization of $\mathfrak R(G_0)$ in the Fock space $\mathbb F$. The inclusion provides the identification of $\mathbb F$ with a subspace of $\mathcal U(\g)'$. Since $\mathbb F$ is locally $\n_+ \oplus \n_+$-nilpotent, Proposition \[thm:matrix elements\] implies that for any $v \in \mathbb F$ there exists a $\g$-module $W \in \mathcal O$ such that $\vartheta_v \in \mathbb M(W).$
Let $W = W_1 \oplus W_2 \oplus \dots \oplus W_m$ be the decomposition of $W$ into a direct sum of indecomposable submodules. Each indecomposable component $W_i, \, i=1,\dots,m,$ is a subfactor of some big projective module $P_{\lambda_i}$. Then $\mathbb M(W_i) \subset \mathbb M(P_{\lambda_i})$, and therefore we have $$\mathbb M(W) = \mathbb M(W_1) + \mathbb M(W_2) + \dots + \mathbb M(W_m) \subset \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda),$$ which shows that $\vartheta(\mathbb F) \subset \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda).$ To prove that in fact $\vartheta(\mathbb F) = \bigoplus_{\lambda\in -\mathbf P^{++}} \mathbb M(P_\lambda)$, we compare the characters of the two spaces, and show that they have the same size.
For any $\lambda \in\mathbf P^+$ the $\ggbar$-module $\mathbb M(P_{-\lambda-2})$ is isomorphic to the quotient of the product $P_{-\lambda-2} \o P_{-\lambda-2}^\star$ by the kernel of the map $$\label{eq:matrix elements map}
\Theta_\lambda: P_{-\lambda-2} \o P_{-\lambda-2}^\star \to \mathcal U(\g)', \qquad \Theta_\lambda(v \o v') = \phi_{v,v'}.$$ Obviously, $I_\lambda = \ker \Theta_\lambda$ is a $\ggbar$-submodule of $P_{-\lambda-2} \o P_{-\lambda-2}^\star$; we describe it more explicitly. It is known that the module $P_{-\lambda-2}$ has a filtration $0 \subset P^{(0)} \subset P^{(1)} \subset P_{-\lambda-2}$ such that $$P^{(0)} \cong V_{-\lambda-2}, \qquad P^{(1)}/P^{(0)} \cong V_\lambda, \qquad P_{-\lambda-2}/P^{(1)} \cong V_{-\lambda-2},$$ and the dual filtration of the module $P_{-\lambda-2}^\star$ is given by $$0 \subset \Ann(P^{(1)}) \subset \Ann(P^{(0)}) \subset P_{-\lambda-2}^\star.$$ They determine a filtration of the tensor product $$\begin{split}
0 \subset P^{(0)} \o \Ann(P^{(1)}) \subset P^{(0)} \o \Ann(P^{(0)}) + P^{(1)} \o \Ann(P^{(1)}) \subset\\
\subset P^{(0)} \o P_{-\lambda-2}^\star + P^{(1)} \o \Ann(P^{(0)}) +
P_{-\lambda-2}\o \Ann(P^{(1)}) \subset\\
\subset P^{(1)} \o P_{-\lambda-2}^\star + P_{-\lambda-2} \o \Ann(P^{(0)})
\subset P_{-\lambda-2} \o P_{-\lambda-2}^\star.
\end{split}$$ If $v\in P^{(0)}$ and $v' \in \Ann(P^{(0)})$, then $\phi_{v,v'}$ is the zero functional, since for any $x \in \mathcal U(\g)$ we have $x\, v \in P^{(0)}$ and $\phi_{v,v'}(x) = \<v', x \, v\> = 0$. Hence the submodule $P^{(0)} \o \Ann(P^{(0)})$ lies in the kernel of the map $\Theta_\lambda$, and similarly does $P^{(1)} \o \Ann(P^{(1)})$. One can easily see that $$\Theta_\lambda \( P^{(1)} \o \Ann(P^{(0)}) \) = \mathbb M(V_\lambda),$$ $$\Theta_\lambda \( P^{(0)} \o P_{-\lambda-2}^\star \) = \Theta_\lambda \( P_{-\lambda-2}\o \Ann(P^{(1)})\) = \mathbb M(V_{-\lambda-2}).$$
It follows that the $\ggbar$-module $\mathbb M(P_{-\lambda-2})$ has a filtration $$0 \subset \mathbb M^{(0)} \subset \mathbb M^{(1)} \subset \mathbb M^{(2)} = \mathbb M(P_{-\lambda-2})$$ such that $$\begin{aligned}
\mathbb M^{(2)}/\mathbb M^{(1)} & \cong V_{-\lambda-2} \o V^\star_{-\lambda-2},\\
\mathbb M^{(1)}/\mathbb M^{(0)} & \cong ( V_\lambda \o V^\star_{-\lambda-2} ) \oplus ( V_{-\lambda-2} \o V^\star_\lambda ),\\
\mathbb M^{(0)} & \cong ( V_\lambda \o V^\star_\lambda ) \oplus ( V_{-\lambda-2} \o V^\star_{-\lambda-2} ).\end{aligned}$$
Thus, the block $\mathbb F(\lambda)$ of is identified with the subspace, spanned by the matrix elements of the big projective module $P_{-\lambda-2}$. Taking direct sums over all $\lambda \in \mathbf P^+$, adding the $\ggbar$-module $\mathbb M(P_{-1}) \cong V_{-1} \o V^\star_{-1}$, and comparing with Theorem \[thm:classical bimodule structure\], we see that $\bigoplus_{\lambda\in - \mathbf P^{++}} \mathbb M(P_\lambda)$ and $\mathbb F$ have the same characters. The statement of the theorem follows.
Cohomology of $\g$ with coefficients in regular representations
---------------------------------------------------------------
The algebra $\mathfrak R(G)$ contains the subalgebra $\mathfrak R(G)^G$ of the conjugation-invariant functions on $G$, which is linearly spanned by the characters of the irreducible finite-dimensional representations. The subalgebra $\mathfrak R(G)^G$ is thus isomorphic to the Grothendieck ring of the finite-dimensional representations of $G$.
There is an isomorphism $\mathfrak R(G)^G \cong \C[\mathbf P]^W$, obtained by restricting the group characters to $\h$ and taking its Fourier expansion. Finally, the algebra $\mathfrak R(G)^G$ also admits a cohomological interpretation, which will be instrumental for further generalizations to the regular representations of the affine and Virasoro algebras. We briefly recall the definition of the cohomology of $\g$.
Let $\boldsymbol\Lambda = \bigwedge \g'$ be the exterior algebra of $\g'$ with unit $\1$. Then
1. The Clifford algebra, generated by $\{\iota(g), \, \eps(g')\}_{g \in \g, g' \in \g'}$ with relations $$\label{eq:Clifford relations}
\{\iota(x), \iota(y) \} = \{ \eps(x'), \eps(y') \} = 0, \qquad
\{\iota(x), \eps(y')\} = \<y',x\>,$$ acts irreducibly on $\boldsymbol\Lambda$, so that for any $\omega \in \boldsymbol\Lambda$ we have $$\iota(g) \1 = 0, \qquad \eps(g') \omega = g'\wedge\omega, \qquad g \in \g,\, g' \in \g', \, \omega \in \boldsymbol \Lambda.$$
2. $\boldsymbol\Lambda$ is a commutative superalgebra, $$\omega_1 \wedge \omega_2 = (-1)^{|\omega_1| \cdot |\omega_2|}\, \omega_2 \wedge \omega_1, \qquad \omega_1,\omega_2 \in \boldsymbol\Lambda,$$ where $|\cdot|$ is the natural grading on $\boldsymbol \Lambda$ satisfying $|\1| = 0, \ |\iota(g)| = -1, \ |\eps(g')| = 1.$
3. The $\g$-module structure on $\boldsymbol\Lambda$ is given by $$\pi_{\boldsymbol\Lambda}(x) = \sum_i \eps(g'_i) \iota([g_i,x]),$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_j\}$ is the corresponding dual basis of $\g'$.
The cohomology $H^\bullet(\g;V)$ of $\g$ with coefficients in a $\g$-module $V$ is the cohomology of the graded complex $C^\bullet(\g;V) = \boldsymbol\Lambda^\bullet \otimes V$, with the differential $$\label{eq:finite differential}
\mathbf d =\sum_i \eps(g'_i) \pi_V(g_i) - \frac12 \sum_{i,j} \eps(g'_i) \eps(g'_j) \iota([g_i,g_j]),$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_i\}$ is the dual basis of $\g'$.
The following is one of the fundamental results in Lie algebra cohomology, (see e.g. [@HS]).
\[thm:vanishing classical cohomology\] For any finite-dimensional $\g$-module $V$ we have $$\label{eq:de Rham}
H^\bullet(\g; V) \cong V^\g \o H_{DR}^\bullet(G),$$ where $H_{DR}^\bullet(G)$ denotes the holomorphic de Rham cohomology $H_{DR}^\bullet(G)$ of the Lie group $G$.
If $V$ is a commutative algebra with a compatible $\g$-action, then its cohomology inherits the multiplication from $V$ and $\boldsymbol\Lambda$, and $H^\bullet(\g;V)$ becomes itself a commutative superalgebra. Moreover, the isomorphism becomes an isomorphism of superalgebras, with respect to the cup product in $H_{DR}^\bullet(G)$.
The diagonal $\g$-action in $\mathbf F$ corresponds to the coadjoint action of $G$ in $\mathfrak R(G)$; thus, we get
There is an isomorphism of commutative superalgebras $$H^\bullet(\g;\mathbf F) = \C[\mathbf P]^W \o H_{DR}^\bullet(G).$$
Our next goal is to study the cohomology of $\g$ with coefficients in the extended regular representation $\mathbb F \cong \mathfrak R(G_0)$. For infinite-dimensional $\g$-modules Theorem \[thm:vanishing classical cohomology\] does not hold, and the cohomology $H^\bullet(\g;\mathbb F)$ does not reduce to $\mathbb F^\g \o H^\bullet_{DR}(G)$. We have instead
\[thm:classical cohomology\] There is an isomorphism of commutative superalgebras $$H^\bullet(\g;\mathbb F) \cong \C[\mathbf P]^W \o \sideset{}{^\bullet}\bigwedge \C^2.$$
It is easy to show using the results of [@W] that for $\lambda \ge -1$ $$H^n(\g;V_\lambda \o V^\star_{-\lambda-2}) = H^n(\g;V_{-\lambda-2} \o V^\star_\lambda) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases}$$ and that for $\lambda \ge0$ we have $H^n(\g;V_{-\lambda-2} \o V^\star_{-\lambda-2}) = 0 $ for all $n$. The spectral sequence associated with the filtration of Theorem \[thm:classical bimodule structure\] can be used to show that $$\label{eq:classical big cohomology}
H^n(\g;\mathbb F(-1)) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases},\qquad
H^n(\g;\mathbb F(\lambda)) =
\begin{cases}
\C, & n=0,2\\
\C^2, & n=1\\
0, & \text{otherwise}
\end{cases}, \quad \lambda \ge0.$$ Also, this spectral sequence shows that we have a natural isomorphism $H^0(\g;\mathbb F) \cong H^0(\g;\mathbf F).$
To explicitly get the generators of the commutative superalgebra $H^\bullet(\g;\mathbb F)$, we pick nonzero elements $$\chi \in H^0(\g;\mathbb F(1)), \qquad
\xi_{-1} \in H^1(\g;\mathbb F(-1)), \qquad
\eta_0 \in H^1(\g;\mathbb F(0)),$$ such that $\eta_0$ is not proportional to $\chi \, \xi_{-1}$. It is known that $H^0(\g;\mathbb F) \cong \C[\mathbf P]^W$ is isomorphic to the polynomial algebra $\C[\chi]$. It is also clear that $H^\bullet(\g;\mathbb F)$ is a free $\C[\chi]$-module. For each $\lambda\ge0$, the set $$B_{\le \lambda} = \{\xi_{-1}, \chi \xi_{-1}, \dots, \chi^{\lambda+1} \, \xi_{-1}\} \bigcup \{\eta_0,\chi \, \eta_0, \dots, \chi^\lambda \, \eta_0\}$$ consists of $2\lambda+3$ linearly independent elements, and in view of is a basis of $H^1(\g;\mathbb F_{\le \lambda})$. Finally, one can check that $\eta_0 \, \xi_{-1} \ne 0$, and thus the elements $\{\eta_0 \, \xi_{-1}, \chi \, \eta_0 \, \xi_{-1},\dots, \chi^{\lambda+1}\, \eta_0 \, \xi_{-1}\}$ give a basis of $H^2(\g;\mathbb F_{\le\lambda})$ for each $\lambda \ge -1$.
It follows that $H^\bullet(\g;\mathbb F) \cong \C[\chi] \otimes \bigwedge^\bullet[\xi_{-1},\eta_0]$, and the theorem is proven.
One of the ingredients in the exterior algebra part of the cohomology $H^\bullet(\g;\mathbb F)$ is the exterior algebra $\bigwedge^\bullet \h$, corresponding to $\bigwedge^\bullet[\eta_0]$ above. It would be interesting to obtain an invariant characterization of the remaining part of $H^\bullet(\g;\mathbb F)$ for arbitrary $\g$.
In each of the two-dimensional spaces $H^1(\g;\mathbb F(\lambda))$ there is a unique up to proportionality cohomology class $\xi_\lambda$ divisible by $\xi_{-1}$; the elements $\frac {\xi_\lambda}{\xi_{-1}}$ constitute a basis of $H^0(\g;\mathbb F) \cong \C[\mathbf P]^W$, associated with the characters of big projective modules (cf. [@La]).
Modified regular representations of the affine Lie algebra $\slhat$.
====================================================================
Regular representations of $\slhat$
-----------------------------------
Let $\hat G$ be the central extension of the loop group $LG$, associated with $G=SL(2,\C)$ (see [@PS]), and let $\ghat$ be the corresponding Lie algebra. As we discussed in the introduction, there is no maximal cell in the affine Bruhat decomposition, and thus we will use the loop version of the finite-dimensional one. An additional advantage is that we get an explicit realization of the left and right regular $\ghat$-actions, analogous to the finite-dimensional case.
The standard basis of $\ghat$ consists of the elements $\{\mathbf e_n, \mathbf h_n, \mathbf f_n\}_{n \in \Z}$ and the central element $\mathbf k$, subject to the commutation relations $$\begin{gathered}
[\mathbf h_m, \mathbf e_n] = 2 \mathbf e_{m+n}, \qquad
[\mathbf h_m, \mathbf f_n] = -2 \mathbf f_{m+n}, \qquad
[\mathbf h_m, \mathbf h_n] = 2 m \delta_{m+n,0} \mathbf k,\\
[\mathbf e_m, \mathbf f_n] = \mathbf h_{m+n} + m \, \delta_{m+n,0} \mathbf k, \qquad
[\mathbf e_m, \mathbf e_n] = [\mathbf f_m, \mathbf f_n] = 0.\end{gathered}$$
The Lie algebra $\ghat$ has a $\Z$-grading $\ghat = \bigoplus_{n \in \Z} \ghat[n]$, determined by $$\deg \mathbf f_n = \deg \mathbf h_n = \deg \mathbf e_n = -n, \qquad \deg \mathbf k = 0,$$ We introduce subalgebras $\ghat_\pm = \bigoplus_{\pm n>0} \g[n]$; the finite-dimensional Lie algebra $\g$ is naturally identified with a subalgebra in $\ghat[0]$.
The element $\mathbf {w_0}$ of the classical Weyl group defines an involution $\hat\omega$ of $\ghat$, such that $$\label{eq:affine Cartan automorphism}
\hat\omega(\mathbf e_n) = - \mathbf f_n, \quad
\hat\omega(\mathbf h_n) = - \mathbf h_n, \quad
\hat\omega(\mathbf f_n) = - \mathbf e_n, \quad
\hat\omega(\mathbf k) = - \mathbf k.$$
We use the loop version of the finite-dimensional Bruhat decomposition , and factorize the central extension $\widehat{LT}$ into the product of loops that extend holomorphically inside and outside of the unit circle. The analogue of is the formal decomposition $$g = \exp \( \sum_{n\in\Z} x_n \mathbf e_n \) \ \mathbf{w_0}\, \tau^\mathbf k \
\exp \( \sum_{m < 0} \zeta_m \mathbf h_m \) \zeta^{\mathbf h_0} \exp \( \sum_{m > 0} \zeta_m \mathbf h_m \) \exp \( \sum_{n\in\Z} y_n \mathbf e_n \).$$ The polynomial algebra $\mathfrak R_0(\hat G_0) = \C[\{x_n\}, \{y_n\}, \{\zeta_{n\ne0}\},\zeta^{\pm1}]$ can be thought of as the algebra of regular functions on the big cell of the loop group, and for $\mathfrak R(\hat G_0)$ we get $$\mathfrak R(\hat G_0) = \mathfrak R_0(\hat G_0) \o \C[\tau^{\pm1}] = \bigoplus_{\varkappa \in \Z} \mathfrak R_\varkappa(\hat G_0), \qquad
\mathfrak R_\varkappa(\hat G_0) = \mathfrak R_0(\hat G_0) \o \C\tau^\varkappa$$
Note that for each $\varkappa$ the subspace $\mathfrak R_\varkappa(\hat G_0)$ is a $\ghatghat$-submodule of $\mathfrak R(\hat G_0)$, but it is not a subalgebra of $\mathfrak R(\hat G_0)$ when $\varkappa \ne 0$ ! It is easy to see that the infinitesimal regular $\ghat$-actions of the central element $\mathbf k$ on $\mathfrak R_\varkappa(\hat G_0)$ are given by $$\label{eq:unmodified central charges}
\pi_l(\mathbf k) = - \varkappa \cdot \Id, \qquad\qquad \pi_r(\mathbf k) = \varkappa \cdot \Id.$$ As vector spaces, all $\mathfrak R_\varkappa(\hat G_0)$ are identified with the same polynomial space, and one can compute the infinitesimal regular actions of $\ghat$ by treating $\varkappa$ as a complex parameter. In particular, the regular actions of $\ghat$ make sense for arbitrary $\varkappa \in \C$. Computations yield the following description, analogous to Proposition \[thm:classical action\].
\[thm:affine action\] The regular actions of $\ghat$ on $\mathfrak R_\varkappa(\hat G_0)$ are given by and $$\label{eq:left affine action}
\begin{split}
\pi_l(\mathbf e_n) &= -\pd{x_n},\\
\pi_l(\mathbf h_n) &= - 2\sum_{i\in\Z} x_i \pd{i_{n+n}} +
\begin{cases}
\pd{\zeta_n} + 2n \varkappa\, \zeta_{-n}, & n>0\\
\zeta \, \pd {\zeta}, & n=0\\
\pd{\zeta_n}, & n<0
\end{cases} ,\\
\pi_l(\mathbf f_n) &= \sum_{i,i' \in \Z} x_i x_{i'} \pd{x_{i+i'+n}} -
\sum_{j<0} x_{j-n}\pd{\zeta_j} - x_{-n} \zeta\, \pd {\zeta} - \sum_{j>0} x_{j-n} \(\pd{\zeta_j} + 2 j \varkappa\, \zeta_{-j} \) - \\
&- \varkappa \; n x_{-n} + \zeta^{-2} \sum_{j,j'>0} \mathrm s_{j'}(-2\zeta_1,-2\zeta_2,\dots) \, \mathrm s_j(-2\zeta_{-1},-2\zeta_{-2},\dots) \pd{y_{n-j+j'}},
\end{split}$$
$$\label{eq:right affine action}
\begin{split}
\pi_r(\mathbf e_n) &= \pd{y_n},\\
\pi_r(\mathbf h_n) &= -2\sum_{i\in\Z} y_i \pd{y_{i+n}} +
\begin{cases} \pd{\zeta_n} & n>0, \\ \zeta \, \pd {\zeta}, & n=0,\\ \pd{\zeta_n} - 2 n \varkappa\, \zeta_{-n}, & n<0. \end{cases} ,\\
\pi_r(\mathbf f_n) &= - \sum_{i,i' \in \Z} y_i y_{i'} \pd{y_{i+i'+n}} +
\sum_{j>0} y_{j-n}\pd{\zeta_j} + y_{-n} \zeta\, \pd {\zeta} + \sum_{j<0} y_{j-n} \(\pd{\zeta_j} - 2 j \varkappa\, \zeta_{-j} \) - \\
&- \varkappa \, n y_{-n} - \zeta^{-2} \sum_{j,j'>0} \mathrm s_{j'}(-2\zeta_{-1},-2\zeta_{-2},\dots) \, \mathrm s_j(-2\zeta_1,-2\zeta_2,\dots) \pd{x_{n+j-j'}},\\
\end{split}$$
where the Schur polynomials $\mathrm s_k(\a_1,\a_2,\dots)$ are defined by $$\mathrm s_m(\a_1,\a_2,\dots) = \sum_{\substack {l_1,l_2, \ldots \ge 0\\ l_1+2l_2 + \ldots = m}}
\frac {\a_1^{l_1} \a_2^{l_2} \dots}{l_1! l_2!\dots}.$$
The presence of the central extension requires the use of some elementary cases of the Campbell-Hausdorff formula in our computations; we use the identity $$\exp(B) \exp(tA) \equivt \exp \(t \sum_{j=0}^\infty \frac 1{j!} \underbrace{[B,\dots,[B,[B,A]]\dots]}_{j \text{ commutators}}\) \exp(B).$$ For example, to derive the last of , we use the formulas: $$\begin{split}
\exp \( \sum_{i\in\Z} y_i \mathbf e_i \) \exp \( t \mathbf f_n \) \equivt &
\exp \(t \mathbf f_n \)
\exp \(- t n y_{-n}\mathbf k + t \sum_{i\in\Z} y_i \mathbf h_{i+n} \) \times \\
&\times \exp \(-t \sum_{i,i'\in\Z} y_i y_{i'} \mathbf e_{i+i'+n} \)
\exp \( \sum_{i\in\Z} y_i \mathbf e_i \),\\
\exp \(\sum_{m>0} \zeta_m \mathbf h_m \) \exp \( t \mathbf f_n \) \equivt &
\exp \( t \sum_{j>0} \mathrm s_j(-2\zeta_1,-2\zeta_2,\dots) \mathbf f_{n+j} \)
\exp \( \sum_{m>0} \zeta_m \mathbf h_m \),\\
\zeta^{\mathbf h_0} \exp \( t \mathbf f_n \) \equivt &
\exp \( t \zeta^{-2}\, \mathbf f_n \) \zeta^{\mathbf h_0},\\
\exp \(\sum_{m<0} \zeta_m \mathbf h_m \) \exp \(t \mathbf f_n \) \equivt &
\exp \(t \sum_{j'>0} \mathrm s_{j'}(-2\zeta_{-1},-2\zeta_{-2},\dots) \mathbf f_{n-j'} \)
\exp \( \sum_{m<0} \zeta_m \mathbf h_m \),\\
\mathbf {w_0} \exp \( t \mathbf f_n \) \equivt &
\exp \(- t \, \mathbf e_n \) \mathbf {w_0}.
\end{split}$$ Combining these equations, we get the desired formulas. We leave the technical calculations to the reader.
Vertex operator algebras: review and useful examples
----------------------------------------------------
We aim to endow $\mathfrak R_\varkappa(\hat G_0)$ (or its modification) with a structure similar to that of an associative commutative algebra. The relevant formalism is provided by the vertex algebra theory.
We recall the definitions of vertex and vertex operator algebras in the most convenient to us form. For more details and equivalent alternative definitions, we refer the reader to the books on the subject [@FLM; @BFr].
Let $\mathcal V$ be a vector space, equipped with a linear correspondence $$\label{eq:state field correspondence}
v \mapsto \mathcal Y(v,z) = \sum_{n \in \Z} v_{(n)} z^{-n-1}, \qquad v_{(n)} \in \End(\mathcal V).$$ We refer to such formal $\End(\mathcal V)$-valued generating functions as ’quantum fields’.
We say that $\mathcal V$ satisfies the locality property, if for any $a,b \in \mathcal V$ $$\label{eq:locality}
(z-w)^N [\mathcal Y(a,z),\mathcal Y(b,w)] = 0 \quad \text{ for } N \gg 0$$ in the ring of $\End(\mathcal V)$-valued formal Laurent series in two variables $z,w$.
A vector $\1 \in \mathcal V$ is called the vacuum vector, if it satisfies $$\label{eq:vacuum}
\mathcal Y(\1,z) = \Id_{\mathcal V}, \qquad \mathcal Y(v,z)\1 \bigr|_{z=0} = v.$$
An element $\mathcal D \in \End(\mathcal V)$, is called the infinitesimal translation operator, if it satisfies $$\label{eq:infinitesimal translation}
\qquad \mathcal D \, \1 = 0, \qquad\qquad [\mathcal D,\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z), \qquad \text{ for all } \ v \in \mathcal V.$$
The space $\mathcal V$ is called a vertex algebra, if it is equipped with a linear map , vacuum vector $\1$, and infinitesimal translation operator $\mathcal D$, satisfying the axioms , , above.
Vertex superalgebras are defined as usual by inserting $\pm$ signs according to parity. A vertex superalgebra $\mathcal V$ is called bi-graded, if it has $\Z$-gradings, $|\cdot|$ and $\deg$, $$\mathcal V = \bigoplus_{m,n\in\Z} \mathcal V^m[n], \qquad
\mathcal V^m[n] = \left\{ v \in\mathcal V \, \biggr| \, |v| = m \text{ and } \deg v = n \right\},$$ such that the parity in superalgebra is determined by $|\cdot|$, and for any homogeneous $v$ $$v \mapsto \mathcal Y(v,z) = \sum_{n\in\Z} v_{(n)} z^{-n-1}, \qquad \text { with } |v_{(n)}| = |v| \text{ and } \deg v_{(n)} = \deg v - n - 1 .$$ In particular, for the vacuum we must have $| \1 | = \deg \1 = 0$. Also, we write $|\mathcal Y(v,z)| = |v|$ and $\deg \mathcal Y(v,z) = \deg v$ for the quantum field $\mathcal Y(v,z)$, if the above conditions are satisfied.
A vertex algebra $\mathcal V$ is called a vertex operator algebra (VOA) of rank $c \in \C$, if there exists an element $\boldsymbol\omega \in \mathcal V$, usually called the Virasoro element, such that the operators $\{\mathcal L_n\}_{n \in \Z}$ defined by $$\mathcal Y(\boldsymbol\omega,z) = \sum_{n \in \Z} \mathcal L_n z^{-n-2},$$ satisfy $\mathcal L_{-1} = \mathcal D$, and the Virasoro commutation relations $$[\mathcal L_m,\mathcal L_n] = (m-n) \mathcal L_{m+n} + \delta_{m+n,0} \frac{m^3-m}{12}\, c.$$
We define the the normal ordered product of two quantum fields $X(z)$ and $Y(z)$ by $$:X(z)Y(w): = X_-(z)Y(w) + Y(w) X_+(z),$$ where $X_\pm(z)$ are the regular and principal parts of $X(z) = \sum_{n \in \Z} X_{(n)} z^{-n-1}$, $$X_+(z) = \sum_{n\ge0} X_{(n)} z^{-n-1}, \qquad X_-(z) = \sum_{n < 0} X_{(n)} z^{-n-1}.$$ For products of three or more quantum fields, the normal ordered product is defined inductively, starting from the left. In general, the normal ordered product is neither commutative nor associative.
The following ’reconstruction theorem’ is an effective tool for constructing vertex algebras.
\[thm:free field construction\] Let $\mathcal V$ be a vector space with a distinguished vector $\1$ and a family of pairwise local $\End(\mathcal V)$-valued quantum fields $\{X^\a(z)= \sum_{n \in \Z}X^\a_{(n)} z^{-n-1}\}_{\a \in \mathfrak I}.$ Suppose $\mathcal V$ is generated from $\1$ by the action of the Laurent coefficients of quantum fields $X^\a(w)$, and that the vectors $\{X^\a(z) \1 \bigr|_{z=0}\}_{\a \in \mathfrak I}$ are linearly independent in $\mathcal V$. Then the operators $$\mathcal Y\(X^{\a_1}_{(-n_1-1)}\dots X^{\a_k}_{(-n_k-1)} \1,z\) =
\ :X^{\a_1}(z)^{(n_1)} \dots X^{\a_k}(z)^{(n_k)}:,$$ where $X(z)^{(n)} = \frac 1{n!} \frac {d^n}{dz^n} X(z)$, satisfy and .
If a linear operator $\mathcal D \in \End(\mathcal V)$ satisfies $\mathcal D \1 = 0$ and $[\mathcal D,X^\a(z)] = \frac d{dz} X^\a(z)$ for every $\a \in \mathfrak I,$ then $[\mathcal D,\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z)$ for any $v \in \mathcal V$.
We say that a vertex algebra $\mathcal V$ has a PBW basis, associated with quantum fields $\{X^\a(z)\}_{\a\in\mathfrak I}$, if the index set $\mathfrak I$ is ordered, and we have a linear basis of $\mathcal V$, formed by the vectors $$\left\{X^{\a_1}_{(-n_1-1)}\dots X^{\a_k}_{(-n_k-1)} \1 \, \biggr| \,
n_1 \ge n_2 \ge \dots \ge n_k \ge 0, \text { and if } n_i=n_{i+1}, \text { then } \a_i\preceq\a_{i+1}\right\}.$$
For two mutually local quantum fields $X(z),Y(w)$ we introduce the operator product expansion (OPE) formalism, and write $$X(z)Y(w) \sim \sum_{j} \frac {C_j(w)}{(z-w)^j},$$ if for a finite collection of quantum fields $\{C_j(w)\}_{j=1,2,\dots}$ we have the equality $$X(z)Y(w) = \sum_{j} \frac {C_j(w)}{(z-w)^j} \; + :X(z)Y(w):$$ where $\frac 1{(z-w)^j}$ should be expanded into the Laurent series in non-negative powers of $\frac wz$. The importance of OPE lies in the fact that all commutators $[X_m,Y_n]$ of Laurent coefficients of quantum fields $X(z),Y(w)$ are completely encoded by the collection $\{C_j(w)\}$.
The remainder of this subsection presents some examples of vertex algebras, which will be used in this paper. All of these algebras are bi-graded and have a PBW basis associated with given quantum fields, for which we specify the OPEs.
We denote by $\hat F(\beta,\gamma)$ the vertex algebra generated by quantum fields $$\begin{aligned}
{5}
\beta(z) &= \sum_{n \in \Z} \beta_n z^{-n}, \qquad &|\beta(z)| &= 0, \qquad \deg \beta(z) &= 0, \\
\gamma(z) &= \sum_{n \in \Z} \gamma_n z^{-n-1}, \qquad &|\gamma(z)| &= 1, \qquad \deg \gamma(z) &= 0,\end{aligned}$$ with the operator product expansions $$\label{eq:beta-gamma OPE}
\beta(z)\gamma(w) \sim \frac 1{z-w}, \qquad \beta(z) \beta(w) \sim \gamma(z)\gamma(w) \sim 0.$$ The commutation relations for the underlying Heisenberg algebra are $$\label{eq:affine beta-gamma commutation}
[\beta_m,\gamma_n] = \delta_{m+n,0}, \qquad [\beta_m,\beta_n] = [\gamma_m,\gamma_n]=0.$$
We denote by $\hat\Lambda(\psi,\psi^*)$ the vertex superalgebra generated by quantum fields $$\begin{aligned}
{5}
\psi(z) &= \sum_{n\in\Z} \psi_n z^{-n-1}, \qquad & |\psi(z)| &= -1, \qquad &\deg \psi(z) &= 1,\\
\psi^*(z) &= \sum_{n\in\Z} \psi^*_n z^{-n}, \qquad & |\psi^*(z)| &= 1, \qquad &\deg \psi^*(z) &= 0, \end{aligned}$$ with the operator product expansions $$\psi(z)\psi(w) \sim \psi^*(z)\psi^*(w) \sim 0, \qquad \psi(z) \psi^*(w) \sim \frac 1{z-w}.$$ The (anti)-commutation relations for the underlying Clifford algebra are $$\label{eq:psi system relations}
\{\psi_m, \psi_n \} = \{ \psi^*_m, \psi^*_n \} = 0, \qquad
\{\psi_m, \psi^*_n\} = \delta_{m+n,0}.$$
We denote by $\ghat_k$ the vertex algebra generated by quantum fields $$X_n = \sum_{n \in \Z} X_n z^{-n-1}, \qquad |X(z)| = 0, \qquad \deg X(z) = 1, \qquad X \in \g,$$ with the operator product expansions $$X(z)Y(w) \sim \frac {[X,Y](w)}{z-w} + k \, \frac {\<X,Y\>}{(z-w)^2},\qquad k \in \C$$ where $\<\cdot,\cdot\>$ is the Killing form on $\g$. The number $k$ is called the level of $\ghat_k$.
We note that a module for the vertex algebra $\ghat_k$ is a $\Z$-graded $\ghat$-module $\hat V = \bigoplus_{n \ge n_0} \hat V[n]$, such that $\pi_{\hat V} (\mathbf k) = k \cdot \Id_{\hat V}$ and $\ghat[m] \hat V[n] \subset \hat V[m+n]$ for any $m,n \in \Z$.
We denote by $\vir_c$ the vertex algebra generated by the quantum field $$L(z) = \sum_{n\in\Z} L_n z^{-n-2}, \qquad |L(z)| = 0, \qquad \deg L(z) = 2,$$ with the operator product expansion $$L(z)L(w) \sim \frac {c/2}{(z-w)^4} + \frac {2 L(w)}{(z-w)^2} + \frac{L'(w)}{z-w}, \qquad c \in \C.$$ The number $c$ is called the central charge of $\vir_c$.
A module for the vertex algebra $\vir_c$ is a $\Z$-graded $\vir$-module $\tilde V = \bigoplus_{n \ge n_0} \tilde V[n]$, such that $\pi_{\tilde V} (\mathbf c) = c \cdot \Id_{\tilde V}$ and $L_{-m} \tilde V[n] \subset \tilde V[m+n]$ for any $m,n \in \Z$.
We denote by $\hat F_\varkappa(a)$ the vertex algebra generated by the quantum field $$a(z) = \sum_{n \in \Z} a_n z^{-n-1}, \qquad |a(z)| = 0, \qquad \deg a(z) = 1,$$ with the operator product expansion $$\label{eq:a OPE}
a(z)a(w) \sim \frac {2\varkappa}{(z-w)^2}, \qquad \varkappa \in \C.$$ The commutation relations for the underlying Heisenberg algebra $\mathcal H(a)$ are $$\label{eq:a commutation}
[a_m, a_n] = 2\varkappa \, m \, \delta_{m+n,0}.$$ Note that the operator $a_0$ is central and kills the vacuum.
Below we give the construction of a vertex algebra, which will be crucial for our future considerations. Let $\hat F_{-\varkappa}(\bar a)$ be defined similarly to $\hat F_\varkappa(a)$, so that $$\label{eq:bar a commutation}
[\bar a_m, \bar a_n] = -2\varkappa \, m \,\delta_{m+n,0}, \qquad
\bar a(z) \bar a(w) \sim - \frac {2\varkappa}{(z-w)^2}.$$
\[thm:dual Fock vertex structure\] Let $\varkappa \ne 0$. The space $\tilde{\mathbb F}_\varkappa = \hat F_\varkappa(a) \o \hat F_{-\varkappa}(\bar a) \o \C[\mathbf P]$ has a vertex algebra structure, extending those of $\hat F_\varkappa(a)$ and $\hat F_{-\varkappa}(\bar a)$, and such that $a_0 \1_\lambda = \bar a_0 \1_\lambda = \lambda \1_\lambda$.
Introduce the quantum fields $\{\mathbb Y(\mu,w)\}_{\mu\in\mathbf P}$ by $$\label{eq:double vertex operator}
\begin{split}
\mathbb Y(\mu,z) & = \exp\( \frac {\mu}{2\varkappa}\sum_{n<0} \frac {a_n}{-n} z^{-n}\)
\exp\(\frac {\mu}{2\varkappa}\sum_{n>0} \frac {a_n}{-n} z^{-n}\) \times\\
& \times
\exp\( -\frac {\mu}{2\varkappa}\sum_{n<0} \frac {\bar a_n}{-n} z^{-n}\)
\exp\( -\frac {\mu}{2\varkappa}\sum_{n>0} \frac {\bar a_n}{-n} z^{-n}\) \, \1_\mu .
\end{split}$$ Straightforward computations lead to the operator product expansions $$a(z) \bar a(w) \sim \bar a(z) a(w) \sim \mathbb Y(\mu,z) \mathbb Y(\nu,w) \sim 0,$$ $$a(z)\mathbb Y(\mu,w)\sim \bar a(z)\mathbb Y(\mu,w) \sim \frac {\mu \, \mathbb Y(\mu,w) }{z-w},$$ and establish mutual pairwise locality for the quantum fields $a(z),\bar a(z), \mathcal Y(\mu,z)$.
The vacuum is, of course, the vector $\1 \o \1 \o \1_0 \in \mathbb F_\varkappa$. We set $\mathcal Y(\1_\lambda,z) = \mathbb Y(\lambda,z)$ for any $\lambda \in \mathbf P$. The spanning and linear independence conditions of Proposition \[thm:free field construction\] are immediate. Finally, we set $\mathcal D \1_\lambda = \frac \lambda{2\varkappa} (a_{-1} - \bar a_{-1}) \1_\lambda$. The conditions on $\mathcal D$ amount to $$\label{eq:double vertex derivative}
\mathbb Y'(\lambda,z) = \frac \lambda{2\varkappa} \, \biggr(:a(z)\mathbb Y(\lambda,z): - :\bar a(z)\mathbb Y(\lambda,z):\biggr),$$ which is checked directly. Applying Proposition \[thm:free field construction\], we get the desired statement.
Theorem \[thm:dual Fock vertex structure\] should be compared with the construction of lattice vertex algebras. It is known that the space $\hat F_\varkappa(a) \o \C[\mathbf P]$ carries a vertex algebra structure only for special values of $\varkappa$, satisfying certain integrality conditions.
Bosonic realizations
--------------------
We now proceed to study the generalizations of the algebra $\mathfrak R(G_0)$. As in the finite-dimensional case, we study modules for the Lie algebra $\ghatghat$, which is equivalent to having two commuting actions of $\ghat$ on the same space.
As in the classical case, the regular $\ghat$-actions on $\mathfrak R_\varkappa(\hat G_0)$, described in Theorem \[thm:affine action\], can be reformulated in terms of representations of Heisenberg algebras. We note that the operators $$\begin{aligned}
\beta_n &= -y_{-n}\\
\gamma_n &= \pd {y_n}
\end{aligned}, \qquad\qquad
a_n = \begin{cases}
\pd {\zeta_n}, & n> 0 \\
\zeta \pd {\zeta}, & n = 0 \\
\pd {\zeta_n} - 2 n \varkappa \, \zeta_{-n}, & n < 0
\end{cases}$$ satisfy the commutation relations ,, and similarly for $$\begin{aligned}
\bar\beta_n &= x_{-n},\\
\bar\gamma_n &= - \pd {x_{n}}
\end{aligned}, \qquad\qquad
\bar a_n = \begin{cases}
\pd {\zeta_n} + 2n \varkappa \, \zeta_{-n} & n> 0 \\
\zeta \pd {\zeta}, & n = 0\\
\pd {\zeta_{n}} & n < 0\\
\end{cases}.$$ Note also that $\C[\zeta^{\pm1}] \cong \C[\mathbf P]$, and $a_0 = \bar a_0 = a$ act on $\C[\mathbf P]$ by derivations $a \1_\lambda = \lambda \1_\lambda$.
The formulas of Theorem \[thm:affine action\] are particularly simple, when written for the generating series $\mathbf e(z), \mathbf h(z), \mathbf f(z)$. For example, becomes
$$\label{eq:unordered bosons}
\begin{split}
\pi_r(\mathbf e(z)) &= \gamma(z),\\
\pi_r(\mathbf h(z)) &= 2 \beta(z)\gamma(z) + a(z),\\
\pi_r(\mathbf f(z)) &= -\beta(z)^2\gamma(z) - \beta(z)a(z) - \varkappa \beta'(z) +
\exp\(\frac 1\varkappa \sum_{n \ne 0} \frac{a_n- \bar a_n}{n} z^{-n} \) \bar \gamma(z) \1_{-2}.
\end{split}$$
Note that in this polynomial realization the constants are annihilated by $\{\mathbf e_n\}_{n \in \Z}$ and $\{\mathbf h_n\}_{n\ge0}$. The vertex algebra formalism requires a different choice of vacuum, and the introduction of normal ordering to make products of quantum fields well-defined. This procedure is well-known in the theory of Wakimoto modules (see [@BFr] and references therein), for which the $\ghat$-action is constructed by modifying the formulas originating from the semi-infinite flag variety. In particular, one expects the shifts of the levels of the representations by the dual Coxeter number $h^\vee = 2$.
The modifications of the formulas leads to the following result.
\[thm:affine bimodule action\] Let $\varkappa \ne 0,$ and let $k = \varkappa - h^\vee$ and $\bar k = -\varkappa - h^\vee$, and let $$\hat{\mathbb F}_\varkappa = \hat F(\beta,\gamma) \o \hat F(\bar\beta,\bar\gamma) \o \tilde{\mathbb F}_\varkappa.$$
1. The space $\hat{\mathbb F}_\varkappa$ has a $\ghatKK$-module structure, defined by $$\label{eq:affine boson left}
\begin{split}
\mathbf e(z) &= \gamma(z),\\
\mathbf h(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k \beta'(z) + \mathbb Y(-2,z) \bar\gamma(z),
\end{split}$$ $$\label{eq:affine boson right}
\begin{split}
\bar {\mathbf e}(z) &= \bar\gamma(z),\\
\bar {\mathbf h}(z) &= 2:\bar\beta(z) \bar\gamma(z): + \bar a(z),\\
\bar {\mathbf f}(z) &= -:\bar\beta(z)^2 \bar\gamma(z): - \bar\beta(z) \bar a(z) - \bar k \bar\beta'(z) + \mathbb Y(-2,z)\gamma(z).
\end{split}$$
2. The space $\hat {\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\hat {\mathbb F}_\varkappa = 6$. (Compatible means that the operators $\mathcal Y(v,z)$ are $\ghatKK$-intertwining operators in the VOA sense).
Similar formulas for the two commuting actions of $\ghat$ were suggested in [@FP], by analogy with the finite-dimensional Gauss decomposition of $G$. However, in order to get a meaningful VOA structure - and the corresponding semi-infinite cohomology theory! - one must incorporate the twist by $\mathbf {w_0}$, built into the Bruhat decomposition.
One can recover the original Wakimoto realization from by properly discarding the ’bar’ variables. We use superscripts ’W’ to distinguish the Wakimoto $\ghat_k$-action from .
The space $\hat W_{\lambda,k} = \hat F(\beta,\gamma)\o \hat F_\varkappa(a) \o \C \1_\lambda$ has the structure of a $\ghat_k$-module with $k = \varkappa - h^\vee$, defined by the formulas $$\begin{split} \label{eq:Wakimoto}
\mathbf e^W(z) &= \gamma(z),\\
\mathbf h^W(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f^W(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k\, \beta'(z).
\end{split}$$ The $\ghat_k$-module $\hat W_{\lambda,k}$ is called the Wakimoto module.
It suffices to show that modifying the standard Wakimoto actions by the extra terms $$\begin{aligned}
\delta\mathbf f(z) & = \mathbf f(z) - \mathbf f^W(z) = \mathbb Y(-2,z) \bar\gamma(z), \\
\overline{\delta\mathbf f}(z) & = \bar{\mathbf f}(z) - \bar{\mathbf f}^W(z) = \mathbb Y(-2,z) \gamma(z),\end{aligned}$$ does not destroy the operator product expansions.
We begin by showing that the commutation relations for $\ghat_k$ hold. Only those involving the modified quantum field $\mathbf f(z)$ need to be considered. We have: $$\begin{aligned}
\mathbf e^W(z) \, \delta\mathbf f(w) &= \gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0,\\
\mathbf h^W(z) \, \delta\mathbf f(w) &= \(2 :\beta(z) \gamma(z): + a(z)\) \mathbb Y(-2,w) \bar\gamma(w) \sim \\
& \sim a(z) \mathbb Y(-2,w) \bar \gamma(w) \sim -\frac {2 \mathbb Y(-2,w)}{z-w} \bar\gamma(w) = - \frac{2 \,\delta\mathbf f(w)}{z-w},\\
\mathbf f^W(z) \, \delta\mathbf f(w) &= -\beta(z) a(z)\gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim \frac {2 \, \mathbb Y(-2,w)}{z-w} \beta(w) \bar\gamma(w),\\
\delta\mathbf f(z) \, \delta\mathbf f(w) &= \mathbb Y(-2,z) \bar\gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0.\end{aligned}$$ Using the operator product expansions above we immediately check that $$\begin{aligned}
\mathbf e(z) \mathbf f(w) &= \mathbf e^W(z) \mathbf f^W(w) + \mathbf e^W(z) \, \delta\mathbf f(w) \sim
\( \frac k{(z-w)^2} + \frac {\mathbf h^W(w)}{z-w} \) + 0 = \frac k{(z-w)^2} + \frac {\mathbf h(w)}{z-w},\\
\mathbf h(z) \mathbf f(w) &= \mathbf h^W(z) \mathbf f^W(w) + \mathbf h^W(z) \, \delta\mathbf f(w) \sim
- \frac {2 \, \mathbf f^W(w)}{z-w} + 0 = - \frac {2 \, \mathbf f(w)}{z-w},\\
\mathbf f(z) \mathbf f(w) &= \mathbf f^W(z) \mathbf f^W(w) + \mathbf f^W(z) \, \delta\mathbf f(w) +
\delta\mathbf f(z) \, \mathbf f^W(w) + \delta\mathbf f(z) \, \delta\mathbf f(w) \sim \\
&\sim 0 + \frac {2 \, \mathbb Y(-2,w)}{z-w} \beta(w) \bar\gamma(w) + \frac {2 \, \mathbb Y(-2,z)}{w-z} \beta(z) \bar\gamma(z) + 0 \sim 0,\end{aligned}$$ and since the operator product expansions not involving $\mathbf f(z)$ are unchanged, we have proved the commutation relations for the (left) $\ghat_k$-action. Similarly, one verifies the commutation relations for the (right) $\ghat_{\bar k}$-action.
We now prove that the two actions of $\ghat_k$ and $\ghat_{\bar k}$ commute. We have $$\begin{aligned}
\bar {\mathbf e}^W(z) \, \delta\mathbf f(w) &= \bar \gamma(z) \mathbb Y(-2,w) \bar\gamma(w) \sim 0,\\
\bar {\mathbf h}^W(z) \, \delta\mathbf f(w) &= \(2 :\bar \beta(z) \bar \gamma(z): + \bar a(z) \) \mathbb Y(-2,w) \bar\gamma(w) \sim \\
& \sim 2 \frac{ \bar\gamma(z) }{z-w} \mathbb Y(-2,w) - \frac {2 \mathbb Y(-2,w)}{z-w} \bar\gamma(w) \sim 0,\end{aligned}$$ which implies that $\bar{\mathbf e}(z) \mathbf f(w) \sim \bar{\mathbf h}(z) \mathbf f(w) \sim 0.$ Finally, we compute $$\begin{aligned}
\bar{\mathbf f}^W(z) \, \delta\mathbf f(w) &= \biggr( -:\bar\beta(z)^2\bar\gamma(z): - \bar k \bar\beta'(z) -
\bar\beta(z) \bar a(z) \biggr) \biggr(\mathbb Y(-2,w) \bar\gamma(w) \biggr) \sim \\
& \sim -2 \, \frac{\bar\beta(z) \bar\gamma(z)}{z-w} \mathbb Y(-2,w) + \frac {\bar k}{(z-w)^2} \mathbb Y(-2,w) -\\
& - \( - \frac {2 \mathbb Y(-2,w)}{z-w} :\bar\beta(z) \bar\gamma(w): +
\frac { :\bar a(w) \mathbb Y(-2,w):}{z-w} - \frac {2 \mathbb Y(-2,w)}{(z-w)^2} \) \sim \\
& \sim \frac {(\bar k+2) \mathbb Y(-2,w)} {(z-w)^2} - \frac {:\bar a(w)\mathbb Y(-2,w):}{z-w} =
- \varkappa \frac {\mathbb Y(-2,w)} {(z-w)^2} - \frac {:\bar a(w)\mathbb Y(-2,w):}{z-w}.\end{aligned}$$ and similarly $$\overline{\delta\mathbf f}(z) \mathbf f^W(w) \sim
\varkappa \frac {\mathbb Y(-2,w)} {(z-w)^2} + \frac {:\bar a(w) \mathbb Y(-2,w):}{z-w} \sim
- \, \bar{\mathbf f}^W(z) \, \delta\mathbf f(w).$$ Therefore, $$\bar{\mathbf f}(z) \mathbf f(w) =
\bar{\mathbf f}^W(z) \mathbf f^W(w) + \bar{\mathbf f}^W(z) \, \delta\mathbf f(w) +
\overline{\delta\mathbf f}(z) \, \mathbf f^W(w) + \overline{\delta\mathbf f}(z) \, \delta\mathbf f(w) \sim 0,$$ and we have established the commutativity of the two $\ghat$-actions.
Proposition \[thm:free field construction\] implies that $\hat {\mathbb F}_\varkappa$ is a vertex algebra. The formulas can be written as $$\begin{split}
\mathbf e(z) &= \mathcal Y(\gamma_{-1}\1_0,z), \\
{\mathbf h}(z) &= \mathcal Y(2\beta_0\gamma_{-1}\1_0 + a_{-1}\1_0,z), \\
\mathbf f(z) &= \mathcal Y(-(\beta_0)^2\gamma_{-1}\1_0 - a_{-1}\beta_0\1_0 - k \beta_{-1}\1_0 - \bar\gamma_{-1}\1_{-2},z),
\end{split}$$ which means that the quantum fields are special cases of the operators $\mathcal Y(\cdot,z)$. The same is true for the quantum fields . Therefore, the vertex algebra structure is compatible (in the vertex algebra sense) with the $\ghatKK$-module structure on $\hat{\mathbb F}_\varkappa$.
To give $\hat{\mathbb F}_\varkappa$ a VOA structure we need to introduce the Virasoro element. The Sugawara construction for the affine algebra $\ghat_k$ gives a Virasoro quantum field with central charge $c = \frac {3k}{k+h^\vee} = 3 - \frac 6\varkappa$: $$\label{eq:Sugawara 1}
\begin{split}
L(z) & = \frac 1{2\varkappa} \( \frac{:\mathbf h^2(z):}2 + :\mathbf e(z)\mathbf f(z): + :\mathbf f(z)\mathbf e(z):\) = \\
& = \frac {:a(z)^2:}{4\varkappa} - \frac {a'(z)}{2\varkappa} \, - :\beta'(z) \gamma(z): +
\frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z).
\end{split}$$ We also note that $$L(z) = L^W(z) - \frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z),$$ where $L^W(z)$ is the Virasoro quantum field given by the Sugawara construction for the standard Wakimoto realization .
Similarly, the affine algebra $\ghat_{\bar k}$ produces another Virasoro quantum field with central charge $\bar c = \frac {3\bar k}{\bar k+h^\vee} = 3 + \frac 6\varkappa$: $$\label{eq:Sugawara 2}
\begin{split}
\bar L(z) & = - \frac 1\varkappa
\( \frac{:\bar {\mathbf h}^2(z):}2 + :\bar{\mathbf e}(z)\bar{\mathbf f}(z): + :\bar{\mathbf f}(z)\bar{\mathbf e}(z):\) = \\
& = - \frac {:\bar a(z)^2:}{4\varkappa} + \frac {\bar a'(z)}{2\varkappa} \, - :\bar \beta'(z) \bar \gamma(z): -
\frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z) \bar\gamma(z).
\end{split}$$
We set $\mathcal L(z) = L(z) + \bar L(z) = L^W(z) + \bar L^W(z)$. To show that $\mathcal L_{-1} = \mathcal D$, we check that $$\label{eq:affine L_{-1}}
\mathcal Y(\mathcal L_{-1} v,z) = \frac d{dz} \mathcal Y(v,z), \quad v \in \hat{\mathbb F}_\varkappa,$$ which for all the generating quantum fields follows from straightforward computations.
Finally, the rank of the VOA $\hat{\mathbb F}_\varkappa$ is equal to $$\operatorname{rank}\hat{\mathbb F}_\varkappa = c + \bar c = \(3 - \frac 6\varkappa\) + \(3 + \frac 6\varkappa\) = 6.$$ This concludes the proof of the theorem.
$\ghatKK$-module structure of $\hat{\mathbb F}_\varkappa$ for generic $\varkappa$.
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We now prove the analogue of the Theorem \[thm:classical bimodule structure\], describing the structure of the $\ghatKK$-module $\hat {\mathbb F}_\varkappa$ for generic values of the parameter $\varkappa$.
For $\lambda \in \h^*, k \in \C$ we denote by $\hat V_{\lambda,k}$ the irreducible $\ghat_k$-module, generated by a vector $\hat v$ satisfying $\g_+ \hat v = \n_+ \hat v = 0$ and $\mathbf h \hat v = \lambda \, \hat v$.
For any $\ghat_k$-module $\hat V$, the restricted dual space $\hat V'$ can be equipped with a $\ghat_k$-action by $$\<g_n \, v', v\> = - \< v', \hat\omega(g_{-n}) v\>, \qquad v \in \hat V,\ v' \in \hat V', \ g \in \g,$$ where $\hat\omega$ is as in . We denote the resulting dual module by $\hat V^\star$.
An important source of $\ghat_k$-modules is the induced module construction. Any $\g$-module $V$ may be regarded as a module for the subalgebra $\mathfrak p = \bigoplus_{n\ge0} \ghat[n]$, with $\g[n]$ acting trivially for $n>0$ and $\mathbf k$ acting as the multiplication by a scalar $k \in \C.$ The induced $\ghat_k$-module $\hat V_k$ is defined as the space $$\label{eq:induced module}
\hat V_k = \mathcal U(\ghat) \o_{\mathcal U(\mathfrak p)} V,$$ with $\ghat_k$ acting by left multiplication.
For the remainder of this section, we will assume that complex numbers $\varkappa, k, \bar k$ satisfy $$\varkappa \notin \mathbb Q, \qquad k = \varkappa - h^\vee, \qquad \bar k = -\varkappa - h^\vee.$$
\[thm:affine bimodule structure\] There exists a filtration $$\label{eq:affine filtration}
0 \subset \hat{\mathbb F}_\varkappa^{(0)} \subset \hat{\mathbb F}_\varkappa^{(1)} \subset \hat{\mathbb F}_\varkappa^{(2)} = \hat{\mathbb F}_\varkappa$$ of $\ghatKK$-submodules of $\hat{\mathbb F}_\varkappa$ such that $$\begin{aligned}
\hat{\mathbb F}_\varkappa^{(2)}/\hat{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \hat V_{-\lambda-2,k} \o \hat V^\star_{-\lambda-2,\bar k}, \label{eq:affine socle F2}\\
\hat{\mathbb F}_\varkappa^{(1)}/\hat{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \hat V_{\lambda,k} \o \hat V^\star_{-\lambda-2,\bar k} \oplus
\hat V_{-\lambda-2,k} \o \hat V^\star_{\lambda,\bar k} \), \label{eq:affine socle F1}\\
\hat{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} \hat V_{\lambda,k} \o \hat V^\star_{\lambda,\bar k}. \label{eq:affine socle F0}\end{aligned}$$
The operator $\mathcal L_0$ determines a $\Z$-grading $\deg$ of $\hat{\mathbb F}_\varkappa$, which is explicitly described by $$\label{eq:affine grading}
\deg \1_\lambda = 0, \qquad \deg X_n = -n \quad \text{ for } X = a,\bar a, \beta,\gamma, \bar\beta,\bar\gamma.$$ The lowest graded subspace $\hat {\mathbb F}_\varkappa[0] = F(\beta_0,\gamma_0) \o F(\bar\beta_0,\bar\gamma_0)\o \C[\mathbf P]$ of the vertex algebra $\hat {\mathbb F}_\varkappa$ is identified with the Fock space $\mathbb F$ for the finite-dimensional Lie algebra $\g$. Moreover, since $\varkappa$ is generic, $\hat{\mathbb F}_\varkappa$ can be constructed as the induced $\ghatKK$-module from the $\ggbar$-module $\mathbb F$: $$\hat {\mathbb F}_\varkappa = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F.$$
We construct the filtration by inducing it from the finite-dimensional one : $$\hat {\mathbb F}_\varkappa^{(0)} = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F^{(0)},
\qquad
\hat {\mathbb F}_\varkappa^{(1)} = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbb F^{(1)}.$$
It is easy to check that ,, respectively imply ,,, which proves the theorem.
The analogue of the Corollary \[thm:classical positive subalgebra\], describing the realization of the subalgebra $\mathfrak R(G) \subset \mathfrak R(G_0)$, is given below.
\[thm:affine positive subalgebra\] There exists a subspace $\hat{\mathbf F}_\varkappa \subset \hat{\mathbb F}_\varkappa$, satisfying
1. $\hat{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\hat{\mathbb F}_\varkappa$, and is generated by the quantum fields , and $\mathbb Y(1,z)$. In particular, $\mathbf F$ is a $\ghatKK$-submodule of $\hat{\mathbb F}_\varkappa$.
2. As a $\ghatKK$-module, $\hat{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\label{eq:affine positive decomposition}
\hat{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \hat V_{\lambda,k} \o \hat V^\star_{\lambda, \bar k}.$$
As before, we identify the lowest graded subspace $\hat {\mathbb F}_\varkappa[0]\subset \hat {\mathbb F}_\varkappa$ with the Fock space $\mathbb F$ for the finite-dimensional Lie algebra $\g$. Recall from Corollary \[thm:classical positive subalgebra\] that the $\ggbar$-module $\mathbb F$ contains the distinguished submodule $\mathbf F$. We define the subspace $\hat {\mathbf F}_\varkappa$ as the $\ghatKK$-submodule of $\hat {\mathbb F}_\varkappa$, induced from $\mathbf F$: $$\hat {\mathbf F}_\varkappa = \mathcal U(\ghatghat) \o_{\mathcal U(\mathfrak p \oplus \mathfrak p)} \mathbf F.$$ It immediately follows from Corollary \[thm:classical positive subalgebra\] that $\hat{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and has the decomposition .
Next, we need to show that $\hat {\mathbf F}_\varkappa$ is a vertex subalgebra. Let $\hat{\mathbf F}_\varkappa'$ denote the space, spanned by the Laurent coefficients of $\hat{\mathbb F}_\varkappa$-valued fields $\mathcal Y(a,z)b$ for all possible $a,b \in \hat {\mathbf F}_\varkappa$. We will establish that $\hat{\mathbf F}_\varkappa' = \hat{\mathbf F}_\varkappa$.
Indeed, $\hat{\mathbf F}_\varkappa'$ is a $\ghatKK$-submodule of $\hat{\mathbb F}_\varkappa$, and can be induced from its lowest graded component $\mathbf F' = \hat{\mathbf F}_\varkappa'[0]$, which is a $\ggbar$-submodule of $\mathbb F$. It suffices to prove that $\mathbf F' = \mathbf F$.
For any $a,b \in \mathbf F$, the lowest graded component of $\mathcal Y(a,z)b$ is equal to the product $a b$ in the algebra $\mathbb F$, and since $\mathbf F$ is a subalgebra, we have $a b \in \mathbf F$. (Note that any element $a \in \mathbf F$ can be obtained this way, for example, by taking $b = \1$). Using the commutation relations with the two copies of $\ghat$, we can prove that the lowest graded component of $\mathcal Y(a,z)b$ lies in $\mathbf F$ for any $a,b \in \hat {\mathbf F}_\varkappa$.
It follows that $\mathbf F' = \mathbf F$ and hence $\hat{\mathbf F}_\varkappa' = \hat{\mathbf F}_\varkappa$, which means that the restrictions of the operators $\mathcal Y(\cdot,z)$, corresponding to the subspace $\hat{\mathbf F}_\varkappa$, are well-defined. Thus $\hat{\mathbf F}_\varkappa$ is a vertex subalgebra of $\hat{\mathbb F}_\varkappa$. It is clear that as a vertex subalgebra $\hat{\mathbf F}_\varkappa$ is generated by the quantum fields , and $\{\mathbb Y(\lambda,z)\}_{\lambda \in \mathbf P^+}$, and the latter are generated by the single operator $\mathbb Y(1,z)$.
Finally, $\hat {\mathbf F}_\varkappa$ contains both $L^W(z)$ and $\bar L^W(z)$ - hence also $\mathcal L(z)$ - and therefore is a vertex operator subalgebra of $\hat {\mathbb F}_\varkappa$.
The vertex operator algebras $\hat{\mathbf F}_\varkappa$ and $\hat{\mathbb F}_\varkappa$ give explicit realizations of the modified regular representations $\mathfrak R'_\varkappa(\hat G)$ and $\mathfrak R'_\varkappa(\hat G_0)$ we discussed in the introduction. It would be interesting to construct them invariantly by using the correlation functions approach [@FZ], interpreting the rational functions $\<\1', \mathcal Y(v_1,z_1) \dots \mathcal Y(v_n,z_n) \1\>$ for $v_1,\dots,v_n \in \mathbb F \cong \hat{\mathbb F}_\varkappa[0]$ as solutions of differential equations similar to the Knizhnik-Zamolodchikov equations.
Semi-infinite cohomology of $\ghat$
-----------------------------------
The fact that the level of the diagonal action of $\ghat$ in the modified regular representations is equal to the special value $-2 h^\vee$ allows us to introduce the semi-infinite cohomology of $\ghat$ with coefficients in $\hat {\mathbb F}_\varkappa$ and in $\hat {\mathbf F}_\varkappa$. In this section we show that for generic values of $\varkappa$ these cohomologies lead to the same algebras of formal characters as in the finite-dimensional case.
We recall the definition of the semi-infinite cohomology [@Fe; @FGZ]. The main new ingredient is the “space of semi-infinite forms” $\boldsymbol{\hat\Lambda}^\semiinfty$, which replaces the finite-dimensional exterior algebra $\boldsymbol\Lambda$. We summarize its properties in the following
Let $\boldsymbol{\hat\Lambda}^\semiinfty = \bigwedge \ghat_- \o \bigwedge (\ghat'_+ \oplus \g')$. Then
1. The Clifford algebra, generated by $\{\iota(g_n), \eps(g'_n)\}_{g \in \g, g' \in \g',n \in \Z}$ with relations $$\label{eq:affine Clifford relations}
\{\iota(x_m), \iota(y_n) \} = \{ \eps(x'_m), \eps(y'_n) \} = 0, \qquad
\{\iota(x_m), \eps(y'_n)\} = \delta_{m,n}\, \<y',x\>.$$ acts irreducibly on $\boldsymbol{\hat\Lambda}^\semiinfty$, so that for any $\omega_- \in \bigwedge \ghat_-,\ \omega_+ \in \bigwedge (\ghat'_+ \oplus \g')$ we have $$\begin{aligned}
\iota(x_n) ( \omega_- \o 1 ) &=
\begin{cases} 0,& n \ge 0 \\ (x_n \wedge \omega_-) \o 1, & n<0 \end{cases}, \qquad
\eps(x'_n) (1 \o \omega_+) &=
\begin{cases} 1 \o (x'_n \wedge \omega_+), & n\ge0\\ 0,& n<0 \end{cases}.\end{aligned}$$
2. $\boldsymbol{\hat\Lambda}^{\semiinfty}$ is a bi-graded vertex superalgebra, with vacuum $\1 = 1 \o 1$, and generated by $$\begin{aligned}
{5}
\iota(x,z) &= \sum_{n\in\Z} \iota(x_n) z^{-n-1}, \qquad | \iota(x,z) | & = & -1, \quad &\deg \iota(x,z) &= 1, \quad & x &\in \g,\\
\eps(x',z) &= \sum_{n\in\Z} \eps(x'_{-n}) z^{-n}, \qquad | \eps(x',z) | & = & 1, \quad &\deg \eps(x',z) &= 0, \quad &x' &\in \g'.\end{aligned}$$
3. $\boldsymbol{\hat\Lambda}^\semiinfty$ has a $\ghat$-module structure on the level $\mathbf k = 2 h^\vee$, defined by $$\pi(x_n) = \sum_{m\in\Z} \sum_i :\eps((g'_i)_m)\iota([g_i,x]_{n+m}):,
\qquad x \in \g.$$
One can think of $\bigwedge \ghat_-$ as the space spanned by formal “semi-infinite” forms $$\omega = \xi'_{i_1} \wedge \xi'_{i_2} \wedge \xi'_{i_3} \wedge \dots,\qquad i_{n+1} = i_n + 1 \text{ for } n\gg0,$$ where $\{\xi_j\}_{j\in \mathbb N}$ is a homogeneous basis of $\ghat_-$. A monomial $\xi_{j_1} \wedge \dots \wedge \xi_{j_m} \in \bigwedge \ghat_-$ is identified with the semi-infinite form with the corresponding factors missing: $$\omega = \pm \ \xi'_1 \wedge \xi'_2 \wedge \dots \wedge \xi'_{j_1-1} \wedge \widehat{\xi'_{j_1}} \wedge \xi'_{j_1+1} \wedge \dots \wedge \xi_{j_m-1} \wedge \widehat{\xi_{j_m}} \wedge \xi_{j_m+1} \wedge\dots.$$ In other words, if $\xi_j \in \ghat$, then $\iota(\xi_j)$ operates as usual by eliminating the factor $\xi'_j$.
The BRST complex, associated with a $\ghat$-module $\hat V$ on the level $k = -2h^\vee$, is the complex $C^{\semiinfty+\bullet} (\ghat, \C\mathbf k;\hat V) = \boldsymbol{\hat\Lambda}^{\semiinfty+\bullet} \otimes \hat V$, with the differential $$\label{eq:affine differential}
\hat{\mathbf d} = \sum_{n\in\Z} \sum_i \eps((g'_i)_n) \pi_{\hat V}((g_i)_n) - \frac12 \sum_{m,n\in \Z} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):,$$ where $\{g_i\}$ is any basis of $\g$, and $\{g'_i\}$ is the dual basis of $\g'$. The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\ghat,\C \mathbf k;\hat V)$.
The BRST complex above gives the relative (to the center) version of the semi-infinite cohomology. We don’t consider any other type of cohomology, and thus simply drop the word ’relative’ everywhere. The condition $k = -2h^\vee$ is equivalent to $\hat{\mathbf d}^2 = 0$.
If $\hat V$ is a vertex algebra, then its semi-infinite cohomology inherits a vertex superalgebra structure [@LZ].
The following theorem is similar to the reduction theorem of [@FGZ] (see also [@Li]), and relates the semi-infinite cohomology for generic values of $\varkappa$ with the classical cohomology of Lie algebras.
\[thm:reduction theorem\] Let $V$ be a $\ggbar$-module, and let $\varkappa \in \C$ be generic. Set $k = \varkappa - h^\vee$ and $\bar k = -\varkappa - h^\vee$, and denote $\hat V$ be the induced $\ghat_k \oplus \ghat_{\bar k}$-module. Then with respect to the diagonal $\g$-action $\hat V$ is a level $\mathbf k = -2 h^\vee$ module, and $$H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V) \cong H^\bullet(\g,V).$$
As a vector space, the module $\hat V$ has a decomposition $$\hat V = \mathcal U(\ghat_-) \o V \o \mathcal U(\ghat_+)',$$ where we identified the factor $\mathcal U(\ghat_-)$, coming from the right induced action of $\ghat_{\bar k}$, with $\mathcal U(\ghat_+)'$ using the non-degenerate (since $\varkappa$ is generic!) contravariant pairing. Therefore, as vector spaces $$\label{eq:affine complex factorization}
C^\bullet(\ghat, \C\mathbf k; \hat V) = C_-^\bullet \o C_0^\bullet \o C_+^\bullet,$$ where $$C_-^\bullet = \bigwedge \ghat_- \o \mathcal U(\ghat_-), \qquad
C_0^\bullet = \bigwedge \g' \o V, \qquad
C_+^\bullet = \bigwedge \ghat'_+ \o \mathcal U(\ghat_+)^*$$
We write the differential $\hat{\mathbf d}$ as $$\hat{\mathbf d} = \mathbf d_- + \mathbf d_0 + \mathbf d_+ + \boldsymbol\delta,$$ where $\mathbf d_\pm$ are the BRST differentials for $\ghat_\pm$, $$\begin{aligned}
\mathbf d_- &= \sum_{n<0} \sum_i \eps((g'_i)_n) \pi_l((g_i)_n) - \frac 12 \sum_{m,n<0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):,\\
\mathbf d_+ &= \sum_{n>0} \sum_i \eps((g'_i)_n) \pi_r((g_i)_n) - \frac12 \sum_{m,n>0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}): ,\end{aligned}$$ the differential $\mathbf d_0$ is defined as in with the $\g$-action $\pi_V$ replaced by $$\label{eq:spectral g-action}
\pi(x) = \pi_{\hat V}(x) + \sum_{n\ne0} :\eps((g_j)'_n) \iota([x,g_j]_n):, \qquad x \in \g,$$ and $\boldsymbol\delta$ includes all the remaining terms: $$\begin{aligned}
\boldsymbol \delta &= \sum_{n>0} \sum_i \eps((g'_i)_n) \pi_l((g_i)_n)
+ \sum_{n<0} \sum_i \eps((g'_i)_n) \pi_r((g_i)_n) - \\
&- \sum_{m>0,n<0} \sum_{i,j} :\eps((g_i)'_m) \eps((g_j)'_n) \iota([g_i,g_j]_{m+n}):.\end{aligned}$$
Following [@FGZ], we introduce the skewed degree $f\deg$ by $$f\deg (w_- \o w_0 \o w_+) = \deg w_+ - \deg w_-, \qquad w_\pm \in C_\pm,\ \ w_0 \in C_0,$$ where the ’$\deg$’ gradings in the complexes $C_\pm$ are inherited from $C^{\semiinfty}(\ghat,\mathbf k;\hat V)$. We set $$\mathfrak B^p = \left\{ v \in C^{\semiinfty}(\ghat,\mathbf k;\hat V) \ \biggr| \ f\deg v \ge p \right\}.$$ One can check that $\mathbf d_\pm$ and $\mathbf d_0$ preserve the filtered degree, and that $\boldsymbol \delta (\mathfrak B^p) \subset \mathfrak B^{p+1}$. Thus, $\{\mathfrak B^p\}_{p\in\Z}$ is a decreasing filtration of the complex $C^{\semiinfty}(\ghat,\mathbf k;\hat V)$, and the associated graded complex has the reduced differential $$\mathbf d_{red} = \mathbf d_- + \mathbf d_0 + \mathbf d_+.$$ We now compute the corresponding reduced cohomology, which will provide a bridge to $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k;\hat V)$.
It is clear that $\mathbf d_\pm^2 = \{\mathbf d_+,\mathbf d_-\} = 0$, and that the differentials $\mathbf d_\pm: C_\pm^\bullet \to C_\pm^{\bullet+1}$ act in their respective factors of . One can also check that $(\mathbf d_0)^2 = \{\mathbf d_0,\mathbf d_\pm\} = 0$; moreover, $$\mathbf d_0 (C_-^\bullet \o C_0^\bullet \o C_+^\bullet) \subset (C_-^\bullet \o C_0^{\bullet+1} \o C_+^\bullet)$$ despite the fact that $\mathbf d_0$ does not act in $C_0^\bullet$.
It is a well-known fact in homological algebra that $$H^n(C_+,\mathbf d_+) = H^n(\ghat_+;\mathcal U(\ghat_+)') = \delta_{n,0} \, \C,$$ with $1 \o 1' \in C_+$ representing the non-trivial cohomology class. Similarly, one has $$H^n(C_-,\mathbf d_-) = H_{-n}(\ghat_-;\mathcal U(\ghat_-)) = \delta_{n,0} \, \C,$$ and $1 \o 1 \in C_-$ represents the non-trivial cohomology. Further, one can check that the subspace $1 \o C_0^\bullet \o 1 \subset C^{\semiinfty+\bullet}(\ghat,\C\mathbf k;\hat V)$ is stabilized by $\mathbf d_0$, and that the $\g$-action on that subspace reduces to the $\g$ action $1 \o \pi_V \o 1$. It follows that $$H^\bullet_{red}(\ghat, \C\mathbf k; \hat V) \cong H^\bullet(1 \o C_0 \o 1,\mathbf d_0) \cong H^\bullet(C_0,\mathbf d) \cong H^\bullet(\g,V).$$
We now return to the cohomology of $C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)$. Since $\hat {\mathbf d}$ preserves the ’$\deg$’ grading, it can be computed separately for each subcomplex $C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[m], m \in \Z$. The filtration $\{\mathfrak B^p[m]\}_{p \in \Z}$ of this complex is finite for each $m$, and leads to a finitely converging spectral sequence with $E_1^{p,q}[m] = H^q_{red}(\mathfrak B^p[m]/ \mathfrak B^{p+1}[m])$.
For $m\ne 0$ we have $H^q_{red}(\mathfrak B^p[m]/ \mathfrak B^{p+1}[m])=0$ for all $p$, hence the spectral sequence is zero, and $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[m]=0$. For $m=0$ we note that $$\mathfrak B^0[0] = C^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[0], \qquad
\mathfrak B^1[0] = 0,$$ which means that $E_1^{p,q}[0] = 0$ unless $p=0$, and the collapsing spectral sequence implies $$H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat V)[0] \cong H^\bullet_{red}(\mathfrak B^0[0]/ \mathfrak B^1[0]) \cong H^\bullet(\g;V).$$
This completes the proof of the theorem.
\[thm:affine cohomology\] The vertex superalgebras $H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbf F}_\varkappa),
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbb F}_\varkappa)$ degenerate into commutative superalgebras. Moreover, we have commutative superalgebra isomorphisms $$\label{eq:affine semiinfinite isomorphisms}
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbf F}_\varkappa) \cong H^\bullet(\g; \mathbf F),\qquad
H^{\semiinfty+\bullet}(\ghat,\C\mathbf k; \hat{\mathbb F}_\varkappa) \cong H^\bullet(\g; \mathbb F).$$ In particular, $$H^{\semiinfty+0}(\ghat,\C \mathbf k;\hat{\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\ghat,\C \mathbf k;\hat{\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
Theorem \[thm:reduction theorem\] gives us isomorphisms on the level of vector spaces. It is also clear from its proof that the semi-infinite cohomology is concentrated in the subspace of $\deg = 0$, and thus the operators $\mathcal Y(\cdot,z)$ on cohomology are reduced to their constant terms. In particular, they are independent of $z$, which means that the vertex superalgebra degenerates into a commutative superalgebra. The multiplication is easily traced back to the multiplications in $\mathbf F \cong \hat{\mathbf F}_\varkappa[0]$ and in the exterior algebra $\boldsymbol\Lambda = \bigwedge \g'$, which shows that are superalgebra isomorphisms.
Modified regular representations of the Virasoro algebra.
=========================================================
Virasoro algebra and the quantum Drinfeld-Sokolov reduction
-----------------------------------------------------------
In this section we present a construction of the regular representation of the Virasoro algebra, which goes in parallel with constructions in the previous sections. However, instead of beginning with a space of functions on the corresponding group (which is, strictly speaking, a semigroup in the complex case), we will use the quantum Drinfeld-Sokolov reduction [@FeFrDS] (see also [@BFr] and references therein), applied to the modified regular representations of $\ghat$ constructed in Section 1. As a result we obtain certain bimodules over the Virasoro algebra, which have the structure similar to their affine counterparts. The result of the quantum Drinfeld-Sokolov reduction applied to the actual regular representation of $\ghat$ should have a standard interpretation in terms of the space of functions on the Virasoro semigroup, but we will not need this fact for our purposes.
Recall that the Virasoro algebra $\vir$ is the infinite-dimensional complex Lie algebra, generated by $\{L_n\}_{n \in \Z}$ and a central element $\mathbf c,$ subject to the commutation relations $$[L_m, L_n] = (m-n) L_{m+n} + \frac {m^3-m}{12}\, \mathbf c.$$
The Virasoro algebra has a $\Z$-grading $\vir = \oplus_{n\in\Z} \vir[n]$, determined by $$\deg L_n = -n, \qquad \deg \mathbf c = 0.$$
There is a functorial correspondence between certain representations of affine Lie algebras and their $\mathcal W$-algebra counterparts, called the quantum Drinfeld-Sokolov reduction [@FeFrDS] (see also [@BFr] and references therein). We review this procedure for the case $\ghat = \slhat$, when the corresponding $\mathcal W$-algebra is identified with the Virasoro algebra.
For any $\ghat_k$-module $\hat V$, the complex $(C_{DS}(\hat V),\mathbf d_{DS})$, $$C_{DS}(\hat V) = \hat V \o \hat\Lambda(\psi,\psi^*), \qquad \mathbf d_{DS} = \sum_{n\in\Z} \psi^*_n \pi_{\hat V}(\mathbf e_n) + \psi^*_1,$$ is called the BRST complex of the quantum Drindeld-Sokolov reduction. The corresponding cohomology is denoted $H_{DS}(\hat V)$.
The BRST complex above is very similar to the semi-infinite cohomology complex for the nilpotent loop algebra $\nhat_+ = \bigoplus_{n\in\Z} \C \mathbf e_n$. Indeed, the corresponding space of semi-infinite forms $\boldsymbol{\Lambda}^\semiinfty(\nhat_+)$ is identified with $\hat\Lambda(\psi,\psi^*)$ by $\iota(\mathbf e_n) \equiv \psi_n, \ \eps(\mathbf e'_n) \equiv \psi^*_{-n}$, and the only modification is the additional term $\psi^*_1$ in the differential.
The BRST complex inherits the gradings $|\cdot|$ and $\deg$ from $\hat\Lambda(\psi,\psi^*)$ and $\hat V$. Since $|\mathbf d_{DS}| = 1$, the grading $|\cdot|$ descends to the cohomology $H_{DS}(\hat V)$. However, with respect to the other grading, the differential $\mathbf d_{DS}$ is not homogeneous. We introduce a modified grading $\deg'$ by $$\begin{gathered}
\deg' \mathbf e(z) = 0, \qquad
\deg' \mathbf h(z) = 1,\qquad
\deg' \mathbf f(z) = 2, \\
\deg' \psi(z) = 0,\qquad
\deg' \psi^*(z) = 1.\end{gathered}$$ The differential $\mathbf d_{DS}$ then satisfies $\deg' \mathbf d_{DS} = 0$, and the grading $\deg'$ descends to $H_{DS}(\hat V)$.
The cohomology $H^0_{DS}(\ghat_k)$ of the vacuum module inherits a vertex algebra structure. We have the following result (details of the proof can be found in [@BFr]).
\[thm:Drinfeld-Sokolov\] For $k \ne -h^\vee$ we have $H^0_{DS}(\ghat_k) \cong \vir_c$, where $c = 1 - \frac{6}{k+h^\vee} - 6k$.
For any $\ghat_k$-module $\hat V$, the vertex algebra $\vir_c \cong H^0_{DS}(\ghat_k)$ acts on $H^0_{DS}(\hat V)$. For $\varkappa \ne 0$, set $\tilde F_{\lambda,\varkappa} = H_{DS}^0(\hat W_{\lambda,\varkappa-h^\vee}).$ The following identifies the $\vir_c$-module structure on $\tilde F_{\lambda,\varkappa}$.
\[thm:Wakimoto to FF\] Let $\varkappa \ne 0$, and let $c = 13 - 6 \varkappa - \frac6\varkappa$. Then $\tilde F_{\lambda,\varkappa} \cong \hat F_\varkappa(a) \o \C \1_\lambda$ as a vector space, and the $\vir_c$-action is given by $$\label{eq:Feigin-Fuks}
L^F(z) = \frac 1{4\, \varkappa} :a(z)^2: + \frac{\varkappa-1}{2\, \varkappa} \, a'(z).$$
In the vector space factorization of $\hat W_{\lambda,\varkappa-h^\vee} = \hat F(\beta,\gamma) \o \hat F_\varkappa(a) \otimes \C \1_\lambda$, the differential $\mathbf d_{DS}$ acts only in the first component. Therefore, we must have $$\tilde F_{\lambda,\varkappa} = H_{DS}(\hat F(\beta,\gamma)) \o \hat F_\varkappa(a) \otimes \C \1_\lambda.$$ A spectral sequence reduces the cohomology $H_{DS}^0(\hat F(\beta,\gamma))$ to the cohomology of the semi-infinite Weil complex $\hat F(\beta,\gamma)\o\hat\Lambda(\psi,\psi^*)$. The latter splits into an infinite product of finite-dimensional Weil complexes, and thus has one-dimensional cohomology, concentrated in degree $0$.
The inclusion of vertex algebras $\ghat_{\varkappa-h^\vee} \hookrightarrow \hat W_{0,\varkappa-h^\vee}$ induces an inclusion $\vir_c \hookrightarrow \tilde F_{0,\varkappa}$, and the explicit formula for $L(z)$ in terms of $a(z)$ is a result of a direct computation.
The realization of Virasoro modules was known long before the quantum Drinfeld-Sokolov reduction, and is called the Feigin-Fuks construction in the literature. We use the superscript “F” to distinguish this standard action from the modified Virasoro actions, which we will be considering later.
Bosonic realization of the regular representation
-------------------------------------------------
The Virasoro analogue of the Peter-Weyl theorem is more subtle than in the case of classical and affine Lie algebras. There is no clear way to calculate the two commuting $\vir$-actions in a way similar to Theorem \[thm:classical bimodule action\] and Theorem \[thm:affine action\]. However, there exists a Fock space realization analogous to Theorem \[thm:affine bimodule action\], which we will call the regular representation of the Virasoro algebra.
\[thm:Virasoro action\] Let $\varkappa \ne 0$, and let $c = 13 - 6 \varkappa - \frac 6\varkappa$ and $\bar c = 13 + 6 \varkappa + \frac 6\varkappa$.
1. The space $\tilde {\mathbb F}_\varkappa$ has a $\virCC$-module structure, defined by $$\begin{aligned}
L(z) &= \frac 1{4\varkappa} :a(z)^2: + \frac{\varkappa-1}{2\varkappa} a'(z)
- \frac 1\varkappa \, \mathbb Y(-2,z), \label{eq:Virasoro action 1}\\
\bar L(z) &= -\frac 1{4\varkappa} :\bar a(z)^2: + \frac{\varkappa+1}{2\varkappa} \bar a'(z)
+ \frac 1\varkappa \, \mathbb Y(-2,z). \label{eq:Virasoro action 2}\end{aligned}$$
2. The space $\tilde {\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\tilde {\mathbb F}_\varkappa = 26$.
The formulas , are nothing else but the result of the two-sided quantum Drinfeld-Sokolov reduction, which consists of two reductions applied separately to the two commuting $\ghat$-actions of Theorem \[thm:affine bimodule action\], cf. formulas , and Proposition \[thm:Wakimoto to FF\].
Rather than give detailed proof of this fact, we choose to verify the commutation relations directly. Introduce notation $$\begin{aligned}
\delta L(z) &= L(z) - L^F(z) = \frac 1\varkappa \, \mathbb Y(-2,z), \\
\overline{\delta L}(z) &= \bar L(z) - \bar L^F(z) = - \frac 1\varkappa \, \mathbb Y(-2,z).\end{aligned}$$ Without the additional terms $\delta L(z), \overline{\delta L}(z)$, both and give two commuting copies of the standard construction with the specified central charges. Therefore, it suffices to show that the presence of these extra terms does not violate the commutation relations for $\virCC$.
Straightforward computations immediately show that $$\delta L(z) \, \delta L(w) \sim \delta L(z) \, \overline{\delta L}(w) \sim \overline{\delta L}(z) \, \overline{\delta L}(w) \sim 0.$$
$$L^F(z) \mathbb Y(-2,w) \sim \frac {\mathbb Y(-2,w)}{(z-w)^2} - \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w},$$ $$\bar L^F(z) \mathbb Y(-2,w) \sim \frac {\mathbb Y(-2,w)}{(z-w)^2} + \frac 1{\varkappa} \frac{:\bar a(w) \mathbb Y(-2,w):}{z-w}.$$
We now prove the commutation relations for the action . We have $$\begin{aligned}
L(z)L(w) & - L^F(z) L^F(w) = L^F(z) \, \delta L(w) + \delta L(z) L^F(w) + \delta L(z) \, \delta L(w) \sim\\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2}
- \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w}\)
+ \frac 1\varkappa \(\frac {\mathbb Y(-2,z)}{(z-w)^2}
+ \frac 1{\varkappa} \frac{:a(z) \mathbb Y(-2,w):}{z-w} \) \sim\\
& \sim \frac 1\varkappa \( \frac {2 \, \mathbb Y(-2,w)}{(z-w)^2} + \frac {\mathbb Y'(-2,w)}{z-w} \) \sim
\frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{z-w} ,\end{aligned}$$ and therefore $$\begin{aligned}
L(z)L(w) & \sim L^F(z) L^F(w) + \frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{z-w} \sim
\( \frac{c/2}{(z-w)^4} + \frac {2L^F(w)}{(z-w)^2} + \frac {(L^F)'(w)}{z-w} \) + \\
& + \frac {2 \, \delta L(w)}{(z-w)^2} + \frac {(\delta L)'(w)}{(z-w)^2} =
\frac{c/2}{(z-w)^4} + \frac {2L(w)}{(z-w)^2} + \frac {L'(w)}{z-w} .\end{aligned}$$ We have established that adding the extra term $\delta L(z)$ to the action preserves the commutation relations for $\vir_c.$ Similarly, the formula gives a representation of $\vir_{\bar c}.$
We now show that the two actions of $\vir_c$ and $\vir_{\bar c}$ commute. Using , we get $$\begin{split}
\delta L(z) \bar L^F(w) & \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,z)}{(z-w)^2} - \frac 1{\varkappa} \frac{:\bar a(z) \mathbb Y(-2,z):}{z-w} \) \sim \\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2} + \frac {\mathbb Y'(-2,w)}{z-w} -
\frac 1{\varkappa} \frac{:\bar a(w) \mathbb Y(-2,w):}{z-w} \) \sim \\
& \sim \frac 1\varkappa \( \frac {\mathbb Y(-2,w)}{(z-w)^2} - \frac 1{\varkappa} \frac{:a(w) \mathbb Y(-2,w):}{z-w} \) .
\end{split}$$ Note that implies $\delta L(z) \bar L^F(w) \sim - L^F(z) \, \overline{\delta L}(w)$, and thus $$L(z)\bar L(w) = L^F(z)\bar L^F(w) + L^F(z) \, \overline{\delta L}(w) + \delta L(z)\bar L^F(w) + \delta L(z)\overline{\delta L}(w) \sim 0,$$ which means that the two Virasoro actions commute.
It is easy to see that the formula can be written as $$L(z) = \mathcal Y\( \frac{(a_{-1})^2}{4\varkappa} \1_0 + \frac{\varkappa-1}{2\varkappa} a_{-2} \1_0 + \frac 1{\varkappa} \,\1_{-2},z\),$$ and similarly for , which means that the vertex algebra structure is compatible with $\virCC$-module structure on $\tilde {\mathbb F}_\varkappa$.
We introduce the VOA structure in $\tilde{\mathbb F}_\varkappa$ by setting $\mathcal L(z) = L(z) + \bar L(z) = L^F(z) + \bar L^F(z)$. One immediately checks that $\mathcal L(z)$ is a Virasoro quantum field with central charge $26,$ and satisfies $\mathcal L_{-1} \1_0 = 0.$ It suffices to check the remaining relation $$\label{eq:Virasoro L_{-1}}
[\mathcal L_{-1},\mathcal Y(v,z)] = \frac d{dz} \mathcal Y(v,z), \quad v \in \tilde{\mathbb F}_\varkappa,$$ for each of the generating quantum fields, which is done by direct computations.
$\virCC$-module structure of $\tilde{\mathbb F}_\varkappa$ for generic $\varkappa$.
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We now describe the socle filtration of the $\virCC$-module $\tilde {\mathbb F}_\varkappa$ for generic $\varkappa$, when it is completely analogous to the finite-dimensional and affine cases, given by Theorem \[thm:classical bimodule structure\] and Theorem \[thm:affine bimodule structure\]. In this subsection we assume that $$\varkappa \notin \mathbb Q, \qquad c = 13 - \frac 6\varkappa - 6\varkappa,\qquad \bar c = 13 + \frac 6\varkappa + 6\varkappa.$$
For a $\vir_c$-module $\tilde V$, the restricted dual space $\tilde V'$ can be equipped with a $\vir_c$-action by $$\<L_n \, v', v\> = \< v', L_{-n} v\>.$$ We denote the resulting dual module by $\tilde V^\star$.
We denote by $\tilde V_{\Delta,c}$ the irreducible $\vir_c$-module, generated by a highest weight vector $\tilde v$ satisfying $L_0 \, \tilde v = \Delta \, \tilde v$ and $L_n \tilde v = 0$ for $n>0$. For any $\lambda \in \h^*$, set $$\Delta(\lambda) = \frac {\lambda(\lambda+2)}{4\varkappa} - \frac \lambda2 , \qquad
\bar \Delta(\lambda) = -\frac {\lambda(\lambda+2)}{4\varkappa} - \frac \lambda2.$$
\[thm:Virasoro bimodule structure\] There exists a filtration $$\label{eq:Virasoro filtration}
0 \subset \tilde{\mathbb F}_\varkappa^{(0)} \subset \tilde{\mathbb F}_\varkappa^{(1)} \subset \tilde{\mathbb F}_\varkappa^{(2)} = \tilde{\mathbb F}_\varkappa$$ of $\virCC$-submodules of $\tilde{\mathbb F}_\varkappa$ such that $$\begin{aligned}
\tilde{\mathbb F}_\varkappa^{(2)}/\tilde{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \tilde V_{\Delta(-\lambda-2),c} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \label{eq:Virasoro socle F2},\\
\tilde{\mathbb F}_\varkappa^{(1)}/\tilde{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \oplus
\tilde V_{\Delta(-\lambda-2),c} \o \tilde V^\star_{\bar \Delta(\lambda),\bar c} \) \label{eq:Virasoro socle F1},\\
\tilde{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in \mathbf P} \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c} \label{eq:Virasoro socle F0}.\end{aligned}$$
One can derive from Proposition \[thm:Wakimoto to FF\] that the for generic $\varkappa$ the reduction sends exact sequences of $\ghat_k$-modules to exact sequences of $\vir_c$-modules, which implies in particular that $$H_{DS}^n(\hat V_{\lambda,k}) = \begin{cases} \tilde V_{\Delta(\lambda),c}, & n=0 \\ 0, & n\ne 0 \end{cases},$$ It is then easy to check that the images $\tilde{\mathbb F}_\varkappa^{(0,1,2)}$ of the $\ghatKK$-submodules $\hat{\mathbb F}_\varkappa^{(0,1,2)}$ from Theorem \[thm:affine bimodule structure\] under the two-sided quantum Drinfeld-Sokolov reduction satisfy the required properties.
An alternative direct approach repeats the steps in the proof of Theorem \[thm:classical bimodule structure\]. In particular, we get a decomposition into blocks, $$\tilde{\mathbb F}_\varkappa = \tilde{\mathbb F}_\varkappa(-1) \oplus \bigoplus_{\lambda\in \mathbf P^+} \tilde{\mathbb F}_\varkappa(\lambda).$$
We also have the following Virasoro analogue of Corollary \[thm:classical positive subalgebra\] and Theorem \[thm:affine positive subalgebra\].
There exists a subspace $\tilde{\mathbf F}_\varkappa \subset \tilde{\mathbb F}_\varkappa$, satisfying
1. $\tilde{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\tilde{\mathbb F}_\varkappa$, and is generated by the quantum fields , and $\mathbb Y(1,z)$. In particular, $\tilde{\mathbf F}_\varkappa$ is a $\virCC$-submodule of $\tilde{\mathbb F}_\varkappa$.
2. As a $\virCC$-module, $\tilde{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\label{eq:Virasoro positive decomposition}
\tilde{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \tilde V_{\Delta(\lambda),c} \o \tilde V^\star_{\bar\Delta(\lambda), \bar c}.$$
The desired subspace $\tilde{\mathbf F}_\varkappa$ is the image of the vertex subalgebra $\hat{\mathbf F}_\varkappa$ under the two-sided quantum Drinfeld-Sokolov reduction. We leave technical details to the reader.
Semi-infinite cohomology of $\vir$
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The central charge for the diagonal action of $\vir$ in the modified regular representations is equal to the special value 26. In this section we study the semi-infinite cohomology of $\vir$ with coefficients in $\tilde {\mathbb F}_\varkappa$ and in $\tilde {\mathbf F}_\varkappa$.
The properties of the appropriate “space of semi-infinite forms” $\boldsymbol{\tilde\Lambda}^\semiinfty$ for the Virasoro algebra are summarized in the following
Set $\boldsymbol{\tilde\Lambda}^\semiinfty = \bigwedge \vir_- \o \bigwedge \vir'_+$, where $\vir_- = \bigoplus_{n\le-2} \C L_n$ and $\vir_+ = \bigoplus_{n\ge-1} \C L_n$. Then
1. The Clifford algebra, generated by $\{b_n, c_n \}_{n \in \Z}$ with relations $$\label{eq:bc system relations}
\{b_m, b_n \} = \{ c_m, c_n \} = 0, \qquad
\{b_m, c_n\} = \delta_{m+n,0}.$$ acts irreducibly on $\boldsymbol{\tilde\Lambda}^\semiinfty$, so that for any $\omega_- \in \bigwedge \vir_-, \, \omega_+ \in\bigwedge \vir'_+$ we have $$\begin{aligned}
b_n (1 \o \omega) &=
\begin{cases} 0,& n \ge -1 \\ 1 \o (L_n\wedge\omega), & n\le-2 \end{cases}, \qquad
c_n (\omega \o 1) &=
\begin{cases} (L'_{-n}\wedge\omega) \o 1, & n\le 1 \\ 0,& n\ge 2 \end{cases}.\end{aligned}$$
2. $\boldsymbol{\tilde\Lambda}^{\semiinfty}$ is a bi-graded vertex superalgebra, with vacuum $\1 = 1 \o 1$, generated by $$\begin{aligned}
{3}
b(z) &= \sum_{n\in\Z} b_n z^{-n-2}, \qquad | b(z) | = -1, \quad &\deg b(z) &= 2,\\
c(z) &= \sum_{n\in\Z} c_n z^{-n+1}, \qquad | c(z) | = 1, \quad &\deg c(z) &= -1.\end{aligned}$$
3. $\boldsymbol{\tilde\Lambda}^\semiinfty$ has a $\vir$-module structure with central charge $c = -26$, defined by $$\pi(L_n) = \sum_{m\in\Z} (m-n) :c_{-m} b_{n+m}:.$$
The BRST complex, associated with a $\vir$-module $\tilde V$ with central charge $c = 26$, is the complex $C^{\semiinfty+\bullet} (\vir, \C\mathbf c;\tilde V) = \boldsymbol{\tilde\Lambda}^{\semiinfty+\bullet} \otimes \tilde V$, with the differential $$\label{eq:Virasoro differential}
\tilde{\mathbf d} = \sum_{n\in\Z} c_{-n} \pi_{\tilde V}(L_n) - \frac 12 \sum_{m,n\in \Z} (m-n) :c_{-m} c_{-n} b_{m+n}:.$$ The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde V)$.
As in the affine case, the special value $c = 26$ of the central charge is required to ensure that $\tilde{\mathbf d}^2 = 0$.
The vertex superalgebras $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde{\mathbb F}_\varkappa)$ degenerate into the commutative superalgebras, and we have commutative algebra isomorphisms $$\label{eq:Virasoro semi-infinite isomorphisms}
\begin{split}
H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde {\mathbf F}_\varkappa) \cong H^{\semiinfty+\bullet}(\ghat, \C\mathbf k;\hat{\mathbf F}_\varkappa), \qquad
H^{\semiinfty+\bullet}(\vir,\C \mathbf c;\tilde {\mathbb F}_\varkappa) \cong H^{\semiinfty+\bullet}(\ghat, \C\mathbf k;\hat{\mathbb F}_\varkappa).
\end{split}$$ In particular, $$H^{\semiinfty+0}(\vir,\C \mathbf c;\tilde {\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\vir,\C \mathbf c;\tilde {\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
The problem of computing the semi-infinite cohomology of $\vir$, as well as its inherited algebra structure, has been extensively studied by mathematicians and physicists working in the string theory. We take advantage of these results, and construct our proof by combining entire blocks from previous papers.
We note that for both $\tilde{\mathbf F}_\varkappa$ and $\tilde{\mathbb F}_\varkappa$ the diagonal action of $\vir$ does not contain additional vertex operator shifts, and is equal to the sum of two standard Feigin-Fuks actions.
The comprehensive answer for the cohomology of tensor products of Feigin-Fuks and/or irreducible modules was given in [@LZ2] for the most difficult case of the central charge $c = c_{p,q}$, corresponding to $\varkappa = \frac pq \in \mathbb Q$. Simplified (for the case of generic $\varkappa$) version of their computations, and the spectral sequence associated with filtrations of Theorem \[thm:Virasoro bimodule structure\], yield
$$H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(\lambda)) =
\begin{cases}
\C, & n=0,3\\
0, & \text{otherwise}
\end{cases},$$ $$H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(-1)) =
\begin{cases}
\C, & n=1,2\\
0, & \text{otherwise}
\end{cases},\qquad
H^{\semiinfty+n}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(\lambda)) =
\begin{cases}
\C, & n=0,2\\
\C^2, & n=1\\
0, & \text{otherwise}
\end{cases},$$ for each $\lambda \in \mathbf P^+$, as well as natural isomorphisms $$H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(\lambda)) \cong H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(\lambda)).$$ The algebra structure of $H^{\semiinfty+0}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$ is in fact independent of $\varkappa$, as can be seen from the change of variables $$p_n = \frac{a_n + \bar a_n}2, \qquad q_n = \frac {a_n - \bar a_n}{2\varkappa}.$$ Indeed, the new commutation relations become $[p_m,p_n] = [q_m,q_n] = 0$ and $[p_m,q_n] = \delta_{m+n,0}$, and the diagonal Virasoro action is given by $$\mathcal L(z) = :p(z)q(z): + p(z) - q(z).$$ For the special case $\varkappa = 1$, corresponding to the pairing of $c=1$ and $\bar c = 25$ modules, the cohomology of a bigger vertex algebra $\mathcal A_{2D} = \bigoplus_{\lambda,\mu \in \Z} \tilde F_{\lambda,1} \o \tilde F_{\mu,-1}$ was identified in [@WZ] with the polynomial algebra $\C[x,y]$ in two variables. The subalgebra $H^{\semiinfty+0}(\vir,\C c; \tilde{\mathbb F}_\varkappa)$ is therefore isomorphic to the polynomial algebra $\C[\chi]$, and we can take any nonzero cohomology class $\chi \in H^{\semiinfty+0}(\vir,\C c; \tilde{\mathbb F}_\varkappa(1))$ as the generator.
The vertex superalgebra structures on $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$ degenerate into commutative superalgebras. It is clear that both are free $\C[\chi]$-modules.
It follows immediately that $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa) \cong \C[\chi] \otimes \bigwedge^\bullet[\eta]$, where we can pick any non-zero element $\eta \in H^{\semiinfty+3}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa(0))$. This settles the case of $\tilde{\mathbf F}_\varkappa$.
To get the generators of $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa)$, we pick non-zero representatives $$\xi_{-1} \in H^{\semiinfty+1}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(-1)), \qquad
\eta_0 \in H^{\semiinfty+1}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa(0)),$$ such that $\eta_0$ is not proportional to $\chi \cdot \xi_{-1}$. One can check that $\eta_0 \xi_{-1} \ne 0$, and as in Theorem \[thm:classical cohomology\] it follows that $H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa) \cong \C[\chi] \otimes \bigwedge^\bullet[\xi_{-1},\eta_0]$. The statement now follows from Theorem \[thm:classical cohomology\] and Corollary \[thm:affine cohomology\].
It would be nice to get a direct proof of isomorphisms by using the techniques of the quantum Drinfeld-Sokolov reduction.
Extensions, generalizations, conjectures
========================================
Heterogeneous vertex operator algebra
-------------------------------------
As we mentioned above, the vertex algebra construction for the Virasoro algebra can be obtained from their affine analogues by applying the two-sided quantum Drinfeld-Sokolov reduction to the left and right $\ghat$-action. One can consider a similar construction where the reduction is only applied to the affine action on one side, thus leading to a vertex operator algebra with two commuting actions of $\ghat_k$ and $\vir_{\bar c}$ with appropriate $k,\bar c$. Indeed, one can see that the following gives a direct realization of such a vertex algebra.
\[thm:dual pair action\] Let $\varkappa \ne 0$. Set $k = \varkappa - h^\vee,\ \bar c = 13 + 6\varkappa + \frac 6\varkappa$, and $\check {\mathbb F}_\varkappa = \hat F(\beta,\gamma) \o \tilde{\mathbb F}_\varkappa.$ Then
1. The space $\check{\mathbb F}_\varkappa$ has a $\ghat_k \oplus \vir_{\bar c}$-module structure, defined by $$\label{eq:heterogeneous affine current}
\begin{split}
\mathbf e(z) &= \gamma(z),\\
\mathbf h(z) &= 2:\beta(z) \gamma(z): + a(z),\\
\mathbf f(z) &= -:\beta(z)^2 \gamma(z): - \beta(z) a(z) - k \beta'(z) - \mathbb Y(-2,z),
\end{split}$$ $$\label{eq:heterogeneous Virasoro current}
\bar L(z) = -\frac {:\bar a(z)^2:}{4\varkappa} + \frac{\varkappa+1}{2\varkappa} \bar a'(z)
+ \frac 1\varkappa \, \mathbb Y(-2,z) \gamma(z).$$
2. The space $\check{\mathbb F}_\varkappa$ has a compatible VOA structure with $\operatorname{rank}\check{\mathbb F}_\varkappa = 28$.
The verification of commutation relations is straightforward. We define the Virasoro quantum field by $$\label{eq:heterogeneous total Virasoro}
\mathcal L(z) = \frac 1{2\varkappa} \(\frac {:\mathbf h(z)^2:}2 + :\mathbf e(z)\mathbf f(z): + :\mathbf f(z)\mathbf e(z):\)
+ \frac {\mathbf h'(z)}2 + \bar L(z)$$ The central charge for the Sugawara construction modified by $\frac {\mathbf h'(z)}2$ is equal to $\frac {3k}{k+h^\vee}-6k$, and we compute $$\operatorname{rank}\check{\mathbb F}_\varkappa = \( \frac{3(\varkappa-h^\vee)}\varkappa - 6(\varkappa-h^\vee) \) + \(13 + \frac 6\varkappa + 6\varkappa \) = 28.$$
We will call the vertex operator algebra of Theorem \[thm:dual pair action\] the heterogeneous VOA. Note (see [@Li] and references therein) that the central charge $c=28$ appears as the critical value in the study of 2D gravity in the light-cone gauge!
The structure of the bimodule $\check {\mathbb F}_\varkappa$ in the generic case is again quite similar to the non-semisimple bimodule $\mathfrak R(G_0)$. From now on we assume that
$$\varkappa \notin \mathbb Q, \qquad k = \varkappa - h^\vee, \qquad \bar c = 13 + \frac 6\varkappa + 6\varkappa.$$
\[thm:dual pair structure\] There exists a filtration $$0 \subset \check{\mathbb F}_\varkappa^{(0)} \subset \check{\mathbb F}_\varkappa^{(1)} \subset \check{\mathbb F}_\varkappa^{(2)} =
\check{\mathbb F}_\varkappa$$ of $\ghat_k \oplus \vir_{\bar c}$-submodules of $\check{\mathbb F}_\varkappa$, such that $$\begin{aligned}
\check{\mathbb F}_\varkappa^{(2)}/\check{\mathbb F}_\varkappa^{(1)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \hat V_{-\lambda-2,k} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c}, \\
\check{\mathbb F}_\varkappa^{(1)}/\check{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P^+} \( \hat V_{\lambda,k} \o \tilde V^\star_{\bar\Delta(-\lambda-2),\bar c} \oplus
\hat V_{-\lambda-2,k} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c} \), \\
\check{\mathbb F}_\varkappa^{(0)} &\cong
\bigoplus_{\lambda\in\mathbf P} \hat V_{\lambda,k} \o \tilde V^\star_{\bar\Delta(\lambda),\bar c}.\end{aligned}$$
The heterogeneous VOA contains a vertex operator subalgebra, analogous to the classical Peter-Weyl subalgebra $\mathfrak R(G) \subset \mathfrak R(G_0)$.
There exists a subspace $\check{\mathbf F}_\varkappa \subset \check{\mathbb F}_\varkappa$, satisfying
1. $\check{\mathbf F}_\varkappa$ is a vertex operator subalgebra of $\check{\mathbb F}_\varkappa$, and is generated by the fields , , and $\mathbb Y(1,z)$. In particular, $\check{\mathbf F}_\varkappa$ is a $\ghat_k \oplus \vir_{\bar c}$-submodule of $\check{\mathbb F}_\varkappa$.
2. As a $\ghat_k \oplus \vir_{\bar c}$-module, $\check{\mathbf F}_\varkappa$ is generated by the vectors $\{\1_\lambda\}_{\lambda \in \mathbf P^+}$, and we have $$\check{\mathbf F}_\varkappa \cong \bigoplus_{\lambda \in \mathbf P^+} \hat V_{\lambda,k} \o \tilde V_{\bar\Delta(\lambda),\bar c}.$$
The proofs of the above theorems are obtained from their affine counterparts by applying the quantum Drinfeld-Sokolov reduction to (right) $\ghat_{\bar k}$-action, similarly to the Virasoro case.
The fact that the ranks of VOAs $\check{\mathbb F}_\varkappa$ and $\check{\mathbf F}_\varkappa$ are equal to 28 naturally leads to the consideration of the semi-infinite cohomology of these modules. We note that although the total Virasoro quantum field $\mathcal L(z)$ does not commute with $\ghat$, the spaces $\check{\mathbb F}_\varkappa$ and $\check{\mathbf F}_\varkappa$ can be regarded as modules over the semi-direct product $\virghat$, such that $$\begin{split}
[\mathcal L_m, \mathbf e_n] &= -(n+m+1)\, \mathbf e_{m+n},\\
[\mathcal L_m, \mathbf f_n] &= (m-n+1)\, \mathbf f_{m+n},\\
[\mathcal L_m, \mathbf h_n] &= -n\, \mathbf h_{m+n} + m(m+1)\delta_{m+n,0}\, k.
\end{split}$$
The semi-infinite cohomology is defined by the BRST complex of the subalgebra $\virnil$, where as before $\nhat_+ = \bigoplus_{n \in \Z} \C \mathbf e_n$ is the nilpotent loop subalgebra of $\ghat$. The condition $c=28$ ensures that the differential squares to zero.
The BRST complex, associated with a $\virnil$-module $\check V$ with Virasoro central charge $c=28$ is the complex $C^{\semiinfty+\bullet} (\vir, \C\mathbf c;\check V) = \boldsymbol{\tilde\Lambda}^{\semiinfty+\bullet} \o \hat\Lambda(\psi,\psi^*) \o \check V$, with the differential $$\label{eq:heterogeneous differential}
\begin{split}
\check{\mathbf d} &= \sum_{n\in\Z} c_{-n} \pi_{\check V}(\mathcal L_n) - \frac 12 \sum_{m,n\in \Z} (m-n) :c_{-m} c_{-n} b_{m+n}: + \\
&+ \sum_{n \in \Z} \psi^*_{-n} \pi_{\check V}(\mathbf e_n) - \sum_{m,n\in\Z} -(m+n+1) c_{-m} :\psi^*_{-n} \psi_{m+n}:
\end{split}$$ The corresponding cohomology is denoted $H^{\semiinfty+\bullet}(\virnil,\C \mathbf c;\check V)$.
The vertex algebras $H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa)$ and $H^{\semiinfty+\bullet}(\virnil,\C\mathbb c; \check{\mathbf F}_\varkappa)$ degenerate into commutative algebra structures, and we have isomorphisms $$H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa) \cong H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbf F}_\varkappa),$$ $$H^{\semiinfty+\bullet}(\virnil,\C\mathbf c; \check{\mathbb F}_\varkappa) \cong H^{\semiinfty+\bullet}(\vir,\C\mathbf c; \tilde{\mathbb F}_\varkappa).$$ In particular, $$H^{\semiinfty+0}(\virnil,\C\mathbf c; \check{\mathbf F}_\varkappa) \cong
H^{\semiinfty+0}(\virnil,\C\mathbf c; \check{\mathbb F}_\varkappa) \cong \C[\mathbf P]^W.$$
We use the technique from [@Li], where similar isomorphisms were established for relative cohomology spaces. Let $\mathbf d_{\n_+} = \sum_{n \in \Z} \psi^*_{-n} \pi(\mathbf e_n)$ be the BRST differential for $\n_+$; one can show that $\mathbf d_{\n_+}^2 = 0 = \{\mathbf d_{\n_+},\check{\mathbf d}\}$, which leads to the spectral sequence associated with decomposition $\check{\mathbf d} = \mathbf d_{\n_+} + (\check{\mathbf d} - \mathbf d_{\n_+})$. Computing the cohomology with respect to $\mathbf d_{\n_+}$ first, and using Proposition \[thm:Wakimoto to FF\], we get the desired statement. For full technical details (there is a slight difference between BRST reduction for $\n_+$ and quantum Drinfeld-Sokolov reduction, but it doesn’t affect the outcome) we refer the reader to [@Li].
General construction of vertex operator algebras and equivalence of categories
------------------------------------------------------------------------------
The vertex operator algebras constructed in the previous sections can be built, like conformal field theories, by pairing the left and right modules from certain equivalent categories of representations of infinite-dimensional Lie algebras. The operators $\mathcal Y(\cdot,z)$ are then constructed by pairing the left and right intertwining operators. There is a unique choice of the structural coefficients for such pairing that would ensure the locality condition for the vertex operator algebras; these coefficients are determined by the tensor structure on the category of representations. Conversely, a natural VOA structure on a bimodule can be used to establish the equivalence of the left and right tensor categories.
The vertex operator algebra constructions in this paper deal with the pairings of different categories of modules. In the affine case, we pair the $\ghat$ modules on levels $k$ and $\bar k = -2h^\vee - k$, symmetric with respect to the critical level $-h^\vee$; the equivalence of the corresponding tensor categories was studied in [@Fi]. In the Virasoro case, we pair the modules with central charges $c$ and $\bar c = 26-c$.
The theorems of Peter-Weyl type can be extended to the quantum group $\mathcal U_q(\g)$, associated with $G$. On one hand, the modules from the category $\mathcal O$ can be $q$-deformed into modules over $\mathcal U_q(\g)$; on the other hand, one can define $q$-deformations $\mathfrak R_q(G)$ and $\mathfrak R_q(G_0)$ of the algebras of regular functions, which have especially simple description for $\g = \sl(2,\C)$. When $q$ is not a root of unity, we have the quantum analogues of isomorphisms , .
The Drinfeld-Kohno theorem establishes an isomorphism of tensor categories of representations of the quantum groups and affine Lie algebras when $q = \exp\(\frac{\pi i}{k+h^\vee}\)$. This equivalence, also extended to $\mathcal W$-algebras, was made explicit in [@S], where intertwining operators for $\mathcal U_q(\g)$ were directly identified with their VOA counterparts for $\ghat_k$ and $\mathcal W(\ghat_k)$; the key ingredients were the geometric results in [@V] on the homology of configuration spaces. The construction in [@S] built conformal field theories, associated to affine Lie algebras and $\mathcal W$-algebras based on their quantum group counterparts, and can be modified to produce the vertex algebras discussed in this paper.
The Drinfeld-Kohno equivalence also allows to couple categories of different types, producing in particular the heterogeneous VOA of the previous subsection. Another important case is the Frenkel-Kac construction, which corresponds to the pairing of modules for $\g$ and $\mathcal W(\ghat)$ with central charge $c=\dim \h$ (see [@F]). However, in general pairings between the quantum group and the affine Lie (or $\mathcal W$-) algebra lead to the generalized vertex algebra structures, satisfying a braided version of the commutativity axiom. An example of such structure was proposed in [@MR].
Integral central charge, semi-infinite cohomology and Verlinde algebras
-----------------------------------------------------------------------
In this work we studied the structure of the generalized Peter-Weyl bimodules for $\ghat$ only for the generic values $k \notin \mathbb Q$ of the central charge. The structure of these bimodules when $k$ is integral is more complex and undoubtedly even more interesting. In the most special case when $k = \bar k = - h^\vee$, we get a regular representation of the affine Lie algebra $\ghat$ at the critical level, which can be viewed as the direct counterpart of the finite-dimensional case. This space admits a realization as a certain space of meromorphic functions on the affine Lie group $\hat G$, and subspaces of spherical functions with respect to conjugation give rise to solutions of the quantum elliptic Calogero-Sutherland system, generalizing the trigonometric analogue in the finite-dimensional case [@EFK].
Another special case is $k = - h^\vee+1, \bar k = -h^\vee - 1$, when the quantum group degenerates into its classical counterpart. In this case the left and right Fock spaces used in our construction each have separate vertex algebra structures, and the operators $\mathbb Y(-2,w)$, which play an important role in this paper, are factored into products of left and right vertex operators used in the basic representations of $\ghat$. The semi-infinite cohomology of the corresponding $\mathcal W$-algebras is fundamental to the string theory, and was studied in [@WZ] for the Virasoro algebra, and in [@BMP] for $\mathcal W_3$.
For positive integral $k$ one expects the existence of truncated versions $\hat{\mathbf F}_{k+h^\vee}$ and $\hat{\mathbb F}_{k+h^\vee}$ of our vertex operators algebras, similar to the truncation in the conformal field theory, where the positive dominant cone $\mathbf P^+$ is replaced by the alcove $\mathbf P^+_k \subset \mathbf P^+$. Then the relative semi-infinite cohomology of $\ghat$ with coefficients in $\hat{\mathbf F}_{k+h^\vee}$ and $\hat{\mathbb F}_{k+h^\vee}$ with respect to the center should be truncated correspondingly. The identification of the zero semi-infinite cohomology groups with the representation ring of $G$ in Corollary \[thm:affine cohomology\] leads to the following conjecture.
\[thm:conjecture\] For positive integral $k$, let $\mathbf V_k(\ghat)$ denote the Verlinde algebra associated with integrable level $k$ representations of $\ghat$, and let $\mathbb V_k(\ghat)$ denote its counterpart associated to the big projective modules (see [@La]). Then we have commutative algebra isomorphisms $$H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbf F}_{k+h^\vee}) \cong \mathbf V_k(\ghat), \qquad \qquad
H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbb F}_{k+h^\vee}) \cong \mathbb V_k(\ghat).$$
In other words, the most essential part of the VOA structure, embodied in the 0th cohomology, is equivalent to the structure of the fusion rules of the tensor category of $\ghat$-modules, encoded in the Verlinde algebra.
It was also realized recently (see [@FHT] and references therein) that the Verlinde algebra $\mathbf V_k(\ghat)$ admits an alternative realization in terms of twisted equivariant K-theory ${}^{k+h^\vee}K_G^{\dim G}(G)$ of a compact simple Lie group $G$ (which, in the notations of [@FHT], is the compact form of the complex Lie group which we denoted by $G$ in this paper).
Thanks to the results of [@FHT], the first isomorphism of our conjecture can be restated in a more invariant form.
We have a natural commutative algebra isomorphism $$H^{\semiinfty+0}(\ghat,\C\mathbf k; \mathfrak R'_{k+h^\vee}(\hat G)) \cong {}^{k+h^\vee}K_G^{\dim G}(G).$$
A conceivable direct geometric proof of the last isomorphism might combine the realization of the left hand side using the works [@GMS] and [@AG] with the interpretation of the right hand side given in the works [@FHT] and [@AS]. A similar K-theoretic interpretation of $H^{\semiinfty+0}(\ghat,\C\mathbf k; \hat{\mathbb F}_{k+h^\vee})$ in our second conjecture might add another twirl to the twisted equivariant K-theory.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper we give an elementary proof of uniqueness of solutions to a gas-disk interaction system with diffusive boundary condition. Existence of near-equilibrium solutions for this type of systems with various boundary conditions has been extensively studied in [@ACMP2008; @CS2014; @CS2015; @CMP2006; @CCM2007; @SR2014; @K2018; @C2007; @K2018-2]. However, the uniqueness has been an open problem, even for solutions near equilibrium. Our work gives the first rigorous proof of the uniqueness among solutions that are only required to be locally Lipschitz; in particular, it holds for solutions far from equilibrium states.
**Keywords:** kinetic equations, integro-differential equations, uniqueness, gas-body interaction, friction.
address:
- 'Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada'
- 'Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada'
author:
- Anton Iatcenko
- Weiran Sun
bibliography:
- '../Global/GasDisk.bib'
title: 'Uniqueness of Solutions to a Gas-Disk Interaction System'
---
Introduction
============
The main goal of this paper is to show uniqueness of solutions to a gas-disk interaction system. This system describes the motion of a disk immersed in a collisionless gas. Among many ways to model the friction between the gas and the disk, the simplest one is to assume that the friction is proportional to the velocity of the disk. In this scenario the velocity of the disk can be found by solving a linear ODE. Here we consider a more realistic model as in [@ACMP2008; @CS2014; @CS2015; @CMP2006; @CCM2007; @SR2014; @K2018; @C2007; @K2018-2; @TA2012; @TA2013; @TA2014], where the evolution of the gas and the disk satisfies a coupled system of integro-differential equations. The coupling is through collisions of gas particles with the disk: these collisions produce a drag force on the disk through momentum exchange and provide a boundary condition for the gas.
In this paper we make a simplifying assumption that the disk is infinite. Together with assumed symmetry this lets us reduce the whole system to one dimension, thus making the disk a single point moving along the horizontal axis. To specify the model we let $f(x, v, t)$ be the density function of the gas that evolves according to the free transport equation away from the disk: $$\begin{aligned}
\label{eq:gas}
\del_t f + v \, \del_x f = 0 , \qquad f(x, v, 0) = \phi_0(v) ,\end{aligned}$$ where $(x, v, t) \in \R \times \R \times \R^+$ are position, velocity, and time respectively. Denote the position of the disk at time $t$ by $\eta(t)$ and its velocity by $p(t)$. The interaction of the gas with the disk is described by a diffusive boundary condition: $$\begin{aligned}
f^+_R(\eta(t), v, t) &= 2 e^{-(v - p(t))^2} \int_{-\infty}^{p(t)} (p(t) - w) f^-_R(\eta(t), w, t) \dw,
\quad v > p(t), \label{BC:R} \\
f^+_L(\eta(t), v, t) &= 2 e^{-(p(t) - v)^2} \int^{ \infty}_{p(t)} \, \ (w - p(t)) f^-_L(\eta(t), w, t) \dw,
\quad v < p(t), \label{BC:L}\end{aligned}$$ where the sub-indices $R$ and $L$ denote the right and left sides of the disk. Throughout the paper superscripts $+$ and $-$ on the density functions denote the postcollisional and precollisional distributions respectively, understood as one-sided limits: $$\begin{aligned}
\label{def:f_pm}
f^\pm(\eta(t), v, t) = \lim_{\Eps \to 0^+} f^\pm(\Eta(t)\pm \Eps v, v, t \pm \Eps) .\end{aligned}$$
The diffusive boundary conditions - essentially state that shape of the outgoing distribution is always Gaussian, with coefficients chosen to ensure the conservation of mass. Therefore, our model considers the case where collisions are instantaneous and the disk does not capture any finite mass of the gas via the collision process.
We assume that the disk is acted on by an external force $F(x, t)$ and the drag force $G_p(t)$ generated through collisions with the gas particles (we have associated the drag force with the sub-index $p$ to emphasize its dependence on the disk velocity $p$). Then the motion of the disk is described by $$\begin{aligned}
{2}
\dot p &= F(\eta(t), t) - G_p(t), \qquad &&p(0) = p_0, \label{eq:disk} \\
\dot \eta &= p(t), &&\eta(0) = 0. \label{eq:disk-eta}\end{aligned}$$ We will write the total drag force as a combination of the drag forces due to particles colliding with the disk from the right and left: $$\begin{aligned}
\label{def:drag-full}
G_p(t) = G_{p, R}(t) - G_{p, L}(t).\end{aligned}$$ The signs are chosen to make both components of the drag positive. Physically speaking, $G_{p, L}$ accelerates the disk and $G_{p, R}$ decelerates it. Their exact expressions are derived from Newton’s Second Law (see [@CS2014] for details): $$\begin{aligned}
G_{p, R}(t) &:= \int^{p(t)}_{-\infty} (p(t) - v)^2 f^-_R(t, \Eta(t), v) \dv +
\int^\infty_{p(t)} (v - p(t))^2 f^+_R(t, \Eta(t), v) \dv \,, \label{def:drag-R}\\
G_{p, L}(t) &:= \int^{p(t)}_{-\infty} (p(t) - v)^2 f^+_L(t, \Eta(t), v) \dv +
\int^\infty_{p(t)} (v - p(t))^2 f^-_L(t, \Eta(t), v) \dv \,. \label{def:drag-L}\end{aligned}$$ The evolution of the complete gas-disk system is governed by equations -. We comment that the derivation of - relies on the Reynolds transport theorem, which assumes that the exchange of momentum between the gas and the disk can only happen through the fluxes of the gas moving into and out of the disk. Hence any particle that stays on the disk does not contribute to the momentum exchange or the drag force. We also note that to have an interaction with the disk the particle to the right (left) of it must be moving slower (faster) than the disk.
Gas-body coupled systems have been extensively studied both numerically and analytically with pure diffusive, specular, and more generally, the Maxwell boundary conditions ([@ACMP2008; @TA2012; @TA2013; @TA2014; @CS2014; @CS2015; @CMP2006; @CCM2007; @SR2014; @K2018; @C2007; @K2018-2]). We refer the reader to a recent paper [@K2018] for a comprehensive list of references. Among the central questions for these systems are their well-posedness and long-time behaviour. Regarding the long-time asymptotics, it is now fairly well-understood that due to the effect of re-collisions, the relaxation of the disks velocity toward its equilibrium state may not be exponential as in the simplified model where re-collisions are ignored. In fact, one may obtain algebraic decay rates [@ACMP2008; @TA2012; @TA2013; @TA2014; @CS2014; @CS2015; @CMP2006; @CCM2007; @SR2014; @K2018; @C2007; @K2018-2]. Moreover, depending on the shape of the body, such rates may or may not depend on the spatial dimension [@C2007; @SR2014].
The well-posedness issue, however, is less understood. To the best of our knowledge, existence of solutions has only been investigated for data near equilibrium states [@ACMP2008; @CS2014; @CS2015; @CMP2006; @CCM2007; @SR2014; @K2018; @C2007; @K2018-2] and uniqueness has been an open question even for these solutions. It is our goal in this paper to give a uniqueness proof for solutions to -, where the disk velocity $p$ only needs to be in the natural space of locally Lipschitz functions. This includes solution spaces considered in [@ACMP2008; @CS2014], as well as more general cases with solutions far from an equilibrium. The main result of this paper is
\[thm:main\] Suppose the initial density $\phi_0 \in L^1 \cap L^\infty (\R)$ and the external force $F(x, t)$ is Lipschitz in $x$ with its Lipschitz coefficient independent of $t$. Then for any $p_0 \in \R$ there exists at most one solution $(\eta, p, f)$ to the system - such that $p$ is locally Lipschitz.
Our main step in proving the main theorem is to show that the drag force due to recollisions, denoted by $G_\text{rec}$, is Lipschitz in the velocity $p$ (Proposition \[prop:Lip-G\]). The main difficulty for such estimate is the dependence the distribution of the recolliding particles on the entire history of the disk motion. We address this issue by taking advantage of the inherently recursive nature of the problem: the distribution of the particles colliding with the disk for the $n^\text{th}$ time at time $t$ is determined by the distribution of the particles colliding with the disk for the $(n-1)^\text{th}$ time at some earlier time $s$. Instead of trying to compute or estimate such $s$ for a given velocity $v$, we use a change of variables $v = v(s, t)$. This allows us to compare the particles that have collided with the disk at the same time in the past instead of comparing particles that have the same velocity at the current time.
Three remarks are in order: first, we have assumed that the initial state of the gas is spatially homogeneous. This assumption can be dropped at the cost of adding more technicalities. Second, due to the essential step of change of variables, so far our technique is only applicable to the case with diffusive boundary conditions. Hence for systems with specular or Maxwell boundary conditions uniqueness is still an open question. Third, this paper only deals with the one-dimensional case with a collisionless gas, but we expect a similar strategy to be applicable in higher dimensions and for systems with simple collisions such as the special Lorenz gas in [@TA2012]. This will be subject to future investigation.
The rest of the paper is laid out as follows: in Section \[sec:Assump\] we state our assumptions, introduce partition of the density function and the change of variables, and reformulate the density function and the drag force into recursive forms. Section \[sec:Assump\] contains the essential ideas and constructions that will be used in various estimates and the uniqueness proof in the later part. In Section \[sec:prelim\_bnds\] we obtain preliminary bounds on the density function using the recursive form. Finally, in Section \[sec:uniq\] we establish the Lipschitz property of the drag force and give a proof of the uniqueness theorem.
Assumptions and Reformulations {#sec:Assump}
==============================
In this section we state all the assumptions used to prove the uniqueness of the solution. We also introduce several reformulations of the density function $f$ as well as the drag term $G$. Most of the discussion here is built upon the understanding of the physics underlying the interactions of the gas particles with the disk.
Throughout this paper we let $T$ be a fixed arbitrary time.
Main Assumptions
----------------
The assumptions on the system are
1. \[a0\] Particles cannot penetrate the disk.
2. \[a1\] *Assumptions on the gas:* the initial distribution $\phi_0 = \phi_0(v)$ satisfies
1. $\phi_0 \in L^\8 (\R)$;
2. The zeroth, first and second moments of $\phi_0$ are finite: $$\begin{aligned}
\int_\R (1 + v^2)\phi_0(v) \dv < \8.\end{aligned}$$
3. \[a2\] *Assumption on the disk:* velocity of the disk is locally Lipschitz with $$\begin{aligned}
\label{const:Lip}
\| p \|_{L^\8(0, T)} + \| \dot p \|_{L^\8(0, T)} \leq M, \qquad \end{aligned}$$ where $M$ may depend on $T$.
Reformulation of the Model {#subsec:reform}
--------------------------
For the rest of the paper we will only consider the gas to the right of the disk since the analysis for the gas to the left of the disk is analogous. This lets us drop the sub-indices $R$ and $L$ in - and -.
We begin by simplifying the expression for the drag forces. The expression for the outgoing density in allows us to write $$\begin{aligned}
\int^\8_{p(t)} (v - p(t))^2 f^+(\Eta(t), v, t) \dv
&= 2 \vpran{\int_{-\8}^{p(t)} (p(t) - v) f^-(\eta(t), v, t) \dv} \int^\8_{p(t)} (v - p(t))^2 e^{-(v - p(t))^2} \dv \\
&= \frac{\sqrt{\pi}}{2} \int_{-\8}^{p(t)} (p(t) - v) f^-(\eta(t), v, t) \dv,\end{aligned}$$ so the expression for the drag force can be written as $$\begin{aligned}
\label{def:drag}
G_p(t) &= \int^{p(t)}_{-\8} \vpran{(p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v)} f^-(\eta(t), v, t) \dv . \end{aligned}$$
### Partition of the density function {#sec:gas_decomp}
To make the drag term more amiable to analysis, we introduce the idea of recursive scattering: for $x\neq \eta(t)$ let $f_n(x, v, t)$ be the density functions of the particles that have collided with the disk exactly $n$ times in the past. Away from the disk they satisfy the same free transport equation as $f$. For $x = \Eta(t)$ we define $f_n^\pm(x, v, t)$ in terms of the one-sided limits similar to those in : $$\begin{aligned}
\label{def:fn_pm}
f_n^\pm(\eta(t), v, t) = \lim_{\Eps \to 0^+} f_n^{\pm}(\Eta(t)\pm \Eps v, v, t \pm \Eps).\end{aligned}$$ The boundary conditions on $f_n$’s are similar to those for the full density function $f$, with the exception that the collision with the disk now increments the sub-index. In particular, for $v > p(t)$ and $n \geq 0$ we write $$\begin{aligned}
\label{def:fn_v}
f_{n+1}^+(\eta(t), w, t) &= 2 e^{-(w - p(t))^2} \int_{-\8}^{p(t)}\ (p(t) - v) f_{n}^-(\eta(t), v, t) \dv .\end{aligned}$$ We also define to be the density function of the particles that have collided with the disk in the past: $$\begin{gathered}
\label{def:frec}
\frec(x, v, t) = \sum_{n \geq 1} f_n(x, v, t) \,.\end{gathered}$$ Thus the full density function is decomposed as $$\begin{aligned}
f(x, v, t) = \phi_0(v) + \frec(x, v, t) \,.\end{aligned}$$ Similarly, we define $G_{p, \text{rec}}$ to be the drag forces due to particles that have collided with the disk in the past: $$\begin{aligned}
\label{def:Grec}
G_{p, \text{rec}}(t) &=
\int^{p(t)}_{-\8} \vpran{(p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v)} \frec(\eta(t), v, t) \dv .\end{aligned}$$
### Average Velocity
We now address the possibility for the particles to collide with the disk multiple times. Throughout the paper we adopt the following notation for the average velocity of the disk on the time interval $[s, t]$: $$\begin{aligned}
\vint{p}_{s, t} = \frac{1}{t-s} \int_s^t p(\tau) \dtau .\end{aligned}$$ It will play a significant role in the precollision conditions and the change of variables. We summarize a few useful properties of the average velocity in the following lemma:
\[lem:technical\] Suppose $t \in (0, T)$ and $s \in (0, t)$. Let $v(\cdot, \cdot)$ be the function defined as $$\begin{gathered}
\label{def:preco-time}
v(s, t) = \vint{p}_{s, t}.\end{gathered}$$ Let $M$ be the Lipschitz constant in . Then
1. \[prop:vst\_Linf\] $v(s, t)$ satisfies the bound $|v(s, t)| \leq M$;
2. \[prop:vst\_deriv\] the derivatives of $v(s, t)$ are $$\begin{aligned}
\label{formula:deriv-v}
\dds{v} = \frac{v - p(s)}{t-s} , \qquad
\dds[t]{v} = \frac{p(t) - v}{t-s} ;\end{aligned}$$
3. \[prop:vst\_deriv\_bnd\] the derivatives of $v$ satisfy the following estimates: $$\begin{aligned}
\label{bnd:dv}
\abs{\dds{v}} \leq \frac{1}{2}M , \qquad
\abs{\dds[t]{v}} \leq \frac{1}{2}M . \end{aligned}$$
Part \[prop:vst\_deriv\] follows from direct computations. Parts \[prop:vst\_Linf\] and \[prop:vst\_deriv\_bnd\] follow from Assumption \[a2\]: $$\begin{gathered}
|v(s, t)| = \abs{ \frac{1}{t-s} \int_s^t p(\tau) \dtau } \leq
\frac{1}{t-s} \int_s^t M \dtau = M \,, \\[3pt]
\abs{\frac{\del v}{\del t}} = \frac{\abs{p(t) - v(s, t)}}{t-s}
\leq \frac{1}{(t-s)^2} \int_s^t \abs{p(t) - p(\tau)} \dtau
\leq \frac{\norm{\dot p}_{L^\infty(0, t)}}{(t-s)^2} \int_s^t (t - \tau) \dtau
= \frac{1}{2} M .\end{gathered}$$ The estimate for $\partial_s v$ is proved via a similar calculation.
### Precollisional Velocities and Precollision Times
In this section we prepare for the key step of change of variables. To illustrate the idea of change of variables, we consider for a moment a simplified case where $\dot p(t) > 0$ for all $t$. Then for each $t \in (0, T)$ the average velocity $\vint{p}_{s, t}$ is strictly increasing in $s$, and thus is a bijection between $[0, t]$ and $[\vint{p}_{0, t}, \, \vint{p}_{t, t}] = [\eta(t)/t, \, p(t)]$. This allows us to use the change of variables $v = v(s, t) = \vint{p}_{s, t}$ in to obtain the following expression for $n \geq 1$ and $w > p(t)$: $$\begin{aligned}
f^+_{n+1}(\eta(t), w, t) &= 2 e^{-(w - p(t))^2} \int_{\eta(t)/t}^{p(t)} (p(t) - v) f^-_n(\eta(t), v, t) \dv
\label{eq:pmon_v} \\[2pt]
&= 2 e^{-(w - p(t))^2} \int_0^t \dds{v} (p(t) - v(s, t)) f^-_n(\eta(t), v(s, t), t) \ds . \label{eq:pmon_s} \end{aligned}$$ One immediate advantage of expression is that it allows us to obtain an explicit recursive relationship between the sequence of outgoing densities $\{f_n^+\}$. Indeed, since the distribution density does not change between collisions, we have $$\begin{aligned}
f^-_n(\eta(t), v(s, t), t) = f^+_n(\eta(s), v(s, t), s).\end{aligned}$$ This in turn implies $$\begin{aligned}
f^+_{n+1}(\eta(t), w, t)
= 2 e^{-(w - p(s))^2} \int_0^t \dds[\tau]{v(s, t)} (p(t) - v(s,t)) f^+_{n}(\eta(s), v(s, t), s) \ds .\end{aligned}$$ In Sections \[sec:prelim\_bnds\] and \[sec:uniq\] we show the full usage of a similar recursive relation for obtaining the estimates for the density function and the drag term.
Without the monotonicity assumption a proper change of variables requires much more work. The main difficulty is the non-injectivity of the mapping $v(\cdot, t)$ defined in . To handle it, we start by identifying that, among all the particles that are to collide with the disk at time $t$, which ones have had a collision in the past. Velocities of such particles will henceforth be called *precollisional*, to signify that the corresponding particles have previously collided with the disk. They must satisfy the following condition:
> There exists time $s \in [0, t)$ such that the particle and the disk have travelled the same distance over $[s, t]$ and $v < p(t)$.
Since the velocity of the particle does not change between consecutive collisions, the above condition can be written as $$\begin{gathered}
\label{cond:same_dist}
(t-s) v = \int_s^t p(\tau) \, \dtau \qquad \text{or} \qquad v = \vint{p}_{s, t} .\end{gathered}$$ Introduce the notation $$\begin{aligned}
\ukt = \min_{\mathsmaller{s \in [0, t]}} \vint{p}_{s, t} \,.\end{aligned}$$ Then the precollisional velocities can be characterized as
Suppose a particle with velocity $v$ is colliding with the disk at time $t$ and $v \neq \ukt$. Then it has collided with the disk in the past if and only if $$\begin{gathered}
\label{precol_cond}
\ukt < v < p(t).\end{gathered}$$
Let $\ukt < v < p(t)$. Since $\vint{p}_{s, t}$ is a continuous function of $s$ for any $t$, it must obtain its minimum at some $s^* \in [0, t]$. Assume $s^* < t$ and suppose for contradiction that the particle with velocity $v$ has not collided with the disk in the past. Let $\omega(s)$ be the position of the particle. Then $$\begin{aligned}
\omega(s) = \eta(t) - (t-s)v.\end{aligned}$$ Since the particle is colliding with the disk from the right and could not have penetrated the disk by assumption \[a0\], it must have been to the right of the disk for all $s \in [0, t)$, that is $$\begin{gathered}
\omega(s) - \eta(s) \geq 0 \quad \forall s \in [0, t).\end{gathered}$$ However, this condition is violated at $s = s^*$ since $$\begin{aligned}
\omega(s^*) - \eta(s^*) &= \big[ \eta(t) - (t-s^*)v \big] - \big[ \eta(t) - (t-s^*)\ukt \big] \nn
\\
&= \big[ \ukt - v \big](t-s^*) < 0, \label{precol_cond_strict_ineq}\end{aligned}$$ which is a contradiction. If $s^* = t$, then $\ukt = \vint{p}_{t, t} = p(t)$. This again violates . The converse is an immediate consequence of .
Denote the set of all possible precollisional velocities by where $$\begin{gathered}
\label{def:Vt}
\Vt = \big( \ukt, \, p(t) \big).\end{gathered}$$
We now identify the times of the precollisions.
\[def:precoll\_time\] Suppose a particle with velocity $v \in \Vt$ is to collide with the disk at time $t$. Then the time $s_v$ is a corresponding *precollision time* if $(v, s_v)$ satisfies and the particle has been ahead of the disk for all $\tau \in (s, t)$, that is, $$\begin{gathered}
\label{cond:ahead}
\eta(t) - (t-\tau)v \geq \eta(\tau) \qquad \fa \tau \in (s, t). \end{gathered}$$ Note that this condition is a mathematical formulation of Assumption \[a0\].
Let be the set of all possible precollision times. To construct a bijection between and we need a more explicit characterization of the latter. To this end, we first rewrite as $$\begin{aligned}
\label{cond:ahead-1}
v \leq \vint{p}_{\tau, t} \qquad \fa \tau \in (s, t) , \\[-25pt] \nn\end{aligned}$$ which in turn implies $$\begin{gathered}
\label{cond:ahead-2}%\label{def:vbar}
v \leq \min_{\tau \in [s, t]}\vint{p}_{\tau, t} .\end{gathered}$$ Motivated by , we define the modified average velocity : $$\begin{gathered}
\label{def:vbar}
\vm := \min_{\tau \in [s, t]} \vint{p}_{\tau, t}.\end{gathered}$$ From the combination of with we conclude that $s_v$ is a precollision time corresponding to $v$ if and only if $v = v(s, t) = \vm$. Consequently, we have $$\begin{gathered}
\label{def:N-t}
\Nt = \set[{s \in [0, t]}]{ v(s, t) = \vm}. \end{gathered}$$ Note the function $\vm[\cdot, t]$ is monotonically, but not necessarily strictly, increasing. It can be thought of as the tightest monotonically increasing lower envelope for $v(s, t)$; the notation had been chosen to reflect that. We give an example of and in Figure \[fig:vm\] to help intuitive understanding of their properties.
*Notation.* For a given $t$, we will use to denote the image of the set $\mc A$ under the map and $\vme^{-1}(\mc B, t)$ to denote the pre-image of the set $\mc B$ under the map . Note that the inversion is only performed in the first variable with the second variable $t$ fixed.
We now establish properties of and . A large part of the analysis is essentially Riesz’s rising sun lemma [@leoniSS] with a sign change.
\[le:Ntc\_open\] Fix $t \in [0, T]$ and let be the set defined in . Then $\Ntc = [0, t] \setminus \Nt$ is open in $[0, t]$.
Note that since $\vm[t, t] = v(t, t) = p(t)$, we have $t \in \Nt$. We write as $$\begin{gathered}
\Ntc = \set[{s \in [0, t)}]{ v(s, t) > \vm }
= \set[{s \in [0, t)}]{ v(s, t) > v(\tau, t) \text{ for some } \tau \in (s, t) }
= \bigcup_{\tau \in [0, t)} \hspace{-2mm} \Or_\tau, \\[-25pt]\end{gathered}$$ where . Since $v(\cdot, t)$ is continuous, the pre-image $\Or_\tau$ is open in the subspace topology on $[0, t]$. Therefore is an open subset of $[0, t]$.
![An example of the average velocity $v(s, t)$ and the corresponding modified average velocity , with $t$ fixed. Note how the difference between the two informs the separation of the time domain into and .[]{data-label="fig:vm"}](vbar.jpg)
\[le:vmprops\] Let $\vm$ be the modified velocity defined in . Then
1. \[le:cont\] $\vm[\cdot, t]: [0, t] \to \Vt$ is continuous.
2. \[le:conn\_comp\] Let $(a, b)$ be a maximal connected component of . Then for all $s \in (a, b)$ we have $$\begin{gathered}
\vm[a, t] = \vm[s, t] = \vm[b, t].\end{gathered}$$ Consequently, $\del_s \vm = 0$ on $\Nt^c$.
3. \[le:range\] Restricting to does not change its range: $\vm[{[0, t]}, t] = \vm[\Nt, t]$.
4. \[le:Lip\_s\] For any fixed $t \in [0,T]$, is Lipschitz in $s$ with $\abs{\del_s \vm[s,t]} \leq M/2$ for almost every $s \in [0, T]$.
5. \[le:Lip\_t\] For any fixed $s \in [0, t]$, is Lipschitz in $t$ with $\abs{\del_t\vm[s, t]} \leq M$ for almost every $t \in [0, T]$.
6. \[le:Dt\] Let $\mc{L}(A)$ be the Lebesgue measure of $A$ and define $$\begin{gathered}
\mc D_t := \set[{s \in [0, t]}]{ \dds{\vm} \text{ exists} }. \\[-28pt]\end{gathered}$$ Then $ \mc{L} ( \vm[ \mc D_t^c, t ] ) = \mc{L} ( v(\mc D_t^c, t ) ) = 0 $.
7. \[le:deriv-vm\] For all $s \in \Nt \cap \mc D_t$ we have $ \del_s \vm = \del_s v(s, t)$.
\[le:cont\] Let $\Eps > 0$ be given. By the uniform continuity of $v(\cdot, t)$ on $[0, t]$, there exists $\delta > 0$ such that $$\begin{aligned}
\label{bound:unif-v-s-t}
|v(s, t) - v(s', t)| < \Eps \quad \text{whenever} \quad |s-s'|<\delta .\end{aligned}$$ Without loss of generality, suppose $0 \leq s - s' \leq \delta$. Then by definition of we have $$\begin{aligned}
0
\leq \vm - \vm[s', t]
&= \vm - \min \{\min_{\tau \in [s', s]}v(\tau, t), \,\, \vm\}
\\
&= \vpran{\vm[s, t] - \min_{\tau \in [s', s]} v(\tau, t)}
\Sgn\Big(\vm[s, t] - \min_{\tau \in [s', s]} v(\tau, t) \Big)
\\
&\leq \Big|v(s, t) - \min_{\tau \in [s', s]} v(\tau, t)\Big| < \Eps ,\end{aligned}$$ where the last inequality follows from .
\[le:conn\_comp\] Since $\vm[\cdot, t]$ is non-decreasing we have $\vm \leq \vm[b, t]$. Suppose for contradiction that $\vm < \vm[b, t]$. Then there must exist $\tau \in [s, b)$ such that $\vm = v(\tau, t)$, which in turn implies that $\vm[\tau, t] = v(\tau, t)$. But then $\tau \in \Nt$ by the definition of $\Nt$, which is a contradiction since $\tau \in (a, b) \subseteq \Ntc$. Equality $\vm[a, t] = \vm$ follows from continuity of .
\[le:range\] Take $s \in \Ntc$ and let $(a, b)$ be the largest connected subset of $\Ntc$ containing it. By part \[le:conn\_comp\], we have $\vm = \vm[b, t]$. Thus $\vm[s, t] \in \vm[\Nt, t]$ since $b \in \Nt$.
\[le:Lip\_s\] For $s \in \Ntc$ let $\left(a_s, b_s \right)$ be the largest connected component of containing $s$. In other words, the lower bound $a_s$ is the largest time less than $s$ such that $\vm[a_s, t] = v(a_s, t)$. Similarly, the upper bound $b_s$ is the smallest time greater than $s$ such that $\vm[b_s, t] = v(b_s, t)$. For $s \in \Nt$ we simply let $a_s = b_s = s$. Then for both cases we have $$\begin{gathered}
v(a_s, t ) = \vm[a_s, t] = \vm[s, t] = \vm[b_s, t] = v(b_s, t ). \end{gathered}$$
Let $\tau, \tau' \in [0, t]$ and assume, without loss of generality, that $\tau \leq \tau'$. If $(\tau, \tau') \ins \Ntc$ then $v(\tau', t) - v(\tau, t) = 0$ by part \[le:conn\_comp\]. Otherwise, by Lemma \[lem:technical\] we have $$\begin{gathered}
|\vm[\tau', t] - \vm[\tau, t] | = \abs{v(a_{\tau'}, t) - v(b_\tau, t) }
\leq \frac{M}{2} \left(a_{\tau'} - b_\tau \right)
\leq \frac{M}{2} \left(\tau' - \tau \right). \end{gathered}$$
\[le:Lip\_t\] Let $t'> t$ and fix $s \in [0, t]$. Since $v(s, t)$ is continuous, $\vm = v(\tau, t)$ for some $\tau \in [s, t]$. We have $$\begin{gathered}
\vm[s, t'] \leq v(\tau, t') \leq v(\tau, t) + \frac{M}{2}|t'-t| \ \implies \ \vm[s, t'] - \vm \leq \frac{M}{2}|t'-t|.\end{gathered}$$ On the other hand, for all $\tau \in [0, t]$ we have $$\begin{gathered}
v(\tau, t')
\geq v(\tau, t) - M|t'-t|
\ \implies \
\vm[s, t'] \geq \vm - M|t'-t|,\end{gathered}$$ Thus the function is Lipschitz it $t$. Consequently, $\partial_t \vm$ exist for almost all $t$ and $\abs{\del_t\vm[s, t]} \leq M$.
\[le:Dt\] Since is Lipschitz in $s$, it is almost everywhere differentiable and thus $\mc{L}(\mc D^c_t) = 0$. Since is absolutely continuous, it possesses the Lusin property: $ \mc{L} ( \vm[ \mc D_t^c, t ] ) = 0 $. The same argument holds for $v(\cdot, t)$.
\[le:deriv-vm\] Let $s \in \Nt \cap \mc D_t$. Then is given by the definition of the classical derivative. Therefore, $$\begin{aligned}
\dds{\vm} &= \lim_{\tau \to s^+} \frac{\vm[\tau, t] - \vm[s, t]}{\tau - s}
= \lim_{\tau \to s^+} \frac{\vm[\tau, t] - v(s, t)}{\tau - s}
\leq \lim_{\tau \to s^+} \frac{v(\tau, t) - v(s, t)}{\tau - s}
= \dds{v(s, t)}, \\[5pt]
\dds{v(s, t)} &= \lim_{\tau \to s^-} \frac{v(s, t) - v(\tau, t)}{s - \tau}
= \lim_{\tau \to s^-} \frac{\vm - v(\tau, t)}{s - \tau}
\leq \lim_{\tau \to s^-} \frac{\vm - \vm[\tau, t]}{s - \tau}
= \dds{\vm} .\end{aligned}$$ It now follows that $ \del_s \vm = \del_s v(s, t)$.
Since the measure of the set $\mc D^c_t$, as well as its images under both and $v(\cdot, t)$, is zero, we can safely ignore it from now on.
From Lemma \[le:vmprops\]\[le:range\] it follows that the map $\vm[\cdot, t] : \Nt \to \Vt$ is a surjection. However, it is not necessarily an injection, so further restriction is required. To show that the restriction we are about to make does not affect the dynamics of the disk we will need the following lemma from [@leoniSS] (page 77):
\[lem:measure\] Let $I \ins \R$ be an interval and let $u: I \to \R$. Assume that there exists a set $E \ins I$ (not necessarily measurable) and $M \geq 0$ such that $u$ is differentiable for all $x \in E$, with $$\begin{gathered}
| u'(x) | \leq M \qquad \text{for all } x \in E.\end{gathered}$$ Then $\mc{L}_\circ (u(E)) \leq M\mc{L}_\circ (E)$, where $\mc{L}_\circ$ denotes the outer Lebesgue measure.
We are now ready to make the restriction and create a bijection.
\[thm:vbar\] For any fixed $t \in [0, T]$, let be the function defined in . Let $$\begin{gathered}
\label{def:Phit_Wt}
\Phi_t := \set[{ s \in [0, t] }]{ \dds{\vm} > 0 } \qquad \text{and} \qquad \mc{W}_t := \vm[\Phi_t , t]. \end{gathered}$$ Then $\vm[s, t] = v(s, t)$ for all $s \in \Phi_t$, the mapping $\vm[\cdot, t]: \Phi_t \to \mc W_t $ is a bijection and is strictly increasing, and $\mc W_t$ contains almost all postcollisional velocities, that is, $\mc{L}(\Vt \! \setminus \! \mc{W}_t) = 0 $.
From Lemma \[le:vmprops\]\[le:conn\_comp\] we know that $\del_s \vm \equiv 0$ for all $s \in \Ntc$, so it must be the case that $\Phi_t \ins \Nt$. Furthermore, since is a monotonically increasing function on the interval $[0, t]$, its restriction to $\Phi_t$ is strictly increasing and thus is a bijection between its domain and range. Choosing $I = [0, t]$, $u = \vm[\cdot, t]$, $E = \Phi_t^c$ and $M=0$ in Lemma \[lem:measure\] gives $$\begin{gathered}
\mc{L}_\circ (\mc{W}_t^c ) = \mc{L}_\circ ( \vm[\Phi_t^c, t] ) = 0.\end{gathered}$$ Hence $\mc{L}_\circ(\Vt \! \setminus \! \mc{W}_t) = 0 $, so $\Vt \! \setminus \! \mc{W}_t$ is measurable and almost all postcollisional velocities are included in $\mc{W}_t$.
We have not yet discussed the relationship between the velocity of the particle that had precollided with the disk at time $s$ and the velocity of the disk itself at time $s$; one would expect the particle to be moving faster in that case. Indeed, combining with yields $$\begin{gathered}
\eta(s) + (\tau-s)v \geq \eta(\tau) \qquad \fa \tau \in (s, t),\end{gathered}$$ which can in turn be rewritten as $ v \geq \vint{p}_{s, \tau}$ for all $\tau \in (s, t)$. Letting $\tau \to s$ gives $v \geq \vint{p}_{s, s} = p(s)$, so the particle is, at least, can not be slower than the disk. The case $v = p(s)$ is the grazing precollision. Since $$\begin{gathered}
\dds{v(s, t)} = \frac{v(s, t) - p(s)}{t-s},\end{gathered}$$ all velocities that have had a grazing precollision and their corresponding (non-unique!) precollision times are collected in $\mc W_t^c$ and $\Phi_t^c$ respectively. Since $ \mc{L} ( \vm[\Phi_t^c, t] ) = 0 $, particles that have had a grazing precollision have no effect on the dynamics of the disk, and thus can be safely excluded.
### Change of Variables
We now make a change variables in : by and Theorem \[thm:vbar\], we have $$\begin{aligned}
G_{p, \text{rec}}(t)
&= \int_{\Vt} \vpran{(p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v)} \frec^-(\eta(t), v, t) \dv \nn
\\[3pt]
%
&= \int_{\mc W_t} \vpran{(p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v)} \frec^-(\eta(t), v, t) \dv \nn
\\[3pt]
&= \int_{\Phi_t} \dds{\vm} \vpran{(p(t) - v(s, t))^2 + \frac{\sqrt{\pi}}{2} (p(t) - v(s, t))}
\frec^-(\eta(t), v(s, t), t) \ds. \label{formula:change-variable-1} %\nn\end{aligned}$$ Furthermore, since vanishes on $\Phi_t^c$, we can write $$\begin{aligned}
G_{p, \text{rec}}(t)
&= \int_{\Phi_t} \dds{\vm} \vpran{(p(t) - v(s, t))^2 + \frac{\sqrt{\pi}}{2} (p(t) - v(s, t))}
\frec^-(\eta(t), v(s, t), t) \ds \nn
\\[3pt]
%
&= \int_{\Phi_t} \ (\ldots) \ \ds + \int_{\Phi_t^c} \ (\ldots) \ \ds \nn
%\label{eq:plus0}
\\[3pt]
%
&= \int_0^t \dds{\vm} \vpran{(p(t) - v(s, t))^2 + \frac{\sqrt{\pi}}{2} (p(t) - v(s, t))} \frec(s, t) \ds ,
\label{eq:vm_once}\end{aligned}$$ where, with a slight abuse of notation, we have written $$\begin{aligned}
\label{def:f-n-abbre}
\frec(s, t) = \sum_{n\geq1} f_n(s, t) = \sum_{n\geq1} f^-_n(\eta(t), v(s, t), t) = \sum_{n\geq1} f^+_n(\eta(s), v(s, t), s) \,.\end{aligned}$$ The last equality in holds because the distribution density does not change between collisions. Note that in , the modified velocity needs to appear only in the derivative since whenever $\partial_s \vm \neq 0$, we have $\vm = v(s, t)$.
Making the same change of variables in for $n\geq 1$ and using the notation in lead us to the key recurrence relation: $$\begin{aligned}
\label{eq:rec}
f_{n+1}(s, t) &= 2 e^{-(v(s, t) - p(s))^2} \int_0^s \dds{\vm[\tau, s]} (p(s) - v(\tau, s)) f_n(\tau, s) \dtau. \end{aligned}$$
For future convenience, we define the density flux of $f_n$ as $$\begin{gathered}
\label{def:j-n}
j_n(s) = \int_0^s \dds{\vm[\tau, s]} (p(s) - v(\tau, s)) f_n(\tau, s) \dtau,\end{gathered}$$ which allows us to write $$\begin{gathered}
\label{eq:rec-1}
f_{n+1}(s, t) = 2 e^{-(v(s, t) - p(s))^2} j_n(s).\end{gathered}$$
The change of variables does not apply to the particles that have not collided with the disk previously; since such particles maintain the initial density distribution, we have $$\begin{aligned}
f_1(s, t) &= 2 e^{-(v(s, t) - p(s))^2} \int^{\vm[0, s]}_{-\8} (p(s) - v) \phi_0(v) \dv, \label{def:f-1}
\\[2pt]
G_{p, 0}(t) &= \int^{\vm[0, t]}_{-\8} \vpran{(p(t) - v)^2 + \frac{\sqrt{\pi}}{2} (p(t) - v)} \phi_0(v) \dv .
\label{def:G-p-0}\end{aligned}$$ Recalling the definition of $G_{p, \text{rec}}$ in , we have constructed a decomposition of the drag force: $$\begin{aligned}
G_p(t) = G_{p, 0}(t) + G_{p, \text{rec}} .\end{aligned}$$
Note that even the particles that had no precollisions obey Assumption \[a0\] or, equivalently, the mathematical formulation in . This is why the effective integration domain in $G_{p, 0}$ and $f_1(s, t)$ is $(-\infty, \vm[0, t])$ instead of $(-\infty, p(t))$: the latter would allow the particles originally in front of the disk to fall behind the disk.
Preliminary Bounds {#sec:prelim_bnds}
==================
For future convenience we define $$\begin{gathered}
\label{def:alpha_n}
\alpha_n(s) = \frac{\left( 2M^2 s\right)^n}{n!}, \qquad \alpha_0(s) \equiv 1, \qquad \alpha_{-1}(s) \equiv 0.\end{gathered}$$
In this section we use the recurrence relation to derive essential bounds on $f_n$ and its derivatives; they are summarized in two propositions.
\[bound:f-n-1\] Let $j_n$ and $f_n$ be the iterative sequences given by and - respectively. Let $M$ be the Lipschitz bound in Assumption \[a2\]. Then there exists a constant $Q_1$ that does not depend on $n$ such that for any $n \geq 1$ we have $$\begin{aligned}
\label{bound:f-n-L-infty}
0 \leq f_n(s, t) \leq Q_1 \alpha_{n-1}(s)
\qquad \text{and} \qquad
0 \leq j_n(s) \leq \frac{1}{2}Q_1 \alpha_n(s) .\end{aligned}$$ Moreover, for each $s \in [0, t]$, the function $f_n(s, \cdot) \in C^1([0, T])$ with the bound $$\begin{gathered}
\label{bound:f-n-C-1}
\abs{\dds[t]{} f_n(s, t)} \leq 4M^2 Q_1 \alpha_{n-1}(s). \end{gathered}$$ As a consequence, the function $\frec$ defined in satisfies $$\begin{aligned}
\label{bound:f-n-C-1}
0 \leq \frec(s, t) \leq Q_1 e^{2M^2 s}
\qquad \text{and} \qquad
\abs{\dds[t]{} \frec(s, t)} \leq 4M^2 Q_1 e^{2M^2 s}. \end{aligned}$$
First we derive the bound . For $n=1$ we use the definition of $f_1$ in to write $$\begin{gathered}
0
\leq f_1(s, t)
\leq 2\int_{-\8}^{\vm[0, s]} \left( p(s) - v \right) \phi_0(v) \dv
\leq 2\int_{-\8}^M \left( M - v \right) \phi_0(v) \dv
=: Q_1 .\end{gathered}$$ For $n \geq 2$ we apply together with the definition of $\alpha_n$ in : $$\begin{aligned}
f_n(s, t) &= 2e^{ -( v(s, t) - p(s) )^2 } \int_0^{s} \dds[\tau]{\vm[\tau, s]} (p(s) - v(\tau, s)) f_{n-1} (\tau, s) \dtau \\
&\leq 2M^2 \int_0^{s} Q_1 \alpha_{n-2} (\tau) \dtau = Q_1\alpha_{n-1}(s). \end{aligned}$$ Note that the above step also gives the bound of $j_n$. Indeed, by its definition, $$\begin{aligned}
j_n(s) \leq \int_0^{s} \abs{\dds[\tau]{\vm[\tau, s]}} \abs{p(s) - v(\tau, s)} f_{n} (\tau, s) \dtau
\leq
M^2 \int_0^s Q_1 \alpha_{n-1}(\tau) \dtau
= \frac{1}{2}Q_1 \alpha_n(s) \,.\end{aligned}$$ The bound follows directly from the definition of $f_n$. Indeed, $$\begin{gathered}
\abs{\dds[t]{}f_n(s, t)}
= 2 \abs{\dds[t]{} e^{ -( v(s, t) - p(s) )^2 } j_{n-1}(s)}
= 2 \abs{v(s, t) - p(s)} \abs{\dds[t]{v(s, t)}} f_n(s, t) \leq 4M^2 Q_1 \alpha_{n-1}(s). \end{gathered}$$ The estimates for $\frec$ and $\del_t \frec$ follow by summing the bounds for $f_n$ and $\del_t f_n$.
\[prop:bound-f-n-s\] For all $t \in [0, T]$ the function $f_n(s, t)$ is Lipschitz in $s$. As a result, it is almost everywhere differentiable in $s \in [0, t]$. Moreover, there exists a positive constant $Q_3$ that does not depend on $n$ such that $$\begin{aligned}
\left| \dds{f_{n+1}(s, t)} \right| & \leq 3^n Q_3 \vpran{\alpha_n(s) + \alpha_{n-1}(s)} ,
\qquad n \geq 0, \label{bound:f-n-deriv-s-simp}\end{aligned}$$ As a consequence, the function $\frec$ defined in satisfies $$\begin{aligned}
\label{bnd:frec_s}
\left| \dds{\frec(s, t)} \right| & \leq 4Q_3e^{6M^2 s}.\end{aligned}$$
Fix $t \in [0, T]$. For $n=0$, the definition of $f_1(s, t)$ in shows it is Lipschitz in $s$ since $v(s, t)$, $p(s)$, and are all Lipschitz in $s$. This allows us to obtain the desired bound by a direct calculation: $$\begin{aligned}
\abs{\dds{}f_1(s, t)}
&= \abs{\dds{}\left( 2e^{ -(v(s, t) - p(s))^2 } \int_{-\8}^{\vm[0, s]} (p(s) - v) \phi_0(v) \dv \right)}
\\[5pt]
&\leq \abs{2(v(s, t) - p(s)) \left( \dds{v(s, t)} - \dot p(s) \right)} f_1(s, t)
\\[5pt]
& \quad \, + 2e^{ -(v(s, t) - p(s))^2} \abs{\dds{\vme(0, s)} \left( p(s) - \vm[0, s] \right)} \phi_0(\vme(0, s))
\\[5pt]
& \quad \, + 2e^{ -(v(s, t) - p(s))^2 } \int_{-\8}^{\vm[0, s]} \abs{\dot p(s)} \phi_0(v) \dv
\\[5pt]
%
&\leq 8M^2 Q_1 + 4M^2 \| \phi_0 \|_\8 + 2M \| \phi_0 \|_1 =: Q_2.\end{aligned}$$ We now proceed by induction. Assume that the conclusion holds for $f_n$. Without loss of generality, assume $0 \leq s < s' \leq t$. Then $$\begin{aligned}
j_n(s) - j_n(s') &= \int_0^{s} \dds[\tau]{\vme(\tau, s)}(p(s) - v(\tau, s)) f_n (\tau, s) \dtau
- \int_0^{s'} \dds[\tau]{\vme(\tau, s')}(p(s') - v(\tau, s')) f_n (\tau, s') \dtau
\\[2pt]
%
&= \int_{s'}^{s} \dds[\tau]{\vme(\tau, s)}(p(s) - v(\tau, s)) f_n (\tau, s) \dtau \\[2pt]
%
& \quad \, + \int_0^{s'} \left[ \dds[\tau]{\vme(\tau, s)} - \dds[\tau]{\vme(\tau, s')} \right]
(p(s) - v(\tau, s)) f_n (\tau, s) \dtau \\[2pt]
%
& \quad \, + \int_0^{s'} \dds[\tau]{\vme(\tau, s')} \big[(p(s) - v(\tau, s)) - (p(s') - v(\tau, s')) \big] f_n (\tau, s) \dtau \\[2pt]
%
& \quad \, + \int_0^{s'} \dds[\tau]{\vme(\tau, s')} (p(s') - v(\tau, s')) \big[ f_n (\tau, s) - f_n (\tau, s') \big]\dtau %\\[5pt]
=: I_1 + I_2 + I_3 + I_4.\end{aligned}$$
By Lemma \[le:vmprops\](d) and Proposition \[bound:f-n-1\], we obtain estimates of $I_1$, $I_3$ and $I_4$ as follows: $$\begin{aligned}
\abs{I_1}
&= \abs{\int_{s'}^{s} \dds[\tau]{\vme(\tau, s)}(p(s)-v(\tau, s)) f_n (\tau, s) \dtau}
\leq |s-s'| M^2 Q_1 \alpha_{n-1}(s), \\%[5pt]
%
\abs{I_3}
&= \abs{\int_0^{s'} \dds[\tau]{\vme(\tau, s')} \left[(p(s) - v(\tau, s)) - (p(s') - v(\tau, s')) \right]
f_n (\tau, s) \dtau} \\%[5pt]
&\leq |s-s'|M^2\int_0^{s'}Q_1 \alpha_{n-1}(\tau) \dtau \leq |s-s'|\frac{Q_1}{2} \alpha_n(s'), %\\[5pt]\end{aligned}$$ and $$\begin{aligned}
%\intertext{and}
%
\hspace{14pt}
\abs{I_4}
&= \abs{\int_0^{s'} \dds[\tau]{\vme(\tau, s')} (p(s') - v(\tau, s')) \left[ f_n (\tau, s) - f_n (\tau, s') \right]\dtau}
\\%[5pt]
&\leq M^2\int_0^{s'} \abs{f_n (\tau, s) - f_n (\tau, s')}\dtau
\leq M^2 |s-s'| \int_0^{s'} 4M^2Q_1 \alpha_{n-1}(\tau) \dtau
\\[2pt]
&\leq |s-s'| 2M^2Q_1 \alpha_n(s').\end{aligned}$$
To bound $I_2$ we note that by Lemma \[le:vmprops\](d) and the induction hypothesis on $f_n$, the integrands $\vm[\cdot, s]$, $\vm[\cdot, s']$ and $(p(s) - v(\cdot, s)) f_n (\cdot, s)$ are all Lipschitz. Hence we can integrate by parts and obtain $$\begin{aligned}
I_2 &= \int_0^{s'} \left[ \dds[\tau]{\vme(\tau, s)} - \dds[\tau]{\vme(\tau, s')} \right]
(p(s) - v(\tau, s)) f_n (\tau, s) \dtau \\[5pt]
%
&= \big[ \left( \vme(\tau, s) - \vme(\tau, s') \right) (p(s) - v(\tau, s)) f_n (\tau, s)\big]_{\tau=0}^{\tau=s} \\[5pt]
%
& \quad \, + \int_0^{s'} \big[ \vme(\tau, s) - \vme(\tau, s') \big] \dds[\tau]{v(\tau, s)} f_n (\tau, s) \dtau \\[5pt]
%
& \quad \, - \int_0^{s'} \big[ \vme(\tau, s) - \vme(\tau, s') \big] (p(s) - v(\tau, s)) \dds[\tau]{f_n (\tau, s)} \dtau \,.\end{aligned}$$ This gives the bound $$\begin{aligned}
\abs{I_2}
&\leq 2 M | \vme(0, s) - \vme(0, s') |Q_1 \delta_{1n} + \frac{M^2}{2}|s-s'|\int_0^{s'} Q_1 \alpha_{n-1}(\tau) \dtau
+ 2 |s-s'|M^2 \int_0^{s'} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau \\[5pt]
%
&\leq 2 |s-s'|M^2 Q_1 \delta_{1n} + \frac{Q_1}{4}|s-s'| \alpha_n(s')
+ 2 |s-s'|M^2 \int_0^{s'} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau,\end{aligned}$$ where $\delta_{1n}$ is the Kronecker delta: $\delta_{1n} = 1$ when $n=1$ and vanishes otherwise. Combining the estimates for $I_1$-$I_4$, we have $$\begin{gathered}
\frac{| j_n(s) - j_n(s') |}{|s-s'|}
\leq
Q_1 \alpha_n(s) \left( \frac{3}{4}
+ 2 M^2 \right)
+ 2 M^2 Q_1 \alpha_{n-1}(s') + M^2 Q_1 \delta_{1n}
+ 2 M^2 \int_0^{s'} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau \,.\end{gathered}$$ The right-hand side of the inequality above is bounded uniformly in $s$ and $s'$ since $\partial_\tau f_n (\tau, s) \in L^\infty(0, s)$ by the induction assumption. Therefore, $j_n(s)$ is Lipschitz, and thus differentiable almost everywhere with $$\begin{aligned}
\left| \dds{j_n(s)} \right|
&= \lim_{s' \to s} \frac{| j_n(s) - j_n(s') |}{|s-s'|}
\\
&\leq
Q_1 \alpha_n(s) \left( \frac{3}{4}
+ 2M^2 \right)
+ M^2 Q_1 \alpha_{n-1}(s)
+ 2 M^2 Q_1 \delta_{1n}
+ 2 M^2 \int_0^{s} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau.\end{aligned}$$ To derive the Lipschitz bound for $f_{n+1}$ we separate the two cases where $n=1$ and $n \geq 2$. For $n=1$ we have $$\begin{aligned}
\abs{\dds{j_1(s)}}
&\leq
Q_1 \alpha_1(s) \left( \frac{3}{4} + 2M^2 \right)
+ M^2 Q_1
+ 2 M^2 Q_1
+ 2 M^2 \int_0^{s} Q_2 \dtau \\[5pt]
%
&\leq
\alpha_{1}(s)
\left( \frac{3}{4} Q_1 + 2M^2 Q_1 + Q_2 \right)
+ 3M^2 Q_1 .
%= \alpha_{1}(s) Q_3 + 3M^2 Q_1.\end{aligned}$$ Applying the bound above in the definition of $f_2$ gives $$\begin{aligned}
\abs{\dds{f_2(s, t)}}
&= \abs{\dds{}\left( 2e^{ -(v(s, t) - p(s))^2 } j_1(s) \right)}
\\[5pt]
%
&\leq 4 \abs{v(s, t) - p(s)} \abs{ \dds{v(s, t)} - \dot p(s)}
e^{ -(v(s, t) - p(s))^2 } \abs{j_1(s)} +
2e^{ -(v(s, t) - p(s))^2} \abs{\dds{j_1(s)}} \\[5pt]
%
&\leq
6 M^2 Q_1
+ \alpha_1(s)
\vpran{ 8M^2Q_1
+ 2 \vpran{\frac{3}{4} Q_1 + 2M^2 Q_1 + Q_2}} .\end{aligned}$$ For $n \geq 2$, by using the bounds for $f_{n+1}$ and $\partial_s j_n$ we have $$\begin{aligned}
\abs{\dds{f_{n+1}(s, t)}}
&= \abs{2 \left( \dds{} e^{-(v(s, t) - p(s))^2 } \right) j_n(s)
+ 2 e^{ -(v(s, t) - p(s))^2 } \dds{} j_n(s)} \\[5pt]
%
&\leq
2 \abs{v(s,t) - p(s)} \abs{ \dds{v(s,t)} - \dot p(s)} f_{n+1}(s,t)
+ 2 e^{ -(v(s,t) - p(s))^2 } \abs{\dds{} j_n(s)} \\[5pt]
%
&\leq
\alpha_n(s) Q_1\left( 16M^2 + \frac{3}{2} \right)
+ 2 M^2 Q_1 \alpha_{n-1}(s)
+ 4 M^2 \int_0^{s} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau . \end{aligned}$$ Using the induction assumption on $f_n$, the last integral term is bounded as $$\begin{aligned}
4 M^2 \int_0^{s} \left| \dds[\tau]{f_n (\tau, s)} \right| \dtau
&\leq
4M^2 Q_3 3^{n-1} \int_0^{s} \vpran{\alpha_{n-1}(\tau) + \alpha_{n-2}(\tau)} \dtau
\\ %[5pt]
&= 2 \cdot 3^{n-1} Q_3 \vpran{\alpha_{n-1}(s) + \alpha_n(s)} .\end{aligned}$$ Hence, if we choose $$\begin{aligned}
Q_3
= Q_1\vpran{16M^2 + \frac{3}{2} + 2 M^2 Q_1} + 2 Q_2 ,\end{aligned}$$ then for $n \geq 2$, we have $$\begin{aligned}
\abs{\dds{f_{n+1}(s, t)}}
\leq
Q_3 \vpran{\alpha_{n-1}(s) + \alpha_n(s)}
+ 2 \cdot 3^{n-1} Q_3 \vpran{\alpha_{n-1}(s) + \alpha_n(s)}
%\\
= 3^n Q_3 \vpran{\alpha_{n-1}(s) + \alpha_n(s)} ,\end{aligned}$$ which finishes the induction proof. Since $Q_3 > Q_2$, the bound above holds for $n=0$ as well.
The Lipschitz estimate follows by summing the bounds : $$\begin{gathered}
\left| \dds{\frec(s, t)} \right| \leq \sum_{n=0}^\8 \left| \dds{f_{n+1}(s, t)} \right|
\leq \sum_{n=0}^\8 3^n Q_3 \vpran{\alpha_{n-1}(s) + \alpha_n(s)} = 4Q_3e^{6M^2 s}. \qedhere\end{gathered}$$
Proof of Uniqueness {#sec:uniq}
===================
In this section we prove the uniqueness theorem. An essential preliminary result is a Lipschitz bound for the density functions corresponding to different disk dynamics. Recall that $\|\cdot\|$ denotes the $L^\infty$-norm unless otherwise specified. We begin by showing that modified average velocity satisfies a Lipschitz bound
\[le:Lip\_p\] Let $p$ and $q$ be two Lipschitz velocity profiles and and be their associated modified velocities. Then $$\begin{aligned}
\abs{\vmp(s, t) - \vmq(s, t)} \leq \norm{p - q} \qquad \text{for all $s, t$.}\end{aligned}$$
For a fixed $s \in [0, t]$ and $t \in [0, T]$ suppose $$\begin{aligned}
\vmp(s, t) = \vint{p}_{\tau_1, t}
\qquad \text{and} \qquad
\vmq(s, t) = \vint{q}_{\tau_2, t} .\end{aligned}$$ Without loss of generality, assume that $\vmp(s, t) \geq \vmq(s, t)$. Then $$\begin{gathered}
\abs{\vmp(s, t) - \vmq(s, t)}
= \vint{p}_{\tau_1, t} - \vint{q}_{\tau_2, t}
\leq \vint{p}_{\tau_2, t} - \vint{q}_{\tau_2, t}
\leq \norm{p - q} . \qedhere\end{gathered}$$
\[prop:Lip-f-rec\] Let $\big\{ p, \Eta^{(p)}, \{ f_n^{(p)}\}_{n=1}^\8, \frec^{(p)} \big\}$ and $\big\{ q, \Eta^{(q)}, \{ f_n^{(q)}\}_{n=1}^\8, \frec^{(q)} \big\}$ be two systems of disk-gas dynamics satisfying Assumptions \[a0\]-\[a2\]. Then there exist a positive constant $Q_6$ that does not depend on $n$ such that the gas densities $ \{ f_n^{(p)} \}_{n=1}^\8$ and $\{ f_n^{(q)} \}_{n=1}^\8 $ satisfy the bound $$\begin{aligned}
\abs{f_{n+1}^{(p)}(s, t) - f_{n+1}^{(q)}(s, t)} &\leq
3^n Q_6 \left( \alpha_n(s) + \alpha_{n-1}(s) \right) \| p - q \| \,, \label{bound:Lip-f-n-p-q}
\qquad n \geq 0 \,.\end{aligned}$$ Consequently, for all $s \in [0, t]$ and $t \in [0, T]$ we have $$\begin{aligned}
\label{cond:Lip-f-rec-p-q}
\abs{\frec^{(p)}(s, t) - \frec^{(q)}(s, t)}
&\leq 4 Q_6 e^{6M^2 s} \| p - q \|. \end{aligned}$$
We show the bounds in by a similar induction proof as for Proposition \[prop:bound-f-n-s\]. First, the difference in density fluxes of $\phi_0$ satisfies $$\begin{aligned}
\abs{j_0^{(p)}(t) - j_0^{(q)}(t)}
&= \abs{\int_{-\8}^{ \vmp(0, t) } \left( p(t) - v \right) \phi_0(v) \dv -
\int_{-\8}^{ \vmq(0, t) } \left( q(t) - v \right) \phi_0(v) \dv}
\\%[2pt]
%
&= \abs{\int_{\vmq(0, t)}^{ \vmp(0, s) } \left( p(t) - v \right) \phi_0(v) \dv -
\int_{-\8}^{ \vmq(0, t) } \big[ \left( p(t) - v \right) - \left( q(t) - v \right) \big] \phi_0(v) \dv}
\\[3pt]
%
&\leq \| \vmq - \vmp \| \| \phi_0 \|_{\8} + \| p - q \| \| \phi \|_1
%
= \left( \| \phi_0 \|_{\8} + \| \phi_0 \|_1 \right) \| p - q \| \,. \end{aligned}$$ Therefore, we have $$\begin{aligned}
\left| f_1^{(p)}(s, t) - f_1^{(q)}(s, t) \right| &=
\left| 2e^{ -( v_p(s, t) - p(s) )^2 } j_0^{(p)}(s) - 2e^{ -( v_q(s, t) - q(s) )^2 } j_0^{(q)}(s)\right| \\[3pt]
%
&\leq 2\left| e^{ -( v_p(s, t) - p(s) )^2 } - e^{ -( v_p(s, t) - p(s) )^2 } \right| \abs{j_0^{(p)}(s)}
+ 2 \left| j_0^{(p)}(s) - j_0^{(q)}(s) \right| \\[4pt]
%
&\leq Q_1 \| p - q \| + 2\| p - q \|\left( \| \phi_0 \|_{\8} + \| \phi_0 \|_1 \right)
\\[3pt]
& = \| p - q \| \left( Q_1 + 2\| \phi_0 \|_{\8} + 2\| \phi_0 \|_1 \right) \,.\end{aligned}$$ Thus by choosing $Q_4 = Q_1 + 2\| \phi_0 \|_{\8} + 2\| \phi_0 \|_1$ we complete the proof for $n=1$. For $n\geq 1$ we have $$\begin{aligned}
f_{n+1}^{(p)}(s, t) - f_{n+1}^{(q)}(s, t)
&= 2e^{-( v(s, t) - p(s))^2 }j_n^{(p)}(s) - 2e^{-( v(s, t) - p(s))^2 }j_n^{(p)}(s) \\[3pt]
%
&\leq 2\left| e^{ -( v_p(s, t) - p(s) )^2 } - e^{ -( v_p(s, t) - p(s) )^2 } \right| j_n^{(p)}(s)
+ 2 \left| j_n^{(p)}(s) - j_n^{(q)}(s) \right| \\[3pt]
%
&\leq \| p - q \| \alpha_{n}(s) + 2 \left| j_n^{(p)}(s) - j_n^{(q)}(s) \right| . \end{aligned}$$ We re-formulate the difference in density fluxes as $$\begin{aligned}
j_n^{(p)}(t) - j_n^{(q)}(t) &= \int_0^t \dds{\vmp(s, t)} (p(t) - v_p(s, t)) f_n^{(p)}(s, t) \ds
- \int_0^t \dds{\vmq(s, t)} (q(t) - v_q(s, t)) f_n^{(q)}(s, t) \ds \\[3pt]
%
&= \int_0^t \left[ \dds{\vmp(s, t)} - \dds{\vmq(s, t)} \right] (p(t) - v_p(s, t)) f_n^{(p)}(s, t) \ds \\[3pt]
%
&\quad \, - \int_0^t \dds{\vmq(s, t)} \Big[ (p(t) - v_p(s, t)) - (q(t) - v_q(s, t)) \Big] f_n^{(p)}(s, t) \ds \\[3pt]
%
&\quad \, - \int_0^t \dds{\vmq(s, t)} ( q(t) - v_q(s, t) ) \Big[f_n^{(p)}(s, t) - f_n^{(q)}(s, t) \Big] \ds %\\[5pt]
%
=: J_1 + J_2 + J_3.\end{aligned}$$ By integration by parts and Proposition \[prop:bound-f-n-s\], we write $J_1$ as $$\begin{aligned}
J_1 &= \int_0^t \dds{} \Big[ \vmp(s, t) - \vmq(s, t) \Big] (p(t) - v_p(s, t)) f_n^{(p)}(s, t) \ds \label{bnd:J1}\\[3pt]
%
&= \left[ \Big( \vmp(s, t) - \vmq(s, t) \Big) (p(t) - v_p(s, t)) f_n^{(p)}(s, t) \right]_{s=0}^{s=t} \nn \\[3pt]
%
&\quad \, + \int_0^t \Big[ \vmp(s, t) - \vmq(s, t) \Big] \dds{v_p(s, t)} f_n^{(p)}(s, t) \ds \nn \\[3pt]
%
&\quad \, - \int_0^t \Big[ \vmp(s, t) - \vmq(s, t) \Big] (p(t) - v_p(s, t)) \dds{ f_n^{(p)}(s, t) } \ds %\\[10pt]
%
=: J_1^1 + J_1^2 - J_1^3. \nn\end{aligned}$$ The boundary terms $J_1^1$ are only nonzero for $n=1$, so we write $$\begin{aligned}
\abs{J_1^1}
&= \abs{\left[ \Big( \vmp(s, t) - \vmq(s, t) \Big) (p(t) - v_p(s, t)) f_n^{(p)}(s, t) \right]_{s=0}^{s=t}}
\\[5pt]
%
&= \abs{\Big( \vmq(0, t) - \vmp(0, t) \Big) (p(t) - v_p(0, t)) f_n^{(p)}(0, t)}
\leq
\| p - q \|2MQ_1 \delta_{1n}.\end{aligned}$$ By Lemma \[le:vmprops\], the second term $J_1^2$ satisfies $$\begin{aligned}
\abs{J_1^2} &= \abs{\int_0^t \Big[ \vmp(s, t) - \vmq(s, t) \Big] \dds{v_p(s, t)} f_n^{(p)}(s, t) \ds} %\\[5pt]
%
\leq \| p - q \| \frac{M}{2} \int_0^t Q_1\alpha_{n-1}(s) \ds = \| p - q \| \frac{Q_1}{4M} \alpha_n (t).\end{aligned}$$ Similarly, the third term $J_1^3$ is bounded as $$\begin{aligned}
\abs{J_1^3}
&= \abs{\int_0^t \Big[ \vmp(s, t) - \vmq(s, t) \Big] (p(t) - v_p(s, t)) \dds{ f_n^{(p)}(s, t) } \ds} \\[5pt]
%
&\leq 2M\| p - q \| \int_0^t 3^{n-1} Q_3 \, \vpran{\alpha_{n-1}(s) + \alpha_{n-2}(s)} \ds \\[5pt]
%
&= \| p - q \|\frac{ 3^{n-1} Q_3}{M} \vpran{\alpha_n (t) + \alpha_{n-1}(t)}.\end{aligned}$$ Combining estimates for $J_1^1$, $J_1^2$ and $J_1^3$ we get $$\begin{aligned}
\abs{J_1} \leq \frac{\| p-q \|}{M}
\left( \frac{Q_1}{4} \alpha_n (t) + 3^{n-1} Q_3 \vpran{\alpha_n (t) + \alpha_{n-1}(t)} + 2M^2Q_1 \delta_{1n} \right).\end{aligned}$$ The second term $J_{2}$ is bounded as $$\begin{aligned}
\abs{J_2}
&= \abs{\int_0^t \dds{\vmq(s, t)} \Big[ (p(t) - v_p(, t)) - (q(t) - v_q(s, t)) \Big] f_n^{(p)}(s, t) \ds} \\[5pt]
%
&\leq \| p - q \| M \int_0^t Q_1 \alpha_{n-1}(s) \ds = \| p - q \| \frac{Q_1}{2M} \alpha_{n}(t).\end{aligned}$$ Using the induction assumption, we derive the bound for $J_3$ as $$\begin{aligned}
\abs{J_3} &= \abs{\int_0^t \dds{\vmq(s, t)} ( q(t) - v_q(s, t) ) \Big[f_n^{(p)}(s, t) - f_n^{(q)}(s, t) \Big] \ds} \\[5pt]
&\leq \| p - q \| M^2 \int_0^t 3^{n-1} Q_5 \left( \alpha_{n-1}(s) + \alpha_{n-2}(s) \right) \ds %\\[5pt]
\\[2pt]
&= \| p - q \| \frac{3^{n-1} Q_5}{2}\left( \alpha_n(t) + \alpha_{n-1}(t) \right) \,.\end{aligned}$$ Let $Q_5 = \frac{3Q_1}{2M} + 1 + \frac{2Q_3}{M} + 4MQ_1$. Then combining the estimates on $J_1$, $J_2$ and $J_3$ gives $$\begin{aligned}
\left| f_{n+1}^{(p)}(s, t) - f_{n+1}^{(q)}(s, t) \right|
&\leq \| p - q \| \alpha_{n}(s) + 2 \left( J_1 + J_2 + J_3 \right) \\[5pt]
&\leq \| p - q \| \left( 2 \cdot 3^{n-1}Q_5 \big( \alpha_n(s) + \alpha_{n-1}(s) \big) + Q_5 \delta_{1n}\right);\end{aligned}$$ for $n=1$ it becomes $$\begin{aligned}
\left| f_{2}^{(p)}(s, t) - f_{2}^{(q)}(s, t) \right|
&\leq \| p - q \|
\left( 2Q_5 \big( \alpha_1(s) + \alpha_0(s) \big) + Q_8 \right)
\leq
3\| p - q \|Q_5 \big( \alpha_1(s) + \alpha_0(s) \big),
%
\intertext{while for $n \geq 2$ we get}
%
\left| f_{n+1}^{(p)}(s, t) - f_{n+1}^{(q)}(s, t) \right| &\leq \| p - q \|
\left( 2 \cdot 3^{n-1}Q_5 \big( \alpha_n(s) + \alpha_{n-1}(s) \big) \right)
\leq \| p - q \| 3^nQ_5 \big( \alpha_n(s) + \alpha_{n-1}(s) \big) .\end{aligned}$$ Choosing $Q_6 = \max\{ Q_4, Q_5\}$ unifies all cases. Estimate follows by summing the bounds in : $$\begin{aligned}
\abs{\frec^{(p)}(t) - \frec^{(q)}(t)} &\leq \sum_{n=1}^\8 \abs{f_n^{(p)}(t) - f_n^{(q)}(t)}
\\
&\leq \vpran{Q_6\sum_{n = 0}^\8 3^n \vpran{\alpha_n(s) + \alpha_{n-1}(s)}} \norm{p - q}
\\
&= 4 Q_6 e^{6M^2 s} \norm{p - q} . \qedhere\end{aligned}$$
The Lipschitz property of the drag force is an immediate consequence of Proposition \[prop:Lip-f-rec\]:
\[prop:Lip-G\] Given two disk velocity profiles $p$ and $q$, let $G_p$ and $G_q$ be the corresponding drag forces defined in . Then for any $T > 0$ there exists a constant $L_T$ such that $$\begin{aligned}
\norm{G_p - G_q} \leq L_T \norm{p - q} . \end{aligned}$$
Recall that we decompose $G_p = G_{p, 0} + G_{p, rec}$ and $G_q = G_{q, 0} + G_{q, rec}$. The Lipschitz bounds for $\abs{G_{p, 0} - G_{q, 0}}$ and $\abs{F(\eta_p(t), t) - F(\eta_q(t), t)}$ can be derived by direct estimates, so we focus on the re-collision part. We only give a sketch of the proof since it is very similar (and at times easier) to the one for . To simplify the notation we let $$\begin{aligned}
A_p(s, t) = (p(t)-v_p(s, t))^2 + \frac{\sqrt{\pi}}{2} (p(t) - v_p(s, t)).\end{aligned}$$ Then the difference becomes $$\begin{aligned}
G_{p, \text{rec}}(t) - G_{q, \text{rec}}(t)
&= \int_0^t \dds{\vmp(s, t)}
A_p(s, t) \, \frec^{(p)}(s, t) \ds
-\int_0^t \dds{\vmq(s, t)} A_q(s, t) \frec^{(q)}(s, t) \ds
\\[2pt]
& = \int_0^t \vpran{\dds{\vmp(s, t)} - \dds{\vmq(s, t)}} A_p(s, t)\frec^{(p)}(s, t) \ds
\\[2pt]
& \quad \,
+ \int_0^t \dds{\vmq(s, t)} \vpran{A_p(s, t) - A_q(s, t)}\frec^{(p)}(s, t) \ds
\\[2pt]
& \quad \,
+ \int_0^t \dds{\vmq(s, t)} A_q(s, t)
\vpran{\frec^{(p)}(s, t) - \frec^{(q)(s, t)}} \ds
= : K_1 + K_2 + K_3 \,.\end{aligned}$$ A Lipschitz bound for $K_1$ follows from the same calculation as in the proof of proposition \[prop:Lip-f-rec\]. A Lipschitz bound for $K_2$ follows from the definition of $v_p$, and a Lipschitz bound for $K_3$ follows from together with the bounds for $\del_s \vmq$ in Lemma \[le:vmprops\]. The combination of the three bounds for $K_1, K_2, K_3$ gives the Lipschitz bound for $G_{\text{rec}}$.
Strictly speaking, the drag force in Proposition \[prop:Lip-G\] is only the contribution from the right-side of the disk. However, as mentioned earlier, a similar Lipschitz property holds for the full drag force defined in , since the estimates for the left side follow from a similar argument. The main modification needed for the left side is to change the definition of the modified average velocity $\vm$ into $$\begin{aligned}
\bar v(s, t) = \max_{\tau \in [s, t]} \vint{p}_{s, t} .\end{aligned}$$ Since replacing minimum with maximum does not affect the properties of the modified average velocity in Lemma \[le:vmprops\], the rest of the estimates remain the same.
We now have all the ingredients to prove the main result of this paper:
For any $T > 0$, by Proposition \[prop:Lip-G\] and the assumption that the external force $F(\cdot, t)$ is Lipschitz, we have $$\begin{aligned}
\| p - q \|_{L^\infty(0, t)} \leq \vpran{L_T + \text{Lip}(F)} \int_0^T \| p - q \|_{L^\infty(0, t)} \dt ,\end{aligned}$$ which gives $p = q$ on $[0, T]$ by Gronwall’s inequality. Meanwhile, for a given $p$, the density function $f$ on the disk can be written explicitly using the decomposition established in Section \[sec:gas\_decomp\]: $$\begin{aligned}
f_R(\eta(t), v, t) = \sum_{n=0}^\infty f_{R, n}(\eta(t), v, t) ,
\qquad
f_L(\eta(t), v, t) = \sum_{n=0}^\infty f_{L, n}(\eta(t), v, t),\end{aligned}$$ together with the initial condition $f(x, v, 0) = \phi_0(v)$. Therefore, the boundary conditions in - are uniquely defined, which combined with the free transport equation gives a unique solution for $f$. We thus obtain a unique solution to the full gas-disk system.
[**Acknowledgements:**]{} The authors want to thank Ralf Wittenberg for fruitful discussions on this problem and pointing out a mistake in an earlier version. The research of W.S. is supported by NSERC Discovery Individual Grant R611626.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The present paper studies the influence of the structural modulation on the low energy physics of the $S\!r_{14-x}C\!a_xC\!u_{24}O_{41}$ oxides, using ab-initio determination of the on-site and nearest neighbor effective parameters. The structural modulations appears to be the key degree of freedom, responsible for the low energy properties, such as the electron localization, the formation of dimers in the $x=0$ compound or the anti-ferromagnetic order in the $x=13.6$ compound.'
author:
- 'Alain Gellé, Marie-Bernadette Lepetit'
title: 'Influence of the incommensurability in $S\!r_{14-x}C\!a_xC\!u_{24}O_{41}$ family compounds.'
---
The family of $S\!r_{14-x}C\!a_xC\!u_{24}O_{41}$ transition-metal oxides has attracted a lot of attention in the last decade. Indeed, the superconducting state observed in the $x>11.5$ compounds under high pressure [@ObsSupra], is supposed to be the realization of the remarkable theoretical prediction of superconductivity in two-legs doped spin ladders [@TheoEchSupra]. Moreover, this family of compounds exhibits a large diversity in electric and/or magnetic properties when chemical — isovalent substitution of $S\!r$ by $C\!a$ — and physical pressure are applied. For instance, under applied pressure, the compounds change from semi-conductor to conductor and finally to superconductor.
This family of compounds possesses a complex layered structure of two alternating subsystems [@struc1]. The layers of the first subsystem are composed of weakly-coupled $C\!uO_2$, spin-$1/2$ chains along the ${\bf c}$ direction. The spins, supported by $3d$ orbitals of the $C\!u^{2+}$ ions, are coupled via two $90^\circ$ $C\!u$–$O$–$C\!u$ bonds. The layers of the second subsystem are composed of weakly-coupled, two-leg spin-1/2 ladders also along the ${\bf c}$ direction. The spins are strongly antiferromagnetically coupled on both legs and rungs, due to the $180^\circ$ $C\!u$–$O$–$C\!u$ bonds. The cell parameters of the two subsystems, in the direction of both chains and ladders, are incommensurate. The compounds have a pseudo-periodicity of 10 chain units for 7 ladder units.
Electro-neutrality analysis shows that these systems are intrinsically doped with six holes by formula unit (f.u.). Similar to high-$T_c$ superconductors, the holes are expected to be mainly supported by the oxygen $2p$ orbitals and to form Zhang-Rice [@ZR] singlets with the associated-copper hole. NEXAFS experiments [@XRay00] have later supported this assumption. A calculation of the Madelung potential [@Mad97] on the concerned oxygen sites suggests that for the undoped compound, $S\!r_{14}C\!u_{24}O_{41}$, the chains exhibit a larger electro-negativity than the ladders, resulting in a localization of all the holes on the former. Different experiments [@COpt97; @XRay00], however suggest that about one hole per f.u. is located on the ladders. Under $C\!a$ substitution, the same experiments, as well as $C\!u$ NMR [@RMN98], show a transfer of part of the holes to the ladders. However the precise number of transferred holes is still under debate. X-ray data [@XRay00] suggest a small hole transfer to the ladders ($1.1$ for $x=12$), while optical conductivity [@COpt97] and $^{63}C\!u$ NMR studies [@RMN98] show a larger hole transfer, of respectively $2.8$ ($x=11$) and $3.5$ ($x=11.5$).
Let us first focus on the undoped system, $S\!r_{14}C\!u_{24}O_{41}$. This is a semiconductor with a $0.18\ eV$ gap. The spin ladders have a singlet ground state with a spin gap of about $35-47\
meV$ [@Neut96; @RMN98; @RMN97]. Surprisingly the spin chains also exhibit a singlet ground state with a spin gap of $11-12\
meV$ [@Magn96B; @RMN97; @RMN98B; @Neut98; @Thermo00]. Since homogeneous spin chains are known to be gap-less in the spin channel, the existence of a gap witnesses their strongly inhomogeneous character. In fact, the electronic structure of the $S\!r_{14}C\!u_{24}O_{41}$ chains is usually understood as formed by weakly interacting dimmers [@Magn96; @Magn96B; @ESR96; @Neut96]. It is now well established that these dimers are formed by second-neighbor spins separated by a Zhang-Rice singlet (ZRS), and order [@RMN98B; @ESR01; @Neut98; @Neut99] according to a pseudo periodicity of 5 sites (one dimer followed by two ZRS).
When $S\!r$ is substituted by $C\!a$ the system becomes more metallic. The doped compound with $x=10$ shows a gap of only $0.023
eV$ [@Magn96B]. This increased conductivity, as a function of the $C\!a$ doping, is usually understood as a consequence of the hole transfer from the chains to the ladders, in which the conduction is supposed to occur. However, a possible enhancement of the holes mobility within the chains is also evoked [@XRay00; @ESR01]. Neutron scattering experiments [@Neut99] showed that the dimerization becomes unstable with $C\!a$ substitution and disappears for $x>8$, although the magnetic interactions within and between the dimers remain unchanged. In parallel, both ESR [@ESR01] and thermal expansion data [@Thermo00] witness a progressive disappearance of the charge order with increasing doping. Finally, at large doping ($x
\ge 11$) and very low temperatures ($<2.5K$) an anti-ferromagnetic phase is observed [@Magn99; @ESR01].
An important aspect of these compounds, which is most of the time neglected, is the modulation of the two subsystems. Indeed, the mutual influence of the two subsystems results in a modulation of each of them with the periodicity of the other. These distortions are particularly large on the chains. Indeed, in the highly doped systems, the $C\!u$–$O$–$C\!u$ angle varies between $89^\circ$ and $99^\circ$, while the $C\!u$–$O$ distance varies with an amplitude of $19\%$. The magnetic interactions being very sensitive to both bond angles and distances between the magnetic sites and the bridging ligands, one can expect that the modulations will be of importance for the low energy physics of the compounds. In the ladder subsystem, the structural distortions are of weaker amplitude since the alkaline-earth counter-ions are attached to it. In addition, the $C\!u$–$O$–$C\!u$ angle varies around $\theta = 180^\circ$ and therefore super-exchange mechanism (that scales as $\cos^4{\theta}\simeq
1-2\theta^2$) should be dominant. Thus, while the effect of the structural modulations on the ladders may be of importance, it can be expected to be much weaker than on the chains sub-system.
The aim of this paper is to study to which extent the modulations influence the chains electronic structure. For this purpose, we performed ab-initio calculations so that to accurately evaluate the influence of the modulations on the magnetic orbital energies (OE) and nearest neighbors (NN) interactions. We choose for these calculations the $S\!r_{14}C\!u_{24}O_{41}$ and $S\!r_{0.4}C\!a_{13.6}C\!u_{24}O_{41}$ compounds in their low temperature phase [@struc].
As the interactions between magnetic sites are essentially local, they can be accurately determined using an embedded fragment ab-initio spectroscopy method [@revue]. The fragment includes the magnetic centers, the bridging oxygens mediating the interactions, and their first coordination shell. Short-range crystal effects are thus treated explicitly while the long-range crystal effects, such as the Madelung potential, are treated within an appropriate bath. The calculations have been performed using the DDCI method [@DDCI], a selected multi-reference single and double configuration interaction that properly treats (i) the strongly correlated character of the system, (ii) the mediation of the interaction via the bridging ligands and (iii) the screening effects on these processes [@bridge]. The reference space has been chosen to be composed by the copper magnetic orbitals. The basis sets are of valence $3\zeta$ [@bases] quality for $C\!u$ and $2\zeta + p$ for $O$. This method has proved to be very accurate in the determination of the the local effective interactions as well as local electronic structures. One can cite, for instance, the remarkable results obtained on the effective exchange and hopping determination in high-$T_c$ parent compounds [@DDCIhtc], copper chains [@DDCIcha] and ladder [@DDCIlad] systems, where the computed values are within experimental accuracy, as well as on the charge ordering in the sodium vanadate low temperature phase [@vana].
Singlet-triplet excitation energies on embedded $C\!u_2 O_6$ fragments will thus yield the effective exchange integrals, while the first doublet-doublet excitation energies and associated wave functions yield both magnetic (hole) OE and hopping effective integrals. In order to study the influence of the incommensurability on these parameters, we performed calculations on 11 fragments associated to successive cells of the chain sub-system.
Let us first look at the hole OE (fig. \[E1\]). One sees immediately that their modulations are very large. Indeed, in the non-doped system, the OE vary within a range of $1.2\ eV$. In the $C\!a$ doped system, the variation is even much larger and spans a $2.2\ eV$ range. Let us notice, that in both systems, the OE variation is larger than the hopping and exchange energy scales (respectively of $150\ meV$ and $20\ meV$, see below). Thus, this is the OE that will dominate the low energy physics through the localization of the magnetic electrons (holes) on the low (high) energy sites. One should notice that the variations of the OE are due to the crystal distortions. Indeed, when using the average crystal structure — where the subsystems distortions have been omitted — the OE exhibit only very small variations [@env] ($<30\ meV$ for the undoped compound).
Before studying the electrons (holes) localization along the chain, let us take a look at the variations of the NN hopping and exchange interactions.
The effective exchange integrals, obtained for the 11 fragments, are reported in fig. \[J1\]. As expected the NN effective exchange for the $x=0$ compound is ferromagnetic and exhibits small modulations around an average value of $21.3\ meV$, the standard deviation being $2.5\ meV$. On the contrary, the $x=13.6$ compound does not follow the expectations. Indeed, the effective exchange varies greatly, going from ferromagnetic values ($20\ meV$) up to anti-ferromagnetic interactions as large as $-10\ meV$. These anti-ferromagnetic interactions are observed either for large $C\!u$–$O$–$C\!u$ angles ($\simeq 98^\circ$) or for large angles between the two magnetic orbitals ($>10^\circ$) associated with short $C\!u$–$C\!u$ distances. In both cases, these strong distortions allow super-exchange mechanism to take place.
The effective hopping integrals between NN copper atoms are expected to be quite small, since the nearly $90^\circ$ $C\!u$–$O$–$C\!u$ angles forbid the through-bridge contribution via the oxygen orbitals. However, this is not what is observed in our calculations (see fig. \[J1\]) since the hopping integrals present very large modulations and can reach values as large as $208\ meV$ for $x=0$ and $266\ meV$ for $x=13.6$. These amplitudes are as large as $1/3$ of the hopping observed in systems with $180^\circ$ $C\!u$–$O$–$C\!u$ angles, such as high-$T_c$ superconductors [@DDCIhtc] or $C\!uO_3$ corner-sharing chains [@DDCIcha].
At this point it is clear that, the systems modulations, and the variations of these modulations according to the $C\!a$ doping, are crucial for the low energy properties of this family of iso-electronic compounds. We will now study whether they can explain the existence of the dimerization that is observed in the weakly doped compounds but not in the highly doped ones. For this purpose we need to extrapolate the values of the OE over the whole chain. It is usual to do so using a Bond Valence Sum (BVS) analysis, that analyzes the distances between the metal atom and its first coordination shell. However, for the $S\!r_{0.4}C\!a_{13.6}C\!u_{24}O_{41}$ system, the BVS analysis yields quite different results from the OE. A further analysis discloses that, in these systems, the Madelung potential on the magnetic centers, and thus the OE, is sensitive to the atomic displacements not only of the first coordination shell, but up to the $8^{\rm th}$ shell of neighbors. This is for this reason that the BVS analysis fails to correctly reproduce the copper valence for this type of compounds.
In order to extrapolate the OE over the whole chain, we used the crystallographic description of the incommensurate structure in a four dimensional space. Each subsystem is thus described by the three $a$, $b$ and $c$ usual spatial dimensions and a fourth coordinate $\tau$ that has the periodicity of the other subsystem and describes the modulations [@struc4]. We fitted the computed values using a Fourier analysis as a function of $\tau$ (see fig. \[four\]). One can note that the doped and undoped compounds present very different OE curves as a function of $\tau$. Let us notice on fig. \[four\]a the half periodicity of the magnetic cell for the $S\!r_{14}C\!u_{24}O_{41}$ compound, observed in neutrons scattering experiments [@Neut98; @Neut99].
Figure \[rempl\] reports the electrons (holes) localization along the chain as derived from the extrapolated on-site orbital energies. In the $x=13.6$ case, a Fourier analysis has also been done on the effective NN exchange in order to predict the sign of the NN interactions for the different positions along the chain. The second neighbor exchange has been considered to be anti-ferromagnetic in agreement with experimental results.
For the undoped $S\!r_{14}C\!u_{24}O_{41}$ system we studied the two types of filling considered in the literature, namely with all the holes on the chains (fig. \[rempl\]a1) and with one hole transfer per f.u. (fig. \[rempl\]a2). When all the holes are on the chains, we retrieve a spin arrangement consistent with the experimental observations [@Neut98; @Neut99; @RMN98], that is dimeric units formed of two second neighbors spins, separated by two ZR singlets. These dimers are clustered by three or four units separated by a free spin, that is a spin with neither first nor second neighbor spin. We found 5 free spins for 100 sites ($0.5$ per f.u.) to be compared with the magnetic susceptibility measurements of $0.55$ free spins per f.u. [@Magn96B]. The number of dimers is 17 ($1.7$ per f.u.) to be compared with the magnetic susceptibility finding of $1.47$ per f.u.. When one hole per f.u. is transfered to the ladders, the picture is totally modified. Indeed, the free spins totally vanish and first neighbor ferro-magnetically coupled spins appear. At the same time the number of (isolated) dimers is strongly reduced, with only 5 for 100 sites, that is $0.5$ per f.u. From this analysis it is clear that if there are holes transferred to the ladders (at low temperature) in the undoped compound, this number is much smaller than 1 per f.u..
For the $x=13.6$ doped system we studied three types of filling where $n=$ 1,2, or 3 holes have been transferred to the ladders. For all fillings, we retrieve arrangements with nearest neighbors spins and no second neighbors dimeric units. The chains are essentially composed of low-spin clusters (either in a singlet or doublet state), with spin arrangements that present weak exchange frustration. Such fillings are thus expected to be specially stable, and should be put into perspective with the anti-ferromagnetic ordering seen in magnetic susceptibility and ESR measurements. For $n=1,2$ one still observes a large number of free spins, while for $n=3$ they have essentially disappeared, in agreement with magnetic susceptibility experiments [@Magn99].
In summary, we have studied the importance of the structural modulations on the low energy physics of the $S\!r_{14-x}C\!a_{x}C\!u_{24}O_{41}$ family. Surprisingly these distortions are not simply responsible for parameters modulations around their average value (except for the NN effective exchange on the undoped compound), but induce very large variations of the orbital energies, NN effective hopping and exchange. This is the variation of the OE (which spans a range of $1.2\ eV$ for $x=0$ and $2.2\ eV$ for $x=13.6$) that is responsible for the low energy properties of the compounds, through the localization of the magnetic electrons. It is in particular responsible of the formation of dimers in the undoped compound. In view of these results one can also suppose that the stabilization of part of the chains sites by larger structural modulations, is responsible for the hole transfer toward the ladders in the doped compounds. In the $x=13.6$ compound, the structural modulation is so large that it even reverse the sign of the NN effective exchange on part of bonds, lifting the exchange frustration that would arise for the large chain filling. In conclusion, one can say that the structural modulation is the key parameter, responsible for the large variation of the low energy properties in this family of compounds.
[**Acknowledgment :**]{} the authors thank Dr. D. Maynau for providing us with the CASDI suite of programs.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Consider a face-to-face parallelohedral tiling of $\mathbb R^d$ and a $(d-k)$-dimensional face $F$ of the tiling. We prove that the valence of $F$ (i.e. the number of tiles containing $F$ as a face) is not greater than $2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay $k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.'
address:
- 'Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkina street, Moscow 119991, Russia'
- 'Yaroslavl State University, 14 Sovetskaya street, Yaroslavl 150000, Russia'
author:
- Alexander Magazinov
title: An upper bound for a valence of a face in a parallelohedral tiling
---
tiling,parallelohedron,Voronoi conjecture
Introduction
============
The central point of the parallelohedra theory is the famous Voronoi conjecture.
\[voronoi\]
Every $d$-dimensional parallelohedron $P$ is affinely equivalent to a Dirichlet-Voronoi domain for some $d$-dimensional lattice.
Although the conjecture was posed in 1909 in the paper [@vor], it has not been proved or disproved so far in the general case. However, several significant partial solutions were obtained [@del; @erd; @ord; @vor; @zhi].
Let $\mathcal T(P)$ be a face-to-face tiling of $\mathbb R^d$ by parallel copies of a parallelohedron $P$. Choose an arbitrary $(d-k)$-dimensional face $F$ of the tiling.
Denote by $\pi$ the orthogonal projection along $\operatorname{\it lin}F$ onto the complementary affine space $(\operatorname{\it lin}F)^\bot$. Then there exists a complete $k$-dimensional cone fan $\operatorname{\it Fan}(F)$ ([*the fan of $F$*]{}) that splits $(\operatorname{\it lin}F)^\bot$ into convex polyhedral cones with vertex $\pi(F)$, and a neighborhood $U = U(\pi(F))$ such that every face $F'\supset F$ corresponds to a cone $C\in \operatorname{\it Fan}(F)$ satisfying $$\pi(F')\cap U = C\cap U.$$ Speaking informally, $\operatorname{\it Fan}(F)$ has the same combinatorics as the transversal section of $\mathcal T(P)$ in a small neighborhood of $F$.
The definition above is equivalent to the definition of a [*star*]{} of a face $F$, introduced by Ryshkov and Rybnikov [@rry], yet seems to be more formal.
Studying the combinatorial structure of such cone fans proved to be an effective tool to verify the Voronoi conjecture in special cases [@del; @ord; @zhi].
For fixed $k$ and varying $d$, it is a complicated problem to classify all possible combinatorial types of $\operatorname{\it Fan}(F)$ if a positive answer to conjecture \[voronoi\] is not preassumed. The classification for $k = 2$ has been obtained in [@min] and consists of 2 combinatorial types of cone fans shown in Figure \[f1\].
\[f1\]
for $k = 3$ Delaunay [@del] proved that every fan of a $(d-3)$-face of a parallelohedral tiling belongs to one of the 5 types shown in Figure \[f2\].
\[f2\]
The case $k = 4$ has been partially considered in [@ord], but the complete classification was not obtained, and for $k>4$ almost nothing is known. The main goal of this paper is to show that a classification is possible [*in principle*]{} for every $k$, i.e. the list of all combinatoral types of $\operatorname{\it Fan}F$, where $F$ is a $(d-k)$-face, is finite. The idea is to obtain an upper bound for the number of tiling parallelohedra incident to a face $F$.
Let $$\nu(F) = card\,\{ P'\in \mathcal T(P) : F\subset P' \}.$$ $\nu(F)$ will be called the [*valence*]{} of the face $F$.
\[2k\]
Let $F$ be a $(d-k)$-dimensional face of a parallelohedral tiling of $\mathbb R^d$. Then $$\nu(F)\leq 2^k.$$
The upper bound $2^k$ is sharp for all integer $d, k$ satisfying $0<k\leq d$; for example, for every $(d-k)$-face $F$ of a cubic tiling of $\mathbb R^d$ holds $$\nu(F) = 2^k.$$
Theorem \[2k\] immediately implies
\[finiteness\]
Given $k\in \mathbb N$, there exists a set of complete $k$-dimensional cone fans $$\{\mathcal C^k_1, \mathcal C^k_2, \ldots, \mathcal C^k_{N(k)} \}$$ such that for every $d$, every $d$-parallelohedron $P$ and every $(d-k)$-face $F$ of $\mathcal T(P)$ the fan of $F$ is isomorphic to some $\mathcal C^k_i$.
Obviously, there is only a finite number of combinatorial types of complete $k$-dimensional cone fans splitting $\mathbb R^k$ into no more than $2^k$ full-dimensional convex polyhedral cones. According to theorem \[2k\], the fan of $F$ necessarily belongs to one of those combinatorial types.
For a centrally symmetric polytope $Q$ denote by $c(Q)$ its center of symmetry.
To proceed with the proof of theorem \[2k\] recall several basic properties of parallelohedral tilings.
1. A parallelohedron $P$ is a centrally symmetric polytope (see [@min]).
2. The set $$\Lambda = \{c(P'): P'\in \mathcal T(P)\}$$ is a lattice (see also [@min]). Under assumption $\mathbf 0 \in \Lambda$, one can also treat $\Lambda$ as a translation group.
3. \[c3\] If $P_1, P_2 \in \mathcal T(P)$ and $P_1\cap P_2 \neq \varnothing$, then $P_1\cap P_2$ is a centrally symmetric face of $\mathcal T(P)$. Moreover, $$c(P_1\cap P_2) = \frac {c(P_1)+c(P_2)}{2}.$$ A face $F$ of $\mathcal T(P)$ representable in the form $F = P_1\cap P_2$, where $P_1, P_2 \in \mathcal T(P)$, is called [*standard*]{} (see [@dol]).
The Voronoi case
================
For a better understanding of the aim, first restrict oneself to the Voronoi case. Start considering this case with a folklore lemma.
\[parity\] Let $P_1, P_2 \in \mathcal T(P)$ be the 2 distinct parallelohedra such that $c(P_1)$ and $c(P_2)$ belong to the same class modulo $2\Lambda$. (In other words, $P_1$ and $P_2$ belong to the same parity class.) Then $P_1\cap P_2 = \varnothing$.
According to property 3 of a parallelohedral tiling (see page ), $$\label{50}
\frac{c(P_1) + c(P_2)}{2} \in (P_1\cap P_2).$$
On the other hand, since $c(P_1)$ and $c(P_2)$ belong to the same class $\mod{2\Lambda}$, $$\frac{c(P_1) + c(P_2)}{2} \in \Lambda.$$ Therefore $\frac{c(P_1) + c(P_2)}{2}$ is a center of some parallelohedron $P_3$ different from $P_1$ and $P_2$. Thus $$\frac{c(P_1) + c(P_2)}{2} \in int\, P_3,$$ which is a contradiction to (\[50\]). Lemma is proved.
The technique used in the proof is rather standard. For example, similar methods were used in [@dsa]. (See also [@vor part 1, p. 277], the description of parallelohedra of a given parity class adjoint to a given parallelohedron by a hyperface.)
Let $\mathcal T_V(\Lambda)$ be the Voronoi tiling for some $d$-dimensional lattice $\Lambda \subset \mathbb R^d$. Then $\mathcal T_V(\Lambda)$ has a dual [*Delaunay*]{} tiling $\mathcal D(\Lambda)$. Every $(d-k)$-face $F$ in $\mathcal T_V(\Lambda)$ has a $k$-dimensional dual face $D(F)$ in $\mathcal D(\Lambda)$ such that $$\label{51}
D(F) = \operatorname{\it conv}\{ c(P): P\in \mathcal T_V(\Lambda)\; \text{and} \; F\in P \}.$$ Moreover, the combinatorial structure of $D(F)$ completely describes the combinatorial structure of $\operatorname{\it Fan}(F)$. In particular, $\nu(F)$ is equal to the number of vertices of $D(F)$.
\[vcase\] The statement of theorem \[2k\] holds for Voronoi tilings.
Following the argument above, it is sufficient to prove that $D(F)$ has at most $2^k$ vertices. According to the properties of a lattice Delaunay tiling, $$dim\, \operatorname{\it aff}D(F) = k,$$ so the set of vertices of $D(F)$ is a subset of some $k$-dimensional lattice $\Lambda(F) \subset \Lambda$.
By lemma \[parity\], every 2 vertices of $D(F)$ belong to different classes $\mod{2\Lambda}$, otherwise 2 parallelohedra of the same parity class had to intersect at $F$. Therefore every 2 vertices of $D(F)$ belong to different classes $\mod{2\Lambda(F)}$. Since $\Lambda(F)$ has exactly $2^k$ classes $\mod{2\Lambda(F)}$, there are at most $2^k$ vertices of $D(F)$. Proposition is proved.
Another approach to estimate the number of vertices of $D(F)$ is described in [@dla Proposition 13.2.8].
Thus, if the Voronoi conjecture (conjecture \[voronoi\]) has a positive answer, theorem \[2k\] is proved.
In the way similar to corollary \[finiteness\], proposition \[vcase\] implies that there are finitely many combinatorial types of $D(F)$ for fixed $k$ and regardless of $d$. Moreover, an algorithm for classification all possible lattice Delaunay $k$-cells is given in [@dsv] An idea of such algorithms certainly belongs to Voronoi [@vor] who constructed an algorithm to classify all possible combinatorial types of Voronoi parallelohedra.
For an arbitrary parallelohedral tiling $\mathcal T(P)$ and its $(d-k)$-face $F$ it is also possible to introduce in the similar way as (\[51\]) the set $$D(F^{d-k}) = \operatorname{\it conv}\{c(P') : P'\in \mathcal T(P) \; \text{and} \; F^{d-k}\subset P' \}.$$ As far as author knows, there are no satisfactory results on $\dim\operatorname{\it aff}D(F)$ in the general case.
In the case $$\label{10}
dim\,\operatorname{\it aff}D(F) \leq k$$ theorem \[2k\] easily follows from lemma \[parity\]. Yet the inequality (\[10\]) remains an open problem.
Outline of the proof
====================
The methods described above essentially involve the assumption that conjecture \[voronoi\] has a positive answer. However, theorem \[2k\] can be proved in non-Voronoi case as well, by exploiting several other ideas.
The proof of theorem \[2k\] consists of the 3 main steps.
1. Construct a set $$\{ F_1 = F, F_2, \ldots, F_m \}$$ of all faces of a parallelohedron $P_0 \in \mathcal T(P)$ such that every 2 faces of the set are equivalent by a $\Lambda$-translation. Prove that $\nu(F_1) = m$.
2. Refine the notion of [*an antipodal set*]{} given in [@dgr]. Prove that the set $$W = \{\pi(F_i) : i = 1, 2, \ldots, m\},$$ is antipodal (here $\pi$ is the above defined projection along $\operatorname{\it lin}F$ onto the complementary space $(\operatorname{\it lin}F)^\bot$).
3. Estimate the cardinality of an arbitrary antipodal set in $\mathbb R^k$.
The third step uses the technique introduced by Danzer and Grünbaum in [@dgr]. However, [@dgr] deals with antipodal full-dimensional point sets in $\mathbb R^k$, i.e. the sets $W$ satisfying $dim \operatorname{\it aff}W = k$. Although Danzer and Grünbaum’s theorem cannot be used directly here, it is not hard to extend the technique to the class of antipodal sets satisfying the refined definition.
Faces equivalent by translation
===============================
To introduce a uniform notation, put $F_1 = F$. Choose a parallelohedron $P_0 \in \mathcal T(P)$ such that $F_1 \subset P_0$. Let $$\{F_1, F_2, \ldots, F_m\}$$ be the set of all faces of $P_0$ equivalent to $F_1$ up to a $\Lambda$-translation.
Denote by $\mathbf t_{ij}$ the vector of $\Lambda$-translation such that $$F_i + \mathbf t_{ij} = F_j.$$ For every $i,j \in \{1, 2, \ldots, m\}$ define $$P_{ij} = P_0 + \mathbf t_{ij}.$$ Clearly, $P_{ij}\in \mathcal T$.
\[valence\]
$\nu(F_1) = m$.
Since $F_i \subset P_0$, one has $$F_1 = F_i + \mathbf t_{i1} \subset P_0 + \mathbf t_{i1} = P_{i1}.$$
Therefore there are at least $m$ parallelohedra of $\mathcal T(P)$ meeting at $F_1$, because the parallelohedra $$P_0 = P_{11}, P_{21}, P_{31}, \ldots, P_{m1}$$ are pairwise different.
Suppose there exists one more parallelohedron $P'\in \mathcal T(P)$ such that $F_1\subset P'$. Thus there is a non-zero $\Lambda$-translation $\mathbf t$ such that $P' = P_0 + \mathbf t$. Obviously, $\mathbf t \neq \mathbf t_{i1}$, so $-\mathbf t\neq \mathbf t_{1i}$ for every $i=1,2,\ldots, m$.
Let $F' = F_1 - \mathbf t$. Since $F_1 \subset P'$, $$F' = F_1 - \mathbf t \subset P' - \mathbf t = P_0.$$ Consequently, $F'$ is a face of $P_0$ equivalent to $F_1$ up to a $\Lambda$-translation, hence $F' = F_i$ for some $i\in \{2,3,\ldots, m\}$. By definition of $\mathbf t_{1i}$, one has $$F_1 - \mathbf t = F' = F_i = F_1 + \mathbf t_{1i}.$$ Therefore $-\mathbf t = \mathbf t_{1i}$, which is a contradiction. So, there are exactly $m$ parallelohedra of $\mathcal T(P)$ meeting at $F_1$, namely $$P_0, P_{21}, P_{31}, \ldots, P_{m1}.$$ Thus $\nu(F_0) = m$.
Constructing an antipodal set
=============================
Call a finite set $W\subset \mathbb R^k$ [*antipodal*]{} if for every pair of distinct points $x, y \in W$ there exists a pair of [*distinct*]{} parallel hyperplanes $\beta, \gamma$ such that $$x\in \beta, \quad y\in \gamma,$$ and $W$ lies between $\beta$ and $\gamma$.
Recall that $\pi$ is a projection along $\operatorname{\it lin}F_1$ onto the complementary space $\mathbb R^k$.
\[antipod\]
Let $w_i = \pi(F_i)$. Then $$W = \{ w_i: i = 1, 2, \ldots, m \}$$ is an antipodal set in $\mathbb R^k$.
It is sufficient to show that for every integer $i,j, 1\leq i<j\leq m$ there exist two distinct parallel hyperplanes $\gamma_{ij}$ and $\gamma_{ji}$ such that $$w_i \in \gamma_{ij}, \quad w_j\in \gamma_{ji},$$ and $W$ lies between these hyperplanes.
In $\mathbb R^d$ take the hyperplane $\Gamma_{ij}$ that separates the parallelohedra $P_0$ and $P_{ji}$ from each other. ($\Gamma_{ij}$ has to be a supporting hyperplane to each of these 2 parallelohedra.) Since $F_j\subset P_0$, $$F_i = F_j + \mathbf t_{ji} \subset P_0 + \mathbf t_{ji} = P_{ji},$$ and therefore $F_i \subset P_0\cap P_{ji} \subset \Gamma_{ij}$.
By definition, put $\Gamma_{ji} = \Gamma_{ij} - \mathbf t_{ji}$. The hyperplane $\Gamma_{ij}$ is supporting to $P_{ji}$, so the hyperplane $\Gamma_{ij}$ is supporting to $P_{ji} - \mathbf t_{ji} = P_0$. Also, since $F_i \subset \Gamma_{ij}$, $$F_j = F_i - \mathbf t_{ji} \subset \Gamma_{ij} - \mathbf t_{ji} = \Gamma_{ji}.$$
Thus $\Gamma_{ij}$ and $\Gamma_{ji}$ are two parallel supporting hyperplanes satisfying $$F_i \subset \Gamma_{ij}, \quad F_j\subset \Gamma_{ji}.$$
Now define $$\gamma_{ij} = \pi (\Gamma_{ij}) \quad \text{and} \quad \gamma_{ji} = \pi (\Gamma_{ji}).$$ Since $P_0$ lies between $\Gamma_{ij}$ and $\Gamma_{ji}$, then $W$ lies between $\gamma_{ij}$ and $\gamma_{ji}$. The hyperplanes $\gamma_{ij}$ and $\gamma_{ji}$ are distinct because the point $\pi(c(P_0))$ lies strictly between them. Thus the required hyperplanes are constructed for every pair of points $(w_i, w_j)$. Lemma \[antipod\] is proved.
Cardinality of an antipodal set
===============================
\[card\]
Let $W = \{w_1, w_2, \ldots, w_m\} \subset \mathbb R^k$ be an antipodal set. Then $m\leq 2^k$.
Denote by $H_x^a$ the homothety with center $x$ and coefficient $a$.
Take an arbitrary $a\in (0, \frac 12)$ and prove that $$H_{w_i}^a(\operatorname{\it conv}W) \cap H_{w_j}^a(\operatorname{\it conv}W) = \varnothing.$$
Indeed, let $\beta_{ij}$ be the hyperplane parallel and equidistant to the hyperplanes $\gamma_{ij}$ and $\gamma_{ji}$ . Then the sets $H_{w_i}^a(\operatorname{\it conv}W)$ and $H_{w_j}^a(\operatorname{\it conv}W)$ lie in different open half-spaces in respect to $\beta_{ij}$ and therefore do not intersect.
Let $k'\leq k$ be the affine dimension of $W$. Since $$H_{w_i}^a(\operatorname{\it conv}W)\subset \operatorname{\it conv}W$$ and because the sets $H_{w_i}^a(W)$ are pairwise non-intersecting, one has $$\label{60}
vol_{k'}(\operatorname{\it conv}W)\geq \sum\limits_{i=1}^m vol_{k'}(H_{w_i}^a(\operatorname{\it conv}W)) = m\cdot a^{k'}\,vol_{k'}(\operatorname{\it conv}W),$$ where $vol_{k'}$ stands for the $k'$-dimensional volume.
Further, because $vol_{k'}(W)>0$, (\[60\]) implies $$\label{61}
m \leq a^{-k'}.$$ The inequality (\[61\]) holds for every $0<a<\frac 12$, so it holds also for $a = \frac 12$: $$m \leq 2^{k'}\leq 2^k,$$
Lemma \[card\] is proved.
Now the statement of theorem \[2k\] is easily obtained by combining lemma \[antipod\] and lemma \[card\].
Acknowledgements {#acknowledgements .unnumbered}
================
There are a number of people who participated in the discussion of ideas and results, namely, R. Erdahl, F. Vallentin, A. Schürmann, A. Garber, A. Gavrilyuk and M. Kozachok.
The research was partially done at Fields Institute, Toronto, Canada during the Thematic Semester of Discrete Geometry and Applications and at Queen’s University, Kingston, Canada with an invitation of prof R. Erdahl. Author also appreciates the effort by the Laboratory of Geometrical Methods in Mathematical Physics at MSU, Moscow to make possible the visit to Canada.
Finally, author acknowledges prof. N. Dolbilin for scientific guidance, an introduction to the theory of parallelohedra and numerous useful comments on the text.
[99]{}
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B. N. Delaunay, N. N. Sandakova, Theory of stereohedra, Proc. of Steklov Math. Inst., 64 (1961), 28 – 51. (In Russian)
M. M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer, Berlin – Heidelberg – New York, 1997.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the recombination of two neutrons and deuteron into neutron and ${}^3$H using realistic nucleon-nucleon potential models. Exact Alt, Grassberger, and Sandhas equations for the four-nucleon transition operators are solved in the momentum-space framework using the complex-energy method with special integration weights. We find that at astrophysical or laboratory neutron densities the production of ${}^3$H via the neutron-neutron-deuteron recombination is much slower as compared to the radiative neutron-deuteron capture. We also calculate neutron-${}^3$H elastic and total cross sections.'
author:
- 'A. Deltuva'
- 'A. C. Fonseca'
title: '${}^3$H production via neutron-neutron-deuteron recombination'
---
Introduction \[sec:intro\]
==========================
The nonrelativistic quantum mechanics solution of the four-nucleon scattering problem has, in the past five years, reached a level of sophistication and numerical accuracy that makes it a natural theoretical laboratory to study nucleon-nucleon (NN) force models with the same confidence as one has used the three-nucleon system in the past. This has been demonstrated in a recent benchmark for ${n\text{-}{}^3\mathrm{H}}$ and ${p\text{-}{}^3\mathrm{He}}$ elastic scattering observables [@viviani:11a], where three different theoretical frameworks have been compared, namely, the hyperspherical harmonics (HH) expansion method [@viviani:01a; @kievsky:08a], the Faddeev-Yakubovsky (FY) equations [@yakubovsky:67] for the wave function components in coordinate space [@lazauskas:04a; @lazauskas:09a], and the Alt, Grassberger and Sandhas (AGS) equations [@grassberger:67] for transition matrices that were solved in the momentum space [@deltuva:07a; @deltuva:07b]. All methods include not only the hadronic NN interaction, but also the Coulomb repulsion between protons. While the first two methods have the advantage of being able to deal with charged-particle reactions at very low energies and include static three-nucleon forces (3NF), the third one is the only method so far to make predictions for multichannel reactions such as $d+d \to d+d$, $d+d \to n+{{}^3\mathrm{He}}$, $d+d \to p+{{}^3\mathrm{H}}$, and $p+{{}^3\mathrm{H}}\to n+{{}^3\mathrm{He}}$ (and the corresponding inverse reactions) [@deltuva:07c; @deltuva:10a].
In a previous publication [@deltuva:12c] a major step was taken in extending the AGS calculations above three- and four-cluster breakup thresholds. Owing to the complicated analytic structure of the four-body kernel above breakup threshold the calculations were performed using the complex energy method [@kamada:03a; @uzu:03a] whose accuracy and practical applicability was greatly improved by a special integration method [@deltuva:12c]. This allowed us to achieve fully converged results for ${n\text{-}{}^3\mathrm{H}}$ elastic scattering with realistic NN interactions. We note that the FY calculations of ${n\text{-}{}^3\mathrm{H}}$ elastic scattering have been recently extended as well to energies above the four-nucleon breakup threshold [@lazauskas:12a], however, using a semi-realistic NN potential limited to $S$-waves.
In the present work we extend the method of Ref. [@deltuva:12c] to calculate the neutron-neutron-deuteron $(nnd)$ recombination into $n+{{}^3\mathrm{H}}$ and its time-reverse reaction, i.e., the three-cluster breakup $n+{{}^3\mathrm{H}}\to n+n+d$. Although breakup reactions are usually measured in nuclear physics, the recombination has the advantage that its rate is finite at threshold where the breakup cross section vanishes due to phase-space factors. Furthermore, $n+n+d \to n+{{}^3\mathrm{H}}$ is the only hadronic recombination reaction in the four-nucleon system that at threshold is not suppressed by the Coulomb barrier (like $n+p+d \to p+{{}^3\mathrm{H}}$) or Pauli repulsion (like $n+n+n+p \to n+{{}^3\mathrm{H}}$). It can take place in any environment with neutrons and deuterons and, with respect to the tritium synthesis, it may be competitive to the electromagnetic capture reaction $n+d \to \gamma+{{}^3\mathrm{H}}$. Thus, one may rise the question at what conditions the $n+n+d \to n+{{}^3\mathrm{H}}$ recombination would dominate over the $n+d \to \gamma+{{}^3\mathrm{H}}$ radiative capture and to what extent it is relevant for astrophysical processes.
In addition, we also present results for the $n+{{}^3\mathrm{H}}$ elastic scattering and study the energy dependence of the total $n+{{}^3\mathrm{H}}$ cross section.
4N scattering Equations \[sec:eq\]
==================================
We use the time-reversal symmetry to relate the $nnd$ recombination amplitude to the three-cluster breakup amplitude of the initial ${n\text{-}{}^3\mathrm{H}}$ state, i.e., $$\label{eq:trev}
\langle \Phi_{1} | T_{13} | \Phi_{3} \rangle =
\langle \Phi_{3} | T_{31} | \Phi_{1} \rangle.$$ Here $| \Phi_{1} \rangle$ is the ${n\text{-}{}^3\mathrm{H}}$ channel state and $| \Phi_{3} \rangle$ is the $nnd$ channel state. The advantage is that the three-cluster breakup amplitude $\langle \Phi_{3} | T_{3 1} | \Phi_{1} \rangle$ is more directly related to the AGS transition operators ${\mathcal{U}}_{\beta \alpha}$ calculated in our previous works [@deltuva:07a; @deltuva:12c]. Since we use the isospin formalism where the nucleons are treated as identical fermions, there are only two distinct two-cluster partitions, namely, $\beta,\alpha=1$ corresponds to the $3+1$ partition (12,3)4 whereas $\beta=2$ corresponds to the $2+2$ partition (12)(34). For the initial ${n\text{-}{}^3\mathrm{H}}$ state we need only ${\mathcal{U}}_{\beta 1}$, i.e., we solve the AGS equations for the four-nucleon transition operators
\[eq:AGS\] $$\begin{aligned}
{\mathcal{U}}_{11} = {}& -(G_0 \, t \, G_0)^{-1} P_{34} -
P_{34} U_1 G_0 \, t \, G_0 \, {\mathcal{U}}_{11} \nonumber \\
{}& + U_2 G_0 \, t \, G_0 \, {\mathcal{U}}_{21}, \label{eq:U11} \\
\label{eq:U21}
{\mathcal{U}}_{21} = {}& (G_0 \, t \, G_0)^{-1} (1 - P_{34})
+ (1 - P_{34}) U_1 G_0 \, t \, G_0 \, {\mathcal{U}}_{11}.\end{aligned}$$
The free resolvent with the complex energy parameter $Z = E+ i\varepsilon$ and the free Hamiltonian $H_0$ is $$\begin{gathered}
\label{eq:G0}
G_0 = (Z - H_0)^{-1}\end{gathered}$$ whereas the pair (12) transition matrix $$\begin{gathered}
\label{eq:t}
t = v + v G_0 t\end{gathered}$$ is derived from the respective potential $v$. The 3+1 and 2+2 subsystem transition operators are obtained from the integral equations $$\begin{gathered}
\label{eq:AGSsub}
U_\alpha = P_\alpha G_0^{-1} + P_\alpha t\, G_0 \, U_\alpha.\end{gathered}$$ The basis states are antisymmetric under exchange of the two nucleons (12). In the $2+2$ partition the basis states have to be antisymmetric also under exchange of the two nucleons (34). The full antisymmetry as required for the four-nucleon system is ensured by the permutation operators $P_{ab}$ of nucleons $a$ and $b$ with $P_1 = P_{12}\, P_{23} + P_{13}\, P_{23}$ and $P_2 = P_{13}\, P_{24}$.
The ${n\text{-}{}^3\mathrm{H}}$ elastic and inelastic reaction amplitudes at the available energy $E = \epsilon_1 + p_1^2/2\mu_1$ are obtained in the limit $\varepsilon \to +0$. Here $\epsilon_1$ is the ${{}^3\mathrm{H}}$ ground state energy, ${{\mathbf{p}}}_1$ is the relative ${n\text{-}{}^3\mathrm{H}}$ momentum, and $\mu_1 = 3m_N/4$, $m_N$ being the nucleon mass. The elastic scattering amplitude is calculated in Refs. [@deltuva:07a; @deltuva:12c]. The amplitude for the $nnd$ breakup is obtained by the antisymmetrization of the general three-cluster breakup amplitude [@deltuva:12e], resulting $$\label{eq:U0}
\begin{split}
\langle \Phi_{3} | T_{3 1} | \Phi_{1} \rangle
= {}& \sqrt{3} \langle \Phi_{3} |
[(1- P_{34}) U_1 G_0 \, t \, G_0 \, {\mathcal{U}}_{11} \\ & {} +
U_2 G_0 \, t \, G_0 \, {\mathcal{U}}_{21} ]
| \phi_{1} \rangle .
\end{split}$$ Here $| \phi_{1} \rangle$ is the Faddeev component of the ${n\text{-}{}^3\mathrm{H}}$ channel state $| \Phi_{1} \rangle = (1+P_1)| \phi_{1} \rangle$; $\epsilon_1$ and $| \phi_{1} \rangle$ are obtained by solving the bound-state Faddeev equation $$\begin{gathered}
\label{eq:phi}
|\phi_{1} \rangle = G_0 t P_1 |\phi_{1} \rangle\end{gathered}$$ at $\varepsilon \to +0$.
We solve the AGS equations in the momentum-space partial-wave framework. The momentum and angular momentum part of the basis states are $ | k_x \, k_y \, k_z
[l_z (\{l_y [(l_x S_x)j_x \, s_y]S_y \} J_y s_z ) S_z] \,\mathcal{JM} \rangle$ for the $3+1$ configuration and $|k_x \, k_y \, k_z (l_z \{ (l_x S_x)j_x\, [l_y (s_y s_z)S_y] j_y \} S_z)
\mathcal{ J M} \rangle $ for the $2+2$. Here $k_x , \, k_y$, and $k_z$ are the four-particle Jacobi momenta as given in Ref. [@deltuva:12a], $l_x$, $l_y$, and $l_z$ are the corresponding orbital angular momenta, $j_x$ and $j_y$ are the total angular momenta of pairs (12) and (34), $J_y$ is the total angular momentum of the (123) subsystem, $s_y$ and $s_z$ are the spins of nucleons 3 and 4, $S_x$, $S_y$, and $S_z$ are the channel spins of two-, three-, and four-particle systems, and $\mathcal{J}$ is the total angular momentum with the projection $\mathcal{M}$. We include a large number of four-nucleon partial waves, $l_x,l_y,l_z,j_x,j_y,J_y \le 4$ and $\mathcal{J} \le 5$, such that the results are well converged. The complex-energy method [@kamada:03a] with special integration weights [@deltuva:12c] is used to treat the singularities of the AGS equations . To obtain accurate results for the breakup amplitude $\langle \Phi_{3} | T_{3 1} | \Phi_{1} \rangle$ near the $nnd$ threshold we had to use $0.1$ MeV $ \le \varepsilon \le 0.4$ MeV that are smaller than $1.0$ MeV $ \le \varepsilon \le 2.0$ MeV used in the elastic scattering calculations of Ref. [@deltuva:12c]. However, the need for relatively small $\varepsilon$ values caused no technical problems since the integration with special weights [@deltuva:12c] provides very accurate treatment of the ${{}^3\mathrm{H}}$ pole whereas the quasi-singularities due to deuteron pole are located in a very narrow region with very small weight, such that about 30 grid points for the discretization of each momentum variable were sufficient.
Results \[sec:res\]
===================
The $nnd$ recombination rate $K_3$ is defined such that the number of recombination events per volume and time is $K_3 \rho_n^2 \rho_d$ with $\rho_n$ ($\rho_d$) being the density of neutrons (deuterons). We calculate it as a function of the relative $nnd$ kinetic energy $E_3 = E - \epsilon_d$, i.e., $$\label{eq:K3}
\begin{split}
K_3 = & {} \frac{ (2\pi)^7 \mu_1 p_1}
{g_3 \pi^2 (\mu_{\alpha y}\mu_{\alpha})^{3/2} E_3^2}
\sum_{m_s} \int d^3k_y \, d^3k_z \\ & {} \times
| \langle \Phi_{3} | T_{3 1} | \Phi_{1} \rangle |^2 \,
\delta \left(E_3 - \frac{k_y^2}{2\mu_{\alpha y}} - \frac{k_z^2}{2\mu_{\alpha}}
\right).
\end{split}$$ Here $\epsilon_d = -2.2246$ MeV is the deuteron bound state energy, $\mu_{\alpha y}$ and $\mu_{\alpha}$ are the reduced masses associated with the four-nucleon Jacobi momenta $k_y$ and $k_z$. For example, in the 2+2 configuration $k_y$ is the relative momentum of the two neutrons while $k_z$ is the relative momentum between the center of mass (c.m.) of the two-neutron subsystem and the deuteron. The $nnd$ state can be represented in both 3+1 and 2+2 configurations equally well; $\mu_{\alpha y}\mu_{\alpha} = m_N^2/2$. The sum in Eq. runs over all initial and final spin projections $m_s$ that are not explicitly indicated in our notation for the channel states while $g_3=12$ takes care of the spin averaging in the initial $nnd$ state. The integral in Eq. , up to a factor, determines also the total cross section $\sigma_3$ for the three-cluster breakup of the initial ${n\text{-}{}^3\mathrm{H}}$ state. Thus, the $nnd$ recombination rate can be related to $\sigma_3$ as $$\label{eq:K3s3}
K_3 = \frac{8 \pi g_1 p_1^2}
{g_3 (\mu_{\alpha y}\mu_{\alpha})^{3/2}\,E_3^2 } \, \sigma_3$$ where $g_1=4$ is the number of ${n\text{-}{}^3\mathrm{H}}$ spin states. Below the four-nucleon breakup threshold $\sigma_3$ can be obtained via the optical theorem as a difference between the total and elastic cross sections. The equation cannot be used right at the $nnd$ threshold where both $E_3$ and $\sigma_3$ vanish. For $E_3 \to 0$ the $nnd$ recombination rate becomes $$\label{eq:K30}
K_3^0 = \frac{ 4\pi}{g_3} (2\pi)^7 \mu_1 p_1
\sum_{m_s} | \langle \Phi_{3}^{0} | T_{3 1} | \Phi_{1} \rangle |^2 ,$$ where for the channel state $|\Phi_{3}^{0} \rangle$ the relative momenta $k_y = k_z = 0$. The most convenient representation for $|\Phi_{3}^{0} \rangle$ is a single-component 2+2 state with $l_y=l_z=S_y=j_y=0$ and $j_x=S_z=\mathcal{J}=1$.
We study the four-nucleon system using realistic high-precision two-nucleon potentials, namely, the inside-nonlocal outside-Yukawa (INOY04) potential by Doleschall [@doleschall:04a; @lazauskas:04a], the Argonne (AV18) potential [@wiringa:95a], the charge-dependent Bonn potential (CD Bonn) [@machleidt:01a], and its extension CD Bonn + $\Delta$ [@deltuva:03c] allowing for an excitation of a nucleon to a $\Delta$ isobar and thereby yielding effective three- and four-nucleon forces. Among these potentials only INOY04 nearly reproduces experimental binding energy of ${{}^3\mathrm{H}}$ (8.48 MeV), while AV18, CD Bonn and CD Bonn + $\Delta$ underbind the ${{}^3\mathrm{H}}$ nucleus by 0.86, 0.48 and 0.20 MeV, respectively.
First we study the ${n\text{-}{}^3\mathrm{H}}$ reactions for existing experimental data. We concentrate on the energy regime relevant for the $nnd$ recombination, i.e., between the three- and four-cluster breakup thresholds. In Fig. \[fig:dcs\] we show the differential cross section for $n$-${{}^3\mathrm{H}}$ elastic scattering at $E_n =9$ MeV neutron energy corresponding to $E_3 = 0.49$ MeV. The predictions agree well with the experimental data of Ref. [@seagrave:72] and are quite insensitive to the choice of the potential. Results for $n$-${{}^3\mathrm{H}}$ elastic scattering above the four-cluster breakup threshold up to $E_n =22.1$ MeV are given in Ref. [@deltuva:12c].
![ \[fig:dcs\] (Color online) Differential cross section for elastic $n$-${{}^3\mathrm{H}}$ scattering at 9 MeV neutron energy as a function of c.m. scattering angle. Results obtained with INOY04 (solid curves) and CD Bonn (dashed-dotted curves) potentials are compared with the experimental data from Ref. [@seagrave:72].](nt9dcs.eps)
In Fig. \[fig:tcs\] we show the total cross section for $n$-${{}^3\mathrm{H}}$ scattering at neutron energies ranging from 0 to 22 MeV and compare it to the data of Refs. [@battat:59; @phillips:80]. The three-cluster (four-cluster) breakup threshold corresponds to $E_n = 8.35$ (11.31) MeV. As already found in Refs. [@lazauskas:04a; @lazauskas:05a; @deltuva:07a], the total $n$-${{}^3\mathrm{H}}$ cross section around the low-energy peak is underpredicted by the traditional two-nucleon potentials while the low-momentum or chiral effective field theory potentials may provide a better description [@deltuva:08b; @viviani:fb19]. Although with increasing energy the predictions approach the experimental data, as already mentioned [@deltuva:07a], the elastic and total cross section data may be inconsistent. In the low-energy regime where the inelastic cross section should vanish for $E_n \le 8.35$ MeV and remain very small at $E_n=9$ MeV, there is in general a better agreement between predictions and experiment for the elastic differential cross section than for the total cross section which is significantly underestimated by theory. A solution to this discrepancy may require new measurements in this energy regime.
![ \[fig:tcs\] (Color online) Total cross section for $n$-${{}^3\mathrm{H}}$ scattering as a function of the neutron lab energy. Results obtained with INOY04 (solid curves) and CD Bonn (dashed-dotted curves) potentials are compared with the experimental data from Refs. [@battat:59; @phillips:80].](nttotz.eps)
In Fig. \[fig:k3\] we study the energy-dependence of the $nnd$ recombination rate in the standard form $N_A^2 K_3$ where $N_A$ is the Avogadro’s number. We show only INOY04 predictions as it is the only used potential with correct $\epsilon_1$ and $p_1$ values. The results at $E_3=0$ are obtained from Eq. while at $E_3 > 0$ it was more convenient to use Eq. where $\sigma_3$ was calculated using optical theorem. Thus, for $E_3 > |\epsilon_d|$ our predictions in Fig. \[fig:k3\] estimate the upper limit for $N_A^2 K_3$ since they assume that the four-cluster breakup cross section is much smaller than the three-cluster breakup cross section. In the relevant energy regime $0 \le E_3 \le |\epsilon_d|$ the $nnd$ recombination rate increases with increasing energy $E_3$ nearly linearly due to the increasing contributions of partial waves with nonzero orbital angular momentum $l_z$. The threshold values $N_A^2 K_3^0$ referring to all employed potentials are collected in Table \[tab:k30\]; they increase with ${{}^3\mathrm{H}}$ binding energy.
![ \[fig:k3\] (Color online) $nnd$ recombination rate $K_3$ as a function of relative kinetic $nnd$ energy $E_3$. Predictions are obtained using the INOY04 potential.](k3.eps)
[l\*[2]{}[c]{}]{} & $|\epsilon_1|$ (MeV) & $N_A^2 \,K_3^0 \; (\mathrm{cm}^6\mathrm{s}^{-1}\mathrm{mol}^{-2})$\
AV18 & 7.62 & $1.31 \times 10^{-5}$\
CD Bonn & 8.00 & $1.41 \times 10^{-5}$\
CD Bonn + $\Delta$ & 8.28 & $1.47 \times 10^{-5}$\
INOY04 & 8.49 & $1.52 \times 10^{-5}$\
Finally we compare the relative importance of the $nnd$ recombination and $nd$ radiative capture. For the latter the number of events, i.e., the number of produced ${{}^3\mathrm{H}}$ nuclei per volume and time is $K_2 \rho_n \rho_d$ with $K_2$ being the $nd$ capture rate. The threshold value for it given in Ref. [@fowler:67] is $N_A K_2^0 = 66.2 \, \mathrm{cm}^3\mathrm{s}^{-1}\mathrm{mol}^{-1}$. The critical density of neutrons at which both processes yield comparable contributions to the ${{}^3\mathrm{H}}$ production in the low-energy (low-temperature) limit is given by $\rho_n^c = K_2^0/K_3^0 \approx 2.6 \times 10^{30} \, \mathrm{cm}^{-3}$. This corresponds to the mass density of $4.4\times 10^{6} \, \mathrm{g}/\mathrm{cm}^{3}$. Thus, one may conclude that at the neutron density available in the laboratories (such as National Ignition Facility with expected $\rho_n \sim 10^{22}$ to $10^{25} \, \mathrm{cm}^{-3}$ [@frenje:pc]) the $nnd$ recombination is entirely irrelevant as well as for the big-bang nucleosynthesis where the estimated baryon density is even lower. On the other hand, the neutron density in core-collapse supernova or neutron stars may be higher than $\rho_n^c$ by several orders of magnitude but the absence of deuterons renders $n+d$ and $n+n+d$ reactions irrelevant. However, based on our results one may conjecture that at such high densities the three-cluster recombination of two neutrons and a heavier nucleus $A$, i.e., $n+n+A \to n + (An)$ might be as important as the corresponding radiative capture $n+A \to \gamma + (An)$. For example, the above reactions with $A$ being ${}^{20}$Ne are relevant for the neon-burning process.\
Summary \[sec:sum\]
===================
We have solved the four-nucleon AGS equations in the energy regime above the three-cluster threshold and studied the rate for the recombination reaction $n+n+d \to n+{{}^3\mathrm{H}}$. The obtained results show that the $nnd$ recombination is not competitive with the radiative $nd$ capture for the production of tritium at neutron densities available in laboratory induced fusion or astrophysical processes. Thus, one may conjecture with a confidence that other nucleon-nucleon-deuteron recombination reactions (for example, $p+p+d \to p + {{}^3\mathrm{He}}$ that could contribute to the hydrogen burning process in stars), being in addition suppressed by the Coulomb repulsion, are inferior to the respective nucleon-deuteron radiative capture reactions at realistic densities, and that four-nucleon recombination reactions are even far less relevant.
In addition, we presented results for the ${n\text{-}{}^3\mathrm{H}}$ elastic differential cross section at $E_n = 9$ MeV and the ${n\text{-}{}^3\mathrm{H}}$ total cross section up to $E_n = 22$ MeV. While the elastic cross section agrees fairly well with the data, there is a disagreement for the total cross section, especially in the low-energy regime. This indicates a possible inconsistencies between the ${n\text{-}{}^3\mathrm{H}}$ elastic and total cross section data and calls for new measurements.
The authors thank J. A. Frenje for discussions.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We compare asynchronous vs. synchronous update of discrete dynamical networks and find that a simple time delay in the nodes may induce a reproducible deterministic dynamics even in the case of asynchronous update in random order. In particular we observe that the dynamics under synchronous parallel update can be reproduced accurately under random asynchronous serial update for a large class of networks. This mechanism points at a possible general principle of how computation in gene regulation networks can be kept in a quasi-deterministic “clockwork mode” in spite of the absence of a central clock. A delay similar to the one occurring in gene regulation causes synchronization in the model. Stability under asynchronous dynamics disfavors topologies containing loops, comparing well with the observed strong suppression of loops in biological regulatory networks.'
author:
- Konstantin Klemm
- Stefan Bornholdt
title: |
Robust gene regulation:\
Deterministic dynamics from asynchronous networks with delay
---
Erwin Schrödinger in his lecture “What is life?” held in 1943 [@Schroedinger] was one of the first to notice that the information processing performed in the living cell has to be extremely robust and therefore requires a quasi-deterministic dynamics (which he called “clockwork mode”). The discovery of a “digital” storage medium for the genetic information, the double-stranded DNA, confirmed one important part of this picture. Today, new experimental techniques allow to observe the dynamics of regulatory genes in great detail, which motivates us to reconsider the other, dynamical part of Schrödinger’s picture of a “clockwork mode”. While the dynamical elements of gene regulation often are known in great detail, the complex dynamical patterns of the vast network of interacting regulatory genes, while highly reproducible between identical cells and organisms under similar conditions, are largely not understood. Most remarkably, these virtually deterministic activation patterns are often generated by asynchronous genetic switches without any central clock. In this Letter we address this astonishing fact with a toy model of gene regulation and study the conditions of when deterministic dynamics could occur in asynchronous circuits. Let us start from the observed dynamics of small circuits of regulatory genes, then derive a discrete dynamical model gene, followed by a study of networks of such genetic switches, with a focus on comparing their asynchronous and synchronous dynamics.
Recently, several small gene regulation circuits have been described in terms of a detailed picture of their dynamics [@Elowitz; @Hes1; @Baltimore; @p53; @Smolen]. A particularly simple motif is the single, self-regulating gene [@Rosenfeld; @Hes1] that allows for a detailed modeling of its dynamics. A set of two differential equations, for the temporal evolution of the concentrations of messenger RNA and protein, respectively, and an explicit time delay for transmission delay provide a quantitative model for the observed dynamics in this minimal circuit [@Jensen03]. The equations of this model take the basic form $$\begin{aligned}
\label{eq:originaldiff1}
\frac {{{\rm d}}c}{{{\rm d}}t} &=& \alpha [f(s(t-\vartheta)) - c(t)] \\
\frac {{{\rm d}}b}{{{\rm d}}t} &=& \beta [c(t)-b(t)]\end{aligned}$$ for the the dynamics of the concentrations $c$ of mRNA and $b$ of protein, with some non-linear transmission function $f(s)$ of an input signal $s$, a time delay $\vartheta$, and the time constants $\alpha$ and $\beta$. In order to define a minimal discrete gene model let us keep the basic features (delay, low pass filter characteristics), omit the second filter, and write the difference equation for one gene $i$ as $$\Delta c_i = \alpha [f(s_i(t-\vartheta)) - c_i(t)] \Delta t~.$$ The non-linear function $f$ is typically a steep sigmoid. We approximate it as a step function $\Theta$ with $\Theta(s)=0$ for $s<0$ and $\Theta(s)=1$ otherwise. Rescaling time with $\epsilon = \alpha \Delta t$ and $\tau = \vartheta / \Delta t$ this reads $$\Delta c_i = \epsilon [\Theta(s_i(t-\tau)) - c_i(t)]~.$$ For simplicity let us update $c_i$ by equidistant steps according to $$\label{eq:cupdatefinal}
\Delta c_i = \left\{
\begin{array}{rl}
+\epsilon, & {\rm if }\;s_i(t-\tau) \ge 0\; {\rm and}\;c_i \le 1-\epsilon \\
-\epsilon, & {\rm if }\;s_i(t-\tau) < 0 \; {\rm and}\;c_i \ge \epsilon \\
0, & {\rm otherwise}
\end{array}
\right.$$ The coupling between nodes is defined by $$\label{eq:xsum}
s_i (t) = \sum_j w_{ij} x_j (t) - a_i~,$$ with discrete output states $x_j (t)$ of the nodes defined as $$\label{eq:xdefinition}
x_j (t) = \Theta(c_j (t) - 1/2)~.$$ The influence of node $j$ on node $i$ can be activating ($w_{ij}=1$), inhibitory ($w_{ij}=-1$), or absent ($w_{ij}=0$). A constant bias $a_i$ is assigned to each node.
In the following let us consider a network model of such nodes. Consider $N$ nodes with concentration variables $c_i$, state variables $x_i$, biases $a_i$ and a coupling matrix $(w_{ij})$. Given initial values $x_i(0)=c_i(0)\in\{0,1\}$ the time-discrete dynamics is obtained by iterating the following update steps:
\(1) Choose a node $i$ at random. (2) Calculate $s_i$ according to Eq. (\[eq:xsum\]). (3) Update $c_i$ according to Eq. (\[eq:cupdatefinal\]).
For $\tau=0$ and $\epsilon=1$ random asynchronous update is recovered. For $\tau>0$ there is an explicit transmission delay from the output of node $j$ to the input of node $i$. To be definite, at $t=0$ we assume that nodes have not flipped during the previous $\tau$ time steps.
Let us first explore the dynamics of a simple but non-trivial interaction network with $N=3$ sites and
non-vanishing couplings $w_{01} = w_{21} = -1$ and $w_{10}= w_{12} = +1$, see Fig. \[fig:combined\]. Note that under [*asynchronous*]{} update the sequence of states reached by the dynamics is not unique. The system may branch off to different configurations depending on node update ordering. This is illustrated in Fig. \[fig:singletsz\_0\](a): Without delay ($\tau=0$) and filter ($\epsilon=1$) the dynamics is irregular, [[*i.e. *]{}]{}non-periodic.
With filter only ($\tau=0$, $\epsilon=0.01$, Fig. \[fig:singletsz\_0\](b)), the dynamics is periodic at times, but also intervals of fast irregular flipping occur. Finally, in the presence of delay ($\tau=100$, $\epsilon=1$, Fig. \[fig:singletsz\_0\](c)) we obtain perfectly ordered dynamics with synchronization of flips. Nodes 0 and 2 change states practically at the same (macro) time, followed by a longer pause until node 1 changes state, etc. With increasing delay time $\tau$ the dynamics under asynchronous update approaches the dynamics under synchronous update (cf. Fig. \[fig:combined\]) when viewed on a coarse-grained (macro) time scale.
Let us further quantify the difference between synchronous and asynchronous dynamics. First, a definition of equivalence between the two dynamical modes has to be given. Let us start from the time series ${{\bf x}}(t)$ of configurations ${{\bf x}}= (x_0,\dots, x_{N-1})$ produced by the asynchronous (random serial) update of the model and the respective time series ${{\bf y}}(u)$ produced by synchronous (parallel) update, using identical initial condition ${{\bf y}}(0) = {{\bf x}}(0)$. These time series live on different time scales, which we call the micro time scale of single site updates in the asynchronous case, and the macro time scale where each time step is an entire sweep of the system. Assume that at time $t_u$ the asynchronous system is in state ${{\bf x}}(t_u) = {{\bf y}}(u)$. In order to follow the synchronous update it has to subsequently reach the state ${{\bf y}}(u+1)$ on a shortest path in phase space. Formally, let us require that there is a micro time $t_{u+1}>t_u$ such that ${{\bf x}}(t_{u+1})= {{\bf y}}(u+1)$ and each node flips at most once in the time interval $[t_u,t_{u+1}]$. Once this is violated we say that an error has occured at the particular macro time step $u$. This error allows to define a numerical measure of discrepancy between asynchronous and synchronous dynamics. Starting from identical initial conditions, the system is iterated in synchronous and asynchronous modes (here for $u_{\rm total} = 10^7$ macro time steps). Whenever the resulting time series are no longer equivalent, an error counter is incremented and the system reset to initial condition. The total error $E$ of the run is the number of errors divided by $u_{\rm total}$.
For the network in Fig. \[fig:combined\] and the initial condition $x_i=c_i=0$ for $i=1,2,3$ the error $E$ is exponentially suppressed with delay time $\tau$ (Fig. \[fig:singlez\_0\]).
The asynchronous dynamics with delay follows the attractor during a time span that increases exponentially with the given delay time. Note that there is only one possibility for the asynchronous dynamics to leave the attractor: When the system is in configuration $(1,1,0)$ or $(0,1,1)$, node $2$ may change state such that the system goes to configuration $(1,0,0)$ or $(0,0,1)$ respectively, whereas the correct next configuration on the attractor is $(0,1,0)$. Consider the case $\epsilon=1$ where $c_i=x_i$ for all $i$. Let us assume that the system is in configuration $(1,1,1)$ and at time $t_0$ node 0 changes state, thereby generating configuration $(0,1,1)$. This decreases the input sum $s_1$ below zero such that for $\tau=0$ node $0$ would change state immediately in its next update. With explicit transmission delay $\tau>0$, however, node 1 still “sees” the input sum $s_i=0$ generated by the configuration $(1,1,1)$ until time step $t_0+\tau$. If node $2$ is chosen for update in this time window $t_0+1,\dots,t_0+\tau$ it changes state immediately and updates are performed in correct order. The opposite case, that node 2 does not receive an update in any of the $\tau$ time steps, happens with probability $(2/3)^\tau$, yielding the correct error decay of the simulation (Fig. \[fig:singlez\_0\]).
Next we demonstrate that there are cases where also low-pass filtering, $\epsilon \ll 1$, is needed for the asynchronous dynamics to follow the deterministic attractor. Consider a network of $N=5$ nodes with bias values $a_0 = a_4 = 0$ and $a_1 = a_2 = a_3 = 1$. The only non-zero couplings are $w_{10} = w_{21} = w_{31} = w_{42} = +1 $ and $w_{01} = w_{43} = -1$. Nodes 0 and 1 form an oscillator, [[*i.e. *]{}]{}$(x_0,x_1)$ iterate the sequence $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. Nodes $2$ and $3$ simply “copy” the state of node $1$ such that under synchronous update always $x_3(t)=x_2(t)=x_1(t-1)$. Consequently, under synchronous update the input sum of node $4$ never changes because the positive contribution from node 2 and the negative contribution from node 3 cancel out. Under asynchronous update, however, the input sum of node 4 may fluctuate because nodes 2 and 3 do not flip precisely at the same time. The effect of the low-pass filter $\epsilon \ll 1$ is to suppress the spreading of such fluctuations on the micro time scale. The influence of the filter is seen in Fig. \[fig:five\_0\].
When $\tau$ is kept constant, the error drops algebraically with decreasing $\epsilon$. An exponential decay $E \sim \exp(-\alpha /\epsilon)$ is obtained when $\tau \propto 1/\epsilon$ (the filter can take full effect only in the presence of sufficient delay).
Let us finally consider an example of a larger network with $N=16$ nodes and $L=48$ non-vanishing couplings (chosen randomly from the off-diagonal elements in the matrix $(w_{ij})$ and assigned values $+1$ or $-1$ with probability $1/2$ each; biases are chosen as $a_i = \sum_j w_{ij}/2$). Simulation runs under pure asynchronous update ($\tau=0$, $\epsilon=1$) typically yield dynamics as in Fig. \[fig:largets\_1\](a).
The time series ${{{\bf x}}(t)}$ is non-periodic and non-reproducible, [[*i.e. *]{}]{}under different order of updates a different series is obtained. For the same initial condition, periodic dynamics is observed in the presence of sufficent transmission delay and filtering, Fig. \[fig:largets\_1\](b). In this case, the system follows precisely the attractor of period 28 found under synchronous update. As seen in Fig. \[fig:largets\_1\](c), the error decays exponentially as a function of the delay time $\tau$.
Let us now turn to the dangers of asynchronous update: There is a fraction of attractors observed under synchronous update that cannot be realized under asynchronous update. Synchronization cannot be sustained if the dynamics is separable. In the trivial case, separability means that the set of nodes can be divided into two subsets that do not interact with each other. Then there is no signal to synchronize one set of nodes with the other and they will go out of phase. In general, synchronization is impossible if the set of flips itself is separable. Consider, as the simplest example, a network of $N=2$ nodes with the couplings $w_{01} = w_{10} = +1$, biases $a_0 = a_1 =
1$ and the initial condition $(y_0(0),y_1(0)) = (0,1)$. Under synchronous update, the state alternates between vector $(0,1)$ and $(1,0)$. Under asynchronous update with delay time $\tau$, the transition of one node $i$ from $x_i=0$ to $x_i=1$ causes the other node $j$ to switch from $x_j=0$ to $x_j=1$ approximately $\tau$ time steps later. The “on”-transitions only trigger subsequent “on”-transitions and, analogously, the “off”-transitions only trigger subsequent “off”-transitions. The dynamics can be divided into two distinct sets of events that do not influence each other. Consequently, synchronization between flips cannot be sustained, as illustrated in Fig. \[fig:cycle\_illu\].
When the phase difference reaches the value $\tau$, on- and off-transitions annihilate. Then the system leaves the attractor and reaches one of the fixed points with $x_0 = x_1$.
These observations have important implications for robust topological motifs in asynchronous networks. First of all, the above example of a small excitatory loop can be quickly generalized to any larger loop with excitatory interactions, as well as to loops with an even number of inhibitory couplings, where in principle similar dynamics could occur. Higher order structures that fail to synchronize include competing modules, e.g. two oscillators (loops with odd number of inhibitory links) that interact with a common target.
In conclusion we find that asynchronously updated networks of autonomous dynamical nodes are able to exhibit a reproducible and quasi-deterministic dynamics under broad conditions if the nodes have transmission delay and low pass filtering as, e.g., observed in regulatory genes. Timing requirements put constraints on the topology of the networks (e.g. suppression of certain loop motifs). With respect to biological gene regulation networks where indeed strong suppression of loop structures is observed [@Shen-Orr02; @Milo02], one may thus speculate about a new constraint on topological motifs of gene regulation: The requirement for deterministic dynamics from asynchronous dynamical networks.
[**Acknowledgements**]{} S.B. thanks D. Chklovskii, M.H. Jensen, S. Maslov, and K. Sneppen for discussions and comments, and the Aspen Center for Physics for hospitality where part of this work has been done.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In the past years, a clear picture of the evolution of outbursts of black-hole X-ray binaries has emerged. While the X-ray properties can be classified into our distinct states, based on spectral and timing properties, the observations in the radio band have shown strong links between accretion and ejection properties. Here I briefly outline the association between X-ray timing and jet properties.'
date: '?? and in revised form ??'
---
Fast time variability
=====================
The fast time variability observed in the X-ray emission from Black-Hole Binaries (BHB) can be extremely strong and complex. It is clearly connected to the spectral evolution throughout their outbursts, which can be described through the use of Hardness-Intensity Diagrams (HID; see [@belloni09 Belloni 2009] and references therein). A total fractional ms variability of $\sim$40% is a major “disturbance” of the accretion flow that can hardly be ignored when trying to understand its properties. Concentrating on the most basic properties, we can identify two categories: [*loud*]{} states (LHS and HIMS in [@belloni09]), characterized by strong flat-top noise components in the power spectra, with total fractional variability 10-40%, and [*quiet*]{} states (SIMS and HSS), with less variability in the form of a power law component. Quasi-Periodic Oscillations (QPO) are observed in all states, with a complex phenomenology. However, the HIMS-SIMS transition is very abrupt and involves the interplay between two very different “flavors” of QPO. This transition can be marked in a HID with a [it QPO line]{} (see Fig. \[fig:figure1\]). At the same time, the high-energy part of the X-ray spectrum undergoes abrupt changes through the transition (see [@motta09 Motta et al. 2009]).
Jet ejection
============
The radio properties of BHB display an evident connection with the X-ray states and transitions (see e.g. [@fender06 Fender 2006]). A relation with the states evolution was presented by [@fbg04 Fender, Belloni & Gallo (2004)] on the basis of four well-studied systems. At its basis, the unified picture of disk-jet coupling presented there identifies two regions of the HID: the hard region where a steady, compact and mildly relativistic jet is observed, and the soft region where there is no evidence of nuclear emission from the binary (see Fig. \[fig:figure1\]). The transition between these two regions marks the ejection of a fast relativistic jet, observed as a bright radio flare or, when imaged, as a superluminal jet. The position of this transition was dubbed “jet line”.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[![Left: schematic HID with the two regions identified through time variability and the ‘QPO line’ between them. Right: same HID, with the radio regions and the jet line’. The two fundamental lines do not coincide.[]{data-label="fig:figure1"}](belloni002.eps "fig:"){width="6.5cm"}]{} [![Left: schematic HID with the two regions identified through time variability and the ‘QPO line’ between them. Right: same HID, with the radio regions and the jet line’. The two fundamental lines do not coincide.[]{data-label="fig:figure1"}](belloni003.eps "fig:"){width="6.5cm"}]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
How do they connect?
====================
[@fbg04 Fender, Belloni & Gallo (2004)] identified the jet line with the QPO line. This identification would lead to the attractive conclusion that the plasma responsible for the noise would be the one ejected inform of a jet. However, [@fhb09 Fender, Homan & Belloni (2009)] recently reported, on the basis of a larger sample of sources, that the two lines do not always coincide, but are close. There seems not to be a direct causal connection, as sometimes one precedes the other and vice versa. However, the close association suggests that both ejection and change in timing properties are the outcome of a complex physical transition that takes place on a longer time scale.
Discussion
==========
The scheme outlined above, based on the HID and fast timing properties can be extended to neutron-star systems and even to white-dwarf binaries (see [@belloni09; @kording08; @fender09; @tudose09 Belloni 2009; K" ording et al. 2008; Fender 2009; Tudose et al. 2009]). It is clear that its properties are intimately connected to the spectral state and to the characteristics of the jet. An important key is the study of the correlated variability at optical wavelengths (see e.g. [@kanbach; @gandhi]), which can shed light on this connection.
2009, in: T.M. Belloni (ed.), *The Jet Paradigm: from Microquasars to Quasars*, Lecture Notes in Physics (Heidelberg: Springer), in press (arXiv:0909.2474)
2006, in: Lewin, W.H.G. & van der Klis, M. (eds.), *Compact stellar X-ray sources*, Cambridge Astrophysics Series, No. 39, Cambridge University Press, p. 381
2009, in: T.M. Belloni (ed.), *The Jet Paradigm: from Microquasars to Quasars*, Lecture Notes in Physics (Heidelberg: Springer), in press (arXiv:0909.2572)
2004, *MNRAS*, 355, 1105
2004, *Nature*, 324, 23
2008, *MNRAS*, 390, L29
2004, *Science*, 320, 1318
2009, *MNRAS* in press (arXiv:0908.2451)
2009, *MNRAS* in press (arXiv:0909.3604)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In the context of solar $\nu$ oscillations among active states, we briefly discuss the current likelihood of Mikheyev-Smirnov-Wolfenstein (MSW) solutions to the solar $\nu$ problem, which appear to be currently favored at large mixing, where small Earth regeneration effects might still be observable in Super-Kamiokande (SK) and in the Sudbury Neutrino Observatory (SNO). We point out that, since such effects are larger at high (low) solar $\nu$ energies for high (low) values of the mass square difference $\delta m^2$, it may be useful to split the night-day rate asymmetry in two separate energy ranges. We show that the difference $\Delta$ of the night-day asymmetry at high and low energy may help to discriminate the two large-mixing solutions at low and high $\delta m^2$ through a sign test, both in SK and in SNO, provided that the sensitivity to $\Delta$ can reach the (sub)percent level.'
address:
- |
$^a$ Dipartimento di Fisica and Sezione INFN di Bari\
Via Amendola 173, I-70126 Bari, Italy\
- |
$^b$ Dipartimento di Scienze dei Materiali dell’Università di Lecce\
Via Arnesano, I-73100 Lecce, Italy\
author:
- 'G.L. Fogli $^a$, E. Lisi $^a$, D. Montanino $^b$, and A. Palazzo $^a$'
title: |
Day-night asymmetry\
of high and low energy solar neutrino events\
in Super-Kamiokande and in the Sudbury Neutrino Observatory
---
Current MSW solutions: LMA, LOW, SMA
====================================
The well-known Mikheyev-Smirnov-Wolfenstein (MSW) [@Wo78] solutions to the solar $\nu$ problem [@Ba89] appear to have a somewhat different likelihood after the most recent data from the Super-Kamiokande (SK) experiment [@Su00], as compared with previous fits [@Fo00; @Ba00; @Go00]. The new SK data, both by themselves [@Su00] and in combination with the results of the chlorine (Cl) [@La00] and gallium (Ga) experiments (SAGE [@Ga00], GALLEX and GNO [@Be00]), tend now to favor the so-called large mixing angle (LMA) and low mass (LOW) regions of the parameter space, with respect to the region at small mixing angle (SMA) [@Su00]. This tendency emerges mainly from a “tension” between the flat (normalized) spectrum observed in SK and the prediction of a nonvanishing spectral distortion in the SMA region favored by total rates [@Su00]. However, a compromise between such two discordant indications is still reached in global fits.
Figure 1 shows the results our global $2\nu$ (active) oscillation fit to the latest 39 solar neutrino data, including the three SK, Cl, and Ga total rates [@Su00; @La00; @Ga00; @Be00], and the two 18-bin day and night SK energy spectra of events [@Su00; @SuPe] above 5.5 MeV. We take as free variables the overall spectrum normalization (to avoid double counting of the total SK rate information), the $\nu$ mass square difference $\delta m^2$, and the mixing angle $\omega$, parametrized in terms of $\tan^2\omega$ to cover the full range $\omega\in [0,\pi/4]$ [@Fo96]. Details of our calculations and of the $\chi^2$ statistical analysis can be found in [@Fo3n] and references therein. Subleading quasi-vacuum effects at low $\delta m^2$ [@Pe88; @Pa90], recently revisited in [@Fr00], are taken into account as described in [@FoQV].
The three local $\chi^2$ minima turn out to be $\chi^2_{\min}=35.1$ (LMA), 38.7 (LOW), and 40.7 (SMA). This implies, from the point of view of [*hypotheses tests*]{} [@PDG0] ($N_{\rm DF}=39-3$), that any of the three solutions is acceptable, since $\chi^2_{\min}/N_{\rm DF}\sim 1$ in any case. From the point of view of mass-mixing neutrino [*parameter estimation*]{} [@PDG0] ($N_{\rm DF}=2$), the relative likelihood is instead governed by $\Delta\chi^2=\chi^2-35.1$. In Fig. 1, $\Delta\chi^2$ contours at 90%, 95%, and 99% C.L. are shown as thin solid, thick solid, and dotted curves, respectively. The LOW solution emerges at 84% C.L. ($\Delta\chi^2=3.6$), while the SMA solution emerges only at 94% C.L. ($\Delta\chi^2=5.1$).
Figure 2 shows, for the sake of completeness, the regions favored by fits to total rates (Cl+Ga+SK), and by the SK day-night spectra, at the same C.L.’s as in Fig. 1. As already mentioned [@Su00], a “tension” emerges between the total rate data (which are highly consistent with the SMA solution) and the SK spectral data (which disfavor such solution).[^1] However, this tension is not enough to exclude the SMA region yet. In fact, a compromise is reached in the global fit of Fig. 1 where, as compared with Fig. 2, the SMA solution survives at smaller mixing angles, corresponding to smaller spectral distortions. The price to pay is an increase in the C.L. at which the SMA solution emerges (94% in Fig. 1), as compared with previous analyses [@Fo00; @Ba00; @Go00].
Summarizing, the LMA and LOW solutions appear to be favored over the SMA solution in the global fit, although it is rather premature to think that the latter is ruled out. In the following, we discuss a possible way to discriminate the two most likely solutions (LMA and LOW) in favorable situations, by separating Earth regeneration effects in two distinct energy ranges.
A possible test to discriminate the LOW and LMA solutions
=========================================================
The slightly positive indication $(\sim 1.3\sigma)$ for an excess of nighttime to daytime events in Super-Kamiokande [@Su00], if confirmed with higher statistical significance, would indicate the occurrence of the Earth regeneration effect for $^8$B solar neutrinos (see, e.g., [@Li97; @Ma00; @Kr97; @Gu99; @Sm00] for recent night-day asymmetry studies in SK). Such indication, by itself, might not be sufficient to discriminate the LMA from the LOW solution, since a slight excess is predicted in both cases. However, it has long been known that the Earth regeneration effect for solar neutrinos is strongly dependent on the neutrino energy [@Ba80; @Mi86]. Such dependence leads to several effects that might be observed in the Super-Kamiokande experiment [@Su00] and in the Sudbury Neutrino Observatory (SNO) [@Mc00], including night-day variations of energy spectrum distortions [@Li97; @Gu99; @Kr00], or variations of the night-day rate asymmetry with the electron energy threshold [@Ma00; @Kr97]. In particular, starting from the simple observation that the Earth regeneration effect is stronger at low energy for the LOW solution, and at high energy for the LMA solution,[^2] we point out that it may be useful to study the night-day asymmetry in two separate energy ranges in both SK and SNO.
For definiteness, we consider the two following representative ranges for the total (measured) energy of recoiling electrons in SK ($\nu$-$e$ scattering) and in SNO ($\nu$-$d$ absorption), $$\begin{aligned}
{\rm Low\ range\ }(L) &=& [5,7.5]{\rm\ MeV}\ ,\\
{\rm High\ range\ }(H) &=& [7.5,20]{\rm\ MeV}\ ,\end{aligned}$$ and calculate the night-day rate asymmetry in such ranges,[^3] $$A_{L,H} = \left( \frac{N-D}{N+D}\right)_{L,H}\ .$$ Since one expects $A_H\gtrsim A_L$ for the LMA solution and $A_H\lesssim A_L$ for the LOW solution, it is useful to introduce the difference $$\Delta = \left( \frac{N-D}{N+D}\right)_{H}-\left( \frac{N-D}{N+D}\right)_{L}\ ,$$ which should change sign when passing from the LMA region $(\Delta \gtrsim 0)$ to the LOW region $(\Delta \lesssim 0)$.
Figures 3 and 4 show the results of our calculations of $\Delta$ (eccentricity effects removed) in SK and SNO, respectively, in the form of isolines at $\Delta\times 100=\pm0.5$, $\pm1$, and $\pm 2$. Notice that the magnitude of $\Delta$ in SNO is typically a factor of two higher than in SK. Figures 3 and 4 confirm that $\Delta>0$ ($<0$) would represent clear evidence in favor of the LMA (LOW) solution, both in SK and SNO, thus allowing a useful “sign discrimination test” to solve the LOW-LMA ambiguity at large mixing. The power of such test decreases as $\Delta\to 0$ in the upper part (lower part) of the LMA (LOW) solution, corresponding to vanishing Earth regeneration effects for $^8$B neutrinos in SK and SNO.
From a comparison of Figs. 1, 3, and 4, it turns out that, in the most favorable case for the LOW solution (i.e., in its upper part), the quantity $\Delta\times 100$ can approximately reach the value $-0.5$ in SK and $-1$ in SNO; analogously, it can reach the value $+1$ (SK) and $+2$ (SNO) for the LMA solution. The separation between the two solutions is thus $\Delta({\rm
LMA})-\Delta({\rm LOW}) \lesssim 1.5\%$ in SK ($\lesssim3\%$ in SNO) and, in its upper range, it seems not too far from the present experimental sensitivity to day-night effects.[^4]
In general, however, one should require for detection of $\Delta\neq 0$ a typical sensitivity at the subpercent level in SK, and at the percent level in SNO, whose viability requires not only a very high statistics, but also a dedicated study of systematics (and of their cancellations in a difference like $\Delta$). Notice also that the energy value separating the $L$ and $H$ ranges does not need to be equal to our representative choice (7.5 MeV) nor to be the same in SK and SNO, and should be tuned to optimize the statistical significance of the $\Delta$-sign test. The test would in any case benefit from a reduction of the SK and SNO energy thresholds, which would both increase the statistics and enhance the sensitivity to Earth effects at low energies.[^5] Such detailed experimental studies are beyond the scope of this work, whose main purpose is to emphasize that the difference of the night-day asymmetry at low and high neutrino energy in SK and SNO might be observable in favorable situations, and help to separate the LOW and LMA solutions. Even in the absence of an accurate measurement of $\Delta$, the simple [*concordance*]{} of the $\Delta$ sign in SK [*and*]{} SNO ($++$ for the LMA solution or $--$ for the LOW solution) would be a valuable information.
Summary and conclusions
=======================
We have presented a global MSW oscillation fit to the solar neutrino data, showing that all the three usual solutions (SMA, LMA, and LOW) emerge at 95% C.L. The LOW and LMA solutions appear to be globally favored, the latter providing the best fit. Within the LMA (LOW) solution, the excess of nighttime events due to $\nu_e$ regeneration in the Earth is larger in the higher (lower) part of the neutrino energy range. Therefore, by taking the difference $\Delta$ of the night-day asymmetry in two separate ranges at high and low energy, a small positive (negative) signal would provide evidence for the LMA (LOW) solution. Detection of $\Delta\neq 0$ requires a (sub)percent sensitivity to day-night effects, which needs to be assessed by further experimental investigations. However, the simple preference (and concordance) of the SK and SNO experiments for a definite $\Delta$ sign would already provide us with valuable information, corroborating other tests envisaged to solve the LOW-LMA ambiguity in such experiments [@Kr00].
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[Fig. 1. Global $2\nu$ fit including total solar neutrino rates (Ga+Cl+SK) [@Su00; @La00; @Ga00; @Be00] and the SK day and night spectra [@Su00; @SuPe].]{} [Fig. 2. Results of the separate fits to total rates and to SK day-night spectra. Total rate data mainly determine the three SMA, LMA, and LOW allowed regions, while the SK day-night spectra exclude a large part of the parameter space where either spectral distortions or day-night effects are large.]{} [Fig. 3. The difference $\Delta$ between the night-day asymmetry calculated in the two ranges $[7.5,20]$ and $[5,7.5]$ MeV for the SK detector ($\times
100$). Solid and dotted curves refer to $\Delta>0$ and $<0$, respectively. The LOW and LMA solutions in Fig. 1 predict opposite signs for $\Delta$.]{} [Fig. 4. As in Fig. 3, but for the SNO detector.]{}
[^1]: Notice that, conversely, the LOW solution is disfavored by total rates, but is highly consistent with SK day-night spectra.
[^2]: See, e.g., Fig. 4 of [@Kr97].
[^3]: The 5 MeV threshold has already been reached in SK, although the $[5,5.5]$ MeV bin is not used yet in SK fits [@Su00].
[^4]: The quoted SK uncertainty on $A=N-D/N+D$, integrated over the full energy range, is $\pm 1.1\%$ (stat.) $\pm0.6\%$ (syst.) [@Su00]. The (larger) total uncertainty of $A_H$ and $A_L$ in the two $H$ and $L$ energy sub-ranges is presumably at the $\sim 2\%$ level, although we have not enough information for a precise estimate.
[^5]: Experiments sensitive to neutrino energies below the SK or SNO threshold can also observe relevant day-night effects in the LOW region [@Ra91] (see also [@Fo00; @Kr97] and references therein).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In static analysis, approximation is typically encoded by abstract domains, providing systematic guidelines for specifying approximate semantic functions and precision assessments. However, it may happen that an abstract domain contains redundant information for the specific purpose of approximating a given semantic function modeling some behavior of a system. This paper introduces correctness kernels of abstract interpretations, a methodology for simplifying abstract domains, i.e.removing abstract values from them, in a maximal way while retaining exactly the same approximate behavior of the system under analysis. We show that, in abstract model checking and predicate abstraction, correctness kernels provide a simplification paradigm of the abstract state space that is guided by examples, meaning that it preserves spuriousness of examples (i.e., abstract paths). In particular, we show how correctness kernels can be integrated with the well-known CEGAR (CounterExample-Guided Abstraction Refinement) methodology.'
author:
- '[Roberto Giacobazzi Francesco Ranzato]{}'
title: Correctness Kernels of Abstract Interpretations
---
Introduction {#intro}
============
In static analysis and verification, model-driven *abstraction refinement* has emerged in the last decade as a fundamental method for improving abstractions towards more precise yet efficient analyses. The basic idea is simple: given an abstraction modeling some observational behavior of the system to analyze, refine the abstraction in order to remove the artificial computations that may appear in the approximate analysis by considering how the concrete system behaves when false alarms or spurious traces are encountered. The general concept of using spurious counterexamples for refining an abstraction stems from the CounterExample-Guided Abstraction Refinement (CEGAR) paradigm [@cgjlv00; @cgjlv03]. The model here drives the automatic identification of prefixes of the counterexample path that do not correspond to an actual trace in the concrete model, by isolating abstract (failure) states that need to be refined in order to eliminate that spurious counterexample. Model-driven refinements, such as CEGAR, provide algorithmic methods for achieving abstractions that are complete (i.e., precise [@gq01; @grs00]) with respect to some given property of the concrete model.
We investigate here the dual problem of *abstraction simplification*. Instead of refining abstractions in order to eliminate spurious traces, our goal is to simplify an abstraction $A$ towards a simpler (ideally, the simplest) model $A_s$ that maintains the same approximate behavior as $A$ does. In abstract model checking, this abstraction simplification has *to keep the same examples* of the concrete system in the following sense. Recall that an abstract path $\pi$ in an abstract transition system ${\mathcal{A}}$ is *spurious* when no real concrete path is abstracted to $\pi$. Assume that a given abstract state space $A$ of a system ${\mathcal{A}}$ gets simplified to $A_s$ and thus gives rise to a more abstract system ${\mathcal{A}}_s$. Then, we say that ${\mathcal{A}}_s$ keeps the same examples of ${\mathcal{A}}$ when the following condition is satisfied: if $\pi_{A_s}$ is a spurious path in the simplified abstract system ${\mathcal{A}}_s$ then there exists a spurious path $\pi_A$ in the original system ${\mathcal{A}}$ that is abstracted to $\pi_{A_s}$. Such a methodology is called EGAS, Example-Guided Abstraction Simplification, since this abstraction simplification does not add spurious paths, namely, it does keep examples, since each spurious path in ${\mathcal{A}}_s$ comes as an abstraction of a spurious path in ${\mathcal{A}}$.
=\[->,>=latex’\] (0,2) node\[name=1\][1]{} (2,5) node\[name=2\][2]{} (2,3) node\[name=3\][3]{} (2,1) node\[name=4\][4]{} (2,-1) node\[name=5\][5]{} (4,3) node\[name=6\][6]{} (4,1) node\[name=7\][7]{} (6,3) node\[name=8\][8]{} (6,1) node\[name=9\][9]{}; (8,2) node\[name=f\][$\Rightarrow$]{}; (0,5) node\[name=a\][$\mathcal{A}$]{}; (1) to (2); (1) to (4); (2) to (6); (3) to (7); (4) to (6); (5) to (7); (6) to (8); (7) to (9);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=b1\] (3.south east) ++(0.1,-0.1) node\[name=b2\]; (b1) rectangle (b2);
(4.north west) ++(-0.1,0.1) node\[name=b3\] (5.south east) ++(0.1,-0.1) node\[name=b4\]; (b3) rectangle (b4);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
(8.north west) ++(-0.1,0.1) node\[name=d1\] (9.south east) ++(0.1,-0.1) node\[name=d2\]; (d1) rectangle (d2);
(10,2) node\[name=1\][1]{} (12,5) node\[name=2\][2]{} (12,3) node\[name=3\][3]{} (12,1) node\[name=4\][4]{} (12,-1) node\[name=5\][5]{} (14,3) node\[name=6\][6]{} (14,1) node\[name=7\][7]{} (16,3) node\[name=8\][8]{} (16,1) node\[name=9\][9]{}; (18,2) node\[name=f\][$\Rightarrow$]{}; (10,5) node\[name=a\][$\mathcal{A}'$]{}; (1) to (2); (1) to (4); (2) to (6); (3) to (7); (4) to (6); (5) to (7); (6) to (8); (7) to (9);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=b1\] (5.south east) ++(0.1,-0.1) node\[name=b2\]; (b1) rectangle (b2);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
(8.north west) ++(-0.1,0.1) node\[name=d1\] (9.south east) ++(0.1,-0.1) node\[name=d2\]; (d1) rectangle (d2);
(20,2) node\[name=1\][1]{} (22,5) node\[name=2\][2]{} (22,3) node\[name=3\][3]{} (22,1) node\[name=4\][4]{} (22,-1) node\[name=5\][5]{} (24,3) node\[name=6\][6]{} (24,1) node\[name=7\][7]{} (26,3) node\[name=8\][8]{} (26,1) node\[name=9\][9]{};
(20,5) node\[name=a\][$\mathcal{A}''$]{}; (1) to (2); (1) to (4); (2) to (6); (3) to (7); (4) to (6); (5) to (7); (6) to (8); (7) to (9);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=b1\] (5.south east) ++(0.1,-0.1) node\[name=b2\]; (b1) rectangle (b2);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (7.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(8.north west) ++(-0.1,0.1) node\[name=d1\] (9.south east) ++(0.1,-0.1) node\[name=d2\]; (d1) rectangle (d2);
Let us illustrate how EGAS works through a simple example. Let us consider the abstract transition system ${\mathcal{A}}$ in Figure \[figure-1\], where concrete states are numbers which are abstracted by blocks of the state partition $\{[1],[2,3],[4,5],[6],[7],[8,9]\}$. The abstract state space of ${\mathcal{A}}$ is simplified by merging the abstract states $[2,3]$ and $[4,5]$: EGAS guarantees that this can be safely done because $\operatorname{pre}^\sharp ([2,3]) = \{[1]\}=
\operatorname{pre}^\sharp([4,5])$ and $\operatorname{post}^\sharp ([2,3]) = \{[6],[7]\}=
\operatorname{post}^\sharp([4,5])$, where $\operatorname{pre}^\sharp$ and $\operatorname{post}^\sharp$ denote, respectively, the abstract predecessor and successor functions in ${\mathcal{A}}$. This abstraction simplification leads to the abstract system ${\mathcal{A}}'$ in Figure \[figure-1\]. Let us observe that the abstract path $\pi = \langle [1], [2,3,4,5], [7], [8,9]\rangle$ in ${\mathcal{A}}'$ is spurious because there is no concrete path whose abstraction in ${\mathcal{A}}'$ is $\pi$, while $\pi$ is instead the abstraction of the spurious path $\langle [1],$ $[4,5],$ $[7], [8,9]\rangle$ in ${\mathcal{A}}$. On the other hand, consider the path $\sigma = \langle [1], [2,3,4,5], [6], [8,9]\rangle$ in ${\mathcal{A}}'$ and observe that all the paths in ${\mathcal{A}}$ that are abstracted to $\pi'$, i.e.$\langle [1],[2,3],[6],[8,9]\rangle$ and $\langle [1],[4,5],[6],[8,9]\rangle$, are not spurious. This is consistent with the fact that $\sigma$ actually is not a spurious path. Likewise, ${\mathcal{A}}'$ can be further simplified to the abstract system ${\mathcal{A}}''$ where the blocks $[6]$ and $[7]$ are merged into a new abstract state $[6,7]$. This transformation also keeps examples because now there is no spurious path in ${\mathcal{A}}''$. Let us also notice that if ${\mathcal{A}}$ would get simplified to an abstract system ${\mathcal{A}}'''$ by merging the blocks $[1]$ and $[2,3]$ into a new abstract state $[1,2,3]$ then this transform would not keep examples because we would obtain the spurious loop path $\tau = \langle [1,2,3], [1,2,3], [1,2,3], ... \rangle$ in ${\mathcal{A}}'''$ (because in ${\mathcal{A}}'''$ $[1,2,3]$ has a self-loop) while there is no corresponding spurious abstract path in ${\mathcal{A}}$ whose abstraction in ${\mathcal{A}}'''$ is $\tau$.
EGAS is formalized within the standard abstract interpretation framework by Cousot and Cousot [@CC77; @CC79]. This ensures that EGAS can be applied both in abstract model checking and in abstract interpretation. Consider for instance the following two basic abstract domains $A_1$ and $A_2$ for sign analysis of an integer variable, so that sets of integer numbers in $\wp(\mathbb{Z})$ is the concrete domain.
(0,0) node\[name=2\] [[$0$]{}]{}; (-1,1) node\[name=3\] [[$\mathbb{Z}_{\leq 0}$]{}]{}; (1,1) node\[name=4\] [[$\mathbb{Z}_{\geq 0}$]{}]{}; (0,2) node\[name=5\] [[$\mathbb{Z}$]{}]{};
(-2.5,1) node [[$A_1$]{}]{};
\(2) – (3); (2) – (4); (3) – (5); (4) – (5);
(5,1) node [[$A_2$]{}]{};
(4,0) node\[name=2\] [[$\mathbb{Z}_{\geq 0}$]{}]{}; (4,2) node\[name=5\] [[$\mathbb{Z}$]{}]{};
\(2) – (5);
Recall that in abstract interpretation the best correct approximation of a semantic function $f$ on an abstract domain $A$ that is defined through abstraction/concretization maps $\alpha$/$\gamma$ is given by $f^A {\triangleq}\alpha \circ f \circ \gamma$. Consider a simple operation of increment $x$++ on an integer variable $x$. In this case, the best correct approximations on $A_1$ and $A_2$ are as follows: $$\begin{gathered}
\text{++}^{A_1} = \{0 \mapsto \mathbb{Z}_{\geq 0},\:
\mathbb{Z}_{\leq 0} \mapsto \mathbb{Z},\:
\mathbb{Z}_{\geq 0} \mapsto \mathbb{Z}_{\geq 0},\:
\mathbb{Z} \mapsto \mathbb{Z}\},\\[-2.5pt]
\text{++}^{A_2} = \{
\mathbb{Z}_{\geq 0} \mapsto \mathbb{Z}_{\geq 0},\:
\mathbb{Z} \mapsto \mathbb{Z}\}.\end{gathered}$$ We observe that the best correct approximations of $\text{++}$ in $A_1$ and $A_2$ encode the same function, meaning that the approximations of $\text{++}$ in $A_1$ and $A_2$ are equivalent, the latter being clearly simpler. In fact, we have that $\gamma_{A_1} \circ \text{++}^{A_1} \circ \alpha_{A_1}$ and $\gamma_{A_2} \circ \text{++}^{A_2} \circ \alpha_{A_2}$ are exactly the same function on $\wp(\mathbb{Z})$. In other terms, the abstract domain $A_1$ contains some “irrelevant” abstract values for approximating the increment operation, that is, $0$ and $\mathbb{Z}_{\leq 0}$. This simplification of an abstract domain relatively to a semantic function is formalized in the most general abstract interpretation setting. This allows us to provide, for generic continuous semantic functions, a systematic and constructive method, that we call *correctness kernel*, for simplifying a given abstraction $A$ relatively to a given semantic function $f$ towards the unique minimal abstract domain that induces an equivalent approximate behavior of $f$ as in $A$. We show how correctness kernels can be embedded within the CEGAR methodology by providing a novel refinement heuristics in a CEGAR iteration step which turns out to be more accurate than the basic refinement heuristics [@cgjlv03]. We also describe how correctness kernels may be applied in predicate abstraction-based model checking [@ddp99; @gs97] for reducing the search space without applying Ball et al.’s [@bpr03] Cartesian abstractions, which typically yield additional loss of precision.
This is an extended and revised version of the conference paper [@gr10] that includes full proofs.
Correctness Kernels
===================
As usual in standard abstract interpretation [@CC77; @CC79], abstract domains (or abstractions) are specified by Galois connections/insertions (GCs/GIs for short) or, equivalently, adjunctions. Concrete and abstract domains, $\tuple{C,\leq_C}$ and $\tuple{A,\leq_A}$, are assumed to be complete lattices which are related by abstraction and concretization maps $\alpha:C{\rightarrow}A$ and $\gamma:A {\rightarrow}C$ that give rise to an adjunction $(\alpha,C,A,\gamma)$, that is, for all $a$ and $c$, $\alpha(c) \leq_A a {\Leftrightarrow}c \leq_C \gamma(a)$. It is known that $\ok{\mu_A {\triangleq}\gamma \circ \alpha: C {\rightarrow}C}$ is an upper closure operator (uco) on $C$, i.e. a monotone, idempotent and increasing function. Also, abstract domains can be equivalently defined as ucos, meaning that any GI $(\alpha,C,A,\gamma)$ induces the uco $\mu_A$, any uco $\mu: C{\rightarrow}C$ induces the GI $(\mu, C, \mu(C), \lambda x.x)$, and these two transforms are the inverse of each other. GIs of a common concrete domain $C$ are preordered w.r.t. their relative precision as usual: ${\mathcal{G}}_1 = (\alpha_1,C,A_1,\gamma_1)\sqsubseteq {\mathcal{G}}_2=(\alpha_2,C,A_2,\gamma_2)$ — i.e. $A_1$/$A_2$ is a refinement/simplification of $A_2$/$A_1$ — iff $\gamma_2 (\alpha_2(C)) \subseteq\gamma_1 (\alpha_1 (C))$. Moreover, ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ are equivalent when ${\mathcal{G}}_1
\sqsubseteq {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$. We denote by $\operatorname{Abs}(C)$ the family of abstract domains of $C$ up to the above equivalence. It is well known that $\tuple{\operatorname{Abs}(C),\sqsubseteq}$ is a complete lattice, so that one can consider the most concrete simplification (i.e., lub $\sqcup$) and the most abstract refinement (i.e., glb $\sqcap$) of any family of abstract domains. Let us recall that the lattice of abstract domains $\tuple{\operatorname{Abs}(C),\sqsubseteq}$ is isomorphic to the lattice of ucos on $C$ $\tuple{\operatorname{uco}(C),\sqsubseteq}$, where $\sqsubseteq$ denotes the pointwise ordering between functions, so that lub’s and glb’s of abstractions can be equivalently characterized in $\operatorname{uco}(C)$. Let us also recall that each $\mu\in \operatorname{uco}(C)$ is uniquely determined by its image $\operatorname{img}(\mu)=\mu(C)$ because $\mu = \lambda
x. \wedge \!\{ y\in C~|~ y\in \mu(C), \, x\leq y\}$. Moreover, a subset $X\subseteq C$ is the image of some uco on $C$ iff $X$ is meet-closed, i.e. $X=\ok{\operatorname{Cl}_\wedge
(X){\triangleq}}\{ \wedge Y~|~ Y\subseteq
X\}$ (note that $\top_C =\wedge \varnothing \in \operatorname{Cl}_\wedge (X)$). Often, we will identify ucos with their images. This does not give rise to ambiguity, since one can distinguish their use as functions or sets according to the context. Hence, if $A,B\in \operatorname{Abs}(C)$ are two abstractions then they can be viewed as images of two ucos on $C$, denoted respectively by $\mu_A$ and $\mu_B$, so that $A$ is more precise than $B$ when $\operatorname{img}(\mu_B) \subseteq \operatorname{img}(\mu_A)$.
Let $f:C{\rightarrow}C$ be some concrete semantic function — for simplicity, we consider 1-ary functions — and let $\ok{f^\sharp:A {\rightarrow}A}$ be a corresponding abstract function defined on some abstraction $A\in \operatorname{Abs}(C)$. Then, $\ok{\tuple{A,f^\sharp}}$ is a sound abstract interpretation when $\ok{\alpha \circ f \sqsubseteq f^\sharp\circ \alpha}$. Moreover, the abstract function $\ok{f^A {\triangleq}\alpha \circ f \circ \gamma: A\rightarrow A}$ is called the *best correct approximation* (b.c.a.) of $f$ on $A$ because any abstract interpretation $\ok{\tuple{A,f^\sharp}}$ is sound iff $\ok{f^A \sqsubseteq f^\sharp}$. Hence, for any abstraction $A$, $\ok{f^A}$ plays the role of the best possible approximation of $f$ on $A$.
The Problem
-----------
Given a semantic function $f:C\rightarrow C$ on some concrete domain $C$ and an abstraction $A \in \operatorname{Abs}(C)$, does there exist the *most abstract domain* that induces the same best correct approximation of $f$ as $A$ does?
Let us formalize the above question. Consider two abstractions $A,B\in \operatorname{Abs}(C)$. We say that $A$ and $B$ induce the same best correct approximation of $f$ when $f^A$ and $f^B$ are the same function up to isomorphic representations of abstract values. If $\mu_A$ and $\mu_B$ are the corresponding ucos then this boils down to: $$\mu_A {\circ}f {\circ}\mu_A = \mu_B {\circ}f {\circ}\mu_B .$$ In order to keep the notation easy, this is denoted simply by $\ok{f^A = f^B}$. Also, if $F\subseteq C{\rightarrow}C$ is a set of concrete functions then $\ok{F^A = F^B}$ means that for any $f\in F$, $\ok{f^A = f^B}$. Hence, given $A\in \operatorname{Abs}(C)$ and by defining $$A_{s} \triangleq \sqcup \{B \in \operatorname{Abs}(C)~|~ F^B = F^A\}$$ the question is whether $F^{A_{s}} = F^A$ holds or not. This leads us to the following notion of correctness kernel.
Given $F\subseteq C{\mbox{\raisebox{0ex}[1ex][1ex]{$
\mathrel{\mathop{
\hspace*{1pt}\longrightarrow\hspace*{1pt}}\limits^{\,_{\mbox{\tiny
\hspace*{-2.2pt}}}}}$}}} C$ define: $\mathscr{K}_F: \operatorname{Abs}(C) \rightarrow \operatorname{Abs}(C)$ as $$\mathscr{K}_F(A) \triangleq \sqcup \{ B \in \operatorname{Abs}(C) ~|~ F^B = F^A\}.$$ If $F^{\mathscr{K}_F(A)} = F^A$ then $\mathscr{K}_F(A)$ is called the *correctness kernel* of $A$ for $F$.
It is worth remarking that the dual question on the existence of the *most concrete domain* that induces the same best correct approximation of $f$ as $A$ has a negative answer, as shown by the following simple example.
\[esempio2\] Consider the lattice $C$ depicted below.
(0,0) node\[name=1\] [$1$]{}; (0,1) node\[name=2\] [$2$]{}; (-1,2) node\[name=3\] [$3$]{}; (1,2) node\[name=4\] [$4$]{}; (0,3) node\[name=5\] [$5$]{};
\(1) – (2); (2) – (3); (2) – (4); (3) – (5); (4) – (5);
Let us also consider the monotonic function $f:C\rightarrow C$ defined as $f\triangleq \{1\mapsto 1,\, 2\mapsto 1,\, 3\mapsto 5,\, 4\mapsto 5,\, 5\mapsto 5\}$ and the abstraction $\mu \in \operatorname{uco}(C)$ whose image is $\mu \triangleq \{1,5\}$. Let us observe that $\mu{\circ}f {\circ}\mu = \{1\mapsto 1,\, 2\mapsto 5,\, 3\mapsto 5,\, 4\mapsto 5,\, 5\mapsto 5\}$. Consider now the abstractions $\rho_1 \triangleq \{1,3,5\}$ and $\rho_2 \triangleq \{1,4,5\}$ and observe that $\rho_i {\circ}f {\circ}\rho_i = \mu{\circ}f {\circ}\mu$. However, we have that $\rho_1 \sqcap \rho_2 = \lambda x.x$, because the image of $\rho_1 \sqcap \rho_2$ is ${\mathcal{M}}(\rho_1 \cup \rho_2) =
\{1,2,3,4,5\}$. Hence, $(\rho_1 \sqcap \rho_2) {\circ}f {\circ}(\rho_1 \sqcap \rho_2) = f \neq \mu{\circ}f {\circ}\mu$. Therefore, if we let $\rho_r = \sqcap \{ \rho \in \operatorname{uco}(C)~|~ \rho \circ f \circ
\rho = \mu \circ f \circ \mu\}$ then $\rho_r = \lambda x.x$. Consequently, the most concrete domain that induces the same best correct approximation of $f$ as $\mu$ does not exist.
The Solution
------------
Our key technical result is the following *constructive* characterization of the property of “having the same b.c.a.” for two comparable abstract domains. In the following, given a poset $A$ and any subset $S\subseteq A$, $\max (S) \triangleq \{
x\in S~|~ \forall y\in S.\; x\leq_A y \Rightarrow x = y\}$ denotes the set of maximal elements of $S$ in $A$.
\[key\] Let $f: C\rightarrow C$ and $A,B \in \operatorname{Abs}(C)$ such that $B\subseteq A$. Suppose that $f{\circ}\mu_A : C {\rightarrow}C$ is continuous (i.e., preserves lub’s of chains in $C$). Then, $$\textstyle
f^B = f^A \;{\Leftrightarrow}\; \operatorname{img}(f^A) \cup \bigcup_{y\in A}
\max(\{x\in A~|~f^A(x) \leq_A y\})
\subseteq B.$$
Let $\mu$ and $\rho$ be the ucos induced by, respectively, the abstractions $A$ and $B$, so that $\mu \sqsubseteq \rho$. Then, observe that $\operatorname{img}(f^A) = \mu(f(\mu(C)))$ and $\{x\in A ~|~ f^A(x) \leq_A y\} = (\mu {\circ}f {\circ}\mu)^{-1}(\downarrow \!y)$. We therefore prove the following equivalent statement which is formalized through ucos: $$\rho {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu
\text{~~iff~~}
\mu(f(\mu(C))) \cup \textstyle \bigcup_{y\in \mu} \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))\subseteq \rho.$$ Let us first prove that $$\rho {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu \:\Leftrightarrow\: \rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\rho\eqno(*)$$\
($\Rightarrow$) On the one hand, $$\begin{aligned}
\mu {\circ}f {\circ}\mu = \rho {\circ}f {\circ}\rho & \Rightarrow
\text{~~~~[by applying $\rho$ to both sides]}
\\
\rho {\circ}\mu {\circ}f {\circ}\mu = \rho{\circ}\rho {\circ}f {\circ}\rho
& \Rightarrow \text{~~~~[because $\rho {\circ}\rho = \rho$ and
$\rho {\circ}\mu = \rho$]} \\
\rho {\circ}f {\circ}\mu = \rho {\circ}f {\circ}\rho &\Rightarrow \\
\rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu &\end{aligned}$$ and on the other hand, $$\begin{aligned}
\mu {\circ}f {\circ}\mu = \rho {\circ}f {\circ}\rho & \Rightarrow
\text{~~~~[by applying $\rho$ in front to both sides]}
\\
\mu {\circ}f {\circ}\mu {\circ}\rho = \rho{\circ}f {\circ}\rho {\circ}\rho
& \Rightarrow
\text{~~~~[because $\rho {\circ}\rho = \rho$ and
$\mu {\circ}\rho = \rho$]} \\
\mu {\circ}f {\circ}\rho = \rho {\circ}f {\circ}\rho &\Rightarrow \\
\mu {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu &\end{aligned}$$ so that $\rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\rho$.\
($\Leftarrow$) We have that: $$\begin{aligned}
\rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\rho & \Rightarrow
\text{~~~~[by applying $\rho$ to both sides]}
\\
\rho {\circ}\rho {\circ}f {\circ}\mu = \rho {\circ}\mu {\circ}f {\circ}\rho & \Rightarrow \text{~~~~[since $\rho {\circ}\rho = \rho$ and
$\rho {\circ}\mu = \rho$]}\\
\rho {\circ}f {\circ}\mu = \rho {\circ}f {\circ}\rho & \Rightarrow\\
\mu {\circ}f {\circ}\mu = \rho {\circ}f {\circ}\rho. &\end{aligned}$$
Let us now observe that $\rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu$: in fact, since $\rho = \rho {\circ}\mu$, this is equivalent to $\rho {\circ}\mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu$, which is obviously equivalent to $\mu (f(\mu(C))) \subseteq \rho$.
Since $\rho = \mu {\circ}\rho$, we have that $\mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\rho$ is equivalent to $\mu {\circ}(f {\circ}\mu) = \mu {\circ}(f{\circ}\mu) {\circ}\rho$. By the characterization of completeness in [@grs00 Lemma 4.2], since, by hypothesis, $f {\circ}\mu$ is continuous, we have that the completeness equation $\mu {\circ}(f {\circ}\mu) = \mu {\circ}(f{\circ}\mu) {\circ}\rho$ is equivalent to $\cup_{y\in \mu} \max((f {\circ}\mu)^{-1}(\downarrow\! y))\subseteq \rho$, which is in turn equivalent to $\cup_{y\in \mu} \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))\subseteq \rho$.
Summing up, we have thus shown that $$\rho {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\mu = \mu {\circ}f {\circ}\rho \:\Leftrightarrow\: \mu (f(\mu(C))) \cup \textstyle
\bigcup_{y\in \mu} \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))\subseteq \rho$$ and this, by the above property $(*)$, implies the thesis.
It is important to remark that the above proof basically consists in reducing the equality $f^A = f^B$ between b.c.a.’s to a standard property of completeness of the abstract domains $A$ and $B$ for the function $f$ and then in exploiting the constructive characterization of completeness of abstract domains by Giacobazzi et al. [@grs00 Section 4]. In this sense, the proof itself is particularly interesting because it provides an unexpected reduction of best correct approximations to a completeness problem.
As a consequence of Lemma \[key\] we obtain the following constructive result of existence for correctness kernels. Recall that if $X\subseteq A$ then $\operatorname{Cl}_\wedge (X)$ denotes the glb-closure of $X$ in $A$, while $\operatorname{Cl}_\vee(X)$ denotes the dual lub-closure.
\[kernel\] Let $A \in \operatorname{Abs}(C)$ and $F\subseteq C\rightarrow C$ such that, for any $f\in F$, $f {\circ}\mu_A$ is continuous. Then, the correctness kernel of $A$ for $F$ exists and it is $$\mathscr{K}_F(A) = \operatorname{Cl_\wedge}\Big(\bigcup_{f\in F}\textstyle \operatorname{img}(f^A) \cup \bigcup_{y\in \operatorname{img}(f^A)}
\max(\{x\in A~|~f^A(x) = y\})\Big).$$
Let $\mu = \mu_A$. We prove the following equivalent statement formalized through ucos: $\operatorname{Cl_\wedge}\left(\bigcup_{f\in F} \bigcup_{y\in \mu(f(\mu(C)))} \left(\{y\} \cup \max(\{x\in \mu~|~\mu(f(x)) = y\})\right)\right)$ is the correctness kernel of $\mu$ for $F$.
Let $\rho_\mu \triangleq
\operatorname{Cl_\wedge}\left( \mu(f(\mu(C))) \cup \textstyle \bigcup_{y\in \mu} \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))\right)$. By Lemma \[key\], we have that $\sqcup \{\rho \in \operatorname{uco}(C)~|~ \rho \sqsupseteq \mu,\:
\rho {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu\} =
\rho_\mu$. Since $\sqcup \{\rho \in \operatorname{uco}(C)~|~ \rho \sqsupseteq \mu,\:
\rho {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu\} =
\sqcup \{\rho \in \operatorname{uco}(C)~|~
\rho {\circ}f {\circ}\rho = \mu {\circ}f {\circ}\mu\}$, as a consequence we also have that $\rho_\mu$ is the correctness kernel of $\mu$ for $F$.\
Therefore, let us prove that $$\begin{gathered}
\operatorname{Cl_\wedge}\left( \mu(f(\mu(C))) \cup \cup_{y\in \mu} \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))\right) =\\
\textstyle \operatorname{Cl_\wedge}\left( \bigcup_{f\in F} \bigcup_{y\in \mu(f(\mu(C)))} \left(\{y\} \cup \max(\{x\in \mu~|~\mu(f(x)) = y\})\right) \right).\end{gathered}$$ Let us first observe that for any $y\in \mu$, if $z\in \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))$ then $z\in\mu$: in fact, $\mu(f(\mu(\mu(z)))) = \mu(f(\mu(z))) \leq y$, so that from $z \leq \mu(z)$, by maximality of $z$, we get $z=\mu(z)$.\
$(\subseteq)$: Consider $y\in \mu$ and $z\in \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))$. Then, it turns out that $z \in \max(\{x\in \mu~|~\mu(f(x)) = \mu(f(\mu(z)))\})$. In fact, since $z=\mu(z)$, we have that $\mu(f(z))=\mu(f(\mu(z)))$. Moreover, if $u\in \{x\in \mu~|~\mu(f(x)) = \mu(f(\mu(z)))\}$ and $z \leq u$ then $\mu(f(\mu(u))) = \mu(f(u)) =
\mu(f(\mu(z))) \leq y$, so that, by maximality of $z$, $z=u$, i.e., $z\in \max(\{x\in \mu~|~\mu(f(x)) = \mu(f(\mu(z)))\})$.\
$(\supseteq)$: Consider $y=\mu(f(\mu(w)))$ and $z\in \max(\{x\in \mu~|~\mu(f(x)) = y\})$. Then, $\mu(f(\mu(z))) = \mu(f(z)) = y$ so that $z\in (\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y)$. If $u\in (\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y)$ and $z \leq u$ then $\mu(f(\mu(z))) \leq \mu(f(\mu(u))) \leq y=\mu(f(\mu(w)))=\mu(f(\mu(z)))$. Hence, since $z\leq u \leq \mu(u)$ and by maximality of $z$, we have that $z=\mu(u)$, and in turn $z=u$. Thus, $z\in \max((\mu {\circ}f {\circ}\mu)^{-1}(\downarrow\! y))$.
Consider sets of integers $\tuple{\wp(\mathbb{Z}),\subseteq}$ as concrete domain domain and the square operation ${\ensuremath{\mathit{sq}}}:\wp(\mathbb{Z}) \rightarrow \wp(\mathbb{Z})$ as concrete function, i.e., ${\ensuremath{\mathit{sq}}}(X) \triangleq \{x^2 ~|~ x\in X\}$, which is obviously additive and therefore continuous. Consider the abstract domain $\operatorname{Sign}\in \operatorname{Abs}(\wp(\mathbb{Z})_\subseteq)$, depicted in the following figure, that represents the sign of an integer variable.
(0,0) node\[name=1\] [[$\varnothing$]{}]{};
(-1,1) node\[name=2\] [[$\mathbb{Z}_{<0}$]{}]{}; (0,1) node\[name=3\] [[$0$]{}]{}; (1,1) node\[name=4\] [[$\mathbb{Z}_{>0}$]{}]{};
(-1,2) node\[name=5\] [[$\mathbb{Z}_{\leq 0}$]{}]{}; (0,2) node\[name=6\] [[$\mathbb{Z}_{\neq 0}$]{}]{}; (1,2) node\[name=7\] [[$\mathbb{Z}_{\geq 0}$]{}]{};
(0,3) node\[name=8\] [[$\mathbb{Z}$]{}]{}; (1) – (2); (1) – (3); (1) – (4); (2) – (5); (2) – (6); (3) – (5); (3) – (7); (4) – (6); (4) – (7); (5) – (8); (6) – (8); (7) – (8);
$\operatorname{Sign}$ induces the following best correct approximation of ${\ensuremath{\mathit{sq}}}$: $$\begin{aligned}
{\ensuremath{\mathit{sq}}}^{\operatorname{Sign}} = \{&\varnothing \mapsto \varnothing, \mathbb{Z}_{<0} \mapsto
\mathbb{Z}_{> 0}, 0 \mapsto 0, \mathbb{Z}_{> 0} \mapsto
\mathbb{Z}_{> 0}, \mathbb{Z}_{\leq 0} \mapsto \mathbb{Z}_{\geq 0}, \\
& \mathbb{Z}_{\neq 0} \mapsto \mathbb{Z}_{>0},
\mathbb{Z}_{\geq 0} \mapsto \mathbb{Z}_{\geq 0},
\mathbb{Z} \mapsto \mathbb{Z}_{\geq 0}\}.\end{aligned}$$ Let us characterize the correctness kernel $\mathscr{K}_{{\ensuremath{\mathit{sq}}}}(\operatorname{Sign})$ by Theorem \[kernel\]. We have that $\operatorname{img}({\ensuremath{\mathit{sq}}}^{\operatorname{Sign}})= \{\varnothing, \mathbb{Z}_{>0}, 0,
\mathbb{Z}_{\geq 0}\}$. Moreover, $$\begin{aligned}
\max (\{x\in \operatorname{Sign}~|~ {\ensuremath{\mathit{sq}}}^{\operatorname{Sign}}(x) = \varnothing\}) &= \{\varnothing\}\\
\max (\{x\in \operatorname{Sign}~|~ {\ensuremath{\mathit{sq}}}^{\operatorname{Sign}}(x) = \mathbb{Z}_{>0}\}) &= \{\mathbb{Z}_{\neq 0}\}\\
\max (\{x\in \operatorname{Sign}~|~ {\ensuremath{\mathit{sq}}}^{\operatorname{Sign}}(x) = 0\}) &= \{0\}\\
\max (\{x\in \operatorname{Sign}~|~ {\ensuremath{\mathit{sq}}}^{\operatorname{Sign}}(x) = \mathbb{Z}_{\geq 0}\}) &= \{\mathbb{Z}\}\end{aligned}$$ Therefore, $\bigcup_{y\in \operatorname{img}({\ensuremath{\mathit{sq}}}^{\operatorname{Sign}})} \max (\{x\in \operatorname{Sign}~|~ {\ensuremath{\mathit{sq}}}^{\operatorname{Sign}}(x) = y\})
= \{\varnothing,\mathbb{Z}_{\neq 0},0,\mathbb{Z}\}$ so that, by Theorem \[kernel\]: $$\mathscr{K}_{{\ensuremath{\mathit{sq}}}}(\operatorname{Sign}) = \operatorname{Cl}_{\cap}(\{\varnothing, \mathbb{Z}_{> 0},
0,\mathbb{Z}_{\geq 0}, \mathbb{Z}_{\neq 0}, \mathbb{Z}\}) = \operatorname{Sign}\smallsetminus
\{\mathbb{Z}_{< 0}, \mathbb{Z}_{\leq 0}\}.$$ Thus, it turns out that we can safely remove the abstract values $\mathbb{Z}_{< 0}$ and $\mathbb{Z}_{\leq 0}$ from $\operatorname{Sign}$ and still preserve the same b.c.a. as $\operatorname{Sign}$ does. Besides, we cannot remove further abstract elements otherwise we do not retain the same b.c.a. as $\operatorname{Sign}$. For example, this means that $\operatorname{Sign}$-based analyses of programs like $$x := k;
\textbf{while}~\text{condition}~ \textbf{do}
~x := x*x;$$ can be carried out by using the simpler domain $\operatorname{Sign}\smallsetminus
\{\mathbb{Z}_{< 0}, \mathbb{Z}_{\leq 0}\}$, yet providing the same input/output abstract behavior.
It is worth remarking that in Theorem \[kernel\] the hypothesis of continuity is crucial for the existence of correctness kernels and this is shown by the following example.
\[esempio1\] Let us consider as concrete domain $C$ the $\omega + 2$ ordinal, i.e., $C \triangleq \{ x\in \mathit{Ord}~|~ x
< \omega\} \cup$, and let $f:C{\rightarrow}C$ be defined as follows: $$f(x) {\triangleq}\left\{
\begin{array}{ll}
\omega & \mbox{ if $x<\omega$;} \\
\omega+1 &\mbox{ otherwise.}
\end{array}
\right.$$ Let $\mu\in \operatorname{uco}(C)$ be the identity $\lambda x.x$ uco, so that $\mu {\circ}f {\circ}\mu = f$. For any $k\geq 0$, consider $\rho_k\in \operatorname{uco}(C)$ defined as $\ok{\rho_k {\triangleq}C\smallsetminus [0,k[}$ and observe that for any $k$, we have that $\rho_k {\circ}f {\circ}\rho_k = f = \mu {\circ}f {\circ}\mu$. However, it turns out that $\sqcup_{k\geq 0} \rho_k = \cap_{k\geq 0}
\operatorname{img}(\rho_k) =
\{\omega, \omega +1\}$. Hence, $(\sqcup_{k\geq 0} \rho_k) {\circ}f {\circ}(\sqcup_{k\geq 0} \rho_k) = \lambda x.\omega +1\neq \mu {\circ}f {\circ}\mu$. Hence, the correctness kernel of $\mu$ for $f$ does not exist. Observe that $\mu {\circ}f=f$ is clearly not continuous and therefore this example is consistent with Theorem \[kernel\].
Correctness Kernels in Abstract Model Checking
==============================================
Following the approach by Ranzato and Tapparo [@rt07], partitions of a state space $\Sigma$ can be viewed as particular abstract domains of the concrete domain $\wp(\Sigma)$. Let $\operatorname{Part}(\Sigma)$ denote the set of partitions of $\Sigma$. Given a partition $P\in \operatorname{Part}(\Sigma)$, the corresponding set of (possibly empty) unions of blocks of $P$, namely $\wp(P)$, is viewed as an abstract domain of $\wp(\Sigma)$ by means of the following Galois insertion $(\alpha_P, \wp(\Sigma)_\subseteq , \wp(P)_\subseteq, \gamma_P)$: $$\alpha_P(S) {\triangleq}\{B\in P~|~ B\cap S \neq \varnothing\}
\text{~~and~~}
\gamma_P(\mathcal{B}) {\triangleq}\cup_{B\in \mathcal{B}} B.$$ Hence, the abstraction $\alpha_P(S)$ provides the minimal over-approximation of a set $S$ of states through blocks of $P$.
Consider a transition system $\mathcal{S}
= \langle \Sigma, {\shortrightarrow}\rangle$ and a corresponding abstract transition system ${\mathcal{A}}=
\langle P, {\shortrightarrow}^\sharp \rangle$ defined over a state partition $P\in \operatorname{Part}(\Sigma)$.[^1] Fixpoint-based verification of a temporal specification on the abstract model ${\mathcal{A}}$ relies on the computation of least/greatest fixpoints of operators which are defined using Boolean connectives (union, intersection, complementation) on abstract states and abstract successor/predecessor functions $\operatorname{post}^\sharp$/$\operatorname{pre}^\sharp$ on the abstract transition system $\langle P,{\shortrightarrow}^\sharp\rangle$. The key point here is that successor/predecessor functions are defined as best correct approximations on the abstract domain $P$ of the corresponding concrete successor/predecessor functions. In standard abstract model checking [@bk08; @cgl94; @cgp99], the abstract transition relation is defined as the existential/existential relation ${\shortrightarrow}^{\exists\exists}$ between blocks of $P$, namely: $$\begin{gathered}
B {\shortrightarrow}^{\exists\exists} C \text{~~~~iff~~~~} \exists x\in B.\exists y\in C.\: x
{\shortrightarrow}y\\
\operatorname{post}^{\exists\exists}(B) {\triangleq}\{C\in P~|~B {\shortrightarrow}^{\exists\exists} C \};~~~~~~
\operatorname{pre}^{\exists\exists}(C) {\triangleq}\{B\in P~|~B {\shortrightarrow}^{\exists\exists} C\}.\end{gathered}$$ As shown in [@rt07], it turns out that $\operatorname{pre}^{\exists\exists}$ and $\operatorname{post}^{\exists\exists}$ are the best correct approximations of, respectively, $\operatorname{pre}$ and $\operatorname{post}$ functions on the above abstraction $(\alpha_P, \wp(\Sigma)_\subseteq , \wp(P)_\subseteq, \gamma_P)$. In fact, for a block $C\in P$, we have that $$\alpha_P(\operatorname{pre}(\gamma_P(C))) = \{B\in P~|~ B\cap \operatorname{pre}(C)\neq \varnothing\}
= \operatorname{pre}^{\exists\exists}(C)$$ and an analogous equation holds for $\operatorname{post}$. We thus have that $\operatorname{pre}^{\exists\exists} = \alpha_P \circ \operatorname{pre}\circ \gamma_P$ and $\operatorname{post}^{\exists\exists} = \alpha_P \circ \operatorname{post}\circ \gamma_P$.
This abstract interpretation-based framework allows us to apply correctness kernels in the context of abstract model checking. The abstract transition system ${\mathcal{A}}= \langle P, {\shortrightarrow}^{\exists\exists} \rangle$ is viewed as an abstract interpretation defined by the abstract domain $(\alpha_P, \wp(\Sigma)_\subseteq ,$ $\wp(P)_\subseteq, \gamma_P)$ and the corresponding abstract functions $\operatorname{pre}^{\exists\exists} = \alpha_P \circ \operatorname{pre}\circ \gamma_P$ and $\operatorname{post}^{\exists\exists} = \alpha_P \circ \operatorname{post}\circ \gamma_P$. Then, the correctness kernel of the abstraction $\wp(P)$ for the concrete predecessor/successor $\{\operatorname{pre},\operatorname{post}\}$, that we denote simply by $\mathscr{K}_{{\shortrightarrow}}(P)$ (by Theorem \[kernel\], this clearly exists since $\operatorname{pre}$ and $\operatorname{post}$ are additive functions), provides a simplification of the abstract domain $\wp(P)$ that preserves the best correct approximations of predecessor and successor functions. This simplification $\mathscr{K}_{{\shortrightarrow}}(P)$ of the abstract state space $P$ works as follows:
\[kernel-part\] $\mathscr{K}_{{\shortrightarrow}}(P)$ merges two blocks $B_1,B_2\in P$ if and only if for any $A\in P$, $A {\shortrightarrow}^{\exists\exists} B_1 \:{\Leftrightarrow}\: A {\shortrightarrow}^{\exists\exists} B_2$ and $B_1 {\shortrightarrow}^{\exists\exists} A \:{\Leftrightarrow}\: B_2 {\shortrightarrow}^{\exists\exists} A$.
By Theorem \[kernel\], we have that the kernel $\mathscr{K}_{{\shortrightarrow}}(P)$ of the abstraction $\wp(P)\in \operatorname{Abs}(\wp(\Sigma))$ for $\operatorname{pre}$ and $\operatorname{post}$ is as follows: $$\begin{gathered}
\mathscr{K}_{{\shortrightarrow}}(P) = \operatorname{Cl}_{\cap}
\Big(\operatorname{img}(\operatorname{pre}^{\exists\exists}) \cup \textstyle \bigcup_{\mathcal{C}\in \operatorname{img}(\operatorname{pre}^{\exists\exists})}
\cup \{\mathcal{B}\in \wp(P)~|~ \operatorname{pre}^{\exists\exists}(\mathcal{C}) = \mathcal{B}\}\\
\bigcup
\operatorname{img}(\operatorname{post}^{\exists\exists}) \cup \textstyle \bigcup_{\mathcal{B}\in \operatorname{img}(\operatorname{post}^{\exists\exists})}
\cup \{\mathcal{C}\in \wp(P)~|~ \operatorname{post}^{\exists\exists}(\mathcal{B}) = \mathcal{C}\}
\Big).\end{gathered}$$ Let us observe that both b.c.a.’s $\operatorname{pre}_{\exists\exists}, \operatorname{post}_{\exists\exists}: \wp(P) {\rightarrow}\wp(P)$ are additive functions, so that for any $\mathcal{C}\in \operatorname{img}(\operatorname{pre}_{\exists\exists})$, $\cup \{\mathcal{B}\in \wp(P)~|~ \operatorname{pre}^{\exists\exists}(\mathcal{C}) = \mathcal{B}\}
\in \operatorname{img}(\operatorname{pre}_{\exists\exists})$ and for any $\mathcal{B}\in \operatorname{img}(\operatorname{post}_{\exists\exists})$, $\cup \{\mathcal{C}\in \wp(P)~|~ \operatorname{post}^{\exists\exists}(\mathcal{B}) = \mathcal{C}\}
\in \operatorname{img}(\operatorname{post}_{\exists\exists})$. Moreover, $\mathscr{K}_{{\shortrightarrow}}(P)$ is closed under arbitrary unions. Hence, the kernel can be simplified as follows: $$\mathscr{K}_{{\shortrightarrow}}(P) = \operatorname{Cl}_{\cap,\cup} (\{\operatorname{pre}^{\exists\exists} (\{C\})~|~
C\in P\} \cup \{\operatorname{post}^{\exists\exists} (\{B\})~|~
B\in P\}).$$ We therefore have that a block $B\in P$ is merged together with all the blocks $B'\in P$ such that for any block $A\in P$, $B \in \operatorname{pre}^{\exists\exists} (\{A\}) {\Leftrightarrow}B' \in \operatorname{pre}^{\exists\exists} (\{A\})$ and $B \in \operatorname{post}^{\exists\exists} (\{A\}) {\Leftrightarrow}B' \in \operatorname{post}^{\exists\exists} (\{A\})$. Thus, the thesis follows.
Reconsider the abstract transition system ${\mathcal{A}}$ in Section \[intro\] and let $P=\{[1],[2,3],[4,5],[6],[7],[8,9]\}$ be the underlying state partition. In this case, we have that $$\begin{array}{l}
\operatorname{img}(\operatorname{pre}^{\exists\exists}) = \operatorname{Cl}_{\cup} \big( \big\{ \{[1]\}, \{[2,3],[4,5]\}, \{[6],[7]\}\big\}\big),\\
\operatorname{img}(\operatorname{post}^{\exists\exists}) = \operatorname{Cl}_{\cup} \big(\big\{\{[2,3],[4,5]\}, \{[6],[7]\},\{[8,9]\}\big\}\big).
\end{array}$$ Hence, by Corollary \[kernel-part\], in the correctness kernel $\mathscr{K}_{{\shortrightarrow}}(P)$ the block $[2,3]$ is merged with $[4,5]$ while $[6]$ is merged with $[7]$. This therefore simplifies the partition $P$ to $P'' = \{[1], [2,3,4,5], [6,7], [8,9]\}$, that is, we obtain the abstract transition system ${\mathcal{A}}''$ in Section \[intro\].
Example Guided Abstraction Simplification {#egas}
=========================================
Let us discuss how correctness kernels give rise to an Example-Guided Abstraction Simplification (EGAS) paradigm in abstract transition systems.
Let us first recall some basic notions of CEGAR [@cgjlv00; @cgjlv03]. Consider an abstract transition system ${\mathcal{A}}=
\langle P, {\shortrightarrow}^{\exists\exists}\rangle$ defined over a state partition $P\in \operatorname{Part}(\Sigma)$ and a finite abstract path $\pi =
\langle B_1,...,B_n\rangle$ in ${\mathcal{A}}$, where each $B_i$ is a block of $P$. Typically, this is a path counterexample to the validity of a temporal formula that has been given as output by a model checker (for simplicity we do not consider here loop path counterexamples). The set of concrete paths that are abstracted to $\pi$ are defined as follows: $$\operatorname{paths}(\pi) {\triangleq}\{\tuple{s_1,...,s_n}\in \Sigma^n~|~\forall i\in [1,n]. s_i\in B_i \;\&\; \forall i\in [1,n). s_i {\shortrightarrow}s_{i+1}\}.$$ The abstract path $\pi$ is *spurious* when it represents no real concrete path, i.e., when $\operatorname{paths}(\pi)=\varnothing$. The sequence of sets of states $\operatorname{sp}(\pi) =
\langle S_1,...,S_n\rangle$ is inductively defined as follows: $S_1 {\triangleq}B_1$; $S_{i+1} {\triangleq}\operatorname{post}(S_i)\cap B_{i+1}$. As shown in [@cgjlv03], it turns out that $\pi$ is spurious iff there exists a least $k\in [1,n-1]$ such that $S_{k+1}=\varnothing$. In such a case, the partition $P$ is refined by splitting the block $B_{k}$. The three following sets partition the states of the block $B_{k}$:
dead-end states: $B_{k}^{\text{dead}} {\triangleq}S_{k}\neq \varnothing$
bad states: $B_{k}^{\text{bad}} {\triangleq}B_{k} \cap \operatorname{pre}(B_{k+1}) \neq \varnothing$
irrelevant states: $B_{k}^{\text{irr}} {\triangleq}B_{k}\smallsetminus
(B_{k}^{\text{dead}} \cup B_{k}^{\text{bad}})$
The split of the block $B_{k}$ must separate dead-end states from bad states, while irrelevant states may be joined indifferently with dead-end or bad states. However, the problem of finding the coarsest refinement of $P$ that separates dead-end and bad states is NP-hard [@cgjlv03] and thus some refinement heuristics are used. According to the basic heuristics of CEGAR [@cgjlv03 Section 4], $B_{k}$ is simply split into $B_{k}^{\text{dead}}$ and $B_{k}^{\text{bad}} \cup B_{k}^{\text{irr}}$.
=\[->,>=latex’\] (0,3) node\[name=1\][1]{} (0,1) node\[name=2\][2]{} (2,4) node\[name=3\][3]{} (2,2) node\[name=4\][4]{} (2,0) node\[name=5\][5]{} (4,3) node\[name=6\][6]{} (4,1) node\[name=7\][7]{}; (6,3.5) node\[name=f,rotate=45\][$\Longrightarrow$]{}; (6,0.5) node\[name=f,rotate=-45\][$\Longrightarrow$]{};
(0,4.5) node\[name=a\][$\mathcal{A}$]{}; (1) to (5); (2) to (4); (2) to (5); (3) to (6); (4) to (7); (5) to (7);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=a3\] (2.south east) ++(0.1,-0.1) node\[name=a4\]; (a3) rectangle (a4);
(3.north west) ++(-0.1,0.1) node\[name=b3\] (5.south east) ++(0.1,-0.1) node\[name=b4\]; (b3) rectangle (b4);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
(8,6) node\[name=1\][1]{} (8,4) node\[name=2\][2]{} (10,7) node\[name=3\][3]{} (10,5) node\[name=4\][4]{} (10,3) node\[name=5\][5]{} (12,6) node\[name=6\][6]{} (12,4) node\[name=7\][7]{}; (14,3.5) node\[name=f,rotate=-45\][$\Longrightarrow$]{}; (8,7.5) node\[name=a\][$\mathcal{A}'$]{};
\(1) to (5); (2) to (4); (2) to (5); (3) to (6); (4) to (7); (5) to (7);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=a3\] (2.south east) ++(0.1,-0.1) node\[name=a4\]; (a3) rectangle (a4);
(3.north west) ++(-0.1,0.1) node\[name=b3\] (4.south east) ++(0.1,-0.1) node\[name=b4\]; (b3) rectangle (b4);
(5.north west) ++(-0.1,0.1) node\[name=b5\] (5.south east) ++(0.1,-0.1) node\[name=b6\]; (b5) rectangle (b6);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
(8,0) node\[name=1\][1]{} (8,-2) node\[name=2\][2]{} (10,1) node\[name=3\][3]{} (10,-1) node\[name=4\][4]{} (10,-3) node\[name=5\][5]{} (12,0) node\[name=6\][6]{} (12,-2) node\[name=7\][7]{}; (8,1.5) node\[name=a\][$\mathcal{A}''$]{};
\(1) to (5); (2) to (4); (2) to (5); (3) to (6); (4) to (7); (5) to (7);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=a3\] (2.south east) ++(0.1,-0.1) node\[name=a4\]; (a3) rectangle (a4);
(3.north west) ++(-0.1,0.1) node\[name=b3\] (3.south east) ++(0.1,-0.1) node\[name=b4\]; (b3) rectangle (b4);
(4.north west) ++(-0.1,0.1) node\[name=b5\] (5.south east) ++(0.1,-0.1) node\[name=b6\]; (b5) rectangle (b6);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
(16,3) node\[name=1\][1]{} (16,1) node\[name=2\][2]{} (18,4) node\[name=3\][3]{} (18,2) node\[name=4\][4]{} (18,0) node\[name=5\][5]{} (20,3) node\[name=6\][6]{} (20,1) node\[name=7\][7]{}; (16,4.5) node\[name=a\][$\mathcal{A}'''$]{};
\(1) to (5); (2) to (4); (2) to (5); (3) to (6); (4) to (7); (5) to (7);
(1.north west) ++(-0.1,0.1) node\[name=a1\] (1.south east) ++(0.1,-0.1) node\[name=a2\]; (a1) rectangle (a2);
(2.north west) ++(-0.1,0.1) node\[name=a3\] (2.south east) ++(0.1,-0.1) node\[name=a4\]; (a3) rectangle (a4);
(3.north west) ++(-0.1,0.1) node\[name=b3\] (3.south east) ++(0.1,-0.1) node\[name=b4\]; (b3) rectangle (b4);
(4.north west) ++(-0.1,0.1) node\[name=b7\] (4.south east) ++(0.1,-0.1) node\[name=b8\]; (b7) rectangle (b8);
(5.north west) ++(-0.1,0.1) node\[name=b5\] (5.south east) ++(0.1,-0.1) node\[name=b6\]; (b5) rectangle (b6);
(6.north west) ++(-0.1,0.1) node\[name=c1\] (6.south east) ++(0.1,-0.1) node\[name=c2\]; (c1) rectangle (c2);
(7.north west) ++(-0.1,0.1) node\[name=c3\] (7.south east) ++(0.1,-0.1) node\[name=c4\]; (c3) rectangle (c4);
Let us see a simple example. Consider the abstract path $\pi=\langle[1], [345], [6]\rangle$ in the abstract transition system ${\mathcal{A}}$ depicted in Figure \[figura-bis\]. This is a spurious path and the block $[345]$ is therefore partitioned as follows: $[5]$ dead-end states, $[3]$ bad states and $[4]$ irrelevant states. The refinement heuristics of CEGAR tells us that irrelevant states are joined with bad states so that ${\mathcal{A}}$ is refined to the abstract transition system ${\mathcal{A}}'$. In turn, consider the spurious path $\pi' = \langle [2], [34], [6] \rangle$ in ${\mathcal{A}}'$, so that CEGAR refines ${\mathcal{A}}'$ to ${\mathcal{A}}'''$ by splitting the block $[34]$. In the first abstraction refinement, let us observe that if irrelevant states would have been joined together with dead-end states rather than with bad states we would have obtained the abstract system ${\mathcal{A}}''$, and ${\mathcal{A}}''$ does not contain spurious paths so that it surely does not need to be further refined.
EGAS can be integrated within the CEGAR loop thanks to the following remark. If $\pi_1$ and $\pi_2$ are paths, respectively, in $\tuple{P_1,{\shortrightarrow}^{\exists\exists}}$ and $\tuple{P_2,{\shortrightarrow}^{\exists\exists}}$, where $P_1,P_2\in \operatorname{Part}(\Sigma)$ and $P_1$ is finer than $P_2$, i.e. $P_1\preceq P_2$, then we say that $\pi_1$ is abstracted to $\pi_2$, denoted by $\pi_1 \sqsubseteq \pi_2$, when $\operatorname{length}(\pi_1)=\operatorname{length}(\pi_2)$ and for any $j\in [1,\operatorname{length}(\pi_1)]$, $\pi_1(j)\subseteq \pi_2(j)$.
\[coro2\] Consider an abstract transition system ${\mathcal{A}}=\tuple{P,{\shortrightarrow}^{\exists\exists}}$ over a partition $P\in \operatorname{Part}(\Sigma)$ and its simplification ${\mathcal{A}}_s=\tuple{\mathscr{K}_{{\shortrightarrow}}(P),{\shortrightarrow}^{\exists\exists}}$ induced by the correctness kernel $\mathscr{K}_{{\shortrightarrow}}(P)$. If $\pi$ is a spurious abstract path in ${\mathcal{A}}_s$ then there exists a spurious abstract path $\pi'$ in ${\mathcal{A}}$ such that $\pi'\sqsubseteq \pi$.
This is a simple consequence of Corollary \[kernel-part\]. Let $\pi = \langle B_1,...,B_n\rangle$, where $B_i \in \mathscr{K}_{{\shortrightarrow}}(P)$, and let $B_k$ be the block of $\pi$ that causes the spuriousness of $\pi$. Then, for each $i\in [1,n]$, we have that $B_i = \cup C_i^{j_i}$, where $C_i^{j_i} \in P$. By Corollary \[kernel-part\], for each $i\in [1,n[$ and $j_i$, $\operatorname{post}^{\exists\exists}
(B_i) = \operatorname{post}^{\exists\exists} (C_i^{j_i})$ and for each $i\in ]1,n]$ and $j_i$, $\operatorname{pre}^{\exists\exists}
(B_i) = \operatorname{pre}^{\exists\exists} (C_i^{j_i})$. Then, in order to define the path $\pi'$ in ${\mathcal{A}}$, for $i \in [1,n]$, one can choose any block $C_i^{j_i}$ in $P$ such that $C_i^{j_i} \subseteq B_i$. The key point to note here is that the definition of the correctness kernel $\mathscr{K}_{{\shortrightarrow}}(P)$ guarantees that $C_k^{j_k}$ causes the spuriousness of $\pi'$ and that $\pi' \sqsubseteq \pi$.
Thus, it turns out that the abstraction simplification induced by the correctness kernel does not add spurious paths. These observations suggest us a new refinement strategy within the CEGAR loop. Let $\pi = \langle B_1,...,B_n\rangle$ be a spurious path in ${\mathcal{A}}$ and $\operatorname{sp}(\pi) =
\langle S_1,...,S_n\rangle$ such that $S_{k+1}=\varnothing$ for some minimum $k\in [1,n-1]$, so that the block $B_{k}$ needs to be split. The set of irrelevant states $B_{k}^{\text{irr}}$ is partitioned as follows. We first define the subset of *bad-irrelevant* states $B_{k}^{\text{bad-irr}}$. Let $\operatorname{pre}^{\exists\exists}(B_{k}^{\text{bad}}) = \{A_1,...,A_j\}$ and $\operatorname{post}^{\exists\exists}(B_{k}^{\text{bad}})= \{C_1,...,C_l\}$. Then, we define: $$B_{k}^{\text{bad-irr}} {\triangleq}\big(\operatorname{post}(A_1 \cup ...\cup A_j) \cap
\operatorname{pre}(C_1 \cup ... \cup C_l)\big) \cap B_{k}^{\text{irr}}.$$ The underlying idea is simple: $B_{k}^{\text{bad-irr}}$ contains the irrelevant states that: (1) can be reached from a block that reaches some bad state and (2) reach a block that is also reached by some bad state. By Corollary \[coro2\], it is therefore clear that by merging $B_{k}^{\text{bad-irr}}$ and $B_{k}^{\text{bad}}$ no spurious path is added w.r.t. the abstract system where they are kept separate. The subset of *dead-irrelevant* states $B_{k}^{\text{dead-irr}}$ is analogosly defined: If $\operatorname{pre}^{\exists\exists}(B_{k}^{\text{dead}}) = \{A_1,...,A_j\}$ and $\operatorname{post}^{\exists\exists}(B_{k}^{\text{dead}})= \{C_1,...,C_l\}$ then $$B_{k}^{\text{dead-irr}} {\triangleq}\big(\operatorname{post}(A_1 \cup ...\cup A_j) \cap
\operatorname{pre}(C_1 \cup ... \cup C_l)\big) \cap B_{k}^{\text{irr}}.$$ It may happen that: (A) an irrelevant state is both bad- and dead-irrelevant; (B) an irrelevant state is neither bad- nor dead-irrelevant. From the viewpoint of EGAS, the states of case (A) can be equivalently merged with bad or dead states since in both cases no spurious path is added. On the other hand, the states of case (B) are called *fully-irrelevant* because EGAS does not provide a merging strategy with bad or dead states. For these states, one could use, for example, the basic refinement heuristics of CEGAR that merge them with bad states.
In the above example, for the spurious path $\tuple{[1],[3,4,5],[6]}$ in ${\mathcal{A}}$, the block $B=[3,4,5]$ needs to be refined: $$B^{\text{bad}}=[3],\; B^{\text{dead}}=[5],\;
B^{\text{irr}}=[4].$$ Here, $4$ is a dead-irrelevant state because $\operatorname{pre}^{\exists\exists}([5]) = \{[1],[2]\}$, $\operatorname{post}^{\exists\exists}([5])= \{[7]\}$ and $(\operatorname{post}([1]\cup [2]) \cap
\operatorname{pre}([7])) \cap [4] = \{4\}$. Hence, according to the EGAS refinement strategy, the dead-irrelevant state $4$ is merged in ${\mathcal{A}}''$ with the dead-end state $5$.
Correctness Kernels in Predicate Abstraction
============================================
Let us discuss how correctness kernels can be also used in the context of predicate abstraction-based model checking [@ddp99; @gs97]. Following Ball et al.’s approach [@bpr03], predicate abstraction can be formalized by abstract interpretation as follows. Let us consider a program $P$ with $k$ integer variables $x_1$,...,$x_k$. The concrete domain of computation of $P$ is $\tuple{\wp(\operatorname{States}),\subseteq}$ where $\operatorname{States}{\triangleq}\{x_1,...,x_k\} \rightarrow \mathbb{Z}$. Values in $\operatorname{States}$ are denoted by tuples $\tuple{z_1,...,z_k}\in \mathbb{Z}^k$. The program $P$ generates a transition system $\tuple{\operatorname{States},{\shortrightarrow}}$ so that the concrete semantics of $P$ is defined by the corresponding successor function $\operatorname{post}:\wp(\operatorname{States}) {\rightarrow}\wp(\operatorname{States})$.
A finite set $\mathcal{P} = \{p_1,...,p_n\}$ of state predicates is considered, where each predicate $p_i$ denotes the subset of states that satisfy $p_i$, i.e. $\{s\in \operatorname{States}~|~ s \models p_i\}$. These predicates give rise to the so-called *Boolean abstraction* $B {\triangleq}\langle \wp(\{0,1\}^n),\subseteq \rangle$ which is related to $\wp(\operatorname{States})$ through the following abstraction and concretization maps (here, $s\models p_i$ is understood in $\{0,1\}$):
$$\begin{aligned}
\alpha_B(S) &{\triangleq}\{\langle s\models p_1,...,s\models p_n \rangle \in
\{0,1\}^n ~|~ s\in S\},\\
\gamma_B(V) &{\triangleq}\{s\in \operatorname{States}~|~ \langle s\models p_1,...,s\models p_n \rangle
\in V\}.\end{aligned}$$
These functions give rise to a disjunctive (i.e., $\gamma$ preserves lub’s) Galois connection $(\alpha_B,
\wp(\operatorname{States})_\subseteq, \wp(\{0,1\}^n)_\subseteq, \gamma_B)$.
Verification of reachability properties based on predicate abstraction consists in computing the least fixpoint of the best correct approximation of $\operatorname{post}$ on the Boolean abstraction $B$, namely, $\operatorname{post}^B {\triangleq}\alpha_B {\circ}\operatorname{post}{\circ}\gamma_B$. As argued in [@bpr03], the Boolean abstraction $B$ may be too costly for the purpose of reachability verification, so that one usually abstracts $B$ through the so-called *Cartesian abstraction*. This latter abstraction formalizes precisely the abstract $\operatorname{post}$ operator computed by the verification algorithm of the c2bp tool in SLAM [@slam02]. However, the Cartesian abstraction of $B$ may cause a loss of precision, so that this abstraction is successively refined by reduced disjunctive completion and the so-called focus operation, and this formalizes the bebop tool in SLAM [@bpr03].
$x$, $y$, $z$, $w$;
$\mathit{foo}$() {
{
$z := 0$; $x:=y$;
$(w)$ { $x$++; $z:=1$; }
} $(!(x = y))$
$(z)$
$(0)$; $(*)$
}
Let us consider the example program in Figure \[exprog\], taken from [@bpr03], where the goal is that of verifying that the assert at line $(*)$ is never reached, regardless of the context in which $\mathit{foo}()$ is called. Ball et al. [@bpr03] consider the following set of predicates $\mathcal{P} {\triangleq}\{p_1 \equiv (z = 0), p_2 \equiv (x=y)\}$ so that the Boolean abstraction is $B = \wp(\{\tuple{0,0},\tuple{0,1},\tuple{1,0},\tuple{1,1}\})_\subseteq$. Clearly, the analysis based on $B$ allows to conclude that line $(*)$ is not reachable. This comes as a consequence of the fact that the least fixpoint computation of the best correct approximation $\operatorname{post}^B$ for the do-while loop provides as result $\{ \tuple{0,0}, \tuple{1,1}\}\in B$ because: $$\begin{aligned}
\varnothing \xrightarrow{z:=0;~ x:=y}\{\tuple{1,1}\}
\xrightarrow{\textbf{if}(w) \{x\text{++};~z:=1;\}}
\{\tuple{1,1}\} \cup \{\tuple{0,0}\} \end{aligned}$$ where, according to a standard approach, the guard of the if statement is simply ignored. Hence, at the exit of the do-while loop one can conclude that $$\begin{aligned}
\{\tuple{1,1},\tuple{0,0}\}\cap p_2 = \{\tuple{1,1},\tuple{0,0}\}\cap
\{\tuple{0,1},\tuple{1,1}\} = \{\tuple{1,1}\}\end{aligned}$$ holds, hence $p_1$ is satisfied, so that $z=0$ and therefore line $(*)$ can never be reached.
Let us characterize the correctness kernel of the Boolean abstraction $B$. Let $S_1 {\triangleq}z:=0;~ x:=y$ and $S_2 {\triangleq}x\text{++};~z:=1$. The best correct approximations of $\operatorname{post}_{S_1}$ and $\operatorname{post}_{S_2}$ on the abstract domain $B$ turn out to be as follows: $$\begin{aligned}
\alpha_B {\circ}\operatorname{post}_{S_1} {\circ}\gamma_B = \Big\{&
\tuple{0,0} \mapsto \{\tuple{1,1}\}, \tuple{0,1} \mapsto \{\tuple{1,1}\},
\tuple{1,0} \mapsto \{\tuple{1,1}\},\\[-5pt]
& \tuple{1,1} \mapsto \{\tuple{1,1}\} \Big\}\\
\alpha_B {\circ}\operatorname{post}_{S_2} {\circ}\gamma_B = \Big\{ &
\tuple{0,0} \mapsto \{\tuple{0,0},\tuple{0,1}\}, \tuple{0,1} \mapsto \{\tuple{0,0}\}, \\[-5pt]
& \tuple{1,0} \mapsto \{\tuple{0,0},\tuple{0,1}\},
\tuple{1,1} \mapsto \{\tuple{0,0}\} \Big\}\end{aligned}$$ Thus, we have that $\operatorname{img}(\alpha_B {\circ}\operatorname{post}_{S_1} {\circ}\gamma_B) = \{\{\tuple{1,1}\}\}$ and $\operatorname{img}(\alpha_B {\circ}\operatorname{post}_{S_2} {\circ}\gamma_B) = \{\{\tuple{0,0},\tuple{0,1}\},
\{\tuple{0,0}\} \}$ so that $$\begin{aligned}
\max\big(\{V\in B~|~ \alpha_B(\operatorname{post}_{S_1}(\gamma_B(V))) = \{\tuple{1,1}\}\}\big) &=
\{\{\tuple{0,0}, \tuple{0,1}, \tuple{1,0}, \tuple{1,1}\}\}\\
\max\big(\{V\in B~|~ \alpha_B(\operatorname{post}_{S_2}(\gamma_B(V))) = \{\tuple{0,0},\tuple{0,1}\}\}\big)
&= \{\{\tuple{0,0}, \tuple{0,1}, \tuple{1,0}, \tuple{1,1}\}\}\\
\max\big(\{V\in B~|~ \alpha_B(\operatorname{post}_{S_2}(\gamma_B(V))) = \{\tuple{0,0}\}\}\big)
&= \{\{\tuple{0,1}, \tuple{1,1}\}\}\end{aligned}$$
Hence, by Theorem \[kernel\], the kernel $\mathscr{K}_F(B)$ of $B$ for $F{\triangleq}\{\operatorname{post}_{S_1},
\operatorname{post}_{S_2}\}$ is: $$\begin{aligned}
\operatorname{Cl}_\cap \Big( \operatorname{Cl}_\cup \big(
\big\{ &
\{\tuple{0,0}\},
\{\tuple{1,1}\},
\{\tuple{0,0},\tuple{0,1}\}, \{\tuple{0,1},\tuple{1,1}\},
\\[-5pt]
&
\{\tuple{0,0},\tuple{0,1},\tuple{1,0},\tuple{1,1}\}
\big\}
\big) \Big) = \operatorname{Cl}_\cup \Big( \big\{
\{\tuple{0,0}\},
\{ \tuple{0,1}\},
\{ \tuple{1,1}\}
\big\} \Big) \end{aligned}$$ where we observe that the set $\{\tuple{0,1}\}$ is obtained as the intersection $\{\tuple{0,0},\tuple{0,1}\} \cap \{\tuple{0,1},\tuple{1,1}\}$. This correctness kernel $\mathscr{K}_F(B)$ can be therefore represented as $$\tuple{\wp(\{\tuple{0,0},\tuple{0,1},\tuple{1,1}\}) \cup \{\tuple{0,0},
\tuple{0,1}, \tuple{1,0}, \tuple{1,1}\},\subseteq}.$$ Thus, it turns out that $\mathscr{K}_F(B)$ is a proper abstraction of the Boolean abstraction $B$ that, for example, is not able to express precisely the property $p_1 \wedge \neg p_2 \equiv (z=0) \wedge (x\neq y)$.
It is interesting to compare this correctness kernel $\mathscr{K}_F(B)$ with Ball et al.’s [@bpr03] Cartesian abstraction of $B$. The Cartesian abstraction is defined as $$C{\triangleq}\tuple{\{0,1,*\}^n \cup \{\bot_C\},\leq}$$ where $\leq$ is the componentwise ordering between tuples of values in $\{0,1,*\}$ ordered by $0 < *$ and $1< *$ ($\bot_C$ is a bottom element that represents the empty set of states). The concretization function $\gamma_{C} :
C\rightarrow \wp(\operatorname{States})$ is as follows: $$\gamma_{C} (\tuple{v_1,...,v_n}) {\triangleq}\{s\in \operatorname{States}~|~
\tuple{s\models p_1,...,s\models p_n} \leq \tuple{v_1,...,v_n}\}.$$ It turns out that these two abstractions are not comparable. For instance, $\tuple{1,0}\in C$ represents $p_1 \wedge \neg p_2$ which is instead not represented by $\mathscr{K}_F(B)$, while $\{\tuple{0,0},\tuple{1,1}\}\in \mathscr{K}_F(B)$ represents $(\neg p_1 \wedge \neg p_2) \vee (p_1 \wedge p_2)$ which is not represented in $C$. However, while the correctness kernel guarantees no loss of information in analyzing the program $P$ (and therefore the analysis with $\mathscr{K}_F(B)$ concludes that $(*)$ cannot be reached), the analysis of $P$ with the Cartesian abstraction $C$ is inconclusive because: $$\begin{aligned}
\bot_C \xrightarrow{z:=0;~ x:=y} \tuple{1,1}
\xrightarrow{\textbf{if}(w) \{x\text{++};~z:=1;\}}
\tuple{0,0} \vee_C \tuple{1,1} = \tuple{*,*} \end{aligned}$$ where $\gamma_C(\tuple{*,*}) =
\operatorname{States}$, so that at the exit of the do-while loop one cannot infer with $C$ that line $(*)$ is unreachable.
Related and Future Work
=======================
Few examples of abstraction simplifications are known. A general notion of domain simplification and compression in abstract interpretation has been introduced in [@fgr96; @gr97] as a formal dual of abstraction refinement. This duality has been further exploited in [@gm08] to include semantics transformations in a general theory for transforming abstractions and semantics based on abstract interpretation. Our domain transformation does not fit directly in this framework. Following [@gr97], given a property $\mathcal{P}$ of abstract domains, the so-called core of an abstract domain $A$, when it exists, provides the most concrete simplification of $A$ that satisfies the property $\mathcal{P}$, while the so-called compressor of $A$, when it exists, provides the most abstract simplification of $A$ that induces the same refined abstraction in $\mathcal{P}$ as $A$ does. Examples of compressors include the least disjuctive basis [@gr98], where $\mathcal{P}$ is the abstract domain property of being disjunctive, and examples of cores include the completeness core [@grs00], where $\mathcal{P}$ is the domain property of being complete for some semantic function. The correctness kernel defined in this paper is neither an instance of a domain core nor an instance of a domain compression. The first because, given an abstraction $A$, the correctness kernel of $A$ characterizes the most abstract domain that induces the same best correct approximation of a function $f$ on $A$, whilst the notion of domain core for the domain property $\mathcal{P}_A$ of inducing the same b.c.a. as $A$ would not be meaningful, as this would trivially yield $A$ itself. The second because there is no (unique) maximal domain refinement of an abstract domain which induces the same property $\mathcal{P}_A$.
The EGAS methodology opens some stimulating directions for future work, such as (1) the formalization of a precise relationship between EGAS and CEGAR and (2) an experimental evaluation of the integration in the CEGAR loop of the EGAS-based refinement strategy of Section \[egas\]. It is here useful to recall that some work formalizing CEGAR in abstract interpretation has already been done [@CGR07; @gq01]. On the one hand, [@gq01] shows that CEGAR corresponds to iteratively compute a so-called complete shell [@grs00] of the underlying abstract model $A$ with respect to the concrete successor transformer, while [@CGR07] formally compares CEGAR with an abstraction refinement strategy based on the computations of abstract fixpoints in an abstract domain. These works can therefore provide a starting point for studying the relationship between EGAS and CEGAR in a common abstract interpretation setting.
#### **Acknowledgements.**
This work was carried out during a visit of the authors to the Equipe “Abstraction” lead by P. and R. Cousot, at École Normale Supérieure, Paris. This work was partially supported by the University of Padova under the Projects AVIAMO and BECOM.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is shown that partial incoherence, in the form of stochastic phase noise, of a Langmuir wave in an unmagnetized plasma gives rise to a Landau-type damping. Starting from the Zakharov equations, which describe the nonlinear interaction between Langmuir and ion-acoustic waves, a kinetic equation is derived for the plasmons by introducing the Wigner-Moyal transform of the complex Langmuir wave field. This equation is then used to analyze the stability properties of small perturbations on a stationary solution consisting of a constant amplitude wave with stochastic phase noise. The concomitant dispersion relation exhibits the phenomenon of Landau-like damping. However, this damping differs from the classical Landau damping in which a Langmuir wave, interacting with the plasma electrons, loses energy. In the present process, the damping is non-dissipative and is caused by the resonant interaction between an instantaneously-produced disturbance, due to the parametric interactions, and a partially incoherent Langmuir wave, which can be considered as a quasi-particle composed of an ensemble of partially incoherent plasmons.'
address:
- |
$^1$ *Dipartimento di Scienze Fisiche, Università di Napoli ”Federico II” and INFN Sezione di Napoli, Complesso Universitario di M.S. Angelo,\
Via Cintia, I-80126 Napoli, Italy*
- |
$^2$ *Institut für Theoretische Physik IV, Ruhr-Universität Bochum\
D-4470 Bochum, Germany*
- |
$^3$ *Dipartimento di Fisica Generale, Università di Torino,\
Via Pietro Giuria 1, 10125 Torino, Italy*
- '$^4$ *Department of Electromagnetics, Chalmers University of Technology, SE-412 96, Göteborg, Sweden*'
author:
- 'R. Fedele$^1$, P.K. Shukla$^2$, M. Onorato$^3$, D. Anderson$^4$, and M. Lisak$^4$'
title: |
[*Submitted*]{}\
Landau damping of partially incoherent Langmuir waves
---
PACS number(s): 52.35.Mw, 52.35.Fp, 52.25.Dg, 03.40.Kf
Keywords: quasi-particles, Langmuir envelopes, Landau damping, modulational instability, nonlinear Schroedinger equation, Zakharov equations.
More than fifty years ago, Landau [@9] discovered a damping of electron plasma waves whose linear dynamics is governed by the Vlasov-Poisson system of equations. This damping was caused by resonant wave-electron interactions. The Landau theory clearly shows that the decay rate of the wave energy due to the interaction with the resonant electrons of the system (satisfying $p=\omega/k$, where $p$ denotes the electron velocity and $\omega$ and $k$ are the frequency and wave number of the electron plasma wave ) is proportional to the first derivative of the equilibrium distribution function $\rho_0 (p)$ of the electrons. Typically the shape of $\rho_0 (p)$ is such that $d\rho_0 (p=\omega/k)/dp<0$, which implies that there are more particles with $p < \omega/k$ (which gain energy from the wave) than with $p
>\omega/k$ (which give energy to the wave). Consequently, the net result is that the wave is damped. However, finite amplitude Langmuir waves in plasmas can be created when some free energy sources, such as electron and laser beams, are available in the system. In the past, several authors [@nishikawa]-[@Zakmushrub] have considered the nonlinear coupling between high-frequency Langmuir and low-frequency ion-acoustic waves. Physically, this coupling occurs because a large amplitude plasma wave, interacting with small density ripples, produces a current which becomes the source for an envelope of Langmuir waves. The low-frequency ponderomotive force of the latter reinforces the density modulation in such a way as to produce a space-charge electric field, which varies on a time scale longer than the electron plasma period. Thus, within the slowly-varying envelope approximation, the dynamics of the Langmuir wavepacket is governed by a nonlinear Schrödinger equation (NLSE), while the non-resonant density ripples follow an ion-acoustic wave equation driven by a ponderomotive force created by the Langmuir wave). In his classic paper, Zakharov [@Zakharov] demonstrated that these nonlinearly coupled equations admit a class of modulationally unstable stationary solutions and also exhibit the phenomenon of wave collapse.\
In this paper, we present a Landau damping of partially coherent Langmuir waves whose dynamics is governed by the Zakharov equations. In fact, we show that the latter can be reduced to a kinetic-like equation for a Wigner-Moyal-like transform for the amplitude of the electric field associated with the Langmuir wave. Based on this kinetic equation we consider the dynamics of small perturbations on a partially coherent background solution consisting of a constant amplitude and a stochastically varying phase. The concomitant dispersion relation depends strongly on the spectral energy density of the Langmuir wave and a Landau-type damping of the perturbations is found due to the resonant interaction between the Langmuir quasi-particle and the instantaneously created ripples.
As it is well known, the nonlinear propagation of a one–dimensional Langmuir wave–packet in an unmagnetized plasma with unperturbed density $n_0$, is governed by the Zakharov equations [@Zakharov] $$i{\partial E\over\partial t}~+~{3 v_{te}^{2}\over
2\omega_{pe}}{\partial^2 E\over\partial
x^2}~-~\omega_{pe}{\delta n\over 2n_0}E~=~0~~~,
\label{zakharov-1}$$ $$\left({\partial^2\over\partial t^2}~-~c_{s}^{2}{\partial^2\over\partial x^2}
\right){\delta n\over 2n_0}~=~-{c_{s}^{2}\over 2}{\partial^2\over\partial x^2}
\left({|E|^2 -|E_0|^2\over 4\pi n_0 T_e}\right)~~~,
\label{zakharov-2}$$ where $E$ is the slowly varying complex electric field amplitude, $\delta n$ is the density fluctuation, $v_{te}$ is the electron thermal velocity, $\omega_{pe}$ is the electron plasma frequency, $T_e$ is the electron temperature, $c_{s}$ is the sound speed, and $E_0$ is the unperturbed electric field amplitude. Moreover, $x$ and $t$ play the role of configurational space coordinate (longitudinal coordinate with respect to the wavepacket centre) and time, respectively. We will show that the nonlinear propagation of a Langmuir wavepacket, governed by the above Zakharov system, can be suitably described in phase space by a kinetic–like equation within a framework similar to the one of the [*Vlasov-Poisson*]{} system. A similar approach has recently been used to study the modulational instability of a large-amplitude electromagnetic wavepacket as well as the coherent instability analysis of intense charged–particle beam dynamics [@5; @6]. Within this framework, we show that a phenomenon similar to the classical Landau damping occurs for partially coherent plasma waves. In particular, this phenomenon tends to suppress the modulational instability. To this end, the above system of equations (\[zakharov-1\]) and (\[zakharov-2\]) can be cast in the following form $$i\alpha{\partial \Psi\over\partial s}~+~{\alpha^{2}\over
2}{\partial^2
\Psi\over\partial
x^2}~-~U~\Psi~=~0~~~,
\label{zakharov-1-bis}$$ $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U~=~-{\partial^2\over\partial x^2}
\left(\langle|\Psi|^2 -|\Psi_0|^2\rangle\right)~~~,
\label{zakharov-2-bis}$$ where the angle brackets account for the statistical ensemble average [*ála*]{} Klimomtovich [@a] due to the partial incoherence of the waves, $s=\sqrt{3}\lambda_{De}\omega_{pe}t$ ($\lambda_{De}\equiv
v_{te}/\omega_{pe}$ is the electron Debye length), $\alpha
=\sqrt{3}\lambda_{De}$, $U=\delta n / 2n_0$, $\mu=\sqrt{m_{e}/3m_{i}}$ ($m_{e}$ and $m_{i}$ are the electron and the ion masses, respectively), and $$\Psi~=~{\mu E\over \sqrt{8\pi n_0 T_e}}~~~,
\label{Psi}$$ $$\Psi_0~=~{\mu E_0\over \sqrt{8\pi n_0 T_e}}~~~.
\label{Psi-zero}$$ Note that (\[Psi-zero\]) can be written as $$\rho_0
\equiv |\Psi_0|^2=\mu^2{\epsilon\over\epsilon_{T}}~~~,
\label{Psi-zero-bis}$$ where $\epsilon\equiv E_{0}^{2}/ 8\pi$ and $\epsilon_T\equiv n_0 T_e$ are the electric energy density and the electron thermal energy density, respectively. According to the theory modelled by the Zakharov system (\[zakharov-1-bis\]) and (\[zakharov-2-bis\]), $\epsilon /\epsilon_T >>1$ (i.e, the ponderomotive effect is larger than the thermal pressure effect). Note also that Eq. (\[zakharov-2-bis\]) implies that $U$ is a functional of $|\Psi|^2$, namely $U=U\left[|\Psi|^2\right]$. Consequently, once (\[zakharov-2-bis\]) is combined with (\[zakharov-1-bis\]), the latter becomes a nonlinear Schrödinger equation (NLSE) with the nonlinear term $U\left[|\Psi|^2\right]$. This implies that, as in Quantum Mechanics, one can introduce the density matrix [@a] as a sort of “two-points correlation function” and then the Wigner-Moyal transform [@b] which is solution of the von Neumann-Weyl equation [@c]. In case of surface gravity waves, whose dynamics and instability is described in deep water by a nonlinear Schrödinger equation, such a kind of approach has been given in the pioneering works by Alber [@e], Crawford et al. [@f] and Janssen [@g]. More recently, a similar approach has been developed for the propagation of electromagnetic wavepackets in nonlinear media where an appropriate kinetic equation is able to show a random version of the modulational instability [@5; @6; @j]. In particular, this approach can be also seen as an application of the Klimontovich statistical average method [@j]. The transition to the phase-space $x-p$ ($p\equiv dx/ds$ is the conjugate variable of $x$) allows us to write a von Neumann equation [@a] for $w(x,p,s)$. In fact, although, we are dealing with a classical system, by following Alber [@e], one can introduce the correlation function (whose corresponding meaning in Quantum Mechanics is nothing but the density matrix for mixed states) $$\rho(x,x',s)= \langle\Psi^*(x,s) \Psi (x',s)\rangle~~~,
\label{wigner}$$ where the statistical ensemble average takes into account the incoherency of the waves. The technique was successfully introduced in the field of statistical quantum mechanics to describe the dynamics of a system in the classical space language [@e]-[@g]. Thus one can define the following Wigner-Weyl transform [@b] $$w(x,p,s) = {1 \over 2 \pi \alpha}
\int_{-\infty}^{\infty}
\rho\left(x + { y \over 2},~ x - { y \over
2}, ~s\right)~\exp\left(i{p y\over \alpha} \right)~dy~~~,
\label{12-0}$$ which allows us to transit from the configuration space to the phase space. In fact, $w(x,p,s)$ is governed by the following kinetic-like equation (von Neumann-like equation [@a] ) $${\partial w\over \partial s} + p {\partial w\over
\partial x}
- \sum_{n=0}^{\infty}{(-1)^{n}\over \left(2n+1\right)!}
\left( { \alpha \over 2} \right)^{2 n}
{\partial^{2n+1}U\over\partial x^{2n+1}}
{\partial^{2n+1}w\over\partial p^{2n+1}}~=~0~~~.
\label{a4}$$ On the other hand, since (\[12-0\]) implies that $$\langle |\Psi|^2\rangle~=~\int_{-\infty}^{\infty}w(x,p,s)~dp~~~,
\label{x-projection}$$ the functional $U\left[|\Psi|^2\right]$ can be expressed as a functional of $w$, namely $U=U\left[w\right]$. Consequently, (\[zakharov-2-bis\]) can be written as $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U~=~-{\partial^2\over\partial x^2}
\left(\int_{-\infty}^{\infty}w~dp -\int_{-\infty}^{\infty}w_0~dp\right)~~~,
\label{zakharov-2-ter}$$ where $w_0$ is the Wigner-Moyal transform of $\Psi_0$ ($\rho_0
=|\Psi_0|^2 = \int_{-\infty}^{\infty}~w_0~dp$). The above scheme allows us to carry out a stability analysis of the stationary Langmuir wave by considering the linear dispersion relation of small amplitude perturbations. We start from the equilibrium state: $w = w_0 (p)$, $U=U_0=0$, and perturb the system according to $$w(x,p,s)=w_0 (p) + w_1 (x,p,s)~~~,
\label{rho-perturbation}$$ $$U(x,s)~=~U_0~+~U_1(x,s)~=~U_1(x,s)~~~,
\label{U-perturbation}$$ where $w_1 (x,p,s)$ and $U_1 (x,s)$ are first-order quantities. Consequently, (\[a4\]) and (\[zakharov-2-ter\]) can be linearized as follows $${\partial w_1\over \partial s} + p {\partial w_1\over
\partial x}
=~\sum_{n=0}^{\infty}{(-1)^{n}\over \left(2n+1\right)!}
\left( { \alpha \over 2} \right)^{2 n}
{\partial^{2n+1}U_1\over\partial x^{2n+1}}
w_{0}^{(2n+1)}~~~,
\label{a4-2}$$ $$\left({\partial^2\over\partial s^2}~-~\mu^{2}{\partial^2\over\partial x^2}
\right)U_1~=~-{\partial^2\over\partial x^2}
\left(\int_{-\infty}^{\infty}w_1~dp\right)~~~,
\label{zakharov-2-linearized}$$ where $w_{0}^{(2n+1)}\equiv d^{2n+1}\rho_0/dp^{2n+1}$. By introducing the Fourier transform of $U_1(x,s)$ and $w_1
(x,p,s)$, i.e. $$U_1(x,s)=\int_{-\infty}^{\infty}~dk~\int_{-\infty}^{\infty}~d\omega~
\widetilde{U_1}(k ,\omega)~\exp\left(ik x-i\omega
s\right)~~~,
\label{U-1-bis}$$ $$w_1(x,p,s)=\int_{-\infty}^{\infty}~dk~\int_{-\infty}^{\infty}~d\omega~
\widetilde{w_1}(k,p,\omega)~\exp\left(ik x-
i\omega s\right)~~~,
\label{w-1}$$ and substituting (\[U-1-bis\]) and (\[w-1\]) into Eq.s (\[a4-2\]) and (\[zakharov-2-linearized\]), we readily obtain the following dispersion relation $$\omega^2~-~\mu^2 k^2~=~k^2\int_{-\infty}^{\infty}~{w_0
\left(p+\alpha k /2\right)~-~w_0
\left(p-\alpha k /2\right)\over \alpha k}~
{dp\over p-\omega /k}~~~.
\label{dispersion-relation}$$
We consider a Lorenzian spectrum of the form: $$w_0(p)=\frac{\rho_0} {\pi} \frac{\Delta} {\Delta^2+p^2},$$ where $\Delta$ is the width of the spectrum. Dispersion relation (\[dispersion-relation\]) becomes: $$\left(\omega^2-\mu^2 k^2\right)\left(\omega^2-\alpha^2 k^4/4 +
2 i k \Delta \omega-k^2 \Delta^2 \right)
~=~\rho_0 k^4~~~,
\label{lorenzian-dispersion-relation}$$ As the simplest case, we take the limit of $\Delta \rightarrow 0$. This case corresponds to a coherent monochromatic wave-train and the dispersion relation reduces to $$\left(\omega^2-\mu^2 k^2\right)\left(\omega^2-\alpha^2 k^4 /4\right)
~=~\rho_0 k^4~~~,
\label{monochromatic-dispersion-relation}$$ which predicts the well known result of a purely growing instability [@Zakharov].
In the following, we concentrate on the limit of small $\alpha k << 1$ and $\omega /k >>1$. We will show that during its nonlinear evolution the Langmuir wavepacket can exhibit a phenomenon, similar to the Landau damping [@Dawson], between parametrically-driven non-resonant density ripples and Langmuir quasi-particles. It is easy to verify that the condition $\alpha k << 1$ (i.e., $\lambda_{De} k<<1$) implies that $${w_0\left(p+\alpha k /2\right)~-~w_0
\left(p-\alpha k /2\right)\over \alpha k}~\approx
dw_{0}/dp~\equiv w_{0}^{'}~~~.
\label{k-small-approximation}$$ Consequently, (\[dispersion-relation\]) becomes $$\omega^2~=~k^2~\int~{w_{0}^{'}
\over p-\omega /k}~dp~~~,
\label{k-small-dispersion-relation}$$ where, using the assumption $\omega /k >>1$, we can neglect the term involving $\mu^2$ because $\mu <<1$. Note that (\[k-small-dispersion-relation\]) is formally identical to the linear dispersion relation that holds for warm plasma waves [@9] or for non-monochromatic charged-particle bunches in circular accelerating machines for an arbitrary complex coupling impedance [@7]. Consequently, the dispersion relation (\[k-small-dispersion-relation\]) allows us to predict a sort of Landau damping [@9], as described in plasma physics for linear plasma waves as well as in charged-particle beam physics. A similar effect, which has been called [*quantum-like Landau damping*]{} (QLLD), has recently been predicted for the nonlinear propagation of a large-amplitude electromagnetic waves in nonlinear media, such as optical fibers and plasmas as well as for the quantum-like description of charged-particle beams in an accelerating machine [@5; @6; @j]. In order to investigate this phenomenon also for large-amplitude Langmuir waves, we apply the standard procedure [@14] used for the Landau damping of a linear plasma wave, which involves integrating the above dispersion relation in the complex plane, evaluating both the Cauchy principle value (PV) and the semi-residue in $p= \omega /k$. The result is $$\omega^2~=~k^2 \int_{PV}~{w_{0}^{'}
\over p-\omega /k}~dp~+
~i\pi k^2 w_{0}^{'}(\beta)~~~,
\label{landau-dispersion-relation}$$ For a symmetric sufficiently smooth stationary background distribution, $w_0 (p)$, we have $$\omega^2~=~\left(1+{3\Delta^{2}
\over \rho_{0}}\right)k^2~+
~i\pi k^2 w_{0}^{'}(\beta)~~~,
\label{reactive-dispersion-relation-2}$$ where $\Delta\equiv \left(\langle p^2 \rangle\right)^{1/2}$ denotes the r.m.s. width of $w_0 (p)$, and $\beta \equiv
\rho_{0}^{1/4}\left(1+3\Delta^{2} / 4\rho_{0}^{2}\right)$. Note that, according to the above hypothesis, we have $\beta >>1$ and $\Delta^2 /\rho_0 <<1$. This implies that $\rho_0 =|\Psi_0|^2
>>1$, and from (\[Psi-zero-bis\]) it follows that $\epsilon
/\epsilon_T >>1/\mu^2$. Eq. (\[reactive-dispersion-relation-2\]) clearly shows that the damping rate is proportional to the derivative of the Wigner distribution $w_0$. This is formally similar to the expression for the Landau damping of a linear plasma wave in a warm unmagnetized plasma. In fact, writing $\omega$ in the complex form $\omega =\omega_R + i\omega_I$, substituting it in (\[landau-dispersion-relation\]) and then separating the real and imaginary parts, we obtain $$\omega_{R}^{2}~-~\omega_{I}^{2}~=~k^2\int_{PV}~{w_{0}^{'}
\over p-\omega /k}~dp~~~,
\label{real-part}$$ and $$\omega_{I}~=~{\pi \over 2} {k^2\over \omega_R}
w_{0}^{'}(\beta)~~~.
\label{imaginary-part}$$ The Landau damping rate of the Langmuir wave in the appropriate unities is $\gamma =\sqrt{3}\omega_{pe}\lambda_{De}\omega_I$. For example, for a Gaussian wavepacket spectrum, i.e. $w_0 (p)= \left(\rho_0
/\sqrt{2\pi \Delta^{2}}\right)\exp\left(-p^2 /2\Delta^{2}
\right)$, one obtains $$\gamma = -\left({3\pi\over 8}\right)^{
1/2}{\mu^{5/2}
\over\Delta^3}
\left({\epsilon\over \epsilon_T}\right)^{5/4}
~\omega_{pe}\lambda_{De}k~\exp\left[-{\mu
\over 2\Delta^{2}}\left({\epsilon\over\epsilon_T}
\right)^{1/2}\right]~~~,
\label{imaginary-part-1}$$ where higher-order terms have been neglected. The present damping mechanism differs from the standard Landau damping in that here $w_0 (p)$ does not represent the equilibrium velocity distribution of the plasma electrons, but it can be considered as the “kinetic” distribution of all Fourier components of the partially incoherent large-amplitude Langmuir wave in the warm plasma. In terms of plasmons, we realize that $w_0 (p)$ represents the distribution of the partially incoherent plasmons that are distributed in $p$-space (i.e. in $k$-space) with a finite “temperature”. Consequently, the QLLD described here is due to the partial incoherence of the wave which corresponds to a finite-width Wigner distribution of the plasmons.
In this paper, we have carried out an analysis for a Landau-type damping in plasmas containing partially incoherent Langmuir waves. Starting from the Zakharov system of equations and employing the Wigner-Moyal quasi-distribution function for the complex Langmuir wave electric field amplitude, it has been shown that the Zakharov system of equations can be converted into a pair of coupled evolution equations in phase space. For a monochromatic and coherent wave ($w_0(p) =\rho_0\delta
(p)$), we have obtained a dispersion relation which exhibits a purely growing instability [@Zakharov]. Furthermore, for a broad-band Langmuir wave spectrum, corresponding to a partially incoherent constant amplitude wave, we have investigated the properties of small perturbations for $\alpha k<<1$ (i.e., $\lambda_{De} k<<1$), and very large dimensionless phase velocity (i.e., $\omega /k
>>1$). Our dispersion relation for this case exhibits a sort of weak Landau damping, very similar to the one predicted for the Vlasov-Poisson system for linear plasma waves. The physical origin of this phenomenon is attributed to the “non-monochromatic” behaviour of the Wigner spectrum of the Langmuir wave, which can be considered as an “ensemble of incoherent plasmons”. Actually, a broad-band spectrum of incoherent plasmons, forming the “wave packet”, interact individually with the self-excited density ripples. Similar to the standard Landau damping, where the electrons interact individually with a linear plasma wave and statistically produce a net transfer of energy from the wave to the particles, the plasmons interact individually with the ripples and produce a change of energy which is more significant around the resonance, which is determined by the condition $p=\omega /k$. Finally, we observe that since the system can be modulationally unstable [@Zakharov], this phenomenon acts in competition with the modulational instability.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using angle-resolved photoemission, we have investigated the development of the electronic structure and the Fermi level pinnning in Ga$_{1-x}$Mn$_{x}$As with Mn concentrations in the range 1–6%. We find that the Mn-induced changes in the valence-band spectra depend strongly on the Mn concentration, suggesting that the interaction between the Mn ions is more complex than assumed in earlier studies. The relative position of the Fermi level is also found to be concentration-dependent. In particular we find that for concentrations around 3.5–5% it is located very close to the valence-band maximum, which is in the range where metallic conductivity has been reported in earlier studies. For concentration outside this range, larger as well as smaller, the Fermi level is found to be pinned at about 0.15 eV higher energy.'
address:
- 'Department of Experimental Physics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden'
- |
Department of Experimental Physics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden\
and Institute of Physics, Polish Academy of Sciences, PL-02-668 Warszawa, Poland
- 'Department of Materials Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden'
author:
- 'H. Åsklund, L. Ilver, and J. Kanski'
- 'J. Sadowski'
- 'R. Mathieu'
date: 'Submitted: May 4, 2001'
title: |
Photoemission studies of Ga$_{1-x}$Mn$_{x}$As:\
Mn-concentration dependent properties
---
[2]{}
Introduction
============
The possibility to include magnetic impurities at relatively high concentrations in GaAs by means of low-temperature molecular beam epitaxy (LT-MBE) has opened new exciting prospects of combining magnetic phenomena with high-speed electronics and optoelectronics. The numerous investigations of Ga$_{1-x}$Mn$_{x}$As alloys that have been carried out so far have revealed interesting material properties, the most notable being carrier-induced ferromagnetism, with reported Curie temperatures as high as 110 K.[@Ohno01] Other interesting properties are anomalous Hall effect, negative magnetoresistance and photoinduced ferromagnetism. Although there is general consensus concerning the importance of Mn-induced holes, the detailed mechanisms behind the ferromagnetic ordering of the Mn spins remain a subject of debate.[@Dietl01; @Inoue00; @Konig00; @Akai98; @Litvinov01; @Schneider87] The electronic state of the Mn ions in samples with high Mn content is also discussed, though at low concentrations (below 1%) the $d^{5}$+hole configuration is established.[@Sanvito00; @Szczytko99] It is clear that further spectroscopic studies related to these problems are strongly motivated. In the present work we have used photoemission to study two key features in the electronic structure of Ga$_{1-x}$Mn$_{x}$As alloys for a range of Mn concentrations: the Mn-related modifications of the electronic structure and the Fermi level position relative to the VBM.
Experiment
==========
The experiments were performed on the toroidal grating monochromator beamline (BL 41) at the MAX I storage ring of the Swedish National Synchrotron Radiation Center MAX-lab, where a dedicated system for molecular beam epitaxy (MBE) is attached to the photoelectron spectrometer. This configuration allows samples to be transferred between the growth and analytical chambers under UHV conditions. In the transfer system the vacuum was in the low 10$^{-9}$ torr range, and in the electron spectrometer in the low 10$^{-10}$ torr range. The ability to transfer samples means that no post-growth treatment was needed to prepare the surfaces for the photoemission measurements. This is a point worth stressing in the present context, as the samples are prepared under rather extreme conditions and change their properties with annealing even at temperatures well below that at which MnAs segregates.[@Hayashi01] Indeed, the spectra presented here are somewhat different from those obtained on sputtered and annealed surfaces.[@Okabayashi99]
The MBE system contains six sources, including an As$_{2}$ valved cracker. It is also equipped with a 10 keV electron gun for reflection high-energy electron diffraction (RHEED). The samples were approximately 10$\times$10 mm$^{2}$ pieces of epi-ready $n$-type GaAs(100) wafers, which were In-glued on tranferrable Mo holders. Each sample preparation started with a 1000 [Å]{} buffer, grown at a substrate temperature ($T_{s}$) of 590 $^{\circ}$C, $T_{s}$ was then lowered to the growth temperature of LT-GaAs and GaMnAs, which was typically 220 $^{\circ}$C. At the low temperature the growth started always with a 200–300 [Å]{} LT-GaAs buffer layer. The As$_{2}$/Ga flux ratio was maintained at values around 10. During deposition of this layer the LT-GaAs growth rate was measured by recording RHEED intensity oscillations. After opening the Mn shutter the RHEED oscillations were observed again during the GaMnAs growth.[@Sadowski00] At this low growth temperature the reevaporation of Mn and Ga from the surface is negligible, so the growth rate increase is proportional to the Mn content. The Mn-concentrations quoted below are estimated to be accurate within 0.5%.
Immediately after transfer the surfaces were checked with low energy electron diffraction (LEED). All Mn-containing samples exhibited 1$\times$2 reconstructed surfaces in RHEED as well in LEED, while the clean reference GaAs sample displayed a $c$(4$\times$4) LEED pattern with sharp integer order and less distinct fractional order spots. Photoemission was excited with mainly $p$-polarized light incident at 45$^{\circ}$ relative to the surface normal, the samples being oriented with the \[110\] azimuth (i.e. the 1-fold periodicity) in the plane of incidence. The electron energy distribution curves were obtained using a hemispherical electron energy analyzer with an angular resolution of 2$^{\circ}$, and the overall energy resolution was around 0.3 eV. A clean Ta foil, in contact with the sample holder, was used to determine the Fermi level position in each case. The counting rates were normalized to the incident beam intensity by means of photocurrent from a gold mesh in the beam path.
Results and discussion
======================
Considering the intrinsic surface sensitivity of photoemission, and that surface compositions in alloy systems often deviate from those in the bulk, it is well motivated to start with a brief comment on this point. The fact that clear and actually unusually persistent RHEED oscillations are observed during growth of GaMnAs, shows that the atoms are still mobile in the surface layer despite the low-temperature conditions. However, once accommodated in lattice sites, further mobility that would lead to phase separation is efficiently inhibited under the low-temperature growth conditions. Thus, it is well motivated to expect that the sample compositions are uniform, including the first atomic layers. We should mention, however, that applying secondary ion mass spectroscopy (SIMS) as well as Auger microprobe analysis on samples exposed to atmospheric pressure, we have observed pronounced enrichment (by a factor of 2) of Mn in the surface layer (and a corresponding depletion in the underlying region), which is clearly associated with oxidation. Typically these redistributions range over a thickness of around 150 [Å]{}. This clearly emphasises the significance of carrying out surface sensitive experiments on [*in situ*]{} prepared samples. Photoemission from Mn 3$d$ states in dilute systems like Ga$_{1-x}$Mn$_{x}$As is easily identified via resonant enhancement of the 3$d$ cross section at the 4$p$ excitation threshold, which occurs at 50 eV photon energy. Since the spectral shape changes quite much in this energy range, and since our aim is to compare spectra recorded from a series of different samples, we have chosen to use a photon energy well above this resonant range (81 eV). Although the absolute cross section of Mn 3$d$ is smaller at 81 eV than that just above 50 eV photon energy, the cross sections of the GaAs valence states are also reduced in a similar way and therefore the Mn 3$d$-induced spectral features are still readily detected. Fig. \[Mn\]a shows a set of such valence band spectra from samples with different Mn contents, together with a reference spectrum from clean LT-GaAs. It is worth pointing out that just like the GaMnAs, such LT-GaAs contains large concentrations of point defects (mostly As antisites), and that spectra from such layers are found to be somewhat different relative those obtained from MBE layers grown at high temperature.[@Aasklund01] Considering that until now only one independent valence-band photoemission study of such materials has been published,[@Okabayashi99] and that the samples are produced under rather extreme growth conditions, it is well motivated to start the discussion with a direct comparison between the present data and the published ones. We note then that the data contain some similarities, but also some significant differences. The main Mn-induced feature is the peak at 3.4 eV below the VBM for the most Mn-rich samples. Its Mn origin is clearly revealed by the resonant enhancement mentioned above. A similar resonant structure was found in Ref. \[RefOkabayashi99\], but at a binding energy of 4.5 eV relative to the Fermi level. Since the Fermi level is located about 0.13 eV above the VBM (see below), there seems to be a discrepancy of almost 1 eV between the two results. Furthermore, the spectra in Ref. \[RefOkabayashi99\] contain a second pronounced peak at about 2.5 eV larger binding energy. This structure is completely missing in our data. The weak asymmetric peak seen in all spectra around 6.5–7 eV in Fig. \[Mn\]a reflects the $X_{3}$ critical point emission. Such density of states (DOS) structures are seen at all photon energies and all emission angles due to diffuse elastic scattering of the direct interband excitation of this state. Altogether we thus find that the present spectra are significantly different from those found in literature, and although the reason for these deviations is not clear, it is natural at this point to suspect that the different surface preparations could be the cause. This would then underline the importance of carrying out these experiments on [*in situ*]{} grown samples.
It is immediately clear in Fig. \[Mn\] that the Mn-induced spectral changes vary with the Mn content. To examine this variation in some more detail, we have generated consecutive difference spectra, as displayed in Fig. \[Mn\]b. The first spectrum in this sequence shows that with 1% Mn the spectral intensity is increased over a range 1–4 eV below the VBM, with a peak centered around 3 eV and a shoulder at 1 eV below the VBM. The main increase coincides with weak structures in the clean GaAs spectrum (around 3 eV and 4 eV below the VBM). As these structures are due to excitations at the high DOS regions at the $X_{5}$ and $\Sigma^{min}_{1}$ points, one could suspect that the Mn-induced changes are in this case caused by disorder-related increase of diffuse scattering. However, from the fact that no corresponding increase is seen for the $X_{3}$ critical point emission, and from the following development of the 3-eV peak, we can safely conclude that these spectral changes do indeed reflect Mn-derived states. With the Mn content raised to 3% we see that the incremental change is somewhat different than the initial one. The peak at 3 eV is increased further, but the range around 1 eV remains essentially unchanged. Increasing the Mn content further results in another change: the main additional spectral contribution appears as an asymmetric peak around 3.8 eV below the VBM, i.e. clearly shifted relative to that found at lower concentrations. Thus, the peak observed at 3.4 eV in the corresponding full spectrum (Fig. \[Mn\]) can be concluded to represent an average of several contributions. Finally, the additional spectral changes with further increase of Mn content are found to be less distinct, the intensity is increased rather uniformly over a range 2–6 eV below the VBM. The next difference spectrum is also essentially a peak centered at 3 eV below the VBM, though it is clearly narrower on the high-energy side.
The important conclusion from the data in Fig. \[Mn\] is that the character of the Mn states in Ga$_{1-x}$Mn$_{x}$As depends on the Mn concentration. Since supplementary X-ray diffraction analysis of our samples shows high degree of perfection in the layers, we have no reason to suspect that the variations seen here are due to varying sample structure quality, but ascribe them to the different Mn contents. No such dependence has been reported in any of the earlier studies. Previous analysis of Mn 3$d$ partial DOS in GaMnAs with 6.9% Mn was successful in modelling the observed spectrum using a configuration interaction model involving Mn 3$d$ and ligand states in a MnAs4 cluster.[@Okabayashi99] Obviously, this kind of model can not account for concentration-dependent properties like those reported here. With an average distance between two impurities of around 25 [Å]{}, it is clear that the explanation must be based on a model in which long range interactions are taken into account.
GaMnAs is also known to exhibit unusual conductivity characteristics:[@Oiwa97] at low Mn concentrations the system is semiconducting, around 4–5% Mn metallic conductivity is reported and with further increase of the Mn content the material becomes again insulating. Interestingly enough, the Curie temperature also exhibits a maximum around the same Mn concentration. These two observations suggest that the density of holes is actually decreasing with Mn concentrations above 5%, and this might be directly reflected by the Fermi level position relative the VBM. In Fig. \[VB\] we show a set of valence-band spectra from samples with varying Mn contents, aligned at the Fermi level. The photon energy used in this case was 38.5 eV, chosen to probe the phase space region around the $X_{3}$ point. This emission is reflected by the prominent peak around 7.5 eV. Considering the high density of defects in LT-GaAs and in GaMnAs, (in the range of 10$^{20}$/cm$^{2}$), it is reasonable to assume that the surface Fermi level does not deviate from that in the bulk. This assumption is supported by the fact that no additional spectral broadening that could be expected due to a emission from a very narrow band bending region was detected in any spectra. Focusing on the $X_{3}$ emission we see that its position is changing with Mn content. This variation is shown more clearly in Fig. \[EF\], where we have plotted the energy separation $E_{\text{F}}-X_{3}$ for a larger set of samples. Starting at a value of 7.35 eV for clean LT-GaAs it is reduced, and settles at a value of 7.1 eV around 1.5% Mn concentration. This pinning position remains stable for Mn concentrations up to around 3.5%, and is likely due to the Mn acceptor level known to be located 113 meV above the VBM.[@Schneider87] With this interpretation we deduce the VBM to be located around 6.95 eV above the $X_{3}$ point, a value well in the range of literature data[@Chiang80] (6.70–7.1 eV) based on angle resolved photoemission and X-ray photoemission. A very interesting feature is observed around 4–5% Mn concentrations, where $E_{\text{F}}$ appears to drop to a position close to the VBM. As already mentioned, samples in this concentration range are reported to exhibit metallic conductivity. The present observations are fully consistent with such behaviour. We also note that the low position of $E_{\text{F}}$ implies an increased density of holes, which in turn may be the explanation for the relatively high Curie temperatures found in this range of Mn concentrations. The most intriguing observation is the shift of $E_{\text{F}}$ back into the band gap region with further increased Mn content. This is consistent with the reported metal-insulator transition,[@Oiwa97] though the present results suggest that the reason for the insulating properties is not impurity scattering, but rather a true reduction of charge carriers.
Conclusions
===========
The present investigations of valence band photoemission from Ga$_{1-x}$Mn$_{x}$As compounds show two new effects. Firstly we find that the spectral changes induced by the Mn atoms depend on the Mn concentration, and secondly we observe that the position of the Fermi level also changes with Mn content. None of these features has been reported previously. The varying shape of the Mn-induced valence-band structures directly shows that the Mn-host interaction cannot be treated with a local model. As to the Fermi level variations, we note that the minimum observed around 3.5–5% Mn content coincides with previously reported metallic conductivity and also with the range of maximum paramagnetic-ferromagnetic transition temperatures.
Acknowledgements {#acknowledgements .unnumbered}
================
We are pleased to acknowledge the technical support of the MAX-lab staff. This work was supportet by grants from the Swedish Natural Science Research Council (NFR), the Swedish Research Council for Engineering Sciences (TFR), and, via co-operation with the Nanometer Structure Consortium in Lund, the Swedish Foundation for Strategic Research (SSF).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space ${\mathscr{W}}^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.'
address:
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA and Department of Mathematics and Statistics, University of Helsinki, Finland'
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA'
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA'
author:
- Tadeusz Iwaniec
- 'Leonid V. Kovalev'
- Jani Onninen
date: 'June 26, 2010'
title: |
Hopf differentials and\
smoothing Sobolev homeomorphisms
---
[^1]
[^2]
[^3]
Introduction
============
It is a fundamental property of Sobolev spaces ${\mathscr{W}}^{1,p}$, $1\le p<\infty$, that any element can be approximated strongly (i.e., in the norm) by $\mathscr C^{\infty}$ smooth functions, or by piecewise affine ones. In the context of vector-valued Sobolev functions, that is, mappings in ${\mathscr{W}}^{1,p}(\Omega,{\mathbb{R}}^n)$, invertibility comes into play. Indeed, the studies of invertible Sobolev mappings are of great importance in nonlinear elasticity [@Ba0; @FG; @MST; @Sv]. The following natural question was put forward by John M. Ball.
[@Ba] If $u \in {\mathscr{W}}^{1,p}(\Omega, {\mathbb{R}}^n)$ is invertible, can $u$ be approximated in $ {\mathscr{W}}^{1,p}$ by piecewise affine invertible mappings?
J. Ball attributes this question to L.C. Evans, who was led to it through his investigation of the partial regularity of minimizers [@Ev] of neohookean energy functionals [@Ba1; @BPO1; @CL; @SiSp]. We provide an affirmative solution of the Ball-Evans problem in the case $p=n=2$. The most general formulation of our result administers Royden algebras ${\mathscr{A}}(\Omega)$ and ${\mathscr{A}}_\circ (\Omega)$, see Section \[secroy\]. We write $${\mathcal{E}}[h]={\mathcal{E}}_\Omega[h]:= {\lVertDh\rVert}^2_{\mathscr L^2(\Omega)}= \int_\Omega {\lvertDh(z)\rvert}^2\, {\textnormal d}z$$ where ${\lvertDh\rvert}$ is the Hilbert-Schmidt norm of the differential.
\[thmapprox\] Let $h \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ be a homeomorphism of Sobolev class $\mathscr W^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$. Then for every $\epsilon >0$ there exist a diffeomorphism $H \colon \Omega{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ such that
(i) $H-h \in {\mathscr{A}}_\circ (\Omega)$
(ii) $ {\lVertH-h\rVert}_{{\mathscr{A}}(\Omega)}\le \epsilon$
(iii) ${\mathcal{E}}[H] \le {\mathcal{E}}[h]$.
Part (iii) is nontirivial only in the finite energy case, ${\mathcal{E}}_\Omega [h]< \infty$. Let us note that the existence of smooth approximation implies the existence of piecewise-affine approximation, since a diffeomorphism can be triangulated. (In the converse direction, a piecewise-affine mapping can be smoothed in dimensions less than four [@Mu], but not in general.) Partial results toward the Ball-Evans problem were obtained in [@Mo] (for planar bi-Sobolev mappings that are smooth outside of a finite set) and in [@BM] (for planar bi-Hölder mappings, with approximation in the Hölder norm). The articles [@Ba; @SS] illustrate the difficulty of preserving invertibility in the process of smoothing a Sobolev homeomorphism.
We also give an application of Theorem \[thmapprox\] to a problem that originated in a series of papers by Eells, Lemaire and Sealey [@EL2; @EL3; @Se]. It concerns the nonlinear differential equation $$\label{heq2}
\frac{\partial}{ \partial \bar z} \left( h_z \overline{h_{\bar z}} \right)=0$$ for mappings defined in a domain in the complex plane ${\mathbb{C}}$. Naturally, the Sobolev space ${\mathscr{W}}^{1,2}_{\operatorname{loc}}(\Omega, {\mathbb{C}})$ should be considered as the domain of definition of equation . This places $ h_z \overline{h_{\bar z}}$ in $\mathscr L^1_{\operatorname{loc}} (\Omega)$, so the complex Cauchy-Riemann derivative $\frac{\partial}{\partial \bar z}$ applies in the sense of distribution. By Weyl’s lemma $h_z \overline{h_{\bar z}}$ is a holomorphic function.
The expression $Q_h:=h_z \overline{h_{\bar z}}\, {\textnormal d}z \otimes {\textnormal d}z$ is known as the Hopf differential of $h$ (named after H. Hopf, who employed a similar device, e.g., in [@Ho Chapter VI]). It is clear that $Q_h$ is a holomorphic quadratic differential whenever $h$ is harmonic, which is a general fact about energy-stationary mappings between Riemannian manifolds [@EL1 (10.5)], [@Job] and [@Stb]. Eells and Lemaire inquired about the possibility of a converse result, e.g., for mappings with finite energy and almost-everywhere positive Jacobian [@EL2 (2.6)]. In this setting a counterexample was provided by Jost [@Jo], who also proved the existence of ${\mathscr{W}}^{1,2}$-solutions of in every homotopy class of mappings between compact Riemann surfaces. A more restricted form of the Eells-Lemaire problem, [@EL3 (5.11)] and [@Se], imposed the additional assumption that $h$ is a quasiconformal homeomorphism, and was settled by Hélein [@He] in the affirmative. Here we dispose with the quasiconformality condition and treat general planar homeomorphisms of finite energy. Since the inverse of such a homeomorphism need not be in any Sobolev class [@HK], some difficulties are to be expected. They shall be overcome with the aid of our approximation theorem \[thmapprox\].
\[thmhopf\] Every continuous, discrete and open mapping $h$ of Sobolev class ${\mathscr{W}}^{1,2}_{\operatorname{loc}}(\Omega, {\mathbb{C}})$ that satisfies equation is harmonic.
The failure of Theorem \[thmhopf\] for uniform limits of homeomorphisms should be mentioned. This is illustrated by Example \[ex\].
Background {#secroy}
==========
Let $\Omega$ be a bounded domain in ${\mathbb{R}}^2 \simeq {\mathbb{C}}$, nonempty open connected set. We consider a class ${\mathscr{A}}(\Omega)$ of uniformly continuous functions $h \colon {\Omega} \to {\mathbb{C}}$ having finite Dirichlet energy, and furnish it with the norm $$\| h\|_{{\mathscr{A}}(\Omega)} = \| h \|_{\mathscr C ({\Omega})} + \| Dh \|_{\mathscr L^2 (\Omega)} < \infty$$ ${\mathscr{A}}(\Omega)$ is a commutative Banach algebra with the usual multiplication of functions in which ${\lVerth_1h_2\rVert}_{{\mathscr{A}}(\Omega)} \le {\lVerth_1\rVert}_{{\mathscr{A}}(\Omega)} {\lVerth_2\rVert}_{{\mathscr{A}}(\Omega)}$. The closure of $\mathscr C^\infty_\circ (\Omega)$ in ${\mathscr{A}}(\Omega)$ will be denoted by ${\mathscr{A}}_\circ (\Omega)$. Suppose, to look at more specific situation, that $\Omega = {\mathbb{U}}$ is a Jordan domain; that is, a simply connected open set whose boundary $\Gamma= \partial {\mathbb{U}}$ is a closed Jordan curve. By goodness of the Carathéodory extension theorem [@Pob p. 18], there is a homeomorphism $\varphi \colon \overline{\mathbb D} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{U}}}$ of the closed unit disk $\overline{{\mathbb{D}}}= \{ \xi \colon {\lvert\xi\rvert} \le 1 \}$ that is conformal in ${\mathbb{D}}$. After the change of variable, $z= \varphi (\xi)$, we obtain a function $H(\xi)= h\big(\varphi (\xi)\big)$ in ${\mathscr{A}}({\mathbb{D}})$. The operation $$\mathbf{T}_\varphi \colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}({\mathbb{D}})$$ so defined is an isometry; ${\lVert\mathbf{T}_\varphi h\rVert}_{{\mathscr{A}}({\mathbb{D}})} = {\lVerth\rVert}_{{\mathscr{A}}({\mathbb{U}})}$. Furthermore, $$\mathbf{T}_\varphi \colon {\mathscr{A}}_\circ ({\mathbb{U}}) \to {\mathscr{A}}_\circ ({\mathbb{D}})$$
\[prpoisson\] Let ${\mathbb{U}}$ be a Jordan domain. There is (unique) bounded linear operator $$\mathbf{P}_{{\mathbb{U}}} \colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}({\mathbb{U}})$$ such that $$\begin{cases} \mathbf{P}_{{\mathbb{U}}} - \mathbf Id\colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}_\circ ({\mathbb{U}})\\
\Delta \circ \mathbf P_{{\mathbb{U}}}=0
\end{cases}$$
We name $\mathbf P_{{\mathbb{U}}}$ the [*Poisson operator*]{}. The energy of $\mathbf P_{{\mathbb{U}}} h$ does not exceed that of $h$. This fact is known as [*Dirichlet’s principle*]{} $$\int_{{\mathbb{U}}} {\lvertD \mathbf P_{{\mathbb{U}}} h\rvert}^2 \le \int_{{\mathbb{U}}} {\lvertD h\rvert}^2$$ The proof of this proposition reduces to the case when ${\mathbb{U}}={\mathbb{D}}$, by conformal change of variables. A routine verification of this case is left to the reader. We only indicate that the less familiar property $\mathbf{P}_{{\mathbb{D}}} h -h \in {\mathscr{A}}_\circ ({\mathbb{D}})$, for $h\in {\mathscr{A}}({\mathbb{D}})$, needs to be justified.
Let $\Omega$ be a domain in ${\mathbb{C}}$ and ${\mathbb{U}}\subset \overline{{\mathbb{U}}} \subset \Omega$ a Jordan domain. There exists (unique) bounded linear operator $$\mathbf R_{{\mathbb{U}}} \colon {\mathscr{A}}(\Omega) \to {\mathscr{A}}(\Omega)$$ such that, for every $h\in {\mathscr{A}}(\Omega)$ $$\begin{cases} \mathbf R_{{\mathbb{U}}}h=h \qquad \mbox{ on }\quad \Omega \setminus {\mathbb{U}}\\
\Delta \mathbf R_{{\mathbb{U}}}h= 0 \qquad \mbox{ in } \quad {\mathbb{U}}\end{cases}$$ The Laplace equation yields ${\mathcal{E}}_{\Omega} [\mathbf R_{{\mathbb{U}}} h] \le {\mathcal{E}}_{\Omega} [h]$. Equality occurs if and only if $h$ is harmonic in ${\mathbb{U}}$.
A short proof of this corollary runs somewhat as follows. The unique harmonic extension of $h \colon \partial {\mathbb{U}}\to {\mathbb{C}}$ inside ${\mathbb{U}}$ given by $\mathbf P_{{\mathbb{U}}} h$ has the property that $\mathbf P_{{\mathbb{U}}} h-h \in {\mathscr{A}}_\circ ({\mathbb{U}})$. Therefore, the zero extension of $\mathbf P_{{\mathbb{U}}} h -h$ outside ${\mathbb{U}}$, denoted by $[\mathbf P_{{\mathbb{U}}}h-h]_\circ$, belongs to ${\mathscr{A}}(\Omega)$. We define $$\mathbf{R}_{{\mathbb{U}}} h:= [\mathbf P_{{\mathbb{U}}} h -h]_\circ +h \in {\mathscr{A}}(\Omega)$$ The desired properties of the operator $\mathbf R_{{\mathbb{U}}}$ so defined are automatically fulfilled in view of Proposition \[prpoisson\].
\[RKC\] Let $\Omega$ be a domain in ${\mathbb{C}}$ and ${\mathbb{U}}\subset \overline{{\mathbb{U}}} \subset \Omega$ a Jordan domain. Suppose that $h\in \mathscr {\mathscr{A}}(\Omega)$ is a homeomorphism of $\Omega$ onto $h(\Omega)$ and $h({\mathbb{U}})$ is convex. Then $\mathbf R_{\mathbb{U}}h$ is homeomorphism in $\Omega$ and is a harmonic diffeomorphism in ${\mathbb{U}}$.
The injectivity of $\mathbf R_{\mathbb{U}}h$ is the content of the Radó-Kneser-Choquet Theorem [@Dub p. 29]. Furthermore, planar harmonic homeomorphisms are $\mathscr C^\infty$-smooth diffeomorphisms according to Lewy’s theorem [@Dub p. 20].
Smoothing Sobolev homeomorphisms, Theorem \[thmapprox\]
=======================================================
We may and do assume that $h$ is not harmonic, since otherwise $H =h$ satisfies the desired properties by Lewy’s theorem (mentioned above). Let $z_\circ \in \Omega$ be a point such that $h$ fails to be harmonic in any neighborhood of $z_\circ$. By choosing the origin of the coordinate system we ensure that $h(z_\circ)$ does not lie on the boundary of any dyadic squares associated with the coordinate system.
Let us choose and fix any $\epsilon >0$. The construction of $H$ proceeds in 5 steps. We construct homeomorphisms $h_k \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$, $k=0, \dots, 5$ such that $h_0=h$, $h_k\in h_{k-1}+{\mathscr{A}}_\circ(\Omega)$, $h_5$ is a diffeomorphism, and ${\lVerth_k-h_{k-1}\rVert}$ is bounded by a multiple of $\epsilon$ for each $k$. In each step we modify the previous construction to gain better regularity. In steps 1, 2 and 4 we use harmonic replacement according to Proposition \[RKC\]. In steps 3 and 5 we smoothen the mapping near the boundaries of the domains in which harmonic replacement was performed. The result of each step is denoted by $h_1,\dots,h_5$. The $h\in {\mathscr{A}}(\Omega)$ requires a few additional details, which are provided at the end of each step.
We begin with a decomposition of the target domain $$\Omega^\ast = \bigcup_{\nu=1}^\infty \overline{{\mathbb{Q}}_{\nu}}$$ into closed nonoverlapping dyadic squares $\overline{{\mathbb{Q}}_{\nu}} \subset \Omega^\ast$. This decomposition is made by selecting the maximal dyadic squares that lie in $\Omega^\ast$. Thus the cover of $\Omega^\ast$ by such squares is locally finite. The preimage of ${\mathbb{Q}}_\nu$ under $h$, denoted by ${\mathbb{U}}_\nu$, is a Jordan domain in $\Omega$. Hereafter ${\mathbb{U}}_\nu$ will be referred to as the curved-square. In fact to every partion of $\Omega^\ast$ into closed squares there will correspond a partition of $\Omega$ into closed curved-squares via the mapping $h \colon \Omega \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \Omega^\ast$, for example: $$\Omega = \bigcup_{\nu=1}^\infty \overline{{\mathbb{U}}_{\nu}}$$ [**Step 1.**]{} For each ${\mathbb{U}}_\nu$ we replace $h \colon \overline{{\mathbb{U}}_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}}$ with a piecewise harmonic homeomorphism $h_1 \colon \overline{{\mathbb{U}}_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}} $ that coincides with $h$ on $\partial \overline{{\mathbb{U}}_{\nu}}$. To this effect we partition the square $ \overline{{\mathbb{Q}}_{\nu}} $, $$\overline{{\mathbb{Q}}_{\nu}} = \overline{{\mathbb{Q}}^1_{\nu}} \cup \overline{{\mathbb{Q}}^2_{\nu}} \cup \dots \cup \overline{{\mathbb{Q}}^n_{\nu}}, \qquad (n=n_\nu=4^{k_\nu})$$ into congruent dyadic squares $ \overline{{\mathbb{Q}}^i_{\nu}}$, $i=1, \dots, n$. The number $n$, depending on $\nu$, will be determined later. For the moment fix $\nu$ and look at the homeomorphisms $$h \colon \overline{{\mathbb{U}}^i_\nu} = h^{-1}(\overline{{\mathbb{Q}}^i_{\nu}}) \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}^i_{\nu}}$$ These mappings belong to the Royden algebra ${\mathscr{A}}( \overline{{\mathbb{U}}^i_\nu})$. With the aid of Propositions \[prpoisson\] and \[RKC\] we replace each $h \colon \overline{{\mathbb{U}}^i_\nu} \to \overline{{\mathbb{Q}}^i_\nu}$ with a harmonic homeomorphism $h^i_\nu \colon \overline{{\mathbb{U}}^i_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}^i_\nu}$ that coincides with $h$ on $\partial \overline{{\mathbb{U}}^i_\nu}$, $i=1,2, \dots , n$. Such mappings are $\mathscr C^\infty$-smooth diffeomorphisms $h^i_\nu \colon {{\mathbb{U}}^i_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} {{\mathbb{Q}}^i_\nu}$. Moreover, $h^i_\nu -h \in {\mathscr{A}}_\circ ({\mathbb{U}}_\nu^i)$ and $$\begin{cases}
{\mathcal{E}}_{{\mathbb{U}}^i_\nu}[h_\nu^i] \le {\mathcal{E}}_{{\mathbb{U}}^i_\nu}[h] \qquad \mbox{for } 1,2, \dots , n \\
{\mathcal{E}}_{\partial {\mathbb{U}}^i_\nu}[h_\nu^i] = {\mathcal{E}}_{\partial {\mathbb{U}}^i_\nu}[h], \quad \mbox{ because } h_\nu^i=h \mbox{ on } \partial {\mathbb{U}}_\nu^i
\end{cases}$$ We obtain a piecewise harmonic homeomorphism by gluing $h^i_\nu$ together along the common boundaries of ${\mathbb{U}}^i_\nu$. Denote it by $$h^n_{\nu} \colon \overline{{\mathbb{U}}_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}}$$ $$h^n_{\nu} \in h + {\mathscr{A}}_\circ ({\mathbb{U}}_\nu)$$ Precisely we define $$h_\nu^n=h+ \sum_{i=1}^n [h_\nu^i-h]_\circ$$ Here and in the sequel the notation $[\varphi]_\circ$ for $\varphi\in {\mathscr{A}}_\circ({\mathbb{U}})$ stands for zero extension of $\varphi$ to the entire domain $\Omega$. Obviously $[\varphi]_\circ \in {\mathscr{A}}_\circ(\Omega)$. The above construction depends on the number $n$. For $\nu$ fixed we actually have a sequence $\{h^n_{\nu}\}_{n=1,2, \dots}$ that is bounded in ${\mathscr{A}}({\mathbb{U}}_\nu)$. However, we have uniform bounds independent of $n$, $${\lVerth^n_{\nu}\rVert}_{\mathscr C(\overline{{\mathbb{U}}_\nu})} \le \operatorname{diam}{\mathbb{Q}}_\nu$$ and $${\mathcal{E}}_{{\mathbb{U}}_\nu} [h^n_{\nu}] \le {\mathcal{E}}_{{\mathbb{U}}_\nu} [h]$$ The key observation is that $$\begin{cases}
h^n_{\nu}-h \in {\mathscr{A}}_\circ ({\mathbb{U}}_\nu)\\
\lim\limits_{n \to \infty} {\lVerth^n_{\nu}-h\rVert}_{{\mathscr{A}}({\mathbb{U}}_\nu)}=0
\end{cases}$$ Indeed, for $z\in \overline{{\mathbb{U}}^i_\nu}$ we have $${\lverth^n_{\nu}(z)-h(z)\rvert} \le \operatorname{diam}{\mathbb{Q}}_\nu^i = \frac{1}{\sqrt{n}} \operatorname{diam}{\mathbb{Q}}_\nu$$ Thus $h^n_{\nu } \rightrightarrows h$ uniformly on $\overline{{\mathbb{U}}_\nu}$ as $n \to \infty$. On the other hand the differential matrices $Dh^n_{\nu}$ are bounded in $\mathscr L^2 ({\mathbb{U}}_\nu, {\mathbb{R}}^{2\times 2})$. Their weak limit exits and is exactly equal to $Dh$, because the mappings converge uniformly to $h$. Therefore, $$\begin{split}
\int_{{\mathbb{U}}_\nu} {\lvertDh^n_{\nu}-Dh\rvert}^2 & = \int_{{\mathbb{U}}_\nu} \Big({\lvertDh^n_{\nu}\rvert}^2 + {\lvertDh\rvert}^2-2 \langle Dh^n_{\nu}, Dh \rangle \Big)\\
& \le 2 \int_{{\mathbb{U}}_\nu} \Big({\lvertDh\rvert}^2 - \langle Dh^n_{\nu}, Dh \rangle \Big) \\
& = 2 \int_{{\mathbb{U}}_\nu} \langle Dh, Dh - Dh^n_{\nu} \rangle \longrightarrow 0
\end{split}$$ We can now determine the number $n=n_\nu$ of congruent dyadic squares in ${\mathbb{Q}}_\nu$, simply requiring that $$\begin{cases}
\operatorname{diam}{\mathbb{Q}}^i_\nu \le {\epsilon} \qquad \mbox{ for every } i=1,2, \dots, n_\nu\\
{\lVertDh^n_{\nu}-Dh\rVert}_{\mathscr L^2 (\overline{{\mathbb{U}}_\nu})} \le \epsilon \cdot 2^{-\nu}
\end{cases}$$ Fix such $n=n_\nu$ and abbreviate the notation for $h^{n_\nu}_{\nu}$ to $h^\nu$. We obtain a homeomorphism $$h_1:=h+ \sum_{\nu =1}^\infty [h^\nu-h]_\circ \in h+ {\mathscr{A}}_\circ (\Omega)$$ where we recall that $ [h^\nu-h]_\circ$ stands for the zero extension of $h^\nu-h$ to the entire domain $\Omega$. Clearly, $h_1$ is harmonic in each ${\mathbb{U}}_\nu^i$, $\nu=1,2, \dots$, $i=1,2, \dots , n_\nu$ and we have $${\lVerth_1-h\rVert}_{\mathscr C (\Omega)} \le \sup \{\operatorname{diam}{\mathbb{Q}}_\nu^i \colon \nu=1,2, \dots, \; \; i=1, \dots, n_\nu \} < {\epsilon}$$ $${\lVerth_1-h\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}+\sum_{\nu =1}^\infty {\lVertDh^\nu-Dh\rVert}_{\mathscr L^2({\mathbb{U}}_\nu)} \le {\epsilon}+ \sum_{\nu=1}^\infty \epsilon \cdot 2^{-\nu} = 2 \epsilon$$ For further considerations it will be convenient to number the squares ${\mathbb{Q}}_\nu^i$ and their preimages ${\mathbb{U}}_\nu^i$ using only one index. These sets will be respectively denoted by ${\mathbb{Q}}^\alpha$ and ${\mathbb{U}}^\alpha$, $\alpha=1,2, \dots$. For the record, $$\label{doeq}
\operatorname{diam}{\mathbb{Q}}^\alpha \le {\epsilon}, \qquad \alpha =1,2, \dots$$ Summing up the energy inequalities for the mappings $h^i_\nu \colon {\mathbb{U}}_\nu^i \to {\mathbb{Q}}_\nu^i$ we see that the total energy of $h_1$ does not exceed the energy of $h$. Even more, since $h$ was assumed to be not harmonic, there is at least one region ${\mathbb{U}}_\nu^i$ for which $h \colon {\mathbb{U}}_\nu^i \to {\mathbb{Q}}_\nu^i$ was not harmonic. Consequently, its harmonic replacement results in strictly smaller energy. Hence $$\label{36p}{\mathcal{E}}_{\Omega} [h_1] < {\mathcal{E}}_{\Omega} [h], \qquad \mbox{so let } \delta= {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - {\lVertDh_1\rVert}_{\mathscr L^2(\Omega)} >0$$
[**Step 2.**]{} Denote by $\mathcal F=\{{\mathbb{Q}}^\alpha \colon \alpha=1,2, \dots\}$ the family of all open squares ${\mathbb{Q}}^\alpha \subset \overline{{\mathbb{Q}}^\alpha} \subset \Omega^\ast$ that are build in Step 1 for the construction of the mapping $h_1 \colon \Omega \to \Omega^\ast$. Let $\mathcal V$ be the set of vertices of these squares. Whenever two squares ${\mathbb{Q}}^\alpha, {\mathbb{Q}}^\beta \in \mathcal F$, $\alpha \ne \beta$, meet along their boundaries the intersection $I^{\alpha , \beta}= \partial {\mathbb{Q}}^\alpha \cap \partial {\mathbb{Q}}^\beta$ is either a point in $\mathcal V$ or a closed interval with endpoints in $\mathcal V$. Denote by $\mathcal J \subset \{I^{\alpha , \beta} \colon \alpha\ne \beta,\; \alpha, \beta=1,2, \dots \}$ the subfamily of all such intersections, excluding empty set and vertices. For each interval $I^{\alpha , \beta} \in \mathcal J$ we shall construct a doubly convex lens-shaped region $\mathbb L^{\alpha, \beta}$ with $I^{\alpha, \beta}$ as its axis of symmetry in the following way. Let $R$ be a number greater than the length of $I^{\alpha, \beta}$ to be chosen later. There exist exactly two open disks of radius $R$ for which $I^{\alpha, \beta}$ is a chord. Let $\mathbb L_R^{\alpha, \beta}$ be their intersection. This is a symmetric doubly convex lens of curvature $\frac{1}{R}$. Thus $\mathbb L_R^{\alpha, \beta}$ is bounded by two circular arcs $\gamma^{\alpha, \beta}= {\mathbb{Q}}^\alpha \cap \partial \mathbb L_R^{\alpha, \beta}$ and $\gamma^{\beta, \alpha}={\mathbb{Q}}^\beta \cap \partial \mathbb L_R^{\alpha, \beta} $. As the curvature of the lens approaches zero the area of $\mathbb L_R^{\alpha, \beta}$ tends to $0$. This allows us to choose $R$ depending on $\alpha$ and $\beta$ so that the lenses $\mathbb L^{\alpha, \beta}=\mathbb L_R^{\alpha, \beta}$ have the following property. $$\label{shouldnumber}
\int_{\mathbb K^{\alpha, \beta}} {\lvertDh_1\rvert}^2 < \frac{\epsilon^2}{2^{\alpha+\beta}}, \quad \mbox{ where } \mathbb K^{\alpha, \beta}=h^{-1}_1(\mathbb L_R^{\alpha, \beta})$$ The lenses $\mathbb L^{\alpha, \beta}$ are disjoint because the opening angle of each lens is at most $\pi/3$ and their axes are either parallel or orthogonal. However, the closures of the lenses considered here may have a common point that lies in $\mathcal V$. On each $\mathbb K^{\alpha, \beta}$ we replace $h_1$ by the harmonic extension of its restriction to $\partial \mathbb K^{\alpha, \beta}$. Thus we obtain a homeomorphism $h_2^{\alpha, \beta} \colon \overline{\mathbb K^{\alpha, \beta}} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\mathbb \overline{L^{\alpha, \beta}}$ of class $h_1+ {\mathscr{A}}_\circ (\mathbb K^{\alpha, \beta})$. By Proposition \[RKC\] the mappings $h_2^{\alpha, \beta} \colon \mathbb K^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\mathbb L^{\alpha, \beta}$ are diffeomorphisms. Finally, we define $$h_2=h_1+\sum_{\alpha, \beta} [h_2^{\alpha, \beta}-h_1]_\circ \in h_1+ {\mathscr{A}}_\circ (\Omega) = h+ {\mathscr{A}}_\circ (\Omega)$$ and observe that, from , $${\lVerth_2-h_1\rVert}_{\mathscr C (\Omega)}\le \sup_{\alpha, \beta} \operatorname{diam}\left( \mathbb L^{\alpha, \beta} \right) \le \epsilon.$$ Also, and Dirichlet’s principle imply $$\begin{split}
\int_{\Omega} {\lvertDh_2-Dh_1\rvert}^2 &\le \sum_{\alpha, \beta} \int_{\mathbb K^{\alpha, \beta}} 2\left({\lvertDh_2\rvert}^2+{\lvertDh_1\rvert}^2\right) \le 4 \sum_{\alpha, \beta =1}^\infty \int_{\mathbb K^{\alpha, \beta}} {\lvertDh_1\rvert}^2 \\&\le 4 \sum_{\alpha, \beta =1}^\infty \frac{\epsilon^2}{2^{\alpha+\beta}} =4{\epsilon^2}
\end{split}$$ Thus $${\lVerth_2-h_1\rVert}_{{\mathscr{A}}(\Omega)} \le \epsilon + 2 \epsilon = 3 \epsilon$$
The boundary of $\mathbb K^{\alpha, \beta}$ consists of two $\mathscr C^\infty$-smooth arcs $\Gamma^{\alpha, \beta}$ and $\Gamma^{\beta, \alpha}$ which share common endpoints, called the apices of $\mathbb K^{\alpha, \beta}$. These are preimages of $\gamma^{\alpha, \beta}$ and $\gamma^{\beta, \alpha}$ under the mapping $h_1$, respectively. Outside of the apices, the homeomorphism $h_2\colon \overline{\mathbb K^{\alpha,\beta}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\overline{\mathbb L^{\alpha,\beta}}$ is $C^{\infty}$ smooth with positive Jacobian. The smoothness is a classical result of Kellogg; [*a harmonic function with $\mathscr C^\infty$-smooth values on a smooth part of the boundary is $\mathscr C^\infty$-smooth up to this part of the boundary*]{} [@GTb Theorem 6.19]. The positivity of the Jacobian on such part of the boundary follows from the convexity of its image, see [@Dub p. 116].
In conclusion, $h_2$ is locally bi-Lipschitz in $\Omega \setminus h^{-1}(\mathcal V)$. The exceptional set $h^{-1}(\mathcal V)$ is discrete because $\mathcal V$ is.\
By we have $$\label{39}
{\lVertDh_2\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh_1\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - \delta$$
[**Step 3.**]{} First we cover the set of vertices $\mathcal V$ by disks $\{\mathbb D_v \colon v \in \mathcal V\}$ centered at $v$ with radii small enough so that $$\label{ddiam}
\operatorname{diam}\mathbb D_v\le \epsilon,$$ and $\{ 3\mathbb D_v \colon v \in \mathcal V \}$ is a disjoint collection of disks in $\Omega^\ast$. Moreover, their preimages under $h_2$ must satisfy $$\label{star}
\sum_{v \in \mathcal V} \int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_2\rvert}^2 < {\epsilon}^2$$ Denote by $\tilde{\Omega}^\ast = \Omega^\ast \setminus \bigcup_{v\in \mathcal V} \overline{{\mathbb{D}}}_v$ and $\tilde{\Omega}= \Omega \setminus \bigcup_{v \in \mathcal V} h_2^{-1} (\overline{{\mathbb{D}}}_v)$. Our focus for a while will be on one of the circular sides of a lens $\mathbb L^{\alpha, \beta}$, say $$\gamma^{\alpha, \beta}= {\mathbb{Q}}^\alpha \cap \partial \mathbb L^{\alpha, \beta} \subset {\mathbb{Q}}^\alpha$$ We truncate it near the endpoints by setting $\tilde{\gamma}^{\alpha, \beta}= \tilde{\Omega} \cap \gamma^{\alpha, \beta}$. Such truncated open arcs are mutually disjoint; even more, their closures are isolated continua in $\Omega^\ast$. This means that there are disjoint neighborhoods of them. We are actually interested in a neighborhood of $\tilde{\gamma}^{\alpha, \beta}$ of the shape of a thin [*concavo-convex lens*]{} that we shall denote by $\tilde{\mathbb L}^{\alpha, \beta}$. By definition, $\tilde{\gamma}^{\alpha, \beta} \subset \tilde{\mathbb L}^{\alpha, \beta} \subset {\mathbb{Q}}^\alpha$. The construction of such lens goes as follows. Let $a$ and $b$ denote the endpoints of $\tilde{\gamma}^{\alpha, \beta}$, we assemble two circular arcs $\tilde{\gamma}_+^{\alpha, \beta}$ and $\tilde{\gamma}_-^{\alpha, \beta}$ with endpoints at $a$ and $b$ to form together with their endpoints a concavo-convex Jordan curve. This Jordan curve constitutes the boundary of a circular lens $\tilde{\mathbb L}^{\alpha, \beta}$. The term concavo-convex lens refers to the configuration in which $\tilde{\mathbb L}^{\alpha, \beta}$ lies in the concave side of the arc $\tilde{\gamma}_-^{\alpha, \beta}$ and convex side of $\tilde{\gamma}_+^{\alpha, \beta}$. It is clear that such lenses can be made arbitrarily thin so that $\tilde{\mathbb L}^{\alpha, \beta} \subset \tilde{\Omega}^\ast$ and the closures of $\tilde{\mathbb L}^{\alpha, \beta}$ will still be isolated continua in $\Omega^\ast$. From now on we fix the family $\{\tilde{\mathbb L}^{\alpha, \beta} \colon \alpha \ne \beta \}$ of such concavo-convex lenses associated with the arcs $\tilde{\gamma}^{\alpha, \beta}$. We then look at their preimages ${\mathbb{U}}^{\alpha, \beta} = h_2^{-1} ( \tilde{\mathbb L}^{\alpha, \beta})$ and the $\mathscr C^\infty$-smooth arcs $\Upsilon^{\alpha, \beta} = h_2^{-1} ( \tilde{\gamma}^{\alpha, \beta})$. The endpoints of $\Upsilon^{\alpha, \beta}$ lie in $\partial {\mathbb{U}}^{\alpha, \beta}$. Moreover, $\Upsilon^{\alpha, \beta}$ splits ${\mathbb{U}}^{\alpha, \beta}$ into two disjoint subdomains ${\mathbb{U}}_+^{\alpha, \beta}$ and ${\mathbb{U}}_-^{\alpha, \beta}$ such that ${\mathbb{U}}^{\alpha, \beta} \setminus \Upsilon^{\alpha, \beta} = {\mathbb{U}}_+^{\alpha, \beta} \cup {\mathbb{U}}_-^{\alpha, \beta}$. Here we have a homeomorphism $h_2\colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta} $ which is $\mathscr C^\infty$-diffeomorphism on $\overline{{\mathbb{U}}}_+^{\alpha, \beta}$ and $\mathscr C^\infty$-diffeomorphism on $\overline{{\mathbb{U}}}_-^{\alpha, \beta}$. Therefore, for some positive number $M_{\alpha, \beta}$, we have pointwise inequlities ${\lvertDh_2\rvert} \le {M_{\alpha, \beta}}$ and $\det Dh_2 \ge \frac{1}{M_{\alpha, \beta}}$ in both $\overline{{\mathbb{U}}}_+^{\alpha, \beta}$ and $\overline{{\mathbb{U}}}_-^{\alpha, \beta}$. Having established such a deformation of lenses and their preimages under $h_2$, we apply Corollary \[muncor\]. We infer that there is also a constant $M'_{\alpha, \beta}>0$ with the following property: to every neighborhood of $\Upsilon^{\alpha, \beta}$, say an open connected set ${\mathbb{U}}_\circ^{\alpha, \beta}\subset {\mathbb{U}}^{\alpha, \beta}$ that contains $\Upsilon^{\alpha, \beta}$, there corresponds a $\mathscr C^\infty$-diffeomorphism, denoted by $h_3 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$, such that $$\begin{cases}
h_3(z)=h_2(z) \quad \mbox{ for } z\in {\mathbb{U}}^{\alpha, \beta}\setminus {{\mathbb{U}}}_\circ^{\alpha, \beta}\\
{\lvertDh_3\rvert} \le {M'_{\alpha, \beta}}\quad \mbox{ and } \quad \det Dh_3 \ge \frac{1}{M'_{\alpha, \beta}} \quad \mbox{in } {\mathbb{U}}^{\alpha, \beta}
\end{cases}$$ We emphasize that $M'_{\alpha, \beta}$ is independent of the neighborhood ${{\mathbb{U}}}_\circ^{\alpha, \beta}$. We choose and fix ${{\mathbb{U}}}_\circ^{\alpha, \beta}$ thin enough to satisfy
- $\overline{{\mathbb{U}}}_\circ^{\alpha, \beta} \subset {{\mathbb{U}}}^{\alpha, \beta} \cup \overline{\Upsilon}^{\alpha, \beta}$
- ${\lvert{{\mathbb{U}}}_\circ^{\alpha, \beta}\rvert} \le [M_{\alpha, \beta}+M'_{\alpha, \beta}]^{-2} \epsilon^2\, 2^{-\alpha - \beta}$
- $\sup_{{\mathbb{U}}^{\alpha, \beta}_\circ} {\lvert{Dh_2}\rvert} \le M_{\alpha, \beta}$
- , we also assume that ${\lvert{{\mathbb{U}}}_\circ^{\alpha, \beta}\rvert} \le [M'_{\alpha, \beta}]^{-2}\delta^2\, 4^{-\alpha - \beta-1}$
Recall that $\delta$ was defined by and later appeared in . This is certainly possible; for instance, take ${{\mathbb{U}}}_\circ^{\alpha, \beta}$ to be the preimage under $h_2$ of a sufficiently thin concavo-convex lens containing $\tilde{\gamma}^{\alpha, \beta}$. We call $h_3 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$ a smoothing of $h_2 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$ associated with a given arc $\Upsilon^{\alpha, \beta}= h_2^{-1}(\tilde{\gamma}^{\alpha, \beta})$. We now define a homeomorphism $h_3\colon \Omega{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^*$ by the rule $$h_3=\begin{cases} \text{smoothing of $h_2$} \quad &\text{in ${\mathbb{U}}^{\alpha,\beta}$} \\
h_2 & \text{in $\Omega\setminus \bigcup_{\alpha,\beta}{\mathbb{U}}^{\alpha,\beta}$}
\end{cases}$$ It belongs to $h_2+{\mathscr{A}}_\circ (\Omega)$. Obviously $h_3$ is a $\mathscr C^{\infty}$-diffeomorphism in $\widetilde{\Omega}$. We have for every $z\in\Omega$ $$\begin{split}
{\lverth_3(z)-h_2(z)\rvert}&\le \begin{cases} \operatorname{diam}\widetilde{\mathbb L}^{\alpha,\beta} \quad &\text{for } z\in{\mathbb{U}}^{\alpha,\beta} \\
0 & \text{otherwise}
\end{cases}
\\ &\le \operatorname{diam}{\mathbb{Q}}^\alpha \le {\epsilon}
\end{split}$$ see . Hence ${\lVerth_3-h_2\rVert}_{\mathscr C({\Omega})}\le{\epsilon}$. As regards the energy of $h_3-h_2$ we find that $$\label{step3chain}
\begin{split}
{\mathcal{E}}_\Omega[h_3-h_2] &= \sum_{\alpha,\beta} \int_{{{\mathbb{U}}}_\circ^{\alpha,\beta}}{\lvertDh_3-Dh_2\rvert}^2
\\
&\le \sum_{\alpha,\beta} {\lvert{{\mathbb{U}}}_\circ^{\alpha,\beta}\rvert}\sup_{{{\mathbb{U}}}_\circ^{\alpha,\beta}}
\left({\lvertDh_3\rvert}+{\lvertDh_2\rvert}\right)^2
\\
& \le \sum_{\alpha,\beta} {\lvert{{\mathbb{U}}}_\circ^{\alpha,\beta}\rvert} [M_{\alpha,\beta}'+M_{\alpha, \beta}]^2\le \sum_{\alpha, \beta =1}^\infty \frac{\epsilon^2}{2^{\alpha+ \beta}} \le {\epsilon^2}
\end{split}$$ These estimates sum up to $${\lVerth_3-h_2\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}+ {\epsilon}= 2 \epsilon$$ Let us record for subsequent use the following estimate, obtained from and . $$\begin{aligned}
\label{mes}
\sum_{v \in \mathcal V} \int_{h_3^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_3\rvert}^2 &\le \sum_{v \in \mathcal V}\left( \int_{h_3^{-1}(3 {\mathbb{D}}_\nu) \setminus h_2^{-1}(3 {\mathbb{D}}_\nu) } {\lvertDh_3\rvert}^2 +\int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_3\rvert}^2 \right) \nonumber \\
\le \int_{\{h_3\ne h_2\}} {\lvertDh_3\rvert}^2 &+ 2\sum_{v \in \mathcal V} \left(\int_{h_2^{-1}(3{\mathbb{D}}_v)} {\lvertDh_3-Dh_2\rvert}^2 + \int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_2\rvert}^2 \right) \nonumber\\
&\le \sum_{\alpha, \beta} \left(M'_{\alpha, \beta} \right)^2 {\lvert{\mathbb{U}}^{\alpha, \beta}_\circ\rvert} + 2\epsilon^2 + 2 \epsilon^2 \le 5 \epsilon^2\end{aligned}$$ For the energy of $h_3$, we observe that $$\begin{split}
{\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} & \le {\lVertDh_3\rVert}_{\mathscr L^2(\Omega\setminus \cup {\mathbb{U}}_\circ^{\alpha, \beta})} + \sum_{\alpha, \beta} {\lVertDh_3\rVert}_{\mathscr L^2({\mathbb{U}}_\circ^{\alpha, \beta})} \\
&= {\lVertDh_2\rVert}_{\mathscr L^2(\Omega\setminus \cup {\mathbb{U}}_\circ^{\alpha, \beta})} + \sum_{\alpha, \beta} {\lVertDh_3\rVert}_{\mathscr L^2({\mathbb{U}}_\circ^{\alpha, \beta})} \\
&\le {\lVertDh_2\rVert}_{\mathscr L^2(\Omega)} + \sum_{\alpha, \beta} {\lvert{\mathbb{U}}_\circ^{\alpha, \beta}\rvert}^{1/2} \sup_{{\mathbb{U}}^{\alpha, \beta}_\circ} {\lvertDh_3\rvert}\\
&\le {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - \delta + \frac{\delta}{2}
\end{split}$$ Thus $$\label{lastadd}
{\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - \frac{\delta}{2}$$
[**Step 4.**]{} We have already upgraded the mapping $h$ to a homeomorphism $h_3 \colon \Omega \to \Omega^\ast$ such that $h_3\in h+{\mathscr{A}}_\circ(\Omega)$ and $$\label{sstep4}
\begin{split}
{\lVerth_3-h\rVert}_{{\mathscr{A}}(\Omega)} & \le {\lVerth_3-h_2\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_2-h_1\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_1-h\rVert}_{{\mathscr{A}}(\Omega)}\\ & \le 2 \epsilon + 3 \epsilon + 2\epsilon = 7 \epsilon
\end{split}$$ Moreover, $h_3$ is a $\mathscr C^\infty$-diffeomorphism on $\Omega \setminus \bigcup_{v\in \mathcal V} h_2^{-1} (\overline{{\mathbb{D}}}_v)$. We now define a homeomorphism $h_4 \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ by performing harmonic replacement of $h_3$ on each set $h_3^{-1}(2 {\mathbb{D}}_v)$. This gives us a $\mathscr C^\infty$-diffeomorphism $h_4 \colon h_3^{-1} (2 \overline{{\mathbb{D}}}_v)\to 2 \overline{{\mathbb{D}}}_v$, see Step 2 for details. For each $z\in \Omega$ $${\lverth_4(z)-h_3(z)\rvert} \le \begin{cases} 2 \operatorname{diam}{\mathbb{D}}_v \quad & \mbox{ if } z\in h_3^{-1} (2{\mathbb{D}}_v) \\
0 & \mbox{ otherwise}\end{cases} \quad \le 2{\epsilon}$$ Hence ${\lVerth_4-h_3\rVert}_{\mathscr C (\Omega)} \le 2{\epsilon}$. Using we estimate the energy as follows. $$\begin{split}
{\mathcal{E}}_\Omega [h_4-h_3] &= \sum_{v\in \mathcal V} {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_4-h_3]\\
& \le 2 \sum_{v\in \mathcal V} \left( {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_4] + {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_3] \right)\\
& \le 4 \sum_{v\in \mathcal V} {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_3] \le 20\epsilon^2
\end{split}$$ Thus, by $$\label{sstep5} {\lVerth_4-h_3\rVert}_{{\mathscr{A}}(\Omega)} \le \epsilon + \sqrt{20} \epsilon \le 6 \epsilon$$ By virtue of Dirichlet’s principle and we have $$\label{wearehungry}
{\lVertDh_4\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)} -
\frac{\delta}{2}$$
[**Step 5.**]{} The final step consists of smoothing $h_4$ in a neighborhood of each smooth Jordan curves $C_v= \partial h_3^{-1}(2{\mathbb{D}}_v)$. We proceed in much the same way as in Step 3, but we appeal to Corollary \[muncircor\] instead of Corollary \[muncor\]. By smoothing $h_4$ in a sufficiently thin neighborhood of each $C_v$ we obtain a $\mathscr C^{\infty}$-diffeomorphism $h_5 \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$, $h_5\in h_4 + {\mathscr{A}}_\circ (\Omega)$ such that $$\label{needsharpen}
{\lVerth_5-h_4\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}$$ We now recapitulate the estimates , and to obtain a $\mathscr C^\infty$-diffeomorphism in $\Omega$ $$H:=h_5\in h+{\mathscr{A}}(\Omega)$$ such that $$\begin{split}
{\lVertH-h\rVert}_{{\mathscr{A}}(\Omega)} & \le {\lVerth_5-h_4\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_4-h_3\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_3-h\rVert}_{{\mathscr{A}}(\Omega)}\\
& \le \epsilon + 6 \epsilon + 7 \epsilon =14 \epsilon
\end{split}$$ which is as strong as (ii) in Theorem \[thmapprox\].\
To obtain the desired energy estimate ${\mathcal{E}}_\Omega [h_5] \le {\mathcal{E}}_\Omega [h]$, we need to sharpen the energy part in . By narrowing further the neighborhoods of $C_v$ we can be make the energy ${\mathcal{E}}_\Omega[h_5-h_4]$ as small as we wish; for example to obtain $${\lVertDh_5-Dh_4\rVert}_{\mathscr L^2 (\Omega)}< \frac{\delta}{2}$$ This is enough to conclude that $${\lVertDH\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)}$$ because of .
Hopf differentials, Theorem \[thmhopf\]
=======================================
A quadratic differential on a domain $\Omega$ in the complex plane ${\mathbb{C}}$ takes the form $Q=F(z)\, {\textnormal d}z \otimes {\textnormal d}z$, where $F$ is a complex function on $\Omega$. Given a conformal change of the variable $z$, $z= \varphi(\xi)$, where $\varphi \colon \Omega' \to \Omega$, the pull back $$\varphi^\sharp (Q)= F\big(\varphi(\xi)\big)\, {\textnormal d}\varphi \otimes {\textnormal d}\varphi = F \big( \varphi (\xi) \big) \dot{\varphi}^2 (\xi) \, {\textnormal d}\xi \otimes {\textnormal d}\xi$$ defines a quadratic differential on $\Omega'$. It is plain that for a complex harmonic function $h \colon \Omega \to {\mathbb{C}}$ the associated Hopf differential $$Q_h= h_z \overline{h_{\bar z}}\, {\textnormal d}z \otimes {\textnormal d}z$$ is holomorphic, meaning that $$\label{heq0}
\frac{\partial}{\partial \bar z} \left(h_z \overline{h_{\bar z}} \right)=0$$ Conversely, if a Hopf differential $Q_h=h_z \overline{h_{\bar z}}\, {\textnormal d}z \otimes {\textnormal d}z$ is holomorphic for some $\mathscr C^1$-mapping $h$, then $h$ is harmonic at the points where the Jacobian determinant $J(z,h):=\det Dh= {\lverth_z\rvert}^2- {\lverth_{\bar z}\rvert}^2 \ne 0$, see [@EL1 10.5] and our Remark \[notredable\]. Here the assumption that $J(z,h) \ne 0$ is critical. Let us illustrate it by the following.
\[ex\] Consider a mapping $h\in \mathscr C^{1,1} ({\mathbb{C}}_\circ)$ defined on the punctured plane ${\mathbb{C}}_\circ = {\mathbb{C}}\setminus \{0\}$ by the rule $$\label{heq1}
h(z)= \begin{cases} \frac{z}{|z|}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
\frac{1}{2} \left(z+ \frac{1}{\bar z}\right)\qquad & \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ Direct computation shows that $$h_z (z)= \begin{cases} \frac{1}{2}|z|^{-1}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
\frac{1}{2} \qquad & \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ and $$h_{\bar z} (z)= \begin{cases} -\frac{1}{2}|z| {\bar z}^{-2}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
-\frac{1}{2}\bar z^{-2} \qquad & \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ Thus $$\label{tristars}
Q_h = - \frac{{\textnormal d}z \otimes {\textnormal d}z}{4z^2} \qquad \mbox{ in } {\mathbb{C}}_\circ$$ It may be worth mentioning that the mapping $h$ in is the unique (up to rotation of $z$) minimizer of the Dirichlet energy $$\mathcal E [H]= \int_{{\mathbb{A}}} {\lvertDH\rvert}^2$$ over the annulus ${\mathbb{A}}= A(r,R)=\{z \colon r< {\lvertz\rvert} < R \}$, $0<r<1<R$, subject to all weak limits of homeomorphisms $H \colon {\mathbb{A}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}A(1, R_\ast)$, where $R_\ast = \frac{1}{2} \left(R + \frac{1}{R}\right)$, see [@AIM]. Note that the Hopf differential of is real along the boundary circles of ${\mathbb{A}}$. The concentric circles are horizontal trajectories of $Q_h$. In fact this is a general property of minimizers [@Job Lemma 1.2.5]. The general pattern is that with the loss of injectivity comes the loss of the Lagrange-Euler equation for the extremal mapping.
Properties of the function $h$ with holomorphic Hopf differential $Q= h_z \overline{h_{\bar z}} \, {\textnormal d}z \otimes {\textnormal d}z$ are of interest in the studies of harmonic mappings [@EL2; @EL3; @Job; @Sc; @Se], minimal surfaces [@DHKWb; @Stb] and Teichmüller theory [@GLb]. In this section we prove Theorem \[thmhopf\] which imposes fairly minimal assumptions that imply harmonicity of $\mathscr W^{1,2}$-solution to the equation . Some elements of the proof go back to [@RS; @RW].
As a consequence of the Stoilow factorization theorem [@AIM p. 56] the branch set of $h$ is discrete, hence removable for continuous harmonic functions. Thus we assume that $h \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ is a homeomorphism of Sobolev class ${\mathscr{W}}^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$ such that $$h_z \overline{h_{\bar z}} =F(z) \quad \mbox{ is holomorphic in } \; \Omega$$ By virtue of Theorem \[thmapprox\], there exists a sequence of diffeomorphisms $h^j \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ converging $c$-uniformly and strongly in ${\mathscr{W}}^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$ to $h$. Denote by $$h_z^j h^j_{\bar z}=:F^j \in \mathscr L_{\operatorname{loc}}^1(\Omega)$$ Thus $F^j \to F$ strongly in $ \mathscr L_{\operatorname{loc}}^1(\Omega)$. Let us first dispose of an easy case.\
[**Case 0.**]{} [*The homogeneous equation $F \equiv 0$*]{}. Since $h^j$ are diffeomorphisms the Jacobian determinant $J(z,h^j)= {\lverth_z^j\rvert}^2-{\lverth_{\bar z}^j\rvert}^2$ is either positive everywhere in $\Omega$ or negative everywhere in $\Omega$. Let us settle the case when $J(z,h^j)>0$ for infinitely many indices $j=1,2,\dots$. For such $j$ we have ${\lverth^j_z\rvert}>{\lverth^j_{\bar z}\rvert}$, which yields ${\lverth^j_{\bar z}\rvert}^2\le {\lverth^j_z h^j_{\bar z}\rvert}$. Passing to the $\mathscr L^1$-limit we obtain $${\lverth_{\bar z}\rvert}^2\le {\lverth_z h_{\bar z}\rvert} = {\lvertF(z)\rvert}\equiv 0.$$ Thus $h$ is holomorphic, by Weyl’s lemma. Similarly, in case $J(z,h^j)<0$ for infinitely many indices $j=1,2,\dots$, we find that $h$ is antiholomorphic.
\[rem\] We observe, based on the above arguments, that for this homogeneous equation $h_z\overline{h_{\bar z}}\equiv 0$ every solution $h\in {\mathscr{W}}_{\operatorname{loc}}^{1,2}(\Omega)$ obtained as the weak ${\mathscr{W}}^{1,2}$-limit of homeomorphisms is either holomorphic or antiholomorphic. The situation is dramatically different if $h_z\overline{h_{\bar z}}\not\equiv 0$; some topological assumption on $h$ are necessary, as illustrated in Example \[ex\].
[**Case 1.**]{} [*Nonhomogeneous equation $F\not\equiv 0$*]{}. The function $F$, being holomorphic, may vanish only at isolated points. Since isolated points are removable for bounded harmonic functions, it suffices to consider the set where $F\ne 0$. Proceeding further in this direction, we may and do assume that $F(z)\equiv 1$ (by a conformal change of the $z$-variable) and $h$ is a ${\mathscr{W}}^{1,2}$-homeomorphism in the closure of the unit square ${\mathbb{Q}}=\{x+iy\colon 0<x<1, 0<y<1\}$. The problem now reduces to establishing that the equation $$\label{eeq3}
h_{z}\overline{h_{\bar z}}\equiv 1$$ implies $\Delta h=0$. This will be proved indirectly by means of the energy-minimizing property $$\label{ineq4}
\mathcal E_{{\mathbb{Q}}}[h]\le \mathcal E_{{\mathbb{Q}}}[H]$$ where $H\colon {\mathbb{Q}}\to h({\mathbb{Q}})$ is any homeomorphism in $h+{\mathscr{A}}_\circ({\mathbb{Q}})$; in particular, $H=h$ on $\partial{\mathbb{Q}}$. Indeed, if $h$ were not harmonic, we would be able to decrease its energy by harmonic replacement (Propositions \[prpoisson\] and \[RKC\]), contradicting .
Proof of the inequality
------------------------
With the aid of the approximation theorem we need only prove for mappings $H\in h+ {\mathscr{A}}_\circ({\mathbb{Q}})$ that are diffeomorphisms on ${\mathbb{Q}}$. From now on we assume that this is the case. Denote ${\mathbb{Q}}^*=h({\mathbb{Q}})=H({\mathbb{Q}})$. We consider a sequence $h^j\in h+{\mathscr{A}}_\circ({\mathbb{Q}})$ of diffeomorphisms $h^j\colon {\mathbb{Q}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{Q}}^*$ converging in ${\mathscr{A}}({\mathbb{Q}})$ to $h$. Moreover we may also assume that $Dh^j\to Dh$ almost everywhere in ${\mathbb{Q}}$ by passing to a subsequence if necessary. Now the sequence $\chi^j\colon\overline{{\mathbb{Q}}}\to\overline{{\mathbb{Q}}}$ of self-homeomorphisms of the closed unit disk given by $\chi^j=H^{-1}\circ h^j$, where $\chi^j=\operatorname{id}$ on $\partial{\mathbb{Q}}$, is converging uniformly on $\overline{{\mathbb{Q}}}$ to $\chi=H^{-1}\circ h$. It is important to observe that $\chi\in {\mathscr{W}}_{\operatorname{loc}}^{1,2}({\mathbb{Q}})$ and $\chi^j$ converges to $\chi$ in ${\mathscr{W}}^{1,2}({\mathbb{Q}}')$ on any compactly contained subdomain ${\mathbb{Q}}'\Subset {\mathbb{Q}}$. Since $h^j$ and $(\chi^j)^{-1}$ are diffeomorphisms on ${\mathbb{Q}}'$ and $\chi^j({\mathbb{Q}}')$, respectively, the chain rule can be applied to the composition $H=h^j\circ (\chi^j)^{-1}$. For $w\in \chi^j({\mathbb{Q}}')$ we have $$\begin{split}
\frac{\partial H(w)}{\partial w} &= h_z^j(z)\frac{\partial (\chi^j)^{-1}}{\partial w}+h_{\bar z}^j(z)\frac{\partial (\chi^j)^{-1}}{\partial \bar w} \\
\frac{\partial H(w)}{\partial \bar w} &= h_z^j(z)\frac{\partial (\chi^j)^{-1}}{\partial \bar w}+h_{\bar z}^j(z)\overline{\frac{\partial (\chi^j)^{-1}}{\partial w}}
\end{split}$$ where $z=(\chi^j)^{-1}(w)$.
The partial derivatives of $(\chi^j)^{-1}$ at $w$ can be expressed in terms of $\chi_{_z}^j(z)$ and $\chi_{\bar z}^j(z)$ by the rules $$\begin{split}
\frac{\partial (\chi^j)^{-1}}{\partial w} &= \frac{\chi_z^j(z)}{J(z,\chi^j)} \\
\frac{\partial (\chi^j)^{-1}}{\partial \bar w} &= - \frac{\chi_{\bar z}^j(z)}{J(z,\chi^j)}
\end{split}$$ where the Jacobian determinant $J(z,\chi^j)$ is strictly positive. This yields $$\begin{split}
\frac{\partial H}{\partial w} & =\frac{h_z^j\overline{\chi_z^j}-h_{\bar z}^j\overline{\chi_{\bar z}^j}}{J(z,\chi^j)} \\
\frac{\partial H}{\partial \bar w} & =\frac{h_{\bar z}^j{\chi_z^j}-h_{\bar z}^j{\chi_{\bar z}^j}}{J(z,\chi^j)}
\end{split}$$ We compute the energy integral of $H$ over the set $\chi^j({\mathbb{Q}}')$ by substitution $w=\chi^j(z)$, $$\begin{split}
\mathcal E_{\chi^j({\mathbb{Q}}')}[H]&=2\int_{\chi^j({\mathbb{Q}}')} \left({\lvertH_w\rvert}^2+{\lvertH_{\bar w}\rvert}^2\right)\,{\textnormal d}w \\
&=2\int_{{\mathbb{Q}}'}\frac{{\lverth_z^j \overline{\chi_z^j} - h_{\bar z}^j \overline{\chi_{\bar z}^j}\rvert}^2+
{\lverth_{\bar z}^j \chi_z^j - h_{z}^j \chi_{z}^j\rvert}^2}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z
\end{split}$$ On the other hand, the energy of $h^j$ over the set ${\mathbb{Q}}'$ is $$\mathcal E_{{\mathbb{Q}}'}[h^j]=2\int_{{\mathbb{Q}}'}\left({\lverth_z^j\rvert}^2+{\lverth_{\bar z}^j\rvert}^2\right)\,{\textnormal d}z$$ Subtract these two integrals to obtain $$\label{cchain}
\begin{split}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j] &\ge \mathcal E_{\chi^j({\mathbb{Q}}')}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j] \\
&= 4\int_{{\mathbb{Q}}'} \frac{\left({\lverth_z^j\rvert}^2+{\lverth_{\bar z}^j\rvert}^2\right)\cdot {\lvert\chi_{\bar z}^j\rvert}^2
-2\operatorname{Re}\left[h_z^j\overline{h_{\bar z}^j} \overline{\chi_z^j}\chi_{\bar z}^j\right]}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z \\
& \ge 4 \int_{{\mathbb{Q}}'} \frac{ 2{\lverth_z^jh_{\bar z}^j\rvert} {\lvert\chi_{\bar z}^j\rvert}^2
-2\operatorname{Re}\left[h_z^j\overline{h_{\bar z}^j} \overline{\chi_z^j}\chi_{\bar z}^j\right]}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z \\
& = 4 \int_{{\mathbb{Q}}'} \left[\frac{{\lvert\chi_z^j-\sigma^j(z) \chi_{\bar z}^j\rvert}^2}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2} -1 \right]
\, {\lverth_z^jh_{\bar z}^j\rvert} \,{\textnormal d}z
\end{split}$$ where we have introduced the notation $$\sigma^j = \sigma^j(z) = \begin{cases}
{h_z^j\overline{h_{\bar z}^j}}{\, {\lverth_z^jh_{\bar z}^j\rvert}^{-1}} \qquad &\text{if } h_z^jh_{\bar z}^j\ne 0 \\
1 & \text{otherwise.}
\end{cases}$$ Note that ${\lvert\sigma^j\rvert}=1$ and $\sigma^j\to 1$ almost everywhere.
Upon using Hölder’s inequality we continue the chain as follows. $$\label{cchain2}
\ge \; 4\frac{\left[\int_{{\mathbb{Q}}'}\left| \chi_z^j-\sigma^j\chi_{\bar z}^j\right| \,\sqrt{{\lverth_z^jh_{\bar z}^j\rvert}}\,{\textnormal d}z\right]^2}{\int_{{\mathbb{Q}}'} J(z,h^j)\,{\textnormal d}z}
-4\int_{{\mathbb{Q}}'} {\lverth_z^jh_{\bar z}^j\rvert}.$$ The denominator in is at most $1$ because $$\int_{{\mathbb{Q}}'} J(z,h^j)\,{\textnormal d}z = {\lvert\chi^j({\mathbb{Q}}')\rvert}\le{\lvert{\mathbb{Q}}\rvert}=1.$$ Therefore, $$\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j]\ge 4\left[\int_{{\mathbb{Q}}'}\left| \chi_z^j-\sigma^j\chi_{\bar z}^j\right|\,\sqrt{{\lverth_z^jh_{\bar z}^j\rvert}}\,{\textnormal d}z \right]^2 -4\int_{{\mathbb{Q}}'}{\lverth_z^jh_{\bar z}^j\rvert}\, {\textnormal d}z.$$ It is at this point that we can pass to the limit as $j\to\infty$, to obtain $$\label{cchain3}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h]\ge 4\left[\int_{{\mathbb{Q}}'}{\lvert\chi_z-\chi_{\bar z}\rvert}\, {\textnormal d}z \right]^2-4{\lvert{\mathbb{Q}}'\rvert}.$$ Since ${\mathbb{Q}}'$ was an arbitrary compactly contained subdomain of ${\mathbb{Q}}$, the estimate remains valid with ${\mathbb{Q}}'$ replaced by ${\mathbb{Q}}$. $$\label{cchain4}
\begin{split}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}}[h] &\ge 4\left[\int_{{\mathbb{Q}}}\left|\frac{\partial\chi}{\partial y}\right|\,{\textnormal d}x\,{\textnormal d}y\right]^2-4 \\
&\ge 4\int_0^1 \left|\int_0^1 \frac{\partial\chi (x,y)}{\partial y}\,{\textnormal d}y\right|\,{\textnormal d}x - 4 \\
&= 4 \int_0^1 {\lvert\chi(x,1)-\chi(x,0)\rvert}\,{\textnormal d}x-4 =4-4 =0
\end{split}$$ as desired.
\[notredable\] When specialized to the case $h \in \mathscr C^1$, Theorem \[thmhopf\] shows that $h$ is harmonic outside of the zero set of its Jacobian.
Auxiliary smoothing results
===========================
Here we present some results concerning smoothing of piecewise differentiable planar homeomorphisms. They can be found in [@Mu] in greater generality, but since we require quantitative control of derivatives, a self-contained proof is in order. Here it is more convenient to use the operator norm of a matrix, denoted by ${\lVert\cdot\rVert}$. Note that ${\lVertA\rVert} \le {\lvertA\rvert} \le 2 {\lVertA\rVert}$ for $2 \times 2$-matrices.
\[mun\] Let ${\mathbb{U}}\subset{\mathbb{R}}^2$ be a domain containing an open segment $I$ with endpoints on the boundary $\partial {\mathbb{U}}$ which splits ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus I= {\mathbb{U}}_1\cup {\mathbb{U}}_2$. Suppose that $f\colon \overline{{\mathbb{U}}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\overline{{\mathbb{U}}^*}\subset{\mathbb{R}}^2$ is a homeomorphism with the following properties:
(i) \[mun2a\] For $j=1,2, \dots$ the restriction of $f$ to $\overline{{\mathbb{U}}_j} $ is $\mathscr C^{\infty}$-smooth, equals the identity on $I$;
(ii) \[mun2\] There is a constant $M>0$ such that for $j=1,2$ the restriction of $f$ to $\overline{{\mathbb{U}}_j}$ satisfies ${\lVertDf\rVert}\le M$ and $\det Df\ge M^{-1}$.
Then for any open set ${\mathbb{U}}_\circ$ with $I\subset {\mathbb{U}}_\circ\subset {\mathbb{U}}$ there is a $\mathscr C^{\infty}$-diffeomorphism $g\colon {\mathbb{U}}\to {\mathbb{U}}^*$ such that
- $g$ agrees with $f$ on ${\mathbb{U}}\setminus {\mathbb{U}}_\circ$ (and also on $I$);
- ${\lVertDg\rVert}\le 20M$ and $\det Dg\ge (20M)^{-1}$ on ${\mathbb{U}}$.
Without loss of generality $I\subset {\mathbb{R}}=\{(x,y) \colon y=0\}$. We write $f$ in components as $(u,v)$ where $u$ and $v$ are functions of $x$ and $y$. Let us introduce a notation; given any $\mathscr C^{\infty}$-smooth function $\beta\colon{\mathbb{R}}\to[0,\infty)$, denote $V(\beta)=\{(x,y)\in {\mathbb{R}}^2\colon {\lverty\rvert}<\beta(x)\}$. We can and do choose $\beta$ so that $I\subset V(\beta)\subset {\mathbb{U}}_\circ$, and further scale it down until the following holds. $$\label{beta}
\begin{split}
{\lvert\beta'(x)\rvert} &\le\frac{1}{40M} \qquad \mbox{ for all } x\in {\mathbb{R}}; \\
{\lvertv_x\rvert}&\le \frac{1}{50M^2} \qquad \text{in }V(\beta)\setminus I, \quad \mbox{ because } v(x,0)=0; \\
{\lvertu_x-1\rvert}&\le \frac{1}{10} \qquad \quad \; \text{ in }V(\beta)\setminus I, \quad \mbox{ because } u(x,0)=x.
\end{split}$$ As a consequence of and , $$\label{beta1}
v_y \ge \frac{M^{-1}-{\lvertu_y v_x\rvert}}{u_x}\ge \frac{1}{2M}.$$ Since $v$ is also $M$-Lipschitz by , the following double inequality holds in $V(\beta)\setminus I$. $$\label{vest}
\frac{1}{2M} \le \frac{v}{y}\le M.$$
Let us fix be a nondecreasing $\mathscr C^{\infty}$ function $\alpha\colon{\mathbb{R}}\to{\mathbb{R}}$ such that $\alpha(t)=0$ for $t\le 1/3$. Let $\alpha(t)=1$ for $t\ge 2/3$. Moreover, $\alpha'(t)\le 4$ for all $t\in{\mathbb{R}}$ and $\alpha(\infty)=1$, by convention. Now we introduce a modification of $u$ on ${\mathbb{U}}$ by setting $$\tilde u := \alpha(t)u+(1-\alpha(t))x \qquad \text{where } t=\begin{cases}\frac{{\lverty\rvert}}{\beta(x)} \quad & \mbox{ if } \beta(x) \ne 0 \\
\infty & \mbox{ otherwise} \end{cases}.$$ Note that $\tilde u=u$ outside of $V(\beta)$. In $V(\beta)\setminus I$ we compute the derivatives as follows. $$\label{d1}
\begin{split}
\tilde u_x &= - t^2 \alpha'(t) \beta'(x) \frac{u-x}{{\lverty\rvert}} + \alpha(t)u_x+1-\alpha(t) \\
\tilde u_y &= t\alpha'(t)\frac{u-x}{y} + \alpha(t) u_y
\end{split}$$ Since $u$ is $M$-Lipschitz by , we have ${\lvertu-x\rvert}\le M{\lverty\rvert}$. From this, and we obtain $$\label{d2}
\frac{8}{10} \le \tilde u_x\le \frac{12}{10},
\quad \text{ and } \quad {\lvert\tilde u_y\rvert}\le 5M,$$ which combined with yields $$\label{d3}
\tilde u_x v_y-\tilde u_y v_x \ge \frac{8}{10}\frac{1}{2M}- \frac{5M}{50M^2}= \frac{3}{10 M}.$$
Next we modify $v$ on ${\mathbb{U}}$. Specifically, $$\tilde v := \alpha(s) v+(1-\alpha(s))\frac{y}{2M} \qquad \text{where } s= \begin{cases} \frac{3{\lverty\rvert}}{\beta(x)} \quad & \mbox{ if } \beta(x) \ne 0 \\
\infty& \mbox{ otherwise} \end{cases}$$ Note that $\tilde{v}=v$ outside of $V(\beta/3)$, and on the set $V(\beta/3)$ we already have $\tilde{u} \equiv x$.
Computations similar to yield (on the set $V(\beta/3)\setminus I$) $$\label{d1v}
\begin{split}
\tilde v_x &= -\frac{1}{3}\alpha'(s) s^2 \frac{v-y}{{\lverty\rvert}} + \alpha(s)v_x; \\
\tilde v_y &= \frac{s\alpha'(s)}{y}\left(v-\frac{y}{2M}\right) + \alpha(s)v_y+\frac{1-\alpha(s)}{2M}.
\end{split}$$ Straightforward estimates based on , and comply $$\label{d4}
\begin{split}
{\lvert\tilde v_x\rvert}& \le \frac{4 M}{3}+\frac{1}{50M^2}<\frac{3M}{2}, \\
\frac{1}{2M}& \le \tilde v_y \le 5M.
\end{split}$$
It remains to check that the mapping $g:=(\tilde u,\tilde v)$, which agrees with $f$ outside of $V(\beta)$, satisfies all the requirements. As regards $\mathscr C^\infty$-smoothness we need only check it on $V(\beta/9)$. But in this neighborhood of $I$ we have a linear mapping, $g(x,y)= \left(x, \frac{y}{2M}\right)$, so $\mathscr C^\infty$-smooth. By virtue of and we have ${\lVertDg\rVert}\le 20M$. The desired lower bound for $\det Dg$ follows from and . Consequently, $g$ is a local diffeomorphism, and since it agrees with $f$ on $\partial V(\beta)$, it is in fact a diffeomorphism, by a topological result: [*a local homeomorphism which shares boundary values with a homeomorphism is injective*]{} [@Mu Lemma 8.2].
We also need a polar version of Proposition \[mun\].
\[muncir\] Let ${\mathbb{U}}\subset{\mathbb{R}}^2$ be a domain containing a circle ${\mathbb{T}}$. Suppose that $f\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^*\subset{\mathbb{R}}^2$ is a homeomorphism with the following properties:
(i) \[munc1\] The restriction of $f$ to ${\mathbb{T}}$ is the identity mapping;
(ii) \[munc2\] There is a constant $M>0$ such that the restriction of $f$ to either component of ${\mathbb{U}}\setminus {\mathbb{T}}$ is $\mathscr C^{\infty}$-smooth with ${\lVertDf\rVert}\le M$ and $\det Df\ge M^{-1}$.
Then for any open set $W$ with ${\mathbb{T}}\subset W\subset {\mathbb{U}}$ there is a $\mathscr C^{\infty}$-diffeomorphism $g\colon {\mathbb{U}}\to {\mathbb{U}}^*$ such that
- $g$ agrees with $f$ on ${\mathbb{U}}\setminus W$ and on ${\mathbb{T}}$;
- ${\lVertDg\rVert}\le 80M$ and $\det Dg\ge (80M)^{-1}$ on ${\mathbb{U}}$.
It is convenient to identify ${\mathbb{R}}^2$ with ${\mathbb{C}}$. Without loss of generality ${\mathbb{T}}=\{z\in{\mathbb{C}}\colon {\lvertz\rvert}=1\}$. Let $\psi(\zeta)=\exp(i\zeta)$. The mapping $F=\psi^{-1}\circ f\circ \psi$ is well-defined in some open horizontal strip $S_h=\{z\in{\mathbb{C}}\colon {\lvert\operatorname{Im}z\rvert}<\epsilon \}$ which we choose thin enough so that $\psi(S_\epsilon)\subset W$ and ${\lvert\psi'\rvert}^2< e^{2\epsilon }\le 2$. Note that $F$ is $2\pi$-periodic and satisfies $${\lVertDF\rVert}\le 2M \quad \text{ and }\quad \det DF\ge (2M)^{-1}.$$
The proof of Proposition \[mun\] applies to $F$ with no changes other than one simplification: $\beta>0$ is now a small positive constant rather than a function. Thus we obtain a diffeomorphism $G$ which agrees with $F$ on ${\mathbb{R}}\cup (S\setminus V(\beta))$ and satisfies ${\lVertDG\rVert}\le 40$ and $\det DG\ge (40M)^{-1}$. Since $F$ was $2\pi$-periodic, so is $G$. Thus, $g:=\psi\circ G\circ \psi^{-1}$ is the desired diffeomorphism.
Our applications require slightly more general versions of Proposition \[mun\] and Corollary \[muncir\], where the separating curve is allowed to have other shapes and $f$ is not required to agree with the identity on the curve.
\[nice\] A parametric curve $\Gamma \colon (0,1) \to {\mathbb{R}}^2$ is [*regular*]{} if $\Gamma$ extends to a bigger interval $(a,b)\supset [0,1]$ so that the extended mapping is a $\mathscr C^\infty$-diffeomorphism onto its image.
Note that a regular curve $\Gamma$ has well-defined endpoints $\Gamma(0)$ and $\Gamma(1)$. Also, $\Gamma$ extends to an injective $\mathscr C^\infty$-mapping $\Phi\colon (0,1)\times (-1,1)\to{\mathbb{R}}^2$ such that ${\lVertD\Phi\rVert}$ and ${\lVert(D\Phi)^{-1}\rVert}$ are bounded. This follows from the existence of a tubular neighborhood of the image of $\Gamma$ [@MRb Theorem 4.26].
Corollaries \[muncor\] and \[muncircor\], given below, generalize Proposition \[mun\] and Corollary \[muncir\] respectively.
\[muncor\] Let ${\mathbb{U}}\subset {\mathbb{R}}^2$ be a domain containing the image of a regular arc $\Gamma$ with endpoints on the boundary $\partial {\mathbb{U}}$ which divides ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus \Gamma={\mathbb{U}}_1 \cup {\mathbb{U}}_2$. Suppose $f \colon \overline{{\mathbb{U}}} \to \overline{{\mathbb{U}}^\ast} \subset {\mathbb{R}}^2$ is a homeomorphism such that $f \circ \Gamma$ is also regular and the restriction of $f$ to each $\overline{{\mathbb{U}}_i}$ is $\mathscr C^\infty$-smooth and satisfies $${\lvertDf(z)\rvert} \le M, \quad \det Df(z) \ge \frac{1}{M} \quad \mbox{for } z\in {\mathbb{U}}_i$$ where $M$ is a positive constant. Then there is a constant $M'>0$ such that to every open set ${\mathbb{U}}' \subset {\mathbb{U}}$ with $\Gamma \subset {\mathbb{U}}'$ there corresponds a $\mathscr C^\infty$-diffeomorphism $g\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^\ast$ with the following properties
- $g(z)=f(z)$ for $z\in {\mathbb{U}}\setminus {\mathbb{U}}'$ (and also on $\Gamma$)
- ${\lvertDg(z)\rvert} \le M'$ and $\det Dg(z) \ge \frac{1}{M'}$ on ${\mathbb{U}}$.
Let ${\mathbb{Q}}=(0,1)\times (-1,1)$. Let $\Phi$ and $\Psi$ be the extensions of $\Gamma$ and $f\circ \Gamma$ to ${\mathbb{Q}}$ as in Definition \[nice\]. There is a domain $\widetilde{{\mathbb{U}}}$ such that $(0,1)\times \{0\}\subset \widetilde{{\mathbb{U}}} \subset {\mathbb{Q}}$, $\Phi(\widetilde{{\mathbb{U}}})\Subset {\mathbb{U}}'$, and the composition $F:=\Psi^{-1}\circ f\circ \Phi$ is defined in $\widetilde{{\mathbb{U}}}$. Note that $F=\operatorname{id}$ on $(0,1)\times \{0\}$. We apply Proposition \[mun\] (with $\widetilde{{\mathbb{U}}}$ in place of ${\mathbb{U}}$ and with $F$ in place of $f$) and obtain a $\mathscr C^\infty$-diffeomorphism $G \colon \widetilde{{\mathbb{U}}} \to F(\widetilde{{\mathbb{U}}})$. Finally, replace $F$ within $\widetilde{{\mathbb{U}}}$ with the diffeomorphism $g=\Psi\circ G\circ \Phi^{-1}$.
\[muncircor\] Let ${\mathbb{U}}\subset {\mathbb{R}}^2$ be a domain containing the image of a $\mathscr C^\infty$-smooth Jordan curve $\Gamma$ which divides ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus \Gamma={\mathbb{U}}_1 \cup {\mathbb{U}}_2$. Suppose $f \colon \overline{{\mathbb{U}}} \to \overline{{\mathbb{U}}^\ast} \subset {\mathbb{R}}^2$ is a homeomorphism such that the restriction of $f$ to each $\overline{{\mathbb{U}}_i}$ is $\mathscr C^\infty$-smooth and satisfies $${\lvertDf(z)\rvert} \le M, \quad \det Df(z) \ge \frac{1}{M} \quad \mbox{for } z\in {\mathbb{U}}_i$$ where $M$ is a positive constant. Then there is a constant $M'>0$ such that to every open set ${\mathbb{U}}' \subset {\mathbb{U}}$ with $\Gamma \subset {\mathbb{U}}'$ there corresponds a $\mathscr C^\infty$-diffeomorphism $g\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^\ast$ with the following properties
- $g(z)=f(z)$ for $z\in {\mathbb{U}}\setminus {\mathbb{U}}'$ (and also on $\Gamma$)
- ${\lvertDg(z)\rvert} \le M'$ and $\det Dg(z) \ge \frac{1}{M'}$ on ${\mathbb{U}}$.
The proof of Corollary \[muncor\] is easily adapted to this case.
Concluding remarks
==================
One may wonder whether the proof of Theorem \[thmapprox\] can be extended to the spaces ${\mathscr{W}}^{1,p}$, $1<p<\infty$, by means of the $p$-harmonic replacement in place of Proposition \[RKC\]. Indeed, $p$-harmonic mappings are $\mathscr C^{1, \alpha}$-smooth [@Ur]. However, the injectivity of $p$-harmonic replacement of a homeomorphism is unclear.
Is there a version of the Radó-Kneser-Choquet theorem for $p$-harmonic mappings? That is, does the $p$-harmonic extension of a homeomorphism onto a convex Jordan curve enjoy the injectivity property?
An attempt to extend Theorem \[thmapprox\] to higher dimensions faces another obstacle: the Radó-Kneser-Choquet theorem fails in dimensions $n\ge 3$ as was proved by Laugesen [@La].
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[^1]: Iwaniec was supported by the NSF grant DMS-0800416.
[^2]: Kovalev was supported by the NSF grant DMS-0968756.
[^3]: Onninen was supported by the NSF grant DMS-1001620.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recent proposals have suggested that a previously unknown decay mode of the neutron into a dark matter particle could solve the long lasting measurement problem of the neutron decay width. We show that, if the dark particle in neutron decay is the major component of the dark matter in the universe, this proposal is in disagreement with modern astro-physical data concerning neutron star masses.'
address:
- 'CSSM and ARC Centre of Excellence for Particle Physics at the Terascale, Department of Physics, University of Adelaide SA 5005 Australia'
- 'IRFU-CEA, Université Paris-Saclay, F91191 Gif sur Yvette, France'
- 'CSSM and ARC Centre of Excellence for Particle Physics at the Terascale, Department of Physics, University of Adelaide SA 5005 Australia'
author:
- 'T.F.Motta'
- 'P.A.M.Guichon'
- 'A. W. Thomas$^*$'
bibliography:
- 'biblio.bib'
title: Neutron to Dark Matter Decay in Neutron Stars
---
Introduction
============
In a recent publication by Fornal *et al.* [@Fornal] a proposal was made to solve the persistent discrepancy between two methods of measuring the neutron life-time. Trapped neutrons in a bottle appear to have a shorter life-time than neutrons in a beam where the decay proton is detected. There is a discrepancy of around 8 seconds (3.5$\sigma$) between the two experimental set-ups. In Ref.[@Fornal] it was suggested that the reason for this difference could lie in a formerly unknown decay channel of the neutron to a dark fermion. This proposal came as an alternative to the previous hypothesis that the experimental disagreement could be caused by the neutron oscillating into it’s mirror counterpart [@Mirror]. Both arguments rely on the proposed existence of a decay channel to a fermion almost degenerate with the neutron.
This proposal attracted the attention of several collaborations and a number of publications followed the original release. In Ref.[@Tang] the authors argue through experimental evidence that in a decay of the form $n \rightarrow \text{DM} + \lambda$, i.e. a dark matter particle plus another decay product $\lambda$, that extra particle could not be a photon ($\lambda \not= \gamma$). Another publication[@Antineutrino] pointed out that this hypothesised decay could also explain a different experimental inconsistency, the “reactor antineutrino anomaly", that is, the $3\sigma$ discrepancy between theory and measurement of the antineutrino flux from a reactor. Finally Czarnecki *et al.* [@Czarnecki] although they did not rule out this explanation, pointed out strong constraints related to the value of the neutron axial charge.
In this publication we argue that allowing the neutron to decay to an almost degenerate dark fermion would mean that inside a neutron star, where the neutrons occupying a Fermi sea can sustain, through degeneracy, very large pressures, a large portion of these neutrons would decay to this dark fermion. This implies a severe decrease in pressure, which means that the maximum mass of neutron stars before gravitational collapse would be drastically lower than the masses of the stars measured so far. This was argued in Refs. [@Motta:2018rxp; @Baym; @Reddy] and will be developed in further detail in this publication.
Framework
=========
Simulating the internal structure of neutron stars ultimately amounts to solving the so-called Tolman-Oppenheimer-Volkof[@TOV] (TOV) equations for several different values of central energy density. The TOV equations give an internal profile for the pressure of the star through ($c=G=\hbar=1$) $$\begin{aligned}
\label{tov}
\frac{dP(r)}{dr}=-\frac{1}{r^2}\left ( \epsilon(r) + P(r) \right )\left ( M(r)+4\pi r^3 P(r) \right )\left ( 1-\frac{2M(r)}{r} \right )^{-1}\end{aligned}$$ and the mass is given by the continuity equation $$\frac{dM(r)}{dr}=4\pi r^2\epsilon(r).
\label{massagain}$$ This set of equations take, as an input, the equation of state (EOS) of the matter of which the star is made. We will adopt, as a model for the core of neutron stars, the infinite nuclear matter EOS from the quark-meson coupling model[@QMC] recently reviewed in Ref.[@QMC_Review]. This model is well established and has been shown to provide an adequate description of high density nuclear matter in several previous calculations[@QMC1; @QMC2].
We compare that equation of state with a modified version of it where the neutron decays to a dark fermion. Since a difference in mass of the order of a few MeV makes absolutely no difference to the mass of a neutron star, we will take this dark fermion to be fully degenerate with the neutron. Ultimately we will show that adding a vector self interaction among the dark fermions can indeed bring the mass up to more acceptable values, as was also shown in Ref. [@Vector]. However, in order for that to happen, the coupling of this vector intermediate particle with the dark fermion has to be simply huge and we will argue that recent publications [@Barbecue; @DAmico] rule out that explanation if the dark particle in neutron decay is the major component of the dark matter in the universe.
Dark Matter
-----------
The proposal by Fornal *et al.* [@Fornal] is based on the decay of the neutron into a dark matter fermion which is almost degenerate with the neutron itself, plus another lighter component to conserve energy.
Their first of three proposals mentioned in the publication is $n \rightarrow \chi + \gamma$, where $\chi$ is (and hereafter refers to) the dark matter fermion. However, as argued above, this model was experimentally excluded by Tang *et al.* [@Tang]. The only viable mode seems to be $$\begin{aligned}
n\rightarrow \chi + \phi\end{aligned}$$ where $\phi$ is a much lighter dark boson. This requires that the energy of the dark particles be in the ranges $$\begin{aligned}
&937.900\text{MeV}<m_\chi<938.543\text{MeV}\\
&937.900\text{MeV}<m_\chi+m_\phi<939.565\text{MeV}.\end{aligned}$$
We argue that
In neutron stars, the presence of this light dark boson $\phi$ is completely irrelevant for it would escape the system very quickly.
All of the proposed models indicate that, in neutron stars, the only change this hypothesis implies is a change in chemical composition from the equilibrium reaction $n \leftrightarrow \chi$, here imposed by the chemical equilibrium equation for the chemical potentials $\mu_n=\mu_\chi$.
QMC
---
The chosen model of nuclear matter interaction is the QMC model[@QMC]. Based on a quark description of the baryons as quark bags interacting directly with mesons (scalar-isoscalar $\sigma$, vector-isoscalar $\omega$, vector-isovector $\rho$) we derive the energy density of the system in Hartree-Fock (HF) approximation.
The Hartree, or mean field, contribution amounts to $$\begin{aligned}
&\epsilon_\text{Hartree}=\frac{m_\sigma^2\sigma^2}{2} + \frac{m_\omega^2\omega^2}{2}+ \frac{m_b^2b^2}{2}& \nonumber\\
&+\frac{1}{\pi^2}\int_{0}^{k_F^n}{k^2}{\sqrt{k^2+M_N^*(\sigma)^2}dk} +
\frac{1}{\pi^2}\int_{0}^{k_F^p}{k^2}{\sqrt{k^2+M_N^*(\sigma)^2}dk} \nonumber\\
&+\frac{1}{\pi^2}\int_{0}^{k_F^e}{k^2}{\sqrt{k^2+m_e^2}dk} +
\frac{1}{\pi^2}\int_{0}^{k_F^\mu}{k^2}{\sqrt{k^2+m_\mu^2}dk}
+\frac{1}{\pi^2}\int_{0}^{k_F^\chi}{k^2}{\sqrt{k^2+m_\chi^2}dk}\end{aligned}$$ where the effective mass of the nucleon is $M_N^*(\sigma)=m_n-g_\sigma\sigma+\frac{d}{2}(g_\sigma\sigma)^2$. The $d$ is what is refered to as scalar polarizability and it is a prominent feature of the QMC model. In our convention $\sigma$, $\omega$, and $b$ refer to the mean field values of the mesons (where $b$ is the mean field value of $\rho$). For each particle the fermi momenta and chemical potentials as functions of the number densities are calculated as $$\begin{aligned}
&k_\varphi^{3}={{3\pi^2n_\varphi}},\quad \varphi=\{p,n,e,\mu,\chi \} \\
&\mu_n= \frac{\partial \epsilon}{\partial n_n}, \quad \mu_p= \frac{\partial \epsilon}{\partial n_p}, \quad \mu_l= \sqrt{k_f(n_l)^2+m_l^2}\end{aligned}$$ And finally the Fock terms $$\begin{aligned}
&\epsilon_\text{Fock}=-G_\omega\frac{1}{(2\pi)^6} \left[
\int_0^{k_F^p}d^3k_1 \int_0^{k_F^p}d^3k_2 \frac{m_\omega^2}{(\vec k_1 - \vec k_2)^2 + m_\omega^2}
+
\int_0^{k_F^n}d^3k_1 \int_0^{k_F^n}d^3k_2 \frac{m_\omega^2}{(\vec k_1 - \vec k_2)^2 + m_\omega^2}\right] \nonumber\\
&-\frac{G_\rho}{4}\frac{1}{(2\pi)^6} \left[
(1)\times\int_0^{k_F^n}d^3k_1 \int_0^{k_F^n}d^3k_2 \frac{m_\rho^2}{(\vec k_1 - \vec k_2)^2 + m_\rho^2} +(1)\times\int_0^{k_F^p}d^3k_1 \int_0^{k_F^p}d^3k_2 \frac{m_\rho^2}{(\vec k_1 - \vec k_2)^2 + m_\rho^2}\right. \nonumber\\
&\left.
+ (2)\times\int_0^{k_F^n}d^3k_1 \int_0^{k_F^p}d^3k_2 \frac{m_\rho^2}{(\vec k_1 - \vec k_2)^2 + m_\rho^2}+ (2)\times\int_0^{k_F^p}d^3k_1 \int_0^{k_F^n}d^3k_2 \frac{m_\rho^2}{(\vec k_1 - \vec k_2)^2 + m_\rho^2} \right]&\nonumber\\
& +\frac{1}{(2\pi)^6}
\int_0^{k_F^p}d^3k_1 \int_0^{k_F^p}d^3k_2 \frac{1}{(\vec k_1 - \vec k_2)^2 + \tilde m_\sigma^2}\times\frac{M_N^*(\sigma)(-g_\sigma C(\sigma))}{\sqrt{M_N^*(\sigma)^2+k_1^2}}
\times\frac{M_N^*(\sigma)(-g_\sigma C(\sigma))}{\sqrt{M_N^*(\sigma)^2+k_2^2}}\nonumber\\
&+\frac{1}{(2\pi)^6}
\int_0^{k_F^n}d^3k_1 \int_0^{k_F^n}d^3k_2 \frac{1}{(\vec k_1 - \vec k_2)^2 + \tilde m_\sigma^2}\times\frac{M_N^*(\sigma)(-g_\sigma C(\sigma))}{\sqrt{M_N^*(\sigma)^2+k_1^2}}
\times\frac{M_N^*(\sigma)(-g_\sigma C(\sigma))}{\sqrt{M_N^*(\sigma)^2+k_2^2}}\nonumber\end{aligned}$$where $$\begin{aligned}
\tilde m_\sigma^2 =m_\sigma^2 + \frac{1}{\pi^2}\sum_{p,n}\int_0^{k_f^n}k^2dk \frac{\partial^2}{\partial \sigma^2} \sqrt{M_N^*(\sigma)^2+k^2}.\end{aligned}$$
The density dependent meson mean field equations in the QMC model are $$\begin{aligned}
&\sigma(n_n,n_p) =- \frac{1}{m_\sigma^2\pi^2}\left( \frac{\partial M_N^*}{\partial\bar\sigma}\right)\left[\sum_{p,n}
\int_{0}^{k_F}k^2dk\frac{M_N^*(\sigma)}{\sqrt{k^2+M_N^*(\sigma)^2}}
\right], \\
&\omega(n_n,n_p)=\frac{g_\omega}{m_\omega^2}\left(n_n+n_p\right), \\
&b(n_n,n_p)=\frac{g_\rho}{m_\rho^2}\left(\frac{n_p}{2} -\frac{n_n}{2}\right).\end{aligned}$$ and finally the pressure is calculated as $P=\sum_f\mu_fn_f-\epsilon$.
In Table \[tab:constants\] we report the constants used to perform the calculations. They are chosen to fit the saturation density at $0.16\text{fm}^{-3}$, the binding energy of symmetric matter at saturation $-15.8\text{MeV}$ and symmetry energy $30\text{MeV}$.
\[tab:constants\]
Neutron Stars
=============
Using the model presented above we calculate the equilibrium densities through the equations $$\begin{aligned}
\text{Neutron $\beta$ decay}\quad&\mu_n=\mu_p+\mu_e\\
\text{Muon $\beta$ decay}\quad&\mu_\mu=\mu_e \\
\text{Charge neutrality}\quad&n_p = n_e+n_\mu \\
\text{Dark matter decay}\quad&\mu_n = \mu_\chi.\end{aligned}$$
Solving these equations we get species fractions that vastly favours the dark matter particle (Fig. \[fig:fraction\]).
![Relative species fraction with dark matter present.[]{data-label="fig:fraction"}](fraction){width="4in"}
That, in turn, leads to a drastic decrease in pressure in the equation of state (Fig.\[fig:eos\]) and as a consequence the Mass versus Radius diagram has a maximum significantly lower than the case without dark matter (Fig.\[fig:massaraio\]).
![Equation of state with dark matter present.[]{data-label="fig:eos"}](eos){width="4in"}
The maximum mass in the diagram with dark matter is of around $0.7$M$_\odot$. The reason for that is that although the central energy density of a star with dark matter is much larger than the star without it, it does not have enough pressure to support itself and therefore the energy density goes down very quickly (Fig.\[fig:edens\])
![Energy density profile as a function of the internal radius of the star. Profile is presented for the maximum mass point of the diagram with dark matter present, an equally heavy star without dark matter and the maximum mass star without dark matter.[]{data-label="fig:edens"}](edens_radius){width="2.5in"}
However, it is unreasonable to assume that even the upper most point in the mass radius diagram of the EOS with dark matter present could ever be reached. Since the star with dark matter has to come from a real star we take the maximum mass star without dark matter and check to which point in the diagram the decay star would occur, that is, which point in the dark matter diagram has a total baryon number plus total dark matter number equal the total baryon number of a star without dark matter. That leads to a maximum mass of $0.58$M$_\odot$ (Fig.\[fig:massaraio\])
![Maximum possible star as an end product of dark matter decay.[]{data-label="fig:massaraio"}](058_1){width="4in"}
Repulsive Vector Interaction
============================
If the dark matter particle were self interacting through a repulsive interaction it is possible that it could build up pressure to sustain larger masses. This approach was used in Ref.[@Vector] and we here perform the same procedure within the framework of QMC. To compare with the neutron-$\omega$ physical system we vary the coupling/mass as multiples of the $n\omega$ vertex couplings, as indicated in the figures. We name this vector intermediate $V$.
The species fraction changes as the $\chi V$ interaction becomes stronger and therefore restores the EOS to it’s previous stiffer version. The greater the strength of the interaction, the less dark matter will be present in the star (Fig.\[fig:vectorfraction\]). That allows the system to support much higher masses.
![Species fraction considering different strengths of vector self-repulsion.[]{data-label="fig:vectorfraction"}](fraction2){width="4in"}
The maximum mass gets to the 2 solar masses value, as all neutron star models must per recent experimental determinations[@Antoniadis; @Demorest], only when the $g_V/m_V$ for the dark matter is 10 times greater than $g_\omega/m_\omega$ (Fig.\[fig:vectorinteraction\]).
![Adding vector self-interactions between the dark fermions through the exchange of a vector boson.[]{data-label="fig:vectorinteraction"}](vectorinteraction){width="4in"}
However, one must consider that Ref. [@DAmico] severely limits the cross-section of such a dark matter particle through astrophysical data recently measured [@Barbecue]. These values of couplings (that is $g_V/m_V$) are way to high to even enter consideration.
Conclusion
==========
We have shown that the addition of this dark matter particle to the composition of neutron stars leads to a giant decrease in maximum mass. The mass versus radius diagram points to $0.7$M$_\odot$ as the mass upper limit for stars with dark matter, however further investigations suggest that, if a star with dark matter is a decay product of a normal neutron star the real maximum mass has to be around $0.58$M$_\odot$. Since most neutron stars measured have masses around $1.5$M$_\odot$ this points to a clear inconsistency of the hypothesis with data. Moreover, a repulsive self-interaction indeed can push the mass limit to an acceptable point only when the ratio coupling/mass of the $\chi V$ interaction is 10 times larger than the $n\omega$ vertex. If this dark matter particle were to correspond with astrophysical dark matter this result would be in clear contradiction to recent astrophysical measurements, as pointed out in Ref. [@DAmico]. Even if it were unconnected with astrophysical dark matter, it would be truly remarkable to have a new kind of matter with self interactions an order of magnitude larger than the familiar strong force.
We therefore state that this decay is simply in contradiction with the data of neutron star masses if the dark particle in neutron decay is a significant component of the dark matter in the universe.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the University of Adelaide and by the Australian Research Council through the ARC Centre of Excellence for Particle Physics at the Terascale (CE110001104) and Discovery Project DP150103164.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, whilst the second two comprise the house-keeping heat. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity.'
author:
- 'Richard E. Spinney and Ian J. Ford'
date: 'December 20, 2011'
title: 'Non-equilibrium thermodynamics of stochastic systems with odd and even variables'
---
For over $100$ years the statement of the second law of thermodynamics stood simply as the Clausius inequality. However in recent years advances in technology have encouraged the thermodynamic consideration of small systems which has led to the generalisation of the concept of entropy production: it may be associated with individual dynamical realisations revealing a wealth of relations valid out of equilibrium. Such extensions had their origins in the dissipation function of Evans et al. for thermostatted systems that led to the Fluctuation Theorem [@Evans93; @Evans95; @Evans02; @Carberry04] with similar, but asymptotic relations for chaotic systems [@Gallavotti95] which were extended to Langevin dynamics [@kurchan] followed by general Markovian stochastic systems [@GCforstochastic]. Crooks and Jarzynski [@Jarzynski97; @crooksoriginal; @Crooks98] then derived work relations for a variety of dynamics which held for finite times. These were followed by similar generalised relations for the entropy production associated with transitions between stationary states [@hatanosasa], the total entropy production [@seifertoriginal] and the heat dissipation required to maintain a stationary state [@IFThousekeeping]. More recently the relationship between the latter quantities has been explored [@Jarpathintegral; @Esposito07; @Ge09; @Ge10] resulting in a formalism involving a division of the total entropy change into two distinct terms, the adiabatic and non-adiabatic entropy productions [@adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2], each of which obeys appropriate fluctuation relations and which map onto the house-keeping and excess heats, respectively, of Oono and Paniconi [@oono]. We seek to take such a formalism and generalise its scope by the explicit inclusion of both even (e.g. spatial) and odd (e.g. momentum) variables that transform differently under time reversal. In doing so we define a new quantity which obeys an integral fluctuation theorem for all time.
Specifically, we consider the dynamics of a general set of variables $\textbf{\em x}=(x^1,x^2,\ldots x^n)$ that behave differently under time reversal such that $\boldsymbol{\varepsilon}\textbf{\em x}=(\varepsilon^1x^1,\varepsilon^2x^2,
\ldots \varepsilon^nx^n)$ where $\varepsilon^i=\pm1$ for even and odd variables $x^i$ respectively. Odd variables arise in the discussion of directional quantities and consequently such a consideration is essential when discussing velocities, from the most simple lattice Boltzmann model to considerations of full phase space. The entropy production of a path of duration $\tau$ depends on two probabilities. The first is the path probability, $P^{\rm F}[\vec{\textbf{\em x}}]$, defined as the probability of the forward trajectory, $\vec{\textbf{\em x}}=\textbf{\em x}(t)$ for $0\leq t\leq \tau$, with a distribution of starting configurations, $P^{\rm F}(\textbf{\em x}(0),0)$, that acts as an initial condition for the general master equation (relevant examples arise, for example, in the context of full phase space [@Brenig; @kampenfluct] and in lattice Boltzmann models): $$\frac{\partial P^{\rm F}(\textbf{\em x},t)}{\partial t}=
\sum_{\textbf{\em x}'}T(\textbf{\em x}|\textbf{\em x}',\lambda^{\rm F}(t))
P^{\rm F}(\textbf{\em x}',t)
\label{master}$$ where $T(\textbf{\em x}|\textbf{\em x}',\lambda^{\rm F}(t))$ is a matrix of transition rates between configurations $\textbf{\em x}'$ and $\textbf{\em x}$, defining the normal dynamics, parameterised by the forward protocol $\lambda^{\rm F}$ at time $t$. We use notation $T(\textbf{\em x}|\textbf{\em x})=-\sum_{\textbf{\em x}'\neq
\textbf{\em x}}T(\textbf{\em x}'|\textbf{\em x})$ which describes the mean escape rate. The path probability of some sequence of $N$ transitions to configurations $\textbf{\em x}_i$ from $\textbf{\em x}_{i-1}$ at times $t_i$, such that $t_0=0$ and $t_{N+1}=\tau$, can then be computed as a function of transition rates and exponential waiting times $$\begin{aligned}
P^{\rm F}[\vec{\textbf{\em x}}]
&=P^{\rm F}(\textbf{\em x}_0,0)e^{\int_{t_{0}}^{t_{1}}dt'T(\textbf{\em x}_{0}|
\textbf{\em x}_{0},\lambda^{\rm F}(t'))}\nonumber\\
&\!\!\!\!\!\!\times\prod_{i=1}^{N} T(\textbf{\em x}_{i}|\textbf{\em x}_{i-1},
\lambda^{\rm F}(t_{i}))dt_i
e^{\int_{t_{i}}^{t_{i+1}}dt'T(\textbf{\em x}_{i}|\textbf{\em x}_{i},
\lambda^{\rm F}(t'))}.
\label{pathprob}\end{aligned}$$ We compare this probability to that of another trajectory $\vec{\textbf{\em x}}^*$, protocol $\lambda^*$, initial condition $P^{*}(\textbf{\em x}^*(0),0)$ and chosen dynamics, denoted $P^{*}$, and write $$A[\vec{\textbf{\em x}}]=\ln\left[{P^{\rm F}[\vec{\textbf{\em x}}]}
/{P^{*}[\vec{\textbf{\em x}}^{*}]}\right].
\label{entform}$$ Such a quantity may obey an integral fluctuation theorem (IFT) which may be derived by explicit summation over all possible paths, $\vec{\textbf{\em x}}$, for which $P^{\rm F}[\vec{\textbf{\em x}}]\neq 0$ as follows $$\begin{aligned}
\langle \exp{\left[-A[\vec{\textbf{\em x}}]\right]}\rangle^{\rm F}
&= \sum_{\vec{\textbf{\em x}}} P^{\rm F}[\vec{\textbf{\em x}}]
\exp{\left[-A[\vec{\textbf{\em x}}]\right]}=\sum_{\vec{\textbf{\em x}}}
P^{\rm F}[\vec{\textbf{\em x}}]\frac{P^{*}[\vec{\textbf{\em x}}^{*}]}
{P^{\rm F}[\vec{\textbf{\em x}}]}\nonumber\\
&=\sum_{\vec{\textbf{\em x}}^{*}} P^{*}[\vec{\textbf{\em x}}^{*}]=1.
\label{IFT}\end{aligned}$$ We assume a one to one mapping between $\vec{\textbf{\em x}}$ and $\vec{\textbf{\em x}}^{*}$ (a condition equivalent to a Jacobian of unity in the transformation) so that we may consider the summation over $\vec{\textbf{\em x}}^{*}$ to be equivalent to that over $\vec{\textbf{\em x}}$. We also require that $P^{*}[\vec{\textbf{\em x}}^{*}]= 0$ for all $P^{\rm F}[\vec{\textbf{\em x}}]= 0$ such that the final summation contains all possible paths $\vec{\textbf{\em x}}^{*}$, meaning the required normalisation of $P^{*}[\vec{\textbf{\em x}}^{*}]$ then yields the result of unity. A key result is the implication $\langle A[\vec{\textbf{\em x}}]\rangle^{\rm F}\geq 0$ by Jensen’s inequality.
A common choice for $P^{*}$, and that used to construct the total entropy production, is that of the normal dynamics under the reversed protocol, denoted $P^{*}=P^{\rm R}$. Given the specification of the normal dynamics we point out that all further specifications, including the choice of protocol, can be systematically derived from the appropriate path transformation $\vec{\textbf{\em x}}^{*}$ which we must choose carefully in conjunction with the dynamics so as to obey the above conditions. At this point we must be clear that given a transition $\textbf{\em x}\to \textbf{\em x}'$ under the normal dynamics, the transition $\textbf{\em x}'\to\textbf{\em x}$ is not, in general, possible under those same dynamics. Explicitly, we can construct models such that $T(\textbf{\em x}'|\textbf{\em x})\neq 0$ whilst $T(\textbf{\em x}|\textbf{\em x}')= 0$ (as an intuitive example: Hamiltonian dynamics cannot produce a negative positional step whilst the velocity is positive). The correct path, $\vec{\textbf{\em x}}^{*}$, to consider is the time reversed trajectory proper which includes a reversal of sign for all odd variables. This is the choice $\textbf{\em x}^*(t)=\textbf{\em x}^{\dagger}(t)=\boldsymbol{\varepsilon}
\textbf{\em x}(\tau\!-\!t)$ and it satisfies the condition $P^{*}[\vec{\textbf{\em x}}^{*}]=
P^{\rm R}[\vec{\textbf{\em x}}^{\dagger}]= 0$ for all $P^{\rm F}[\vec{\textbf{\em x}}]= 0$ required for an IFT. The reversed protocol $\lambda^{*}=\lambda^{\rm R}$ may be similarly obtained from the forward protocol, which may be treated as an even dynamical variable, meaning it transforms to yield $\lambda^{*}(t)=\varepsilon\lambda^{\rm F}(\tau\!-\!t)
=\lambda^{\rm F}(\tau\!-\!t)=\lambda^{\rm R}(t)$. And finally we require the choice of initial condition for the reverse path. This may be informed physically: we seek to characterise the irreversibility of the forward path and so initiate the reverse behaviour by time reversing the coordinates, $\textbf{\em x}(\tau)$, and distribution, $P^{\rm F}(\textbf{\em x}
(\tau),\tau)$, at the end of the forward process and evolve forward in time from there. The distribution can also be found by applying the transformation rules used to obtain the trajectory $\vec{\textbf{\em x}}^{\dagger}$ from $\vec{\textbf{\em x}}$ such that $P^{*}(\textbf{\em x}^{*}(0),0)=P^{R}(\textbf{\em x}^{\dagger}(0),0)
=\boldsymbol{\hat{\varepsilon}}P^{\rm F}(\boldsymbol{\varepsilon}
\textbf{\em x}(\tau),\tau)=P^{\rm F}(\boldsymbol{\varepsilon}
\boldsymbol{\varepsilon}
\textbf{\em x}(\tau),\tau)=P^{\rm F}(\textbf{\em x}(\tau),\tau)$ where $\boldsymbol{\hat{\varepsilon}}$ denotes the time reversal operation on the distribution. In this instance the path probability is therefore $$\begin{aligned}
P^{\rm R}[\vec{\textbf{\em x}}^{\dagger}]
&=P^{\rm R}(\textbf{\em x}^{\dagger}_0,0)e^{\int_{t_{0}}^{t_{1}}dt'
T(\textbf{\em x}^{\dagger}_{0}|\textbf{\em x}^{\dagger}_{0},
\lambda^{\rm R}(t'))}\nonumber\\
&\!\!\!\!\!\!\!\!\!\!\!\!\times\prod_{i=1}^{N} T(\textbf{\em x}^{\dagger}_{i}|
\textbf{\em x}^{\dagger}_{i-1},\lambda^{\rm R}(t_{i}))dt_i
e^{\int_{t_{i}}^{t_{i+1}}dt'
T(\textbf{\em x}^{\dagger}_{i}|\textbf{\em x}^{\dagger}_{i},
\lambda^{\rm R}(t'))}.\end{aligned}$$ We have $\textbf{\em x}_i^{\dagger}=\boldsymbol{\varepsilon}\textbf{\em x}_{N-i}$ so we may rearrange to give $$\begin{aligned}
&P^{\rm R}[\vec{\textbf{\em x}}^{\dagger}]=P^{\rm F}(\textbf{\em x}_N,\tau)
e^{\int_{t_{N}}^{t_{N+1}}dt'T(\boldsymbol{\varepsilon}\textbf{\em x}_{0}|
\boldsymbol{\varepsilon}\textbf{\em x}_{0},\lambda^{\rm R}(t'))}\\
&\!\!\!\!\!\!\!\!\!\times\prod_{i=1}^{N} e^{\int_{t_{N\!-\!i}}^{t_{N\!-\!i\!+\!1}}dt'
T(\boldsymbol{\varepsilon}
\textbf{\em x}_{i}|\boldsymbol{\varepsilon}
\textbf{\em x}_{i},\lambda^{\rm R}(t'))}
T(\boldsymbol{\varepsilon}\textbf{\em x}_{i\!-\!1}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm R}(t_{N\!-\!i\!+\!1}))dt_i\nonumber.\end{aligned}$$ We then perform a change of variable $t'\to \tau-t'$ and use $\lambda^{\rm R}(t_i)=\lambda^{\rm F}(t_{N-i+1})$ such that $$\begin{aligned}
&P^{\rm R}[\vec{\textbf{\em x}}^{\dagger}]=P^{\rm F}(\textbf{\em x}_N,\tau)
e^{-\int_{t_{1}}^{t_{0}}dt'T(\boldsymbol{\varepsilon}\textbf{\em x}_{0}|
\boldsymbol{\varepsilon}\textbf{\em x}_{0},\lambda^{\rm F}(t'))}\\
&\times\prod_{i=1}^{N} e^{-\int_{t_{i+1}}^{t_{i}}dt'T(\boldsymbol{\varepsilon}
\textbf{\em x}_{i}|\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t'))}
T(\boldsymbol{\varepsilon}\textbf{\em x}_{i-1}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t_{i}))dt_i\nonumber.\end{aligned}$$ A comparison of $P^{\rm F}[\textbf{\em x}]$ and $P^{\rm R}[\textbf{\em x}^{\dagger}]$ characterises the irreversibility of the forward path and defines the total entropy production (using units $k_B=1$) $$\begin{aligned}
\Delta S_{\rm tot}&=\ln{{P^{\rm F}[\vec{\textbf{\em x}}]}}-\ln{P^{\rm R}[
\vec{\textbf{\em x}}^{\dagger}]}\nonumber\\
&=\ln{\!\frac{P^{\rm F}(\textbf{\em x}_0,0)}
{P^{\rm F}(\textbf{\em x}_N,\tau)}}+
\sum_{i=0}^{N}\ln{\frac{e^{\int_{t_{i}}^{t_{i+1}}dt'T(\textbf{\em x}_{i}|
\textbf{\em x}_{i},
\lambda^{\rm F}(t'))}}{e^{\int_{t_{i}}^{t_{i+1}}dt'T(\boldsymbol{\varepsilon}
\textbf{\em x}_{i}|\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t'))}}}\nonumber\\
&+\sum_{i=1}^{N}\ln{\!\frac{T(\textbf{\em x}_{i}|\textbf{\em x}_{i-1},
\lambda^{\rm F}(t_i))}{T(\boldsymbol{\varepsilon}\textbf{\em x}_{i-1}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},\lambda^{\rm F}(t_i))}}
\label{Stot}\end{aligned}$$ which by its definition and Eq. (\[IFT\]) obeys [@seifertoriginal] $$\langle \exp{[-\Delta S_{\rm tot}]}\rangle^{\rm F}=1.$$ We find that this form of $\Delta S_{\rm tot}$ is more complicated than previous descriptions [@Harris07; @adiabaticnonadiabatic0] unless $\boldsymbol{\varepsilon}\textbf{\em x}=\textbf{\em x}$. Note that if detailed balance holds, such that $P^{\rm eq}(\textbf{\em x})
T(\textbf{\em x}'|\textbf{\em x})=P^{\rm eq}(\boldsymbol{\varepsilon}
\textbf{\em x}')T(\boldsymbol{\varepsilon}\textbf{\em x}|\boldsymbol{
\varepsilon}\textbf{\em x}')$, we expect $P^{\rm eq}$, the equilibrium state for a given $\lambda^{\rm F}(t)$, to satisfy $P^{\rm eq}(
\textbf{\em x})=P^{\rm eq}(\boldsymbol{\varepsilon}\textbf{\em x})$ due to time-reversal invariance, along with $T(\textbf{\em x}|\textbf{\em x})
=T(\boldsymbol{\varepsilon}\textbf{\em x}|\boldsymbol{\varepsilon}
\textbf{\em x})$. For a system in equilibrium, we therefore conclude that $\Delta S_{\rm tot}=0$ for all paths.
Next we consider alternative specifications of $P^{*}$. We consider the adjoint dynamics which lead to the same stationary state, $P^{\rm st}
(\textbf{\em x},\lambda^{\rm F}(t))$, as the normal dynamics, but generate flux of the opposite sign in that stationary state. It can be shown [@adiabaticnonadiabatic0; @Harris07; @Jarpathintegral] that this requires an adjoint transition rate matrix $T^{\rm ad}$ described by $$T^{\rm ad}(\textbf{\em x}|\textbf{\em x}',\lambda^{\rm F}(t))
=T(\textbf{\em x}'|\textbf{\em x},
\lambda^{\rm F}(t))\frac{P^{\rm st}(\textbf{\em x},\lambda^{\rm F}(t))}{
P^{\rm st}(\textbf{\em x}',\lambda^{\rm F}(t))}.
\label{adj1}$$ However, in the same way that the normal dynamics may not, in general, permit transitions $\textbf{\em x}'\to\textbf{\em x}$ or $\boldsymbol{\varepsilon}\textbf{\em x}\to
\boldsymbol{\varepsilon}\textbf{\em x}'$, similarly the adjoint dynamics may not, in general, permit transitions $\textbf{\em x}\to\textbf{\em x}'$ or $\boldsymbol{\varepsilon}\textbf{\em x}'\to
\boldsymbol{\varepsilon}\textbf{\em x}$. Thus we must consider the representation of the adjoint dynamics as either Eq. (\[adj1\]) or $$\!\!\!\!\!T^{\rm ad}(\boldsymbol{\varepsilon}\textbf{\em x}'|\boldsymbol{\varepsilon}
\textbf{\em x},\lambda^{\rm F}(t))=T(\boldsymbol{\varepsilon}\textbf{\em x}|
\boldsymbol{\varepsilon}\textbf{\em x}',\lambda^{\rm F}(t))\!\frac{P^{\rm st}(
\boldsymbol{\varepsilon}\textbf{\em x}',\lambda^{\rm F}(t))}{P^{\rm st}(
\boldsymbol{\varepsilon}\textbf{\em x},\lambda^{\rm F}(t))}
\label{adj2}$$ depending on the specific transition being considered. Explicitly, when choosing $P^{*}[\vec{\textbf{\em x}}^*]$, we should not consider $P^{\rm ad}[\vec{\textbf{\em x}}]$ or $P^{\rm ad}[\vec{\textbf{\em x}}^{\dagger}]$ since these might violate the required condition $P^{*}[\vec{\textbf{\em x}}^{*}]= 0$ for all $P^{\rm F}[\vec{\textbf{\em x}}]= 0$, required for an IFT.
Under the adjoint dynamics, however, an appropriate transformation of $\vec{\textbf{\em x}}$ is $\textbf{\em x}^{*}(t)=
\textbf{\em x}^{\rm R}(t)=\textbf{\em x}(\tau\!-\!t)$. Applying the transformation rules used to obtain $\vec{\textbf{\em x}}^{\rm R}$ yields the reverse protocol as before $\lambda^{*}(t)=\lambda^{\rm F}(\tau\!-\!t)=\lambda^{\rm R}(t)$ and the initial distribution $P^{*}(\textbf{\em x}^{*}(0),0)=
P^{\rm ad,R}(\textbf{\em x}^{\rm R}(0),0)=
P^{\rm F}(\textbf{\em x}(\tau),\tau)$. The path probability is then $$\begin{aligned}
&P^{\rm ad,R}[\vec{\textbf{\em x}}^{\rm R}]
=P^{\rm ad,R}(\textbf{\em x}^{\rm R}_0,0)
e^{\int_{t_{0}}^{t_{1}}dt'T^{\rm ad}(\textbf{\em x}^{\rm R}_{0}|
\textbf{\em x}^{\rm R}_{0},\lambda^{\rm R}(t'))}\nonumber\\
&\!\!\!\times\prod_{i=1}^{N} T^{\rm ad}(\textbf{\em x}^{\rm R}_{i}|
\textbf{\em x}^{\rm R}_{i-1},
\lambda^{\rm R}(t_{i}))dt_ie^{\int_{t_{i}}^{t_{i+1}}dt'T^{\rm ad}(
\textbf{\em x}^{\rm R}_{i}|
\textbf{\em x}^{\rm R}_{i},\lambda^{\rm R}(t'))}\nonumber\\
&=P^{\rm F}(\textbf{\em x}_N,\tau)e^{-\int_{t_{1}}^{t_{0}}dt'
T^{\rm ad}(\textbf{\em x}_{0}|
\textbf{\em x}_{0},\lambda^{\rm F}(t'))}\\
&\!\!\!\times\prod_{i=1}^{N} e^{-\int_{t_{i+1}}^{t_{i}}dt'T^{\rm ad}(
\textbf{\em x}_{i}|\textbf{\em x}_{i},\lambda^{\rm F}(t'))}
T^{\rm ad}(\textbf{\em x}_{i-1}|\textbf{\em x}_{i},
\lambda^{\rm F}(t_{i}))dt_i\nonumber.\end{aligned}$$ We then construct a quantity of the form given in Eq. (\[entform\]), utilise Eq. (\[adj1\]) and the property $T^{\rm ad}(\textbf{\em x}|\textbf{\em x})=
T(\textbf{\em x}|\textbf{\em x})$, valid by means of balance, to obtain $$\begin{aligned}
\Delta S_{\rm 1}&=\ln{{P^{\rm F}[\vec{\textbf{\em x}}]}}-\ln{P^{\rm ad,R}[
\vec{\textbf{\em x}}^{\rm R}]}\nonumber\\
%&=\ln{\!\frac{P^{\rm F}(\textbf{\em x}_0,0)}
%{P^{\rm F}(\textbf{\em x}_N,\tau)}}+
%\sum_{i=1}^{N}\ln{\!\frac{T(\textbf{\em x}_{i}|\textbf{\em x}_{i-1},
%\lambda^{\rm F}(t_i))}{T^{\rm ad}(\textbf{\em x}_{i-1}|\textbf{\em x}_{i},
%\lambda^{\rm F}(t_i))}}\nonumber\\
&=\ln{\!\frac{P^{\rm F}(\textbf{\em x}_0,0)}
{P^{\rm F}(\textbf{\em x}_N,\tau)}}
+\sum_{i=1}^{N}\ln{\!\frac{P^{\rm st}(\textbf{\em x}_{i},
\lambda^{\rm F}(t_i))}
{P^{\rm st}(\textbf{\em x}_{i-1},\lambda^{\rm F}(t_i))}}\end{aligned}$$ which through its definition and Eq. (\[IFT\]) obeys $$\langle \exp{[-\Delta S_{1}]}\rangle^{\rm F}=1$$ which exists in the literature as the Hatano-Sasa relation [@hatanosasa] or IFT for the non-adiabatic entropy production [@adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2]. Let us now consider, once again under the adjoint dynamics, the path transformation choice $\textbf{\em x}^{*}(t)=\textbf{\em x}^{\rm T}(t)
=\boldsymbol{\varepsilon}\textbf{\em x}(t)$. Applying the transformation rules we obtain the protocol $\lambda^{*}(t)=
\varepsilon\lambda^{\rm F}(t)=\lambda^{\rm F}(t)$ and initial distribution $P^{*}(\textbf{\em x}^{*}(0),0)=P^{\rm ad,F}(\textbf{\em x}^{\rm T}(0),0)=
\boldsymbol{\hat{\varepsilon}}
P^{\rm F}(\boldsymbol{\varepsilon}\textbf{\em x}(0),0)
=P^{\rm F}(\textbf{\em x}(0),0)$. The path probability for this case is therefore $$\begin{aligned}
&P^{\rm ad,F}[\vec{\textbf{\em x}}^{\rm T}]=
P^{\rm ad,F}(\textbf{\em x}^{\rm T}_0,0)
e^{\int_{t_{0}}^{t_{1}}dt'
T^{\rm ad}(\textbf{\em x}^{\rm T}_{0}|\textbf{\em x}^{\rm T}_{0},
\lambda^{\rm F}(t'))}\nonumber\\
&\quad\times\prod_{i=1}^{N}
T^{\rm ad}(\textbf{\em x}^{\rm T}_{i}|\textbf{\em x}^{\rm T}_{i-1},
\lambda^{\rm F}(t_{i}))dt_ie^{\int_{t_{i}}^{t_{i+1}}dt'
T^{\rm ad}(\textbf{\em x}^{\rm T}_{i}|
\textbf{\em x}^{\rm T}_{i},\lambda^{\rm F}(t'))}\nonumber\\
&=P^{\rm F}(\textbf{\em x}_0,0)e^{\int_{t_{0}}^{t_{1}}dt'T^{\rm ad}(
\boldsymbol{\varepsilon}\textbf{\em x}_{0}|
\boldsymbol{\varepsilon}\textbf{\em x}_{0},
\lambda^{\rm F}(t'))}\\
&\quad\times\prod_{i=1}^{N}
T^{\rm ad}(\boldsymbol{\varepsilon}\textbf{\em x}_{i}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i-1},\lambda^{\rm F}(t_{i}))dt_i
e^{\int_{t_{i}}^{t_{i+1}}dt'
T^{\rm ad}(\boldsymbol{\varepsilon}\textbf{\em x}_{i}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},\lambda^{\rm F}(t'))}\nonumber.\end{aligned}$$ By Eq. (\[entform\]) this then allows us to define $$\begin{aligned}
&\Delta S_{\rm 2}=\ln{{P^{\rm F}[\vec{\textbf{\em x}}]}}-\ln{P^{\rm ad,F}
[\vec{\textbf{\em x}}^{\rm T}]}\nonumber\\
&=\sum_{i=0}^{N}\ln{\frac{e^{\int_{t_{i}}^{t_{i+1}}dt'T(\textbf{\em x}_{i}|
\textbf{\em x}_{i},\lambda^{\rm F}(t'))}}{e^{\int_{t_{i}}^{t_{i+1}}dt'
T(\boldsymbol{\varepsilon}\textbf{\em x}_{i}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t'))}}}\nonumber\\
&+\sum_{i=1}^{N}\ln{\!\frac{P^{\rm st}(\boldsymbol{\varepsilon}
\textbf{\em x}_{i-1},
\lambda^{\rm F}(t_i))}{P^{\rm st}(\boldsymbol{\varepsilon}\textbf{\em x}_i,
\lambda^{\rm F}(t_i))}\frac{T(\textbf{\em x}_{i}|
\textbf{\em x}_{i-1},\lambda^{\rm F}(t_i))}{
T(\boldsymbol{\varepsilon}\textbf{\em x}_{i-1}|
\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t_i))}}
\label{S2}\end{aligned}$$ which similarly must obey $$\langle \exp{[-\Delta S_2]}\rangle^{\rm F}=1.
\label{S2IFT}$$ Unlike $\Delta S_1$, the quantity $\Delta S_2$ is new in the literature. We must immediately recognise that $\Delta S_{\rm tot}\neq \Delta S_1+\Delta S_2$ differing by a quantity $$\Delta S_3=\sum_{i=1}^{N}\ln{\!{\frac{P^{\rm st}(\textbf{\em x}_{i-1},
\lambda^{\rm F}(t_i))P^{\rm st}(\boldsymbol{\varepsilon}\textbf{\em x}_{i},
\lambda^{\rm F}(t_i))}{P^{\rm st}(\textbf{\em x}_i,\lambda^{\rm F}(t_i))
P^{\rm st}(\boldsymbol{\varepsilon}\textbf{\em x}_{i-1},
\lambda^{\rm F}(t_i))}}}
\label{S3}$$ such that $\Delta S_{\rm tot}=\Delta S_1+\Delta S_2+\Delta S_3$. If $\boldsymbol{\varepsilon}\textbf{\em x}=\textbf{\em x}$ then $\Delta S_3=0$ and $\Delta S_2$ reduces to the adiabatic entropy production appearing in [@adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2]. More importantly we must recognise that $\Delta S_{\rm tot}\!-\!\Delta S_1\!=
\!\Delta S_2\!+\!\Delta S_3=\ln{P^{\rm ad,R}[\vec{\textbf{\em x}}^{\rm R}]}-
\ln{P^{\rm R}[\vec{\textbf{\em x}}^{\rm \dagger}]}$ or $\Delta S_{\rm tot}\!-
\!\Delta S_2\!=\!\Delta S_1\!+\!\Delta S_3=
\ln{P^{\rm ad,F}[\vec{\textbf{\em x}}^{\rm T}]}-
\ln{P^{\rm R}[\vec{\textbf{\em x}}^{\rm \dagger}]}$ cannot be written in the form required for Eq. (\[IFT\]) and so do not obey an IFT and do not necessarily have any bounds on the sign of their mean. We proceed by following the formalism of Seifert [@seifertoriginal; @seifertprinciples] and write $$\Delta S_{\rm tot}=\ln{\frac{P^{\rm F}(\textbf{\em x}(0),0)}
{P^{\rm F}(\textbf{\em x}(\tau),\tau)}}+\frac{\Delta Q}{T_{\rm env}}=
\Delta S_{\rm sys}+\frac{\Delta Q}{T_{\rm env}},$$ where $T_{\rm env}$ is the temperature of the environment, and that of Oono and Paniconi, such that total heat transfer to the environment, $\Delta Q$, is the sum of the excess heat and house-keeping heat $\Delta Q=
\Delta Q_{\rm ex}+\Delta Q_{\rm hk}$ [@oono]. The house-keeping heat is associated with the entropy production in stationary states and arises from a non-equilibrium constraint that breaks detailed balance. The sum $\Delta S_2
+\Delta S_3$ is manifestly the entropy production in the stationary state and since we are considering Markov systems, both $\Delta S_2$ and $\Delta S_3$ are only non-zero when detailed balance is broken. Hence it is sensible to associate $\Delta S_2 +\Delta S_3$ with the house-keeping heat such that $$\Delta Q_{\rm hk}=(\Delta S_2\!+\!\Delta S_3)T_{\rm env}.$$ $\Delta S_1$ is zero for all trajectories in the stationary state consolidating the definition of the excess heat as the heat transfer associated with an entropy flow that exactly cancels the change in system entropy in the stationary state such that $$\Delta Q_{\rm ex}=(\Delta S_1\!-\!\Delta S_{\rm sys})T_{\rm env}.$$ However, the prevailing definition of the house-keeping heat does not make clear its properties when the system is not in a stationary state. A reported formalism suggests that it is associated with the adiabatic entropy production which serves as a general measure of the breakage of detailed balance [@adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2]. When considering cases where $\boldsymbol{\varepsilon}\textbf{\em x}=\textbf{\em x}$, this is a consistent approach and the mean house-keeping heat obeys strict positivity requirements suggesting the entropy additively increases due to non-equilibrium constraints and a lack of detailed balance on top of that arising from relaxation. However, with the inclusion of odd variables this simple picture no longer holds, with an ambiguity illustrated by the fact that any of $\Delta S_2$, $\Delta S_3$ or $\Delta S_2\!+\!\Delta S_3$ could be argued to be a measure of the departure from detailed balance. In the light of Eq. (\[S2IFT\]) we propose that it is sensible to divide the house-keeping heat into two quantities which map onto $\Delta S_2$ and $\Delta S_3$. It is important to observe that, on average, the rate of change of $\Delta S_3$ vanishes in the stationary state by means of balance: the path integral over an increment in $\Delta S_3$ explicitly vanishes. Consequently we define the ‘transient house-keeping heat’ and the ‘generalised house-keeping heat’ $$\Delta Q_{\rm hk,T}=\Delta S_3T_{\rm env}\quad
\quad\Delta Q_{\rm hk,G}=\Delta S_2T_{\rm env}$$ such that $\Delta Q_{\rm hk}=\Delta Q_{\rm hk,T}\!+\!\Delta Q_{\rm hk,G}$. Since $\langle d\Delta S_3/d\tau\rangle ^{\rm F,st}\!=\!0$, the generalised house-keeping heat, when averaged, has the mean properties previously attributed to the house-keeping heat: it describes the heat flow required to maintain a non-equilibrium stationary state and is rigorously non-negative. Our central result therefore is $$\langle \exp{[-\Delta Q_{\rm hk,G}/T_{\rm env}]}\rangle^{\rm F}=1$$ so $\langle\Delta Q_{\rm hk,G}\rangle^{\rm F}\geq 0$ for all times, protocols and initial conditions. As a corollary we also state that in general $$\langle \exp{[-\Delta Q_{\rm hk}/T_{\rm env}]}\rangle^{\rm F}\neq 1
\label{noIFT}$$ providing no bounds on $\langle\Delta Q_{\rm hk}\rangle^{\rm F}$ except in the stationary state when $\Delta S_1=0$ and $\Delta Q_{\rm hk}/T_{\rm env}=\Delta S_{\rm tot}$ or generally when $P^{\rm st}(\boldsymbol{\varepsilon}\textbf{\em x},
\lambda^{\rm F}(t))=P^{\rm st}(\textbf{\em x},\lambda^{\rm F}(t))$. As such the view that the mean rate of entropy production is the sum of two specific non-negative contributions as in [@adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2], is incomplete. The contribution associated with a non-equilibrium constraint requires further unravelling, particularly when out of stationarity.
![\[lattice\]Allowed moves between positions $X_i$ and $\pm$ velocity states are shown by arrows, with associated rates $T$. Periodic boundaries allow jumps from $X_L+$ to $X_1+$ and $X_{1}-$ to $X_{L}-$. A given path contributes to the transient and generalised house-keeping heats, $T_{\rm env}\Delta S_3$ and $T_{\rm env}
\Delta S_2$, respectively, due to transitions between, and residence times $\Delta t$ at, each phase space point, as indicated. These correspond to individual terms in the summations in Eqs. (\[S2\]) and (\[S3\]). ](lattice){width="\columnwidth"}
To explore the nature of the house-keeping heat we consider its behaviour in the approach to the stationary state of a simple model of particle dynamics on a ring. The phase space consists of $L$ identical spatial positions $X_{1},X_{2}\ldots X_{L}$ and two velocities labelled $+$ and $-$ as shown in Fig. \[lattice\] with the time reversal properties $\boldsymbol{\varepsilon}X_i\pm=X_i\mp$ necessitated by the one-way nature of many of the transitions. The stationary state probabilities that arise from these dynamics are $P^{\rm st}(X_i+)=A/(L(A+B))$ and $P^{\rm st}(X_i-)=B/(L(A+B))$. Any difference between the velocity reversal rates $A$ and $B$ gives rise to a non-equilibrium stationary state by providing a stationary particle current, which for $A>B$ runs from left to right. Such dynamics amount to a very simple lattice Boltzmann model. Contributions $\Delta S_2$ and $\Delta S_3$ associated with particle behaviour consisting of instantaneous transitions and waiting periods are indicated. We consider particle behaviour over a small time interval $dt$, and compute the mean entropy production rates to leading order in $dt$. Examining the path probability in Eq. (\[pathprob\]) we need only consider $N=0$ or $N=1$ transitions. Identifying leading order terms in the products of $P$, $T$, exponentiated waiting times and $\Delta S_3$ that make up the average of the form given in Eq. (\[IFT\]) yields $$\frac{d\langle\Delta S_3\rangle^{\rm F}}{dt}=\sum_{i=1}^{L}
2P(X_i+)B\ln{\frac{A}{B}}+2P(X_i-)A\ln{\frac{B}{A}}.
\label{S3dt}$$ For non-stationary $P$ its sign is unbounded: for example if all the probability were uniformly distributed initially amongst the $+$ velocity states it would equal $2B\ln(A/B)$, whilst if it were distributed over the $-$ states it would be $-2A\ln(A/B)$ instead. Such non-zero contributions to $\Delta S_3$ require an asymmetric stationary state in odd variables which thus explains their absence when the stationary velocity distribution is assumed to be symmetric, such as in overdamped Langevin descriptions (see [@IFThousekeeping] and examples in [@adiabaticnonadiabatic2]). However, in the stationary state with $P=P^{\rm st}$, $d\langle\Delta
S_3\rangle^{\rm F}/dt$ is demonstrably equal to zero as claimed. By similar means $$\begin{aligned}
\frac{d\langle\Delta S_2\rangle^{\rm F}}{dt}=\sum_{i=1}^{L}&P(X_i+)
\left[A-B-B\ln{\frac{A}{B}}\right]\nonumber\\
&+P(X_i-)\left[B-A-A\ln{\frac{B}{A}}\right]
\label{S2dt}\end{aligned}$$ which is positive for all positive $A$ and $B$ and reduces to $d\langle\Delta S_2\rangle^{\rm F,st}/dt=(A-B)^2/(A+B)$ in the stationary state. We note that the sum of Eqs. (\[S3dt\]) and (\[S2dt\]) has no bound on its sign and relates to the inequality in Eq. (\[noIFT\]). Further, $d\langle\exp{[-\Delta S_{2}]}
\rangle^{\rm F}/dt=0$ and $\langle \exp{[-\Delta S_2(t=0)]}
\rangle^{\rm F}=1$ which explicitly demonstrates the expected IFT for any normalised $P(X_i\pm)$. Finally, we note that for $A=B$, all contributions vanish in detail as this corresponds to equilibrium where there is no entropy production.
We have extended the formalism found in [@hatanosasa; @IFThousekeeping; @adiabaticnonadiabatic0; @adiabaticnonadiabatic1; @adiabaticnonadiabatic2] and split the total entropy production into two rigorously positive contributions and a third contribution which has no bounds on its sign. We have argued that this final quantity is, in the mean, a transient contribution to the house-keeping heat and it is the mean generalised house-keeping heat that is rigorously positive for all times. It is not straightforward to consolidate this with the two causes of time reversal asymmetry namely relaxation to the stationary state and imposed non-equilibrium constraints: $\Delta S_3$ exists only in the presence the latter, but is, in the mean, its own measure of relaxation to the stationary state. It could be argued that the non-adiabatic entropy production and Hatano-Sasa relation do not fully capture the entropy production due to transitions between stationary states, but associating $\Delta S_3$ with one or other form of entropy production is not entirely satisfactory as it occurs when the line between them is blurred. Nevertheless, either interpretation elucidates a new layer of complexity in the theory of entropy production in stochastic systems. Further exploration in the context of continuous stochastic processes is to be reported elsewhere [@SpinneyFord]. The authors acknowledge financial support from EPSRC. \[1\]\[1\][\#1]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The kinetics of the self-assembly of supramolecular polymers is dictated by how monomers, dimers, trimers etc., attach to and detach from each other. It is for this reasons that researchers have proposed a plethora of pathways to explain the kinetics of various self-assembling supramolecules, including sulfur, linear micelles, living polymers and protein fibrils. Recent observations hint at the importance of a hitherto ignored molecular aggregation pathway that we refer to as “*body evaporation and addition*”. In this pathway, monomers can enter at or dissociate from any point along the backbone of the polymer. In this paper, we compare predictions for the well-established *end evaporation and addition* pathway with those that we obtained for the newly proposed *body evaporation and addition* model. We quantify the lag time, characteristic of nucleated reversible polymerisation, in terms of the time it takes to obtain half of the steady-state polymerised fraction and the apparent growth rate at that point, and obtain power laws for both as a function of the total monomer concentration. We find, perhaps not entirely unexpectedly, that the *body evaporation and addition* pathway speeds up the relaxation of the polymerised monomeric mass relative to that of the *end evaporation and addition*. However, the presence of the *body evaporation and addition* pathway does not affect the dependence of the lag time on the total monomer concentration and it remains the same as that for the case of *end evaporation and addition*. The scaling of the lag time with the forward rate is different for the two models, suggesting that they may be distinguished experimentally.'
author:
- 'Nitin S. Tiwari'
- Paul van der Schoot
title: 'On the Kinetics of Body *versus* End Evaporation and Addition of Supramolecular Polymers'
---
Introduction {#sec1}
============
Supramolecular polymerisation processes are of immense importance in biology and in chemistry. [@tomdegreef] Some classic examples in biology include actin and microtubule self-assembly that play important roles in the context of the mechanics of the cell, and $\beta-$amyloid and prion protein aggregation implicated in neuro-degenerative diseases. [@takalo; @blanchoin; @fletcher; @chiti; @murphy] Similarly, supramolecular polymerisation has major applications in the chemistry of medicine and in molecular electronics. [@aida] In this light, it is not surprising that researchers have long studied the thermodynamic and kinetics of self-assembly. As the time evolution of self-assembly is very much system specific, in particular the early time kinetics, itself the most extensively studied aspect of reversible polymerisation, a whole host of molecular pathways of supramolecular self-assembly have been proposed. [@oosawa; @cates_all; @cohen]
It is generally believed that Oosawa was the first to suggest a model in the context of the polymerisation of actin filaments, where a monomer can be added to or removed from the ends. [@oosawa; @cohen] Oosawa’s model has one important ingredient, known as *nucleation*. This means that a stable critical nucleus of $n_c \ge 1$ monomers must be formed before polymer growth commences. Although Oosawa’s model of self-assembly is in agreement with experimental data in the context of actin polymerisation, it fails to describe many other protein aggregation processes. [@curve_fitting] Indeed, the prevalent molecular pathway for self-assembly is dictated by the complex molecular structure of the monomers involved, as well as by the type of bonding between monomers that form a polymer. This results in molecular pathways that are more complex than the simple pathway proposed by Oosawa, which is sometimes also referred to as *end evaporation and addition* [@dubbeldam; @cates_eea; @semenov_eea].
In the context of the living polymerisation or linear polymers, researchers proposed a plethora of pathways by which self-assembly can occur, e.g., polymer *scission and recombination* [@cates_sr1; @semenov_sr; @cates_sr], *secondary nucleation* [@cohen; @tuomas_review] and *two-stage nucleation*. [@knowles_two_stage] Further work shows that the kinetics of self-assembly is strongly dependent on which of the above mentioned pathways are active in the assembly process. [@curve_fitting] The influence molecular aggregation pathways have on the early time kinetics of linear self-assembly, which is the most studied aspect of the problem in hand, motivates researchers to study and characterize all possible pathways. [@tuomas_science; @cohen; @knowles_two_stage; @Tiwari_2016; @tuomas_review]
With the aim to probe the molecular pathway responsible for linear self-assembly in the context of supramolecular polymerisation, Albertazzi *et al.* recently performed experiments with a self-assembling molecule known as 1,3,5 benzenetricarboxamide or BTA for short. [@albertazzi] By imaging the supramolecular polymers at different assembly times, they were able to investigate monomer mixing on the scale of individual polymers. Because their observations could not be explained by any of the hitherto known molecular pathways, they suggested the need to revisit theoretically and experimentally the dynamic behaviour of supramolecular polymers. From their observations, they conclude that the molecular pathway responsible for self-assembly of BTAs is the one in which the monomer can be removed from and inserted anywhere along the polymer backbone.
We have schematically depicted the novel pathway in Fig. \[fig1\]. At time $t=0$ the solution contains only two types of supramolecular homopolymer, and as time progresses mixing of monomers occurs at the supramolecular level. However, contrary to crowding of differently labeled monomers at the ends, which is to be expected if monomers can only attach on and detach from the ends, the mixing takes place homogeneously along the polymer backbone. We call their proposed pathway the “*body evaporation and addition*” pathway to contrast it with the conventional *end evaporation and addition* pathway, and study theoretically the kinetics of this pathway in the presence of the *scission and recombination* pathway and *primary nucleation*. [@oosawa] The molecular pathway *scission and recombination* allows polymers to break at any point on the backbone resulting into two smaller polymers and recombining two polymers into a longer polymer. *Primary nucleation* is the mechanism by which a critical number of monomers spontaneously self-assemble to form the shortest stable polymer.[@oosawa] Many researchers have concluded that the time evolution of the length distribution of living polymers is typically a mixture of molecular pathways, which are switched on and off dependent on the system of interest. [@tuomas_science; @hong; @hong_moment_closure] It is for this reason that we study the kinetics of the newly proposed pathway in combination with the molecular pathways already referred to. At first glance, *body evaporation and addition* pathway looks similar to *end evaporation and addition*, and naively one would perhaps presume that a simple renormalisation of the rate constants can account for the former. However, a closer look at the problem reveals that for *end evaporation and addition* every polymer has only two ends, resulting into a probability of addition or removal of monomer at the ends that plausibly is independent of the length of the polymer. In contrast, in the case of *body evaporation and addition* the probability of adding or removing a monomer is proportional to the number of bonds in a polymer in which a monomer is being added. Hence, the addition or removal of a monomer along the polymer backbone in the *body evaporation and addition* depends on the size of that particular polymer. As the size of an individual polymer changes as a function of time, it is not possible to simply renormalise the rate constants associated with *body evaporation and addition* pathway and expect it to behave like *end evaporation and addition*.
To study the kinetics of linear self-assembly we start by writing the discrete rate equations. However, the rate equations are highly nonlinear and have so far eluded exact analytical solution except in a few limiting cases. [@cohen; @hong] Hence, we study the kinetics of self-assembly by closing the discrete reaction rate equations by insisting on plausible approximations. This way, we obtain dynamical equations for the first two moments of the polymer length distribution. These are the number of polymers and the polymerised monomeric mass, of which the latter is primarily probed in assembly experiments. [@tuomas_science; @hellstrand; @tuomas_review] We obtain asymptotic analytical solutions of the resulting dynamical equations. From our analytical solutions we quantify the early time kinetics and show that the scaling of the lag time with the total monomer concentration is identical to that of the standard *end evaporation and addition* pathway even in the presence of the proposed *body evaporation and addition*. However, the lag time significantly does decreases with increasing predominance of *body evaporation and addition* pathway. We also show that when only one of the two addition and evaporation pathways is present, the half-time and the apparent growth rate differ in their scaling with the forward rate constants of monomer insertion.
![Schematic showing the effect of *body evaporation and addition* pathway on the mixing and growth of differently dye-labeled but otherwise identical monomers at time $t=0$. Note the homogeneous insertion of black monomers along the green polymer backbone and *vice versa* via the free monomers in the solution.[]{data-label="fig1"}](Fig1.eps){width="6.5in"}
The remainder of this paper is organised as follows: In Section \[sec2\], we introduce the moment equations for the generalized molecular pathway and study the equilibrium properties of the two moments of the length distribution. In Section \[sec3\], the dynamical equations for the moments are solved in the absence of polymer recombination and in the limit of vanishing fragmentation and nucleation rate constants. The analytical solutions are then compared with numerical solutions, and used to calculate the half-time and the apparent growth rate for the polymerisation kinetics. Finally, in Section \[sec4\], we discuss the results and findings of our theoretical analysis.
Master Equations and Moment Equations {#sec2}
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The molecular pathways that we include in our study are i) *primary nucleation*, ii) *end evaporation and addition*, iii) *body evaporation and addition* and iv) *scission and recombination*. [@cates_all; @knowles_two_stage; @Tiwari_2016] Our moment equations, derived from the generalized rate equations, allow us to probe the quantitative role of the various molecular pathways involved. The molecular pathway of interest can in principle always be made dominant by switching off other pathways completely or asymptotically. In order to study the kinetics of nucleated polymerisation we first transform the molecular pathways into the corresponding reaction representation. $$\begin{aligned}
\label{eq1}
& \text{i) primary nucleation} \hspace{85pt} n_c x \xrightleftharpoons[k_n^-]{k_n^+} y_{n_c}, \\
& \text{ii) end evaporation and addition} \hspace{26pt} y_i + x \xrightleftharpoons[2 k_e^-]{2 k_e^+} y_{i+1}, \\
& \text{iii) body evaporation and addition} \hspace{19pt} y_i + x \xrightleftharpoons[k_b^-(i-1)]{k_b^+(i-1)} y_{i+1}, \\
& \text{iv) scission and recombination} \hspace{40pt} y_i + y_j \xrightleftharpoons[k_f^-]{k_f^+} y_{i+j} \quad \text{for} \quad i,j \le n_c,\end{aligned}$$ where $x$ and $y_i$ are the concentration of monomers and that of polymers of degree of polymerisation $i$, respectively, and $n_c$ is the critical nucleus size, i.e., the degree of polymerisation of smallest stable polymer. Furthermore, $k_n^+$, $k_n^-$, $k_e^+$, $k_e^-$, $k_b^+$, $k_b^-$, $k_f^+$ and $k_f^-$ are the rates of nucleus formation and disintegration (subscript n), monomer addition and removal from the ends (subscript e), the rates of monomer addition and removal from the polymer backbone excluding the ends (subscript b) and polymer recombination and scission (subscript f). The factor of $i-1$ in reaction iii), describing body evaporation and addition, accounts for the fact that a monomer can be added in $i-1$ places on the backbone a polymer of size $i$, and that any one of $i-1$ monomers can be removed from a polymer of size $i+1$ because removal from the ends is forbidden for this pathway.
The reactions are assumed to be reaction limited rather that diffusion limited, implying that the reaction rates are constant in time. The indices $i$ and $j$ for reaction iv) obey $i,j \ge n_c$, where $n_c \ge 2$. We consider the case $n_c \ge 2$ in order to be able to close the sums in the master equation and obtain the dynamical equations for the first two principal moments of the polymer length distribution. In the case $n_c \ge 2$, a monomer is not counted as a polymer, whereas for $n_c=1$, a monomer can be an active monomer, i.e., a polymer of size one, or an inactive monomer. Hence, the master equations for $n_c \ge 2$ are fundamentally different from that for $n_c=1$ polymerisation.
One additional assumption that we employ in order to close the sums in our master equations is that we assume $k_n^-=k_e^-$. Later we will see that this approximation does not alter our results because in order to be able to close our discrete master equation, we neglect disintegration of a nuclei via the *end evaporation and addition* pathway. Our approximation of irreversible nucleus formation has been employed in the past by several researchers and the results have been quantitatively compared with the experimental data on protein polymerisation, justifying our approximation. [@tuomas_science; @hong; @tuomas_review]
Before we delve deeper into our analysis, a few remarks should be made. In principle, we consider four mechanisms that are responsible for the time evolution of the length distribution: the *primary nucleation*, the *end evaporation and addition*, the *body evaporation and addition* and the *scission and recombination* pathway. Our goal in this work is very specific and is to compare the *early time* kinetics of the two pathways of interest, which are the *end evaporation and addition* and the *body evaporation and addition*. However, if we do not include the *scission and recombination* pathway, the resulting dynamical equations will be singular, meaning that we can make a parameter corresponding to this pathway small but never put it to zero. Hence, it is for purely mathematical reasons that we make use of the most general description that formally includes all the aforementioned pathways. This is also the reason why we implement *primary nucleation*. To make the problem mathematically tractable we work in the limit of strongly nucleated systems, where for the time domain of our interest the primary nucleation step is not functional and hence can be ignored for all practical purposes. This said, we will explain in detail all of our approximations and limitations, as and when they come in this paper.
To derive a closed form of the moment equations, we start with the discrete master equation for the reaction schemes defined above. For the polymers, this yields \[eq2\] &=& k\_n\^+ x(t)\^[n\_c]{} \_[i,n\_c]{} + 2 x(t) y\_[i-1]{}(t) - 2 x(t) y\_[i]{}(t) + 2 y\_[i+1]{}(t) - 2 y\_i(t)\
& & + (i-2) x(t) y\_[i-1]{} - (i-1) x(t) y\_[i]{} + (i-1) y\_[i+1]{} - (i-2) y\_i\
& & - (i- 2 n\_c +1) y\_i(t) + 2 \_[j=i+n\_c]{}\^ y\_j(t) + \_[k+l=i]{} y\_k(t) y\_l(t)\
& & - 2 y\_i (t) \_[j=n\_c]{}\^ y\_j, where the first term on the right-hand-side of Eq. \[eq2\] accounts for the formation of the critical nucleus, the next four terms stem from the *end evaporation and addition* pathway, and terms six to nine represent the *body evaporation and addition*. The last four terms result from *scission and recombination*. Here, $\delta_{i,n_c}$ denotes Kronecker delta that acquires value of 1 when $i=n_c$ and is zero otherwise. Notice that Eq. (5) is missing the term for nucleus disintegration. This is due to our approximation of $k_n^- = k_e^-$ that allows nucleus to disintegrate via monomer removal from an end hence the nucleus disintegration term gets absorbed in the end evaporation terms.
The factor of $(i- 2 n_c +1)$ in the tenth term on the right-hand side of Eq. \[eq2\] accounts for the number of bonds allowed to break such that the fragmenting filaments are larger than the nucleus size $n_c$. It should be mentioned that the same term, in principle, should include a factor of $\theta(i-2 n_c)$. However, this factor would prevent us from closing the summations and obtaining the dynamical equations for the first two moments of the full polymer length distribution. Hence, we choose not to include it in our master equation. As a consequence, an inconsistency arises for the dimers and trimers, at least if we focus on the case of $n_c=2$. Indeed, this choice would in principle allow a dimer and a trimer to break via the polymer scission mechanism, yet this is disallowed in our way of implementation of reaction schemes. The reason is that in our final analysis we set the limit of $k_f^- \rightarrow 0$, justifying our approximation. Lastly, in the eleventh term, i.e., polymer scission term, the lower limit $i+n_c$ makes sure that two fragments post-scission are stable polymers of size greater than or equal to $n_c$. The condition of the conservation of mass finally results into a time-dependent equation for the monomers \[eq3\] = -( \_[i=n\_c]{}\^ i y\_i(t) ).
Eq. \[eq2\] is different from previously obtained master equations on account of the additional terms that describe the contribution of the body evaporation $\kmb$ and addition $\kpb$. [@cohen; @hong] We also implement the *scission and recombination* pathway to allow for polymer fragmentation resulting only into fragments greater than or equal to the critical size; the recombining polymers also have to be of the size $i \ge n_c$. The reason for this is that we assume that any fragment of size $l \le n_c$ is highly unstable and quickly disintegrates to $l$ monomers. This prohibits the recombination of fragments smaller than the critical nucleus, as they do not exist in a polymeric state. Additionally, by disallowing the fragmentation that results in a fragment smaller than the critical size we prevent this step from contributing to the free monomeric pool. As a consequence, we completely decouple the *end evaporation and addition* from the *scission and recombination* pathway, i.e., one of the pathways can be switched on or off without affecting the other. [@cohen]
Our master equation as those very much like it discussed at length in the literature are being highly non-linear equations, and have so far eluded exact analytical solution. [@tuomas_review] Hence, a standard practice in the field is not to study the full length distribution, but only the first two moments of it. [@oosawa; @cohen] These are the polymer concentration, $P$, and the polymerised monomeric mass, $M$. The dimensionality of both are in moles per liter if the rate constants are given in molar units. Of these two quantities, the latter quantity is readily measured by means of, e.g., circular dichroism or fluorescence microscopy. [@hellstrand; @kelly_cd] The number concentration of polymers can in principle be quantified by measuring the mean size of the polymers, using techniques such as static and dynamic light scattering, and calculating the ratio of the polymerised mass to the mean degree of polymerisation. [@zhang_rev]
The two principal moments expressed in our variables read \[eq4\] P = \_[i=n\_c]{}\^ y\_i, for the polymer concentration and \[eq5\] M = \_[i=n\_c]{}\^ i y\_i, for the polymerised mass. We obtain the dynamical equations for $P$ and $M$ by extracting the first two principal moments from the full polymer length distribution described by Eq. \[eq2\]. In the process of deducing the dynamical equations for the moments, the only approximation we employ is that we neglect all terms arising from the disintegration of nuclei, which for early times are negligible in number anyway, at least in the limit $k_n^+ \rightarrow 0$. See Appendix A for details.
The dynamical equation for the number of polymers $P(t)$ that we obtain reads \[eq6\] &=& - P(t)\^2 + + k\_n\^+ \^[n\_c]{}, and for the time evolution of the the polymerised monomeric mass $M(t)$ we find \[eq7\] &=& 2 + (-M(t)) (M(t)+P(t))\
& &+ (2 P(t)-M(t))+ n\_c k\_n\^+ (-M(t))\^[n\_c]{}, where $\mtot$ is the total concentration of monomers in the system. A detailed derivation of Eqs. \[eq6\] and \[eq7\] from Eqs. \[eq2\] and \[eq3\] is provided in Appendix A. For this general set of equations the initial polymerised monomeric mass can have any value between and including, 0 and $\mtot$, i.e., $0\le M(0) \le \mtot$. The same holds true for $P(0)$ and $M(t) \ge P(t)$ for all times, where the equality holds only when all the polymers are critical nuclei.
In the absence of the *body evaporation and addition* terms, Eqs. \[eq6\] and \[eq7\] have been compared with kinetic Monte Carlo simulations that do allow for the disintegration of nuclei. [@Tiwari_2016] As expected, the time evolution of the moments obtained from the simulation are in quantitative agreement with Eqs. \[eq6\] and \[eq7\] in the appropriate limit of strongly nucleated polymerisation, justifying our approximation. With regard to the domain of validity of the Eqs. \[eq6\] and \[eq7\], they produce nonzero and positive $M(\infty)$ and $P(\infty)$ only above the so-called critical concentration of monomers. [@paul_review] In the absence of other pathways, except *end evaporation and addition*, the critical concentration obtained from our kinetic equations has a simple analogy to that of the thermodynamic theory of linear self-assembly. [@paul_review] However, in the presence of more complex pathways, the mapping is not so trivial as the kinetic theory demands the introduction of additional energy scales associated with the various pathways. For example, an energy scale for monomer removal or addition along the backbone of a polymer would be needed to characterise the *body evaporation and addition* in addition to an energy scale associated with the monomer addition to or removal from the ends of a polymer.
Notice that Eqs. \[eq6\] and \[eq7\] are highly nonlinear offering little hope of exact analytical solution. This implies that we have not improved the state of affairs significantly in the context of the various simplifications of the equations that we have already implemented. Of course, one may linearise Eqs. \[eq6\] and \[eq7\] and obtain linear solutions, but this is of limited help as a linear solution can never result in sigmoidal kinetics. Hence, with the aim to reduce the nonlinearity of the equations but preserve the most important and generic behaviour of the kinetics of self-assembly, we employ several additional approximations. Our first approximation is to restrict our analysis to the limit $k_n^+ \rightarrow 0$. This way we reduce the degree of the polynomial, which is essential to obtain an analytical solution. In addition, we also neglect the polymer recombination $\kpf P(t)^2$ term in Eq. \[eq6\]. By doing so, we break the reversibility condition and hence do not expect the system to follow the corresponding law of mass action. However, earlier studies on nucleated self-assembly have shown that in the context of the early time kinetics we focus attention on, polymer recombination does not play a significant role and hence can be ignored as in fact we shall also make plausible below. [@knowles_two_stage] Indeed, by identifying the most dominant terms in the dynamical equations for the polymerisation kinetics, the predicted lag phase has been shown to be in a quantitative agreement with theoretical models, at least in the absence of *body evaporation and addition*. [@tuomas_science; @knowles_two_stage; @hong; @hong_moment_closure; @subramaniam] This motivates us to do the same for our reaction pathway, despite it potentially being inaccurate in the long-time limit.
The steady-state solution of Eqs. \[eq6\] and \[eq7\] sheds some light on the role of molecular pathways and how our approximations, i.e., $k_n^+ \rightarrow 0$ and $k_f^+ P^2=0$, impact upon the long-time behaviour. In the steady state, the time derivatives of the two moments are zero and we are left with algebraic equations. If we define $K_f=\kpf/\kmf$ to be the fragmentation equilibrium constant, we can switch off the effect of *scission and recombination*, at least in equilibrium, by taking the asymptotic limit $K_f=0$. If $k_f^+=0$ this will be true for any value of $k_f^- \neq 0$. Within this approximation we can compare the effect of *end evaporation and addition* and *body evaporation and addition* explicitly. In this limit the equilibrium polymerised monomeric mass for a nucleus size of $n_c=2$ becomes \[eq8\] M() = -. In the absence of *body evaporation and addition*, i.e., for $\kpb=\kmb=0$, the polymerised monomeric mass becomes $M(\infty)=\mtot-\kme/\kpe=\mtot-K_e^{-1}$, where $K_e \equiv \kpe/\kme$. In the absence of *body evaporation and addition*, the ratio of rate constants $K_e$ can be mapped onto the equilibrium constant used in the thermodynamic theory of linear self-assembly. [@paul_review] In that case, in the polymerised regime, $y_1(\infty)= K_e^{-1}$, remains equal to the critical concentration. Hence, in this limit the polymerised monomeric mass is in agreement with the thermodynamic theory of self-assembly. [@paul_review] However, in the presence of *body evaporation and addition* pathway the effective elongation constant has to be renormalized to account for the free energy associated with monomer addition to or removal from the ends and that associated with monomer addition or removal along the backbone of the polymer.
In the limit $K_f \rightarrow 0$ the steady-state number of polymers $P$ and average degree of polymerisation $L$ are given by \[eq9\] P() &=& , and \[eq10\] L() &=& = 2 n\_c -1 for all values of $\kpb$ and $\kmb$. Note that the equilibrium average degree of polymerisation $L(\infty)$ only depends on the critical nucleus size, $n_c \ge 2$, not on the concentration. This, of course, disagrees with the thermodynamic theory but, as we shall see below, this does not preclude very large values of $L(t)$ for intermediate times. [@paul_review] Similar results were obtained by Cohen *et al.* in the absence of *body evaporation and addition*. [@knowles_two_stage]
Lag Time Analysis {#sec3}
=================
The steady-state solutions of the moment equations clearly indicate that our generalized reaction schemes do not result into the long polymers in the limit of $t \rightarrow \infty$ to be expected for nucleated polymerisation. However, as shown in Fig. \[fig2\], for intermediate times the mean polymer length overshoots and attains very large values. Also, Fig. \[fig2\] shows that the mean polymer length for very large and very small values of recombination rate constant, $k_f^+=10^{8}, 10^{-8}$, $M s^{-1}$ the term $k_f^+ P^2$ has no effect on the early time kinetics for any reasonable value of $k_f^+$. This justifies our approximation of neglecting the term representing the contribution of polymer recombination as in this paper we are mainly interested in comparing the early time kinetics of *end evaporation and addition* and *body evaporation and addition*. The early time kinetics is characterised by the lag time, i.e., the time intercept of the tangent at the inflection point of the polymerisation curve, i.e., $M(t)$. [@hellstrand] For the sake of simplicity, we limit ourselves to strongly nucleated polymerisation, and set $k_n^+ \rightarrow 0$. This also reduces the degree of polynomial on the right hand side of Eqs. \[eq6\] and \[eq7\], enabling us to obtain analytical solutions. In the limit $k_n^+ \rightarrow 0$, the polymerisation process has to be seeded, i.e., some initial polymerised mass, $M(0)=P(0) \neq 0$, has to be provided, otherwise the system stays in the initial state $M(0)=P(0)=0$.
![Time evolution of the mean length of the polymer $L(t)=M(t)/P(t)$, obtained by numerically solving Eqs. \[eq6\] and \[eq7\] for two values of recombination rate $\kpf=10^{8}$ and $10^{-8} M s^{-1}$. The remainder of the system parameters are $k_n^+=10^{-5} s^{-1}$, $\kpe=5 \times 10^{5} M^{-1} s^{-1}$, $\kme=10^{-2} s^{-1}$, $\kpb=5 \times 10^{5} M^{-1} s^{-1}$, $\kmb=10^{-2} s^{-1}$, $\kmf=10^{-4} s^{-1}$, $\mtot=10^{-5} M$, $M(0)=\mtot \times 10^{-4}$ and $P(0)=\mtot \times 10^{-6}$, where $M$ is moles per liter and $s$ is in seconds. The values of the chosen rate constants do not correspond to any particular experiment. However, the order of magnitude is similar to the parameters found in the literature of protein polymerisation. [@tuomas_review] The initial conditions are chosen to be small but non-zero, because of the necessity to seed the polymerisation process in the limit of $k_n^+ \rightarrow 0$.[]{data-label="fig2"}](Fig2.eps){width="4in"}
The resulting dynamical equations become, after employing the approximations and rearranging terms in Eqs \[eq6\] and \[eq7\], \[eq11\] &=& ( M(t)-(2 n\_c -1) P(t) ), for the polymer number concentration, and \[eq12\] &=& -M(t) (+ M(t)) + P(t) (- M(t)), for the polymerised mass, where we introduce new dynamical constants $\alpha \equiv \kmb -\kpb \mtot$, $\beta \equiv 2(\kmb-\kme)+\kpb\mtot+2\kpe \mtot$ and $\gamma \equiv \kpb+2\kpe$. It should be emphasized that we define the parameters $\alpha$, $\beta$ and $\gamma$ merely for notational simplicity and do not associate any physical meaning to them. Eqs. \[eq11\] and \[eq12\] are nonlinear in nature, hence, the first trivial analysis demands the linearisation of these equations. Researchers have analysed similar equations for variety of pathways and obtained $t^2$ or $t^3$ time dependence for the polymerised mass fraction. [@ferron_early] However, for our particular choice of molecular pathways, the resulting linearised equations always yield simple exponential time dependence that in turn result into a linear time dependence of early times.
![Time evolution of the polymerised mass fraction $M(t)/M(\infty)$ and the polymer concentration $P(t)/P(\infty)$ obtained by numerically solving Eqs. \[eq6\] and \[eq7\] for $\kpf=0$ $M^{-1} s^{-1}$. The remainder of the system parameters are $k_n^+=10^{-5} s^{-1}$, $\kpe=10^{5} M^{-1} s^{-1}$, $\kme=10^{-2} s^{-1}$, $\kpb=10^{5} M^{-1} s^{-1}$, $\kmb=10^{-2} s^{-1}$, $\kmf=10^{-4} s^{-1}$, $\mtot=10^{-5} M$, $M(0)=\mtot \times 10^{-5}$ and $P(0)=\mtot \times 10^{-6}$, where, $M$ is moles per liter and $s$ is in seconds. The inset shows the early time behaviour of the number of polymers $P(t)$, which remains essentially constant during the relaxation of the polymerised monomeric mass, $M(t)$, and only relaxes after that. This figure highlights our claim of separation of time scales between the time evolution of the polymerised mass fraction and that of the polymer concentration. See the main text.[]{data-label="fig3"}](Fig3.eps){width="6.5in"}
Although the dynamical equations Eqs. \[eq11\] and \[eq12\] contain the polymer scission term, we can reduce the effect of polymer scission on the kinetics of the polymerised monomeric mass, $M$, by choosing a very small value for the scission rate constant $\kmf$. The value of $\kmf=10^{-4}$ that we chose is small enough to diminish the effect of scission for the early time kinetics of the polymerised mass, as we find and also show below that this is true for our choice of set of parameters. For this small value of $\kmf$, the typical trajectory of the two moments is shown in Fig. \[fig3\]. We notice that the polymerised mass $M$ evolves much faster in time than the number density of polymers $P$. The inset in Fig. \[fig3\]b shows that in the asymptotic limit of vanishing scission rate $\kmf$, the lag phase for both moments is characterised by an approximately equal time scale. However, after the lag phase, $M$ relaxes to its equilibrium value much faster than $P$ does. Hence, for the early time kinetics of the polymerised monomeric mass, the number concentration of polymers remains effectively constant, i.e., equal to initial value $P(t)=P(0)$.
Making use of this, we can solve Eq. \[eq11\] for $M(t)$ for early times, yielding the explicit solution, \[eq13\] M(t) &=& - , where for the purpose of notational simplicity we define $\sigma \equiv \alpha+\gamma P(0)$ and $\eta \equiv \sqrt{\sigma ^2+4 \beta \kpb P(0)}$. In Fig. \[fig4\] we show that Eq. \[eq13\] agrees with the numerical results obtained in the limit $\kmf \rightarrow 0$. Below we will analyse the effect of molecular pathways by calculating the lag time from Eq. \[eq13\]. For now, the main conclusion is that Eq. \[eq13\] predicts sigmoidal kinetics, i.e., a lag phase followed by exponential growth and subsequent saturation of the solution for $M(t)$.
Next, we solve for $P(t)$ in the long-time limit when $M(t)$ has achieved its equilibrium value $M(\infty)$. For long times, i.e., post-lag phase of the polymerised mass fraction, $P(t)$ is given by \[eq14\] P(t) &=& . Notice that for the analytical solution of the number concentration of polymers, the time scale of the evolution of $P(t)$ depends only on the scission rate constant $\kmf$. This is due to our choice of reaction scheme, where the *scission and recombination* affects only the number of polymers. In contrast, *end evaporation and addition* and *body evaporation and addition* only affect the exchange of monomers between the free monomer pool and the polymer pool.
Having obtained a closed analytical expression for the time evolution of the polymerised mass, $M(t)$, we can now calculate the lag time, $\tlag$. This is achieved by calculating the inflection point or the point of maximum growth rate, i.e., the time at which the second derivative of Eq. \[eq13\] is equal to zero. We then calculate the time intercept of the tangent at the inflection point, resulting in the analytical expression for the lag time. In general, for nucleated polymerisation kinetics the lag time $\tlag$ is a linear combination of two terms. [@hong_moment_closure] These terms represent two important characteristics of the polymerisation kinetics: the half-time $\thalf$ and apparent growth rate $\kapp$. The half-time $\thalf$ is the time at which the polymerised monomeric mass is exactly half of its steady-state (long-time) value and from Eq. \[eq13\] we find that it is equal to the inflection point. Furthermore, the apparent growth rate $\kapp$ is the growth rate of the polymerisation curve at the inflection point (so, the time derivative of $M(t)$ at $t=\thalf$).
Hence, we find \[eq15\] = - , where \[eq16\] =, denotes the half-time and \[eq17\] = - , the apparent growth rate.
To investigate the influence of the overall monomer concentration on the lag time, let us assume that all the other parameters, i.e., the rate constants, are constant and do not depend on the monomer concentration. This also ties in with our assumption that our polymerisation process is reaction-limited and not diffusion-limited. Let us first focus on the case $\kpb=\kpe$ and $\kmb=\kme$, i.e., which is true if *body evaporation and addition* kinetics and *end evaporation and addition* are equally likely. We see that $\alpha = \kmb -\kpb \mtot \approx -\kpb \mtot$ where we have $\kpb \mtot \gg \kmb$ for polymers to exist. The other parameters hence become $\beta = 2(\kmb-\kme)+\kpb\mtot+2\kpe \mtot \approx 3 \kpb \mtot$ and $\gamma = \kpb+2\kpe = 3\kpb \approx $ constant, i.e., independent of total monomer concentration. This further implies that $\sigma =\alpha + \gamma P(0) \approx -\kpb \mtot$, because $\alpha \sim \kpb \mtot \gg \gamma P(0) = 3 \kpb P(0)$ as $\mtot \gg P(0)$. Finally, the denominator in the expression for half-time is $\eta = \sqrt{\sigma^2 + 4 \beta \kpb P(0)} \approx \sqrt{(\kpb \mtot)^2+ 12 \kpb \mtot P(0)}$.
Again, as $P(0) \ll \mtot$, we infer that $\eta \approx \kpb \mtot$. From Eq. \[eq16\] the half-time has a logarithmic numerator resulting in a weak dependence of the numerator on $\mtot$, and effectively we have $\thalf \sim \eta^{-1} \sim \mtot^{-1}$. Similarly, $\kapp=\eta^2/2 \sigma$, where $\eta \approx \kpb \mtot$, and hence the apparent growth rate scales as $\kapp \sim \mtot$. In Fig. \[fig5\], we show, by fitting the concentration dependence of the half-time Eq. \[eq17\] on a double logarithmic scale that our predictions for the power laws are indeed correct. It confirms that the logarithmic correction is indeed negligible. These power laws for the half-time and the apparent growth rate have same exponents as for the case of *end evaporation and addition* in the limit of weak scission obtained before. [@hong]
Let us now examine how the half-time and the apparent growth rate depend on the forward rates of the *body evaporation and addition* and the *end evaporation and addition* pathways, when only one of them is present. In the absence of *end evaporation and addition* pathway, i.e., $\kpe=\kme=0$, we have $\eta \sim \kpb \mtot$, as shown above. This, in combination with the fact that the half-time scales as $\thalf \sim \eta^{-1}$ and that the apparent growth rate scales as $\kapp \sim \sigma/\eta^{-1}$, where $\sigma \sim - \kpb \mtot$, results in scaling of $\thalf \sim (\kpb)^{-1}$ and $\kapp \sim \kpb$. This is confirmed in Fig. \[fig6\].
In the absence of *body evaporation and addition* only primary nucleation and the *end evaporation and addition* pathways are active, in the limit where *scission and recombination* is sufficiently weak. In that limit, our equations are exactly the same as those presented in earlier works. [@hong] From the work of Hong et al., we already know that for weak scission, both the half time and the reciprocal apparent growth rate scale linearly with the total monomer concentration, which is the same as what we found for *body evaporation and addition*. [@hong] In other words, from the monomer concentration dependence of the lag time we cannot distinguish body from end evaporation and addition. However, differences do show up when considering the dependence of the lag time on the forward rate constants. For the *end evaporation and addition*, the half time and the reciprocal apparent growth rate scale as the square root of the forward rate constant of monomer addition, i.e., $\sqrt{\kpe}$, [@hong] which contrasts with what we found for *body evaporation and addition*. For the latter we found a linear scaling with the forward rate constant.
In conclusion, although the scaling of the lag time with the total monomer concentration is the same for both pathways, there is a difference in the scaling with the forward rate constants, or, equivalently, with the equilibrium constants for the two pathways. This in principle provides us with a probe to inspect the existence of *body evaporation and addition* pathway in the polymerisation process, at least if we knew how to control them.
Discussion and Conclusions {#sec4}
==========================
In our model calculations, we compare predictions based on the newly discovered kinetic pathway of *body evaporation and addition* with the well studied pathway of *end evaporation and addition*, in the context of strongly nucleated reversible polymerisation. To be able to do that, we rely on a kinetic scheme that includes a third pathway, known as *scission and recombination*, that in the end we switch off asymptotically. [@cates_sr; @tuomas_review] The presence of this third pathway is needed solely for mathematical reasons, to achieve our purpose. In practice, we cannot exclude the presence of *scission and recombination*, and our set of equations, which focus on the two moments of the full distribution, allow for that.
The way the *scission and recombination* and *end evaporation and addition* pathways are implemented in the literature creates a conundrum, in which a process where a monomer is removed from the end can either be seen as a polymer scission or as an end evaporation event. [@hong; @tuomas_review] To remedy this, we have implemented the *scission and recombination* pathway for strongly nucleated linear reversible polymerisation in such a way that a polymer can break, resulting into the formation of two polymers of size greater than or equal to the *critical nucleus*, which in our model has to be bigger than or equal to a dimer. We modelled polymer recombination, so the merging of two short polymers into a longer one the same way. Also we exclude the case $n_c=1$. This prescription prevents the *scission and recombination* pathway from interfering with the *end evaporation and addition*. So in our prescription *scission and recombination* does not directly affect the monomer pool.
In our view, our alternative implementation of the two pathways is the more sensible one, because detachment of a monomer is inherently different from the breaking of a polymer. Indeed, the latter process should strongly depend on the length of the polymer, whilst the former arguably should not provided the polymer is sufficiently long. [@tom_length_dependence] In addition, our implementation of the *scission and recombination* pathway reproduces the thermodynamically consistent law of mass action for the amount of polymerised material under conditions of equilibrium and in the presence of the *end evaporation and addition* pathway. The amount of polymerised material of course is only one of the moments of the full length distribution. As for the other moments, such as the number of polymers in solution, they suffer from thermodynamically inconsistent long-time behaviour, a drawback that we share with previous studies. [@cohen; @hong; @tuomas_science]
However, if we focus on the short-time behaviour of the reversible polymerisation reactions, we could argue, as in fact is tacitly done in the literature, that the long-time behaviour of the system is inconsequential. [@hong; @tuomas_review] Indeed, many thermodynamically inconsistent pathways have been shown to be in quantitative agreement with experimental findings regarding the early time kinetics, which is the prime focus of almost all experimental works. [@tuomas_science; @knowles_two_stage; @hong; @hong_moment_closure; @subramaniam] For this reason, we also focus on the early time kinetics, when comparing body and end evaporation and addition, in the limit of asymptotically weak scission. Because polymer recombination has been shown theoretically not to substantially alter the early-time polymerisation kinetics, we suppress the recombination step completely for mathematical expedience in order to obtain closed-form solutions, in the final steps of our analysis. [@knowles_two_stage]
In this limit, we find that the typical polymerisation kinetics of *body evaporation and addition* is characterised by a separation of time scales between the polymerised monomeric mass $M$ and the polymer number concentration $P$. This is obvious from the difference in the lag times for these two quantities, i.e., the time scales required to get a substantial growth. Such separation of time scales has also been found for the *end evaporation and addition* pathway in combination with weak polymer scission and no recombination albeit that it is is not as strong. [@hong_moment_closure] To illustrate this, in Fig. \[fig7\], we show the polymerised mass fraction and the polymer concentration as a function of time, where we vary the forward rate of body addition at a fixed value of the forward rate of end addition. We cover the full spectrum from predominant body to predominant end evaporation and addition.
Comparing Fig. 7a and 7b, it is clear that the separation of time scales is many orders of magnitude larger for *body evaporation and addition* than that for *end evaporation and addition*. Indeed, the polymerised mass fraction evolves much faster in time than the number of polymers does, if the *body evaporation and addition* pathway is active. It is this vast separation of time scales that allows us to obtain first order perturbative solutions to the two moment equations that are in quantitative agreement with numerical results for vanishing scission rate constant.
Our perturbative solutions for the polymerised mass fraction and the polymer concentration, are sigmoidal as a function of time and provide us with the closed form solution for the lag time. The lag time associated with the polymerised mass fraction is the one that is experimentally readily accessible, and hence we focus on that. We find, within the limit where it is valid, that the lag time produces only a weak dependence on the initial polymerised monomeric mass. This contrasts with other studies where they do not include *body evaporation and addition* in the reaction scheme, and is a result of the extremely fast kinetics connected with that pathway. [@oosawa; @cohen; @hong; @hong_moment_closure]
In the usual definition of the lag time, it takes the form of the sum of a half-time $\thalf$ and a reciprocal of an apparent growth rate $\kapp$. [@tuomas_review] For *body evaporation and addition* we find the same linear scaling of the the half time and the reciprocal apparent growth rate with the total monomer concentration, $\mtot$, as was found in the literature for the *end evaporation and addition*. [@hong] In fact, for the *bulk evaporation and addition* pathway it is the product $\kpb \mtot$ dictates the scaling of the lag time, implying that both time scales are also linearly dependent on the forward rate for the body addition, $\kpb$. In contrast, for the *end evaporation and addition* pathway both time scale are proportional to the square root of the forward rate constant of the end addition, $\sqrt{\kpe}$. [@hong]
It seems that the newly proposed pathway, although it speeds up the growth of the polymerised mass by providing a larger number of places to insert or remove a monomer from a polymer, does not affect the dependence of the lag time on the total monomer concentration. However, it does affect how the relevant forward rate constant influences that time scale. This means that to distinguish end from body evaporation and addition experimentally by probing the dependence of the lag time on the system parameters, we would need to be able to control this quantity. This is not trivial. It may well be that highly specialised techniques, such as those used in reference [@albertazzi], are required to actually observe it. In fact, this may even be the reason, why it had not been considered before.
Acknowledgements
================
We thank Mariana Oshima Menegon, Shari Finner and Stefan Paquay for a critical reading of the manuscript. This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek through Project No. 712.012.007.
Derivation of Moment Equations from Discrete Master Equation
============================================================
We start by writing down the discrete master equation, Eq. \[eq2\], for the reaction schemes described in the main text. Next, we define the principal moments, the number of polymers $P$ and the polymerised monomeric mass $M$, Eqs. (7) and (8).
After substituting Eq. \[eq4\] in Eq. \[eq5\], the equation for the number of polymers $P$ is given by, &=& \_[i=n\_c]{}\^ k\_n\^+ x(t)\^[n\_c]{} \_[i,n\_c]{} + 2 \_[i=n\_c]{}\^ x(t) y\_[i-1]{}(t) - 2 x(t) \_[i=n\_c]{}\^ y\_[i]{}(t) + 2 \_[i=n\_c]{}\^ y\_[i+1]{}(t) - 2 \_[i=n\_c]{}\^ y\_i(t)\
& & + \_[i=n\_c]{}\^ (i-2) y\_[i-1]{}(t) - \_[i=n\_c]{}\^ (i-1) x(t) y\_[i]{}(t) + \_[i=n\_c]{}\^ (i-1) y\_[i+1]{}(t) - \_[i=n\_c]{}\^ (i-2) y\_i(t)\
& & - \_[i=n\_c]{}\^ (i- 2 n\_c +1) y\_i(t) + 2 \_[i=n\_c]{}\^ \_[j=i+n\_c]{}\^ y\_j(t)\
& & + \_[i=n\_c]{}\^ \_[k+l=i]{} y\_k(t) y\_l(t) - 2 \_[i=n\_c]{}\^ y\_i (t) \_[j=n\_c]{}\^ y\_j. To obtain a kinetic equation without any summation signs, we collect and simplify the equation term by term. Let us start with the terms accounting for monomer addition at the end, & & ( \_[i=n\_c]{}\^ y\_[i-1]{} - \_[i=n\_c]{}\^ y\_[i]{} ) = ( \_[i=n\_c-1]{}\^ y\_[i]{} - \_[i=n\_c]{}\^ y\_[i]{} ) = y\_[n\_c-1]{} = 0. Notice that $y_{n_c-1}=0$, because the smallest stable polymer is the critical nucleus of size $n_c$. Similarly for end evaporation, & & ( \_[i=n\_c]{}\^ y\_[i+1]{} - \_[i=n\_c]{}\^ y\_[i]{} ) = ( \_[i=n\_c+1]{}\^ y\_[i]{} - \_[i=n\_c]{}\^ y\_[i]{} ) = - y\_[n\_c]{}. The analysis in the main text assumes the strongly nucleated polymerisation, i.e., $k_n^+ \rightarrow 0$. In this limit the critical nuclei are highly unstable and can be neglected.
For terms six and seven in Eq. (A1) that represent monomer addition on the polymer backbone, we obtain \_[i=n\_c]{}\^ ( (i-2) y\_[i-1]{} - (i-1) y\_i ) = \_[i=n\_c-1]{}\^ (i-1) y\_i - \_[i=n\_c]{}\^ (i-1) y\_i = 0. Similarly, terms eight and nine in Eq. (A1) accounting for monomer removal from the polymer backbone, simplify to \_[i=n\_c]{}\^ (i-1) y\_[i+1]{} - \_[i=n\_c]{}\^ (i-2) y\_i = \_[i=n\_c]{}\^ (i-1) y\_[i+1]{} - \_[i=n\_c-1]{}\^ (i-1) y\_i = - (n\_c-1) y\_[n\_c]{}. Once again under the assumption of strongly nucleated polymerisation, we neglect $y_{n_c}$.
The contribution from polymer scission can be rewritten in terms of a theta function as - \_[i=n\_c]{}\^ (i- 2 n\_c +1) y\_i + 2 \_[i=n\_c]{}\^ \_[j=n\_c]{}\^ y\_j (i-j-n\_c)\
- \_[i=n\_c]{}\^ (i- 2 n\_c +1) y\_i + 2 \_[i=n\_c]{}\^ (1+j-2 n\_c) y\_j = (M-(2 n\_c -1) P), and the contribution of polymeric recombination is \_[k=n\_c]{}\^ \_[l=n\_c]{}\^ y\_k y\_l - 2 \_[i=n\_c]{}\^ y\_i \_[j=n\_c]{}\^ y\_j = ( \_[k=n\_c]{}\^ y\_k ) ( \_[l=n\_c]{}\^ y\_l ) - 2 ( \_[i=n\_c]{}\^ y\_i ) ( \_[j=n\_c]{}\^ y\_j ) = -P\^2 This gives us following equation for the number of polymers $P$, = - P(t)\^2 + ( M(t)-(2 n\_c -1) P(t) ) + k\_n\^+ x(t)\^[n\_c]{} For the polymerised monomeric mass $M(t)$, the evolution equation is, &=& \_[i=n\_c]{}\^ k\_n\^+ i x(t)\^[n\_c]{} \_[i,n\_c]{} + 2 x(t) \_[i=n\_c]{}\^ i y\_[i-1]{}(t) - 2 x(t) \_[i=n\_c]{}\^ i y\_[i]{}(t) + 2 \_[i=n\_c]{}\^ i y\_[i+1]{}(t) - 2 \_[i=n\_c]{}\^ i y\_i(t)\
& & + \_[i=n\_c]{}\^ i (i-2) y\_[i-1]{}(t) - \_[i=n\_c]{}\^ i (i-1) x(t) y\_[i]{}(t) + \_[i=n\_c]{}\^ i (i-1) y\_[i+1]{}(t) - \_[i=n\_c]{}\^ i (i-2) y\_i(t)\
& & - \_[i=n\_c]{}\^ i (i- 2 n\_c +1) y\_i(t) + 2 \_[i=n\_c]{}\^ \_[j=i+n\_c]{}\^ i y\_j(t) + \_[i=n\_c]{}\^ \_[k+l=i]{} i y\_k(t) y\_l(t)\
& & - 2 \_[i=n\_c]{}\^ i y\_i (t) \_[j=n\_c]{}\^ y\_j(t). The terms involving monomer addition at the ends are 2 x(t) \_[i=n\_c]{}\^ i ( y\_[i-1]{}(t)-y\_i(t) ) = 2 ( \_[i=n\_c-1]{}\^ (i+1) y\_i - \_[i=n\_c]{}\^ i y\_i ) = 2 x(t) P(t), and the term arising from monomer removal at the ends are 2 \_[i=n\_c]{}\^ i ( y\_[i+1]{}(t)-y\_i(t) ) = 2 ( \_[i=n\_c+1]{}\^ (i-1) y\_i - \_[i=n\_c]{}\^ i y\_i ) = 2 (-P(t) - n\_c y\_[n\_c]{}). Where we neglect the contribution, $n_c y_{n_c}$.
Next the terms contributed by the monomer addition in the bulk becomes & & \_[i=n\_c]{}\^ ( i (i-2) y\_[i-1]{} - i (i-1) y\_i )\
&=& ( \_[i=n\_c-1]{}\^ (i+1)(i-1) y\_i - \_[i=n\_c]{}\^ i (i-1) y\_i )\
&=& ( \_[i=n\_c]{}\^ (i\^2-1) y\_i + \_[i=n\_c]{}\^ i y\_i - \_[i=n\_c]{}\^ i\^2 y\_i )\
&=& x(t) (M(t) - P(t)), and the terms from monomer removal from the bulk are & & ( \_[i=n\_c]{}\^ i (i-1) y\_[i+1]{} - \_[i=n\_c]{}\^ i (i-2) y\_i )\
&=& ( \_[i=n\_c+1]{}\^ (i-1) (i-2) y\_i - \_[i=n\_c]{}\^ i (i-2) y\_i )\
&=& - \_[i=n\_c+1]{}\^ (i-2) y\_i = (- n\_c y\_[n\_c]{} + 2 P(t)- M(t)). This completes our closure of discrete master equation to obtain moment equations, that are given by, &=& - P(t)\^2 + ( M(t)-(2 n\_c -1) P(t) ) + k\_n\^+ x(t)\^[n\_c]{}, and &=& 2 ( x(t) P(t) - P(t) ) + x(t) (M(t)+P(t)) + (2 P(t)-M(t))+ n\_c k\_n\^+ x(t)\^[n\_c]{}.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a general framework to represent discrete configuration systems using hypergraphs. This representation allows one to transfer combinatorial removal lemmas to their analogues for configuration systems. These removal lemmas claim that a system without many configurations can be made configuration-free by removing a few of its constituent elements. As applications of this approach we give; an alternative proof of the removal lemma for permutations by Cooper [@cooper06], a general version of a removal lemma for linear systems in finite abelian groups, an interpretation of the mentioned removal lemma in terms of subgroups, and an alternative proof of the counting version of the multidimensional Szemerédi theorem in abelian groups with generalizations.'
author:
- 'Lluís Vena[^1]'
title: On the removal lemma for linear configurations in finite abelian groups
---
Introduction {#s.intro}
============
In 1976 Ruzsa and Szemerédi introduced the Triangle Removal Lemma [@ruzsze78], which roughly states that in a graph with not many triangles (copies of the complete graph on $3$ vertices), we can remove a small proportion of the edges to leave the graph with no copies of the triangle at all.[^2] They gave a short proof of Roth’s theorem [@rot53] as an application: any subset of the integers with positive (upper) density contains non-trivial $3$-term arithmetic progressions.[^3] The case for $k$-term arithmetic progressions, conjectured by Erdős and Turán [@tur41], was established by Szemerédi in 1975 and is now called Szemerédi’s Theorem.
Both the Triangle Removal Lemma and Szemerédi’s Theorem use Szemerédi’s Regularity Lemma [@sze78] in their original proofs.[^4] However, Szemerédi’s Theorem does not seem to easily follow from the Triangle Removal Lemma (or the general Removal Lemma for Graphs [@erdfranrod86; @fur95].) Indeed, while the Triangle Removal Lemma follows almost immediately from the Regularity Lemma, the proof of Szemerédi’s Theorem is more involved and the Regularity Lemma is used in only one, yet crucial, step.
In [@frarod02] Frankl and Rödl showed that a $k$-uniform hypergraph version of the Removal Lemma suffices to establish the existence of non-trivial $(k+1)$-term arithmetic progressions in subsets of the integers with positive density.[^5] The argument is similar to the one used by Ruzsa and Szemerédi [@ruzsze78] to show the $3$-term case using the Triangle Removal Lemma.[^6] This simple proof of Szemerédi’s Theorem, along with the many applications that the Regularity Lemma for Graphs has (see the surveys [@komsim96; @komshosimsze02]), were the main motivations to extend the Regularity and the Removal lemmas from graphs to hypergraphs. This extension was done by several authors following different approaches: a combinatorial approach from Nagle, Rödl, Schacht, and Skokan [@nagrodsch06; @rodsko04], an approach using quasirandomness by Gowers [@gow07], a probabilistic approach from Tao [@tao06], and a non-standard measure-theoretic approach by Elek and Szegedy [@elesze12].
Arithmetic removal lemmas
-------------------------
In [@gre05], Green used Fourier analysis techniques to establish a regularity lemma and a removal lemma for linear equations in finite abelian groups. These statements are analogous to their combinatorial counterparts. The Group Removal Lemma [@gre05 Theorem 1.5] ensures that for every $\varepsilon>0$ and every positive integer $m$, there exists a $\delta=\delta(\epsilon,m)>0$ such that, for any finite abelian group $G$, if $$\label{eq.1}
x_1+\cdots+x_m=0$$ has less than $\delta |G|^{m-1}$ solutions with $x_i\in X_i\subset G$, then we can, by removing less than $\varepsilon |G|$ elements in each $X_i$, create sets $X_i''$ for which there is no solution to (\[eq.1\]) with $x_i\in X_i''$ for all $i\in[1,m]$.
Král’, Serra and the author [@kraserven09] gave an alternative proof of the above removal lemma, showed by Green as [@gre05 Theorem 1.5], using the removal lemma for directed graphs [@alosha04]. With this alternative approach, the result was extended to any finite group, eliminating the need of commutativity.
While a removal lemma for linear systems for some $0/1$-matrices was shown to hold in [@kraserven09] using graphs, the work of Frankl and Rödl [@frarod02] suggested that the hypergraph setting might provide the right tools to extend the removal lemma for one equation to a linear system. Indeed, Shapira [@sha10], and independently Král’, Serra, and the author [@kraserven12], used the Removal Lemma for Hypergraphs to obtain a removal lemma for linear systems over finite fields and proved a conjecture by Green [@gre05 Conjecture 9.4] regarding a removal lemma for linear systems in the integers. A partial result for finite fields was obtained by Král’, Serra, and the author [@kraserven08], and also independently by Candela [@can_tesis_09].
In addition to showing the removal lemma for finite fields, Shapira [@sha10] raised the issue of whether an analogous result holds for linear systems over finite abelian groups. In [@ksv13], Král’, Serra, and the author answered the question affirmatively provided that the determinantal[^7] of the integer matrix that defines the system is coprime to the cardinality of the finite abelian group. See [@ksv13 Theorem 1] or Theorem \[t.rem\_lem\_ksv13\] for the result.
In a different direction, Candela and Sisask [@cansis13] proved that a removal lemma for integer linear systems holds over certain compact abelian groups. The main result in [@cansis13] has been recently extended by Candela, Szegedy and the author [@canszeven14+] to any compact abelian group provided that the integer matrix has determinantal $1$.
#### Previous combinatorial arguments.
The proof schemes of the previous arithmetic removal lemmas in [@kraserven09; @kraserven12; @ksv13; @sha10] are inspired by the approaches of [@ruzsze78] and [@frarod02], and can also be found in [@gow07; @sol04; @sze10; @tao12]. Concisely, the main argument involves constructing a pair of graphs (or hypergraphs) $(K,H)$ so as to make it possible to transfer the removal lemma from the graph/hypergraph combinatorial setting to an arithmetic context. The pair $(K,H)$ is said to be a representation of the system and usually satisfies the property that each copy of $H$ in $K$ is associated with a solution of our system. Moreover, these copies of $H$ should be evenly distributed throughout the edges of $K$. The notion of representability of a system by a hypergraph has been formalized by Shapira in [@sha10 Definition 2.4].
Main notions and results {#s.main_notion_and_results}
------------------------
This paper is built around two main pieces. The first one is the generalization of the combinatorial representability notion introduced by Shapira in [@sha10]. This generalization is stated as Definition \[d.rep\_sys\]. The second piece is a removal lemma for homomorphisms of finite abelian groups; given a finite abelian group $G$ and a homomorphism $A$ from $G^m$ to $G^k$, if a set $X=X_1\times\cdots\times X_m$ does not contain many $\mathbf{x}\in X$ with $A\mathbf{x}=0$, then $X$ can be made solution-free by removing a small proportion of each $X_i$. The detailed statement can be found below as Theorem \[t.rem\_lem\_ab\_gr\]. Additionally, we provide an interpretation of Theorem \[t.rem\_lem\_ab\_gr\] in terms of subgroups, which is stated as Theorem \[t.rem\_lem\_subgroups\], and can be seen as a removal lemma for finite abelian subgroups.
### Systems of configurations and representability
#### Systems of configurations.
Let us introduce the notion of a (finite) system of configurations. Let $m$ be a positive integer and let $G$ be a set. A system of degree $m$ over $G$ consists of a pair $(A,G)$, where $A$ is a property on the configurations of $G^m$, $A: G^m\to \{0,1\}$. If $x\in G^m$ is such that $A(x)=1$, then $x$ is said to be a *solution* to $(A,G)$. In this paper all the sets $G$ considered are finite.
#### Representable systems.
We introduce as Definition \[d.rep\_sys\] a more general notion of representability for systems than the one given by Shapira in [@sha10 Definition 2.4]. Although rather technical, Definition \[d.rep\_sys\] asks for the existence of a pair of hypergraphs $(K,H)$, associated to a finite system of configurations $(A,G)$, with the following summary of properties (in parenthesis appear references to the properties described in Definition \[d.rep\_sys\]):
- Each copy of $H$ in $K$ is associated to a solution of $(A,G)$ (domain and range of $r$ in RP\[prop\_rep2\].) The edges are associated with elements of $G$ (RP\[prop\_rep1\], third point.)
- Given a solution of $(A,G)$, there are many copies of $H$ in $K$ associated to such solution (cardinality of $r^{-1}(\mathbf{x},q)$ in RP\[prop\_rep2\].) Those copies are well (evenly) distributed through the edges associated to the elements that configure the solution (RP\[prop\_rep3\] and RP\[prop\_rep4\].)
- The number of vertices in $H$ is bounded (first point in RP\[prop\_rep1\].)
All these points are sufficient to prove a removal lemma for representable systems, which has Theorem \[t.rep\_sys\_rem\_lem\] as the precise statement.
Let us mention that the representations used in [@kraserven09; @kraserven12; @ksv13; @sha10] can be seen to give a representation according to Definition \[d.rep\_sys\]. In [@ksv13], the authors used Shapira’s definition, with an extra post-processing, to show a removal lemma for integer linear systems with determinantal $1$ and where the sets to be removed are small with respect to the total group, [@ksv13 Theorem 1]. The additional features of Definition \[d.rep\_sys\] with respect to the representation notion given by Shapira in [@sha10] allow us to extend the result [@ksv13 Theorem 1] (Theorem \[t.rem\_lem\_ksv13\] in this paper) to Theorem \[t.rem\_lem\_ab\_gr\] in the following way.
- The set $Q$ is used to remove the determinantal condition and to extend the result to any homomorphism $A$ with domain $G^m$ and image in $G^k$.
- The vector $\gamma$ or, more precisely, the vector of proportions $(1/\gamma_1, \ldots, 1/\gamma_m)$, allows us to claim that the $i$-th removed set is an $\epsilon$-proportion of the projection of the whole solution set onto the $i$-th coordinate. The projection of the solution set is, in general, smaller than the whole abelian group $G$.[^8] See Theorem \[t.rep\_sys\_rem\_lem\] for further details.
### Removal lemma for representable systems.
Let $I\subset \mathbb{N}$ be a set of indices. Consider $(\mathcal{A},\mathcal{G},m)=\{(A_i,G_i)\}_{i\in I}$ a family of systems of degree $m$. Let $S(A,G)$ denote the set of solutions for $(A,G)\in (\mathcal{A},\mathcal{G},m)$ and let $S(A,G,X)$ denote the subset of $\mathbf{x}\in S(A,G)$ with $\mathbf{x}\in X\subset G^m$.
Let $\Gamma=\{\gamma(i)\}_{i\in I}=\{(\gamma_1(i),\ldots,\gamma_m(i))\}_{i\in I}$ be a family of $m$-tuples of positive real numbers indexed by $I$. The family $(\mathcal{A},\mathcal{G},m)$ of systems is said to be $\Gamma$-representable, and the system $(A_i,G_i)$ is said to be associated with $\gamma_i$, if Definition \[d.rep\_sys\] in Section \[s.representables\_systems\] holds. The representability property suffices to show a removal lemma for configuration systems.
\[t.rep\_sys\_rem\_lem\] Let $(\mathcal{A},\mathcal{G},m)$ be a $\Gamma$-representable family of systems. Let $(A,G)$ be an element in the family associated to $\gamma=(\gamma_1,\ldots,\gamma_m)$. Let $X_1,\ldots,X_m$ be subsets of $G$ and let $X=X_1\times \cdots \times X_m$.
For every $\varepsilon>0$ there exists a $\delta=\delta(m,\varepsilon)>0$, universal for all the members of the family, such that if $$|S(A,G,X)|<\delta |S(A,G)|,$$ then there are sets $X_i'\subset X_i$ with $|X_i'|<\varepsilon |G|/ \gamma_i$ for which $$S\left(A,G,X\setminus X'\right)=\emptyset,$$ where $X\setminus X'=(X_1\setminus X'_1)\times \cdots \times (X_m\setminus X'_m)$.
Let us notice that, if a family of systems is $\Gamma$-representable, then the conclusions of Theorem \[t.rep\_sys\_rem\_lem\] holds with smaller $\gamma_j(i)$’s as the restrictions on the theorem decrease with the $\gamma$’s. In the representability notion of Shapira [@sha10], as well as in other works like [@kraserven09; @kraserven12; @ksv13], the $\gamma_j(i)$ are all $1$. Thus, the notion of representability Definition \[d.rep\_sys\] is an extension of [@sha10 Definition 2.4].
### Removal lemma for finite abelian groups.
Let $G$ be a finite abelian group. Given $\mathbf{b}\in G^k$, a homomorphism $A$ from $G^m$ to $G^k$ induces a property $(A,\mathbf{b})$ in $G^m$ given by: $\mathbf{x}\in S((A,\mathbf{b}),G)$ if and only if $A(\mathbf{x})=b$. Let $(\mathcal{A},\mathcal{G},m)$ be the family of systems given by the homomorphisms $(A,\mathbf{b})$ with fixed $m$. These are the configuration systems that we consider for most of the paper, especially in Section \[s.equiv\_and\_rep\] and onwards.
The set of homomorphisms $A:G^m\to G^k$ are in bijection with $k\times m$ homomorphism matrices $(\vartheta_{i,j})$ for some homomorphisms $\vartheta_{i,j}:G\to G$ depending on $A$.[^9] In particular, given $\mathbf{b}=(b_1,\ldots,b_k)^{\top}\in G^k$, $(x_1,\ldots,x_m)\in S\left(\left(A,\mathbf{b}\right),G\right)$ if and only if $$\left(
\begin{array}{ccc}
\vartheta_{1,1} & \cdots & \vartheta_{1,m} \\
\vdots &\ddots & \vdots \\
\vartheta_{k,1} & \cdots & \vartheta_{k,m} \\
\end{array}\right)
\left(\begin{array}{c}
x_1 \\
\vdots \\
x_m \\
\end{array}\right)
= \left(\begin{array}{c}
b_1 \\
\vdots \\
b_k \\
\end{array}\right)
\iff
\sum_{i=1}^m \vartheta_{j,i}(x_i)=b_j, \; \forall j\in [1,k].$$ Thus, we may use the term *$k\times m$ homomorphism system on $G$* to refer to the system induced by a homomorphism from $G^m$ to $G^k$.
Let $S_i((A,\mathbf{b}),G)$, $i\in [1,m]$ denote the projection of the solution set $S((A,\mathbf{b}),G)$ to the $i$-th coordinate of $G^m$, $S_i((A,\mathbf{b}),G)=\pi_i(S((A,\mathbf{b}),G))$. The solution set $S((A,\mathbf{b}),G)$ is denoted by $S(A,G)$ when $\mathbf{b}=0$ or understood by the context.
In the sections \[s.proof\_rl-lsg-1\] and \[s.representability\_product\_cyclics\] of this paper, we show that the family of homomorphisms of finite abelian groups is $\Gamma$-representable with $\gamma_i=|G|/|S_i((A,\mathbf{b}),G)|$ when $m\geq k+2$. Hence Theorem \[t.rep\_sys\_rem\_lem\], together with the additional comments to the construction presented in Section \[s.finish\_rem\_lem\_dkA1\], implies Theorem \[t.rem\_lem\_ab\_gr\].
\[t.rem\_lem\_ab\_gr\] Let $G$ be a finite abelian group and let $m, k$ be two positive integers. Let $A$ be a group homomorphism from $G^m$ to $G^k$. Let $\mathbf{b}\in G^k$. Let $X_i\subset G$ for $i=[1,m]$, and $X=X_1\times \cdots \times X_m$.
For every $\varepsilon>0$ there exists a $\delta=\delta(m,\varepsilon)>0$ such that, if $$\left|S((A,\mathbf{b}),G,X)\right|<\delta \left|S((A,\mathbf{b}),G)\right|,$$ then there are sets $X_i'\subset X_i\cap S_i((A,\mathbf{b}),G)$ with $|X_i'|<\varepsilon |S_i((A,\mathbf{b}),G)|$ and $$S\left((A,\mathbf{b}),G, X\setminus X'\right)=\emptyset, \text{ where } X\setminus X'=(X_1\setminus X'_1)\times \cdots \times (X_m\setminus X'_m).$$
Also, let $I\subset [1,m]$ be such that $X_i\supset S_i((A,\mathbf{b}),G)$ for $i\in I$. The previous statement holds with the extra condition that $X_i'=\emptyset$ for $i\in I$.
Let us state the known arithmetic removal lemma for finite abelian groups [@ksv13 Theorem 1].
\[t.rem\_lem\_ksv13\] Let $G$ be a finite abelian group and let $m, k$ be two positive integers. Let $A$ be a $k\times m$ integer matrix with determinantal coprime with the order of the group $|G|$.[^10] Let $\mathbf{b}\in G^k$. Let $X_i\subset G$ for $i=[1,m]$, and $X=X_1\times \cdots \times X_m$. For every $\varepsilon>0$ there exists a $\delta=\delta(m,\varepsilon)>0$ such that, if $$\left|S((A,\mathbf{b}),G,X)\right|<\delta \left|S((A,\mathbf{b}),G)\right|=\delta |G^{m-k}|,$$ then there are sets $X_i'\subset X_i\cap S_i((A,\mathbf{b}),G)$ with $|X_i'|<\varepsilon |G|$ and $$S\left((A,\mathbf{b}),G, X\setminus X'\right)=\emptyset,
\text{ where } X\setminus X'=(X_1\setminus X'_1)\times \cdots \times (X_m\setminus X'_m).$$
We can see that Theorem \[t.rem\_lem\_ab\_gr\] extends Theorem \[t.rem\_lem\_ksv13\] in three ways:
- The coprimality condition between the determinantal of $A$ and the order of the group is not needed.
- The systems induced by homomorphisms are more general than the systems induced by integer matrices. In particular, if $G=\prod_{i=1}^t{\mathbb Z}_{n_i}$, $n_{i+1}|n_i$, and we let $x=(x_1,\ldots,x_t)\in G$ with $x_i\in {\mathbb Z}_{n_i}$, then the homomorphism systems allow for linear equations between the components $x_i$ and $x_j$. This fact is used to prove the multidimensional version of Szemerédi’s Theorem. See Section \[s.intro\_applic\].
- The sizes of the deleted sets $X_i'$ are an $\varepsilon$-proportion of $|S_i((A,\mathbf{b}),G)|$ and not of $|G|$. In particular, if $|S_i((A,\mathbf{b}),G)|=1$, $\varepsilon<1$, and $X_i$ contains that element, then $X_i'=\emptyset$. This makes the result best possible in the following sense: if $S((A,\mathbf{b}),G,X)>\delta |S((A,\mathbf{b}),G)|$, then, in order to delete all the solutions, there should exist an $i$ with $|X_i'|> |S_i((A,\mathbf{b}),G)| \delta/m$. Therefore, we remove, at most, an $\varepsilon$-proportion of the right order of magnitude.
- The set of variables $x_i$ with $X_i=S_i(A,G)$ and for which no element should be removed is arbitrary. The argument leading to [@ksv13 Theorem 1], allows the existence of a set of indices $I$ of full sets from which no element is removed. However, the argument from [@ksv13] imposes an upper bound on the size of $I$. The argument in Section \[s.adding\_variables\] remove those bounds on $I$.[^11]
### Removal lemma for finite abelian subgroups
As Theorem \[t.rem\_lem\_ab\_gr\] can be applied to any finite abelian group $G$ and any homomorphism, we can rephrase the result in terms of subgroups.
\[t.rem\_lem\_subgroups\] For every $\epsilon>0$ and every positive integer $m$, there exists a $\delta=\delta(\epsilon,m)>0$ such that the following holds. Let $G_1,\ldots,G_m$ be finite abelian groups. Let $S$ be a subgroup of $G_1\times \cdots\times G_m$ and let $s\in \prod_{i\in[m]} G_i$. Let $X_i$ be a subset of $G_i$ for each $i\in[1,m]$. If $|\left[X_1\times\cdots\times X_m\right] \cap s+S|<\delta |S|$ then there exist $X_1',\ldots,X_m'$, with $|X_i'|<\epsilon \pi_i (S)$ for all $i\in[1,m]$, such that $\{\left[X_1\setminus X_1' \times\cdots\times X_m\setminus X_m'\right] \cap s+S\}=\emptyset$. If $\pi_i (s+S)\subset X_i$, for some $i\in[1,m]$ then we can assume $X_i'=\emptyset$.
Indeed, any subgroup of a finite abelian group is the kernel of a homomorphism (namely the quotient map $\prod_{i\in[m]} G_i\to \left[\prod_{i\in[m]} G_i\right]/S$, $S$ being our subgroup of interest.) Notice also that, instead of $\prod_{i\in[m]} G_i$, we could consider our domain to be a supergroup $G^m>\prod_{i\in[m]} G_i$, for some suitable finite abelian group $G$, as $S$ is also a subgroup of $G^m$. Moreover, we can assume $G^m/S<G^k$ for some $k$ (take, for instance $k=m$.) Therefore, Theorem \[t.rem\_lem\_ab\_gr\] suffices to show Theorem \[t.rem\_lem\_subgroups\]. Since the kernel of a homomorphism generates a subgroup of $G^m$, Theorem \[t.rem\_lem\_subgroups\] implies Theorem \[t.rem\_lem\_ab\_gr\]. Hence the version of the result for subgroups Theorem \[t.rem\_lem\_ab\_gr\], and the version for systems of homomorphisms Theorem \[t.rem\_lem\_subgroups\], are equivalent.
### Removal lemma for permutations {#s.intro_permutations}
In [@cooper06], Cooper introduced a regularity lemma and a removal lemma for permutations. Let $\mathcal{S}(i)$ denote the set of bijective maps from $[0,i-1]$ to $[0,i-1]$. Slightly modifying the notation in [@cooper06], let $\Lambda^{\tau}(\sigma)$, for $\tau\in \mathcal{S}(m)$ and $\sigma\in \mathcal{S}(n)$, be the set of occurrences of the pattern $\tau$ in $\sigma$. That is to say, the set of index sets $\{x_0<\cdots < x_{m-1}\}\subset [0,n-1]$ such that $\sigma(x_i)<\sigma(x_j)$ if and only if $\tau(i)<\tau(j)$.
\[p.rem\_lem\_permutations\] Suppose that $\sigma\in \mathcal{S}(n)$, $\tau\in \mathcal{S}(m)$. For every $\epsilon>0$ there exist a $\delta=\delta(\epsilon,m)>0$ such that, if $|\Lambda^{\tau}(\sigma)|<\delta n^m$, then we may delete at most $\epsilon n^2$ index pairs to destroy all copies of $\tau$ in $\sigma$.
In Section \[s.perm\_rep\_and\_rem\_lem\] we can find a representation where the valid configurations are given by the set $\Lambda^{\tau}(\sigma)$, and where we shall delete pairs of indices to destroy them. Hence Proposition \[p.rem\_lem\_permutations\] follows from Theorem \[t.rep\_sys\_rem\_lem\].
Applications {#s.intro_applic}
------------
#### Multidimensional Szemerédi.
One of the main applications of Theorem \[t.rem\_lem\_ab\_gr\] is a new proof of the counting version of the multidimensional Szemerédi’s Theorem for finite abelian groups.
The original proof of the multidimensional Szemerédi theorem for the integers was given by Furstenberg and Katznelson [@furskatz78] and uses ergodic theory. Solymosi [@sol04] observed that a removal lemma for hypergraphs would imply the multidimensional Szemerédi theorem (a detailed construction can be found in [@gow07]). Solymosi’s geometric argument uses hypergraphs and follows the lines of the argument by Ruzsa and Szemerédi [@ruzsze78] to obtain Roth’s Theorem [@rot53] from the Triangle Removal Lemma. With the development of the Regularity Method for Hypergraphs [@gow07; @nagrodsch06; @rodsko04; @tao06] and the corresponding Removal Lemma for Hypergraphs [@gow07; @nagrodsch06; @tao06], a combinatorial proof of the multidimensional version of Szemerédi’s Theorem for the integers could be pushed forward [@gow07; @sol04].
In [@tao12], Tao uses the same construction as Solymosi [@sol04] to show [@tao12 Theorem B.2] and a lifting trick to obtain a generalized version of the multidimensional Szemerédi theorem for finite abelian groups, [@tao12 Theorem B.1]. Theorem \[t.rem\_lem\_ab\_gr\] can be used to prove both [@tao12 Theorem B.1] and [@tao12 Theorem B.2].
The argument to deduce [@tao12 Theorem B.2] (Theorem \[t.multi\_szem\_ab\_gr\] in this paper) from Theorem \[t.rem\_lem\_ab\_gr\] explicitly shows that the dependencies in [@tao12 Theorem B.2] and in [@tao12 Theorem B.1] are independent of the dimension of the space and depend only on the number of points required in the configuration. On the other hand, the relation between $\delta$ and $\epsilon$ obtained using Theorem \[t.rem\_lem\_ab\_gr\] is worse than the direct construction of [@tao12 Theorem B.2] due to the larger uniformity of the hypergraph used.
\[t.multi\_szem\_ab\_gr\] Let $\varepsilon >0$. Let $G^m$ be a finite abelian group and let $S\subset G^m$ be such that $|S|/|G^m|\geq \varepsilon$. There exists $\delta=\delta(\varepsilon,m+1)>0$ such that the number of configurations of the type $\{(x_1,\ldots,x_m),(x_1+a,x_2,\ldots,x_m),\ldots,(x_1,x_2,\ldots,x_m+a)\}\subset S$, for some $a\in G$, is at least $\delta |G|^{m+1}$.
Consider the abelian group $P=G^m$, $X_i=S\subset P$, for $i\in[1,m+1]$. Let $\mathbf{x}_i=(x_{i,1},\ldots,x_{i,m})$, $i\in[1,m+1]$, be the variables of the homomorphism system that can be derived from the following linear equations: $$\label{eq.system_multidim}
\left\{\begin{array}{cl}
x_{1,1}-x_{2,1}= x_{1,j}-x_{j+1,j} & \text{ for } j\in[2,m] \\
x_{1,j}=x_{i,j} & \text{ for all }(i,j)\in[1,m+1]\times[1,m], i\neq j+1.\\
\end{array}\right.$$ Indeed, $\mathbf{x}_1$ is thought of as the centre of the configuration. The first equations state that the difference between the $j$-th coordinate of $\mathbf{x}_{j+1}$ and the $j$-th coordinate of $\mathbf{x}_1$ is the same regardless of $j$; this is achieved by setting all the differences to be equal to the difference between the first coordinate of $\mathbf{x}_2$ and the first coordinate of $\mathbf{x}_1$. The second set of equations treat the other coordinates, imposing that all the other coordinates of $\mathbf{x}_{j+1}$, except the $j$-th, should be equal to those of $\mathbf{x}_1$. Therefore, $(\mathbf{x_1},\ldots,\mathbf{x}_{m+1})$ is a solution to the system defined by (\[eq.system\_multidim\]) if and only if $\mathbf{x}_1=(y_1,\ldots,y_m)$, $\mathbf{x}_2=(y_1+a,\ldots,y_m)$, …, $\mathbf{x}_{m+1}=(y_1,\ldots,y_m+a)$ for some $y_1,\ldots,y_m,a\in G$.
By adding some trivial equations, like $0=0$, the system induces a homomorphism $A:P^{m+1} \to P^{m}$, with $S(A,P)\cong G^{m+1}$. Observe that $S_i(A,P)\cong G^m$ as any point in $P=G^m$ can be the $i$-th element in the configuration.
Consider the $\delta=\delta_{\text{Theorem~\ref{t.rem_lem_ab_gr}}}(m+1,\varepsilon/(m+1))$ coming from Theorem \[t.rem\_lem\_ab\_gr\] applied with $\varepsilon/(m+1)$ and $m+1$. Let us proceed by contradiction and assume that the number of solutions is less than $\delta S(A,P)=\delta |G|^{m+1}$. Now we apply Theorem \[t.rem\_lem\_ab\_gr\] and find sets $X_1',\ldots,X_{m+1}'$ with $|X_i'|<|G^m|\varepsilon/(m+1)$ such that the sets $X_i=S\setminus X_i'$ bear none of the desired configurations.
Observe that any point $\mathbf{x}\in S\subset G^m$ generates a solution to the linear system as $(\mathbf{x},\ldots,\mathbf{x})\in P^{m+1}$ is a valid configuration with $a=0_{G}$. Consider $S'=S\setminus (\cup X_i')$. Since $|S|>\varepsilon|G^m|$ and $|X_i'|<|G^m|\varepsilon/(m+1)$, there exists an element $\mathbf{s}$ in $S'$, as $S'$ is non-empty. Therefore $\mathbf{s}\in X_i$ for every $i\in[1,m+1]$. Thus $(\mathbf{s},\ldots,\mathbf{s})\in P^{m+1}$ is a solution that still exists after removing the sets $X_i'$ of size at most $\varepsilon/(m+1)$ from $S$. This contradicts Theorem \[t.rem\_lem\_ab\_gr\]. Therefore, we conclude that at least $\delta |G|^{m+1}$ solutions exist.
#### Other linear configurations.
More generally, we can show the following corollary of Theorem \[t.rem\_lem\_ab\_gr\].
\[c.homotetic\_solutions2\] Let $G$ be a finite abelian group, let $A$ be a $k\times m$ homomorphism for $G$ and let $\mathbf{b}\in G^k$. Assume that $S(A,G)=S((A,\mathbf{b}),G)\subset G^m$ contains a set $R$ satisfying the following conditions.
(i) The projection of $R$ onto the $i$-th coordinate of $G^m$ is $S_i(A,G)$. This is, $\pi_i(R)=S_i(A,G)$.
(ii) For each $i\in[1,m]$ and for each pair $g_1,g_2\in S_i(A,G)$, $|\pi_i^{-1}(g_1)\cap R|=|\pi_i^{-1}(g_2)\cap R|$.
Then, for every $\varepsilon>0$ there exists a $\delta=\delta(\varepsilon,m)>0$ such that, for any $S\subset G$ with $|S^m\cap R|>\varepsilon |R|$, we have $\left|S(A,G,S^m)\right|\geq\delta \left|S(A,G)\right|$.
We proceed by contradiction. Choose $\delta=\delta_{\text{Theorem~\ref{t.rem_lem_ab_gr}}}(m,\varepsilon/(m+1)$ and assume that $\left|S(A,G,S^m)\right|<\delta \left|S(A,G)\right|$. Then there are sets $X_i'$, with $|X_i'|< \varepsilon/(m+1)$, such that $S(A,G,\prod_{i=1}^m S\setminus X_i')=\emptyset$. However, by $(i)$ and $(ii)$ we delete at most $\epsilon\frac{m}{m+1}|R|$ hence $R\cap \prod_{i=1}^m S\setminus X_i'\neq \emptyset$, thus $S(A,G,\prod_{i=1}^m S\setminus X_i')\neq \emptyset$ reaching a contradiction.
In particular, if the linear system $(A,G)$ satisfies $S_i(A,G)=G$ for all $i\in[1,m]$ and $(x,\ldots,x)\in S(A,G)$ for each $x\in G$, then Corollary \[c.homotetic\_solutions\] shows that any set $S\subset G$ with $|S|\geq \epsilon |G|$, satisfies that $|S(A,G,S^m)|>\delta |S(A,G)|$ for some $\delta>0$ depending on $\epsilon$ and $m$. That is, any set with positive density will contain a positive proportion of the solutions. Corollary \[c.homotetic\_solutions2\] can be particularized as Corollary \[c.homotetic\_solutions\] which presents a perhaps more directly applicable form.
\[c.homotetic\_solutions\] Let $G$ be a finite abelian group, let $G_1,\ldots G_s$ be subgroups of $G$. Let $\Phi_1,\ldots,\Phi_t$ be group homomorphisms $$\begin{aligned}
\Phi_i: G_1\times \cdots\times G_s &\to G \nonumber \\
(x_1,\ldots,x_s) &\mapsto \Phi_i(x_1,\ldots,x_s). \nonumber\end{aligned}$$ For every $\epsilon>0$ there exists a $\delta=\delta(\epsilon,t)>0$ such that, for every $S\subset G$ with $S\geq \epsilon |G|$, $$\begin{gathered}
\left|\left\{x\in G, \mathbf{x}\in \prod_{i=1}^s G_i \; |\; (x+\Phi_1(\mathbf{x}),
\ldots,x+\Phi_t(\mathbf{x}))\in S^t\right\}\right|>\\
\delta
\left|\left\{x\in G, \mathbf{x}\in \prod_{i=1}^s G_i \;|\; (x+\Phi_1(\mathbf{x}),
\ldots,x+\Phi_t(\mathbf{x}))\in G^t\right\}\right|. \nonumber\end{gathered}$$
Consider $R=\{(x,\ldots,x)\}_{x\in G}$. Observe that the configuration set $\{x\in G, \mathbf{x}\in \prod_{i=1}^s G_i \;|\; (x+\Phi_1(\mathbf{x}),
\ldots,x+\Phi_t(\mathbf{x}))\in G^t\}$ is a subgroup of $G^t$, whence there exists a homomorphism $A$ such that $S(A,G)$ is the configuration set and, in this case, $S_i(A,G)=G=\pi_i(R)$ for all $i$. Thus the hypotheses of Corollary \[c.homotetic\_solutions2\] are fulfilled and the result follows.
Corollary \[c.homotetic\_solutions\] encompasses the simplex-like configurations from the multidimensional version of Szemerédi’s theorem with the evaluation: $G={\mathbb Z}_p^k$, $G_1={\mathbb Z}_p$, $\Phi_t=0$ and $\Phi_1,\ldots,\Phi_{k}$ being the coordinate homomorphisms $$\begin{aligned}
\Phi_i: {\mathbb Z}_p&\to {\mathbb Z}_p^t \nonumber \\
x&\mapsto (0,\ldots,0,\overbrace{x}^{i},0,\ldots,0). \nonumber\end{aligned}$$
Additionally Corollary \[c.homotetic\_solutions\] generalizes [@tao12 Theorem B.1] which asserts that given a finite abelian group $G$, for every $\epsilon>0$ and $t,m$ positive integers, there are, in any set $S\subset G^m$ with $|S|>\epsilon |G|^m$, $\delta(\epsilon,t,m) |G|^{m+1}$ configurations $$\{y\in G^m, x\in G \; | \; (y+\Phi_1(x),\ldots,y+\Phi_{(2t+1)^m}(x))\in S^{(2t+1)^m}\}$$ with $$\begin{aligned}
\Phi_i(x)=(\chi_1(i) x,\ldots,\chi_m(i) x) \nonumber\end{aligned}$$ where $(\chi_1(i),\ldots,\chi_m(i))$ are the components of $i$ in base $2t+1$ shifted by $-t$ so their values lie in $[-t,t]$ instead of the usual $[0,2t]$. An example of an extra configuration that Corollary \[c.homotetic\_solutions\] covers are the “rectangles” $(x,x+x_1,x+x_2,x+x_1+x_2)\in S^4$, for $S\subset G={\mathbb Z}_3^n$, with $x_1\in G_1$ and $x_2\in G_2$ two subgroups of $G$, isomorphic to ${\mathbb Z}_3^{n-\log(n)}$ and ${\mathbb Z}_3^{\sqrt{n}}$ respectively, and such that $G_1+G_2=G$.
The arguments to show Corollary \[c.homotetic\_solutions\], Corollary \[c.homotetic\_solutions2\], or Theorem \[t.multi\_szem\_ab\_gr\] exemplify that Theorem \[t.rem\_lem\_ab\_gr\] presents a comprehensive approach to the asymptotic counting of homothetic-to-a-point structures found in dense sets of products of finite abelian groups. More precisely, the constants involved in the lower bound of the number of configurations depend only on the number of points of the configuration and on the density of the set, but not on the configuration itself nor on the structure of the finite abelian group.
If we ask for configurations in the integers, the constant does depend on the configuration as we are not interested in solutions that occur due to the cyclic nature of the components of the finite abelian group. Therefore, we should reduce the density of the sets to allow only the desired solutions. This affects the total number of configurations found in the finite abelian group.
#### Monochromatic solutions.
Theorem \[t.rem\_lem\_ab\_gr\] also allows us to extend the results in [@serven14] regarding a counting statement for the monochromatic solutions of bounded torsion groups. In particular, we ensure that there are $\Omega\left(|S(A,G)|\right)$ monochromatic solutions, thus improving the asymptotic behaviour $\Omega\left(|G|^{m-k}\right)$ stated in [@serven14]. Here $S(A,G)$ represents the solution set of $A\mathbf{x}=0$, $\mathbf{x}\in G^m$, when $A$ is a $k\times m$ full rank integer matrix and the asymptotic behaviour depends on the number of colours.
#### Hypergraph containers.
Using the hypergraph containers tools from [@saxtho13+], Theorem \[t.rem\_lem\_ab\_gr\] can be used to extend [@saxtho13+ Theorem 10.3] or [@saxtho13+2 Theorem 2.10], regarding the number of subsets free of solutions of a given system of equations, and show for instance Theorem \[t.saxtom+\], where homomorphism systems are considered. Following the notation in [@saxtho13+], a homomorphism system $A$ is said to be *full rank* if there exists a solution to $Ax=\mathbf{b}$ for any $\mathbf{b}\in G^k$. A full rank $k\times m$ homomorphism system $A$ (or with coefficients over a finite field) is said to be *abundant* if any $k\times m-2$ subsystem of $A$ formed using $m-2$ columns of homomorphisms also has full rank. Given a set $Z\subset G^m$ of discounted solutions and $\mathbf{b}\in G^k$, a set $X\subset G$ is said to be $Z$-solution-free if there is no $x\in X^m-Z$ with $Ax=\mathbf{b}$. Let $\text{ex}(A,\mathbf{b},G)$ denote the size of the maximum $Z$-solution-free set.
\[t.saxtom+\] Let $\{G_i\}_{i\in I}$ be a sequence of finite abelian groups. Let $A_i$ be a sequence of abundant $k\times m$ homomorphism systems and $\mathbf{b}_i\in G_i^k$ a sequence of independent vectors such that $|S((A_i,\mathbf{b}_i),G_i)|=|G_i|^{m-k}$. Let $Z$ be such that $Z\subset S((A_i,\mathbf{b}_i),G_i)$ and $|Z|=o(|G_i|^{m-k})$. Then the number of $Z$-solution-free subsets of $G_i$ is $2^{\text{ex}(A_i,\mathbf{b}_i,G_i)+o(|G_i|)}$.
Outline of the paper
--------------------
The main results of the paper are Theorem \[t.rep\_sys\_rem\_lem\] and Theorem \[t.rem\_lem\_ab\_gr\]. To prove Theorem \[t.rep\_sys\_rem\_lem\], we observe that the notion of representation, Definition \[d.rep\_sys\], is sufficient to transfer the hypergraph removal lemma, Theorem \[t.rem\_lem\_edge\_color\_hyper\] in this paper, to the representable setting. The argument can be found in Section \[s.representables\_systems\]. Some examples of representable systems and their correspondent removal lemmas are presented.
In Section \[s.equiv\_and\_rep\] we introduce the notion of $\mu$-equivalent linear systems (see Section \[s.equiv\_systems\]). In Section \[s.oper\_between\_representable\], we show some relations between the representability of the systems $(A_1,G_1)$ and $(A_2,G_2)$ whenever $(A_2,G_2)$ is $\mu$-equivalent to $(A_1,G_1)$. These results are used in the proof of Theorem \[t.rem\_lem\_ab\_gr\]. Indeed, the strategy of the proof can be summarized as finding a suitable sequence of $\mu$-equivalent systems, from the system of our interest, to a representable one. As Section \[s.oper\_between\_representable\] shows, we can then find a representation for our original system.
In Section \[s.proof\_rl-lsg-1\] and Section \[s.representability\_product\_cyclics\] we prove Theorem \[t.rem\_lem\_ab\_gr\] by arguing that the systems involved in the statement of the theorem are representable. Section \[s.finish\_rem\_lem\_dkA1\] is devoted to show the cases where $m\leq k+1$ and to prove the second part of the result involving the sets $X_i$ for which $X_i\supset\pi_i(S(A,G))$.
The sketch of the construction for the representation is as follows. Given $G=\prod_{i=1}^t {\mathbb Z}_{n_i}$ with $n=n_1$ and $n_i|n_j$ for $i\geq j$, we interpret the homomorphism $A:G^m\to G^t$ as a homomorphism $A'$ from $({\mathbb Z}_{n}^t)^m$ to $({\mathbb Z}_{n}^t)^k$ in a natural way. Then any solution of $S(A,G)$ is related to $|S(A',{\mathbb Z}_{n}^t)|/|S(A,G)|$ solutions of $S(A',{\mathbb Z}_{n}^t)$. This reduction process is detailed in Section \[s.proof\_rl-lsg-1\].
As Section \[s.hom\_mat\_to\_integer\_mat\] shows, the homomorphism matrix $A'$ can be thought of as an integer matrix from ${\mathbb Z}_{n}^{t m}$ to ${\mathbb Z}_{n}^{t k}$ with $t m$ variables and $t k$ equations in ${\mathbb Z}_{n}$. This interpretation as an integer matrix allows for the construction of the representation by using the ideas in the proof of [@ksv13 Lemma 4]. The construction is detailed in Section \[s.representability\_product\_cyclics\] and involves several transformations to the pair $(A',G')$ to address the different issues like the determinantal being larger than $1$. The main characteristics of those transformations are described in the statements of Section \[s.oper\_between\_representable\]. The $\Gamma$-representability with $\gamma_j(i)> 1$ involves the generation of several systems. The construction of such systems is detailed in Section \[s.gamma-effective\] and they are combined in Section \[s.final\_composition\] to create a single $1$-strongly-representable system.[^12] A summary of all the transformations can be found in a table in Section \[s.unwrap\_const\].
Representable systems {#s.representables_systems}
=====================
In this work $[a,b]$ stands for the integers between $a$ and $b$, both included. If $\mathbf{x}\in G^m$, then $(\mathbf{x})_i$ denotes the $i$-th component of $\mathbf{x}$. Let us recall some notions regarding hypergraphs. Given a hypergraph $K=(V,E)$, $V=V(K)$ denotes the vertex set, $E=E(K)$ denotes the edge set and $|K|=|V(K)|$ denotes the size of the vertex set. A hypergraph $K$ with vertex set $V=V(K)$ and edge set $E=E(K)$ is said to be $s$-uniform if each edge in $E$ contains precisely $s$ vertices. Throughout this paper, we consider hypergraphs with edges coloured by integers. A hypergraph $K$ is said to be $m$-coloured if each edge in $K$ bears a colour in $[1,m]$. If $K$ is an $m$-coloured hypergraph, $E_i(K)$ denotes the set of edges coloured $i\in[1,m]$ in $K$. By a copy of $H$ in $K$ we understand an injective homomorphism of colored hypergraphs of $H$ into $K$ respecting the colors of the edges (the map is from vertices to vertices, injective, and maps edges colored $i$ to edges with color $i$). We use $C(H,K)$ to denote the set of colored copies of $H$ in $K$. If $H$ has $m$ edges $\{e_1,\ldots,e_m\}$ with $e_i$ colored $i$ then $H$ can be identified with $(e_1,\ldots,e_m)$.
Representability {#s.rep_tech_def}
----------------
The definition of a representable system, Definition \[d.rep\_sys\], is a generalized notion of the one formalized in [@sha10] that suffices to obtain a removal lemma; in our case Theorem \[t.rep\_sys\_rem\_lem\]. These representability notions have been used in several works like [@can_tesis_09; @kraserven08; @kraserven09; @kraserven12; @ksv13; @sha10; @sze10] to translate the conclusion of the removal lemma for graphs or hypergraphs to linear systems of equations. The representable system notion could potentially be used in more general contexts than the homomorphism systems described in this work.
Recall that a system $(A,G)$ is a pair given by a finite set $G$ and a property $A:G^m\to \{0,1\}$. $S(A,G)$ denotes the preimage of $1$ by $A$. $\gamma$ denotes a tuple of $m$ positive real numbers $(\gamma_1,\ldots,\gamma_m)$. $(\mathcal{A},\mathcal{G},m)$ denotes a family of systems and $\Gamma$ a collection of $\gamma$’s, one for each system.
\[d.rep\_sys\] The family of finite systems $(\mathcal{A},\mathcal{G},m)$ is said to be $\Gamma$-representable if there are positive real numbers $\chi_1, \chi_2$, depending on the family $(\mathcal{A},\mathcal{G},m)$, and for each $(A,G)\in (\mathcal{A},\mathcal{G},m)$ and the $\gamma=(\gamma_1,\ldots,\gamma_m)\in \Gamma$ associated with $(A,G)$, there exists a pair of coloured hypergraphs $(K,H)$ with the following properties RP\[prop\_rep1\], RP\[prop\_rep2\] and RP\[prop\_rep3\].
1. \[prop\_rep1\]
- $K$ and $H$ are $s$-uniform $m$-colored hypergraphs.
- $H$ has $m$ different edges $\{e_1,\ldots,e_m\}$ and the edge $e_i$ is coloured $i$. Moreover $\chi_1\geq|V(H)|=h>s\geq2$.
- Each edge in $K$ bears a label in $G$ given by $l:E(K)\to G$.
2. \[prop\_rep2\] There exist a positive integer $p$, a set $Q$, and a surjective map $r$ $$\begin{aligned}
r:C(H,K) &\longrightarrow S(A,G)\times Q\nonumber \\
H=\{e_1,\ldots,e_m\}&\longmapsto (r_0(H),r_q(H)) \nonumber
\end{aligned}$$ such that $r_0(H)=(l(e_1),\ldots,l(e_m))$, and, for any given $\mathbf{x}\in S(A,G)$ and $q\in Q$, the set $r^{-1}(\mathbf{x},q)$ has size $$|r^{-1}(\mathbf{x},q)|=p \lambda \prod_{i=1}^m \gamma_i \text{ with } \lambda= c\frac{|K|^s}{|G|}$$ for some $c\geq \chi_2$.
3. \[prop\_rep3\] If $e_i$ is an edge coloured $i$ in a copy $H \in r^{-1}(\mathbf{x},q)$, then $p\frac{\prod_{j=1}^m\gamma_j}{\gamma_i}$ copies of $H$ in $r^{-1}(\mathbf{x},q)$ contain $e_i$.
If, additionally,
1. \[prop\_rep4\] For any edge $e_i$ coloured $i$ and $l(e_i)=\mathbf{x}_i$, there exists a copy of $H\in r^{-1}(\mathbf{x},q)$, with $(\mathbf{x})_i=\mathbf{x}_i$, containing $e_i$,
then the family is said to be strongly $\Gamma$-representable.
If $H\in r^{-1}(\mathbf{x},q)$ we say that $H$ is related to $\mathbf{x}$ through $q$. If a system $(A,G)$ belongs to a $\Gamma$-representable family of systems and has $\gamma$ as its associated parameters then $(A,G)$ is said to be $\gamma$-representable. If $\gamma_1=\cdots=\gamma_m=1$ we say that the system is $1$-representable. The vector $(K,H,\gamma,l,r,Q,p,c)$ defines the $\gamma$-representation and the key parameters are $\chi_1$ and $\chi_2$.
#### Comments on Definition \[d.rep\_sys\]. {#s.rep_oper_def}
In the definition, the hypergraphs $H$ and $K$ could have also been asked to be directed.
By choosing $Q=\{1\}$, $p=1$ and $\gamma_1=\cdots=\gamma_m=1$ for all the systems $(A,G)$, Definition \[d.rep\_sys\] covers the representation notions in [@kraserven09; @ksv13; @kraserven12; @sha10]. The main purpose of the introduction of the set $Q$ is to accommodate the determinantal condition from [@ksv13 Theorem 1]. The different $p$ and $\gamma$ allow for removing different proportions for different sets $S_i(A,G)$, the projections of the solution set to the coordinates of $G^m$.
Asking for the bounds on $s$, $h$ and $c$ to depend on $m$ and on the family of systems as a whole is one of the key points in the representability notion. The existence of $r$ in RP\[prop\_rep2\] and the definition of $r_0(q)$, imply that the labels of the edges of each copy of $H$ in $K$, ordered by colours, form a solution of the system $(A,G)$.
For each solution $\mathbf{x}=(x_1,\ldots,x_m)\in S(A,G)$, the set $Q$ equipartitions the copies of $H$ in $K$ related to $\mathbf{x}$. The conditions RP\[prop\_rep2\] and RP\[prop\_rep3\] guarantee, for each $\mathbf{x}$, $q$ and $i\in[1,m]$, the existence of a set of $i$-colored edges with size $$|E_i(\mathbf{x},q)|=\lambda \gamma_i=c\frac{|K|^s}{|G|} \gamma_i,$$ where $c$ is lower bounded by a function of $m$ such that the following holds. For each edge $e\in E_i(\mathbf{x},q)$, there are $p \frac{\prod_{j=1}^m \gamma_j}{\gamma_i}$ copies of $H$ in $K$ related to $(\mathbf{x},q)$ containing $e$. $p$ is independent on $\mathbf{x}$, $i$, $q$ or $e$. By the existence of $r$ in RP\[prop\_rep2\], any copy of $H$ in $K$ related to $\mathbf{x}$ through $q$ intersects $E_i(\mathbf{x},q)$ for all $i\in [1,m]$.
If the system is strongly representable, then $E_{i}(\mathbf{x},q)$ is the set of edges labelled with $(\mathbf{x})_i$.
In Definition \[d.rep\_sys\], we could have made the constants $c$ to depend on the pair $(\mathbf{x},q)$ as long as $c_{\mathbf{x},q}\geq \chi_2$ for any $(\mathbf{x},q)\in S(A,G)\times Q$. The proof of Theorem \[t.rep\_sys\_rem\_lem\] in Section \[s.rep\_sys\_proof\] can be adapted to this case by using the bound $\chi_2$ instead of $c$.
If the system is $\gamma$-strongly-representable, then the new set $Q$ can be considered to be $\{1\}$ at the expense of increasing the value of $p$ to $p|Q|$. Indeed, for any $q$, the set of hypergraphs $H$ in $K$ related to $(\mathbf{x},q)$ contains all the edges labelled $(\mathbf{x})_i$. Therefore any edge labelled $(\mathbf{x})_i$ contains $p\frac{\prod_{j=1}^m\gamma_j}{\gamma_i}|Q|$ copies of $H$ related to $\mathbf{x}$ in $\cup_{q\in Q} \;r^{-1}(\mathbf{x},q)$.
Representable systems and the removal lemma {#s.rep_sys_proof}
-------------------------------------------
The proof of the removal lemma for representable systems, Theorem \[t.rep\_sys\_rem\_lem\], uses the coloured version of the hypergraph removal lemma, Theorem \[t.rem\_lem\_edge\_color\_hyper\] in this work. Theorem \[t.rem\_lem\_edge\_color\_hyper\] can be deduced from Austin and Tao’s [@austao10 Theorem 1.5]. Alternatively, the coloured version of the hypergraph removal lemma can be proved using the arguments that lead to the colourless version of the hypergraph removal lemma [@elesze12; @gow07; @rodletal; @tao06], or it can be found in Ishigami’s [@ishi09].
\[t.rem\_lem\_edge\_color\_hyper\] For any positive integers $r$, $h$, $s$ with $h\ge s\ge 2$ and every $\varepsilon >0$ there exists $\delta>0$ depending on $r$, $h$, $s$ and $\varepsilon$ such that the following holds.
Let $H$ and $K$ be $r$-colored $s$-uniform hypergraphs with $h=|V(H)|$ and $M=|V(K)|$ vertices respectively. If the number of copies of $H$ in $K$ (preserving the colors of the edges) is at most $\delta M^h$, then there is a set $E'\subseteq
E(K)$ of size at most $\varepsilon M^{s}$ such that the hypergraph $K'$ with edge set $E(K)\setminus E'$ is $H$–free.
Let $(K,H)$ be the hypergraph pair that $\gamma$-represents the system $(A,G)$, with $\gamma=(\gamma_1,\ldots,\gamma_m)$. Let us denote the labelling by $l:E(K)\to G$ and the representation function by $r:C(H,K)\to S(A,G)\times Q$. The components of $r$ are given by $r_0:C(H,K)\to S(A,G)$ and $r_q:C(H,K)\to Q$. Recall that, by Definition \[d.rep\_sys\], if $H_0=\{e_1,\ldots,e_m\}$ is a copy of $H$ in $K$, then $r_0(H)=(l(e_1),\dots,l(e_m))$. Let $K_X$ be the subhypergraph of $K$ with the same vertex set as $K$ and the edges belonging to $r_0^{-1}(S(A,G,X))$. In other words, $K_X\subset K$ is the hypergraph containing only the edges whose labels belong to the restricted solution set.
By the property RP\[prop\_rep2\] of the $\gamma$-representability of the system, the total number of copies of $H$ in $K$ is, for the $c$ and $p$ provided by the representation, at most $$c\frac{|K|^s}{ |G|} p |Q||S(A, G)| \prod_{i=1}^m \gamma_i.
$$ Let $\lambda=c\frac{|K|^s}{ |G|}$. Since $H$ has $h$ vertices, it follows that $$\lambda p |Q| |S(A, G)| \prod_{i=1}^m \gamma_i<|K|^h.$$ On the other hand, the hypothesis $|S(A,G,X)|<\delta |S(A,G)|$, $\delta$ to be determined later, implies that the total number of copies of $H$ in $K_X$ is at most $$\lambda p |Q| |S(A, G, X)| \prod_{i=1}^m \gamma_i < \delta\lambda p|Q| |S(A, G)| \prod_{i=1}^m \gamma_i < \delta |K|^h.$$ We apply the Removal Lemma for colored hypergraphs, Theorem \[t.rem\_lem\_edge\_color\_hyper\], with $\varepsilon'=c\varepsilon/m$. By setting $\delta$ according to $\varepsilon'$ and $H$ in Theorem \[t.rem\_lem\_edge\_color\_hyper\], we obtain a set of edges $E'\subset E(K_X)$ with cardinality at most $\varepsilon'|K|^s$ such that $K_X \setminus E'$ has no copy of $H$. We note that $\delta$ depends on $s,h,m$ and $\varepsilon'$, which in our context and by the representability, all depend on $m$ and $\varepsilon$.
We next define the sets $X'_i\subset X_i$ as follows. The element $x$ is in $X_i'$ ($x$ is removed from $X_i$) if $E'$ contains at least $\lambda \gamma_i /m $ edges labelled $x$ and colored $i$. We observe that $$|X'_i|\le \frac{|E'|}{(\lambda\gamma_i/m)} =\frac{m|G|}{c|K|^s}|E'| \frac{1}{\gamma_i}\le \varepsilon \frac{|G|}{\gamma_i}.$$
We claim that $S(A,G,X\setminus X')$, with $X\setminus X'=\prod_{i=1}^m X_i\setminus X_i'$, is empty. Indeed, pick one element $\mathbf{x}=(x_1,\ldots ,x_m)\in S(A,G,X)$ and $q\in Q$. By RP\[prop\_rep2\] there are $p\lambda \prod_{i=1}^m \gamma_i$ copies of $H$ in $r^{-1}(\mathbf{x},q)$. Since $\mathbf{x}\in S(A,G,X)$, all these copies belong to $K_X$. On the other hand, by RP\[prop\_rep3\], every edge of $K$ coloured $i$ is contained in at most $p\prod_{j\in[1,m]\setminus \{i\}} \gamma_j$ copies of $H$ in $r^{-1}(\mathbf{x},q)$. Let $E'_{i,x_i}$ denote the set of edges in $E'$ labelled with $x_i$ and colored $i$. Then $$\sum_{i=1}^m \left[|E'_{i,x_i}| p\prod_{j\in[1,m]\setminus \{i\}} \gamma_j \right]\geq p \lambda\prod_{j\in[1,m]} \gamma_j$$ as there are no copies related to $(\mathbf{x},q)$ after $E'$ has been removed. By the pigeonhole principle, at least one of the sets $E'_{i,x_i}$ is such that $|E'_{i,x_i}|>\lambda \gamma_i/m$. By the definition of $X'_i$, the element $x_i$ belongs to $X'_i$ and thus $\mathbf{x}\not\in X\setminus X' \supset S(A,G,X\setminus X')$. This proves the claim and finishes the proof of the result.
Examples of representable systems and removal lemmas {#s.some_rep_syst}
----------------------------------------------------
### Subhypergraph copies
As expected, the coloured hypergraph removal lemma can be retrieved from Theorem \[t.rem\_lem\_edge\_color\_hyper\]. The system of configurations induced by “the copies of an $r$-coloured $k$-uniform hypergraph $H_0$ in an $r$-coloured $k$-uniform hypergraph $K_0$” can be represented by Definition \[d.rep\_sys\] as follows. Order the edges of $H$ arbitrarily. $H=H_0$ and $K=K_0$ as the pair of hypergraphs that represents the system. The property $A$ is the map from $E(K)^{|E(H)|}$ to $\{0,1\}$ such that $A(e_1,\ldots,e_{|E(H)|})=1$ if and only if the edges $(e_1,\ldots,e_{|E(H)|})$ conform a copy of $H$ in $K$ in which $e_i$ is the $i$-th edge of $H$ with the chosen order. The map $l$ is given by the identity map of the edge in $K$, $r_0$ is the identity map induced by the property $A$, $Q=\{1\}$, $\lambda=c=\gamma_i=1$. The sets $X_i$ in the removal lemma Theorem \[t.rep\_sys\_rem\_lem\] are the edges in $K_0$ coloured using the colour of the $i$-th edge in $H$.
### Permutations {#s.perm_rep_and_rem_lem}
The copies of $\tau\in \mathcal{S}(t)$ in $\sigma\in \mathcal{S}(n)$, as defined by the set $\Lambda^{\tau}(\sigma)$ in Section \[s.intro\_permutations\], can be represented using directed and coloured graphs $H$ and $K$ in Definition \[d.rep\_sys\] as follows. Given a finite set $V$, let ${V\choose i}$ denote the set of subsets of $i$ different elements of $V$. Let $A$ be the property $A:{[0,n-1]\choose 2}^{\left|{[0,t-1] \choose 2}\right|} \to \{0,1\}$ such that $A(e_1,\ldots,e_{t(t-1)/2})=1$ if and only if the collection of endpoints of the edges $\{e_1,\ldots,e_{t(t-1)/2}\}$ configure an $m$-element set $\{x_0<\cdots<x_{t-1}\}$ in $[0,n-1]$ belonging to $\Lambda^{\tau}(\sigma)$.
Given a permutation $\sigma\in \mathcal{S}(n)$, let us define the loopless bicolored directed graph $G_{\sigma}$ as follows. The vertex set $V(G_{\sigma})$ is given by the $n$-element set $[0,n-1]$. The directed edge $e=(i,j)$ or $e=\{i\to j\}$, from $i$ to $j$, belongs to $E(G_{\sigma})$ if and only if $\sigma(i)<\sigma(j)$. The edge $e=\{i\to j\}$ is painted blue if $i<j$ and painted red if $i>j$. Observe that $|E(G_{\sigma})|={n \choose 2}$.
We claim that the system for the permutations involved in Proposition \[p.rem\_lem\_permutations\] is representable with $A$ as above, $H=G_{\tau}$, $K=G_{\sigma}$, $m={t \choose 2}$, $Q=\{1\}$, $\gamma_i=c=\lambda=1$ and $r_0$ given as follows. If $\{x_0<\ldots<x_{t-1}\}\subset [0,n-1]=V(K)$ is a set of indices that generates a copy of $H$ in $K$, then $r_0(\{x_0<\ldots<x_{t-1}\})=\{x_0<\ldots<x_{t-1}\}$.
\[cl.0\] If $H_0$, with $V(H_0)=\{x_0<\cdots<x_{t-1}\}$, is a copy of $G_{\tau}=([0,t-1],E(G_{\tau}))$ in $G_{\sigma}$, then the only map (homomorphism) from $f:V(G_{\tau}) \to V(H_0)$ with the property “if $e=\{i\to j\}\in E(G_{\tau})$ and is coloured $c$, then $\{f(i)\to f(j)\}\in E(G_{\sigma}{|V(H_0)})$ and is coloured $c$” is the map $f(i)=x_i$ for all $i\in [0,t-1]$.
The map $f$ must be bijective. Indeed, since $G_{\tau}$ is a complete graph if $f$ were not bijective, then the graph induced by $V(H_0)$ would contain a loop as $f$ is a homomorphism, but $G_{\sigma}$ is loopless.
If $f$ is not the map $f(i)=x_i$, then there exist a pair $i,j\in[1,t-1]$ with $i<j$ but $f(i)>f(j)$. If the edge between $i$ and $j$ is $e=\{i\to j\}$, then $f(e)=\{f(i)\to f(j)\}$. In such case, $e$ is painted blue as $i<j$ and $f(e)$ is painted red as $f(i)>f(j)$, hence $f$ is not an homomorphism. If the edge between $i$ and $j$ is $e'=\{j\to i\}$, then $e$ is coloured red but $f(e')$ is blue. Therefore, if $f$ is a homomorphism, it has to be the isomorphism with $f(i)=x_i$.
\[cl.1\] If $H_0$, with $V(H_0)=\{x_0<\cdots<x_{t-1}\}$ is a copy of $G_{\tau}$ in $G_{\sigma}$ where $x_i\in V(G_{\sigma})$ corresponds to the $i$-th vertex of $G_{\tau}$, then $\{x_0<\cdots <x_{m-1}\}\in \Lambda^{\tau}(\sigma)$.
Let $e=\{x_i\to x_j\}$ be an edge in $G_{\sigma}$, then $\sigma(x_i)<\sigma(x_j)$. Since $H_0$ is a copy of $G_{\tau}$ where $x_i$ corresponds to the $i$-th vertex of $G_{\tau}$, $e'=\{i\to j\}$ is an edge in $G_{\tau}$ meaning that $\tau(i)<\tau(j)$ as wanted. Since the reverse implication also holds, the result is shown.
\[cl.2\] If $\{x_0<\cdots <x_{t-1}\}\in \Lambda^{\tau}(\sigma)$ then the graph induced by $x_0,x_1,\ldots,x_{t-1}$ in $G_{\sigma}$ is a copy of $G_{\tau}$ with the map from $V(G_{\tau})=[0,t-1]$ to $\{x_0,x_1,\ldots,x_{t-1}\}\subset V(G_{\sigma})$ given by $i\mapsto x_i$ for $i\in[0,t-1]$.
Assume the pair $\{i,j\}\in {[0,t-1]\choose 2}$, with $i<j$, is such that $\sigma(x_i)<\sigma(x_j)$. By the construction of $G_{\sigma}$ we have the edge $\{x_i\to x_j\}$ and is painted blue (as $x_i<x_j$). Since $\{x_0<\cdots <x_{t-1}\}\in \Lambda^{\tau}(\sigma)$, then $\tau(i)<\tau(j)$. Hence $G_\tau$ has the edge $\{i\to j\}$ coloured blue (as $i<j$).
Assume now that the pair $\{i,j\}\in {[0,t-1]\choose 2}$, with $i<j$, is such that $\sigma(x_i)>\sigma(x_j)$. $G_{\sigma}$ contains the edge $\{x_j\to x_i\}$ painted red (as $x_j>x_i$). On the other side we have $\tau(i)>\tau(j)$ as $\{x_0<\cdots <x_{t-1}\}\in \Lambda^{\tau}(\sigma)$. Hence $G_\tau$ has the edge $\{j\to i\}$ coloured red (as $j>i$).
Therefore, the map $i\mapsto x_i$, for $i\in[0,t-1]$, is a graph homomorphism preserving the colours and the directions of the edges as claimed.
Combining claims \[cl.0\]-\[cl.2\], we observe that $r_0$ is well defined and the representation of $\Lambda^{\tau}(\sigma)$ is given by the pair $(G_{\tau},G_{\sigma})$ with the parameters described above. Proposition \[p.rem\_lem\_permutations\] is shown by using Theorem \[t.rep\_sys\_rem\_lem\] with $X_i={[0,n-1] \choose 2}$ for all $i\in [t(t-1)/2]$. In this case the proof of Theorem \[t.rep\_sys\_rem\_lem\] should use, instead of Theorem \[t.rem\_lem\_edge\_color\_hyper\], a removal lemma for directed and coloured graphs that can by obtained by combining the arguments from [@alosha04 Lemma 4.1] with [@komsim96 Theorem 1.18].[^13]
Equivalent systems and representability {#s.equiv_and_rep}
=======================================
In this section we assume that the systems are defined by a homomorphism. The definition for $\mu$-equivalent systems is introduced in Section \[s.equiv\_systems\] and in Section \[s.oper\_between\_representable\] the relations between $\mu$-equivalent systems and their representations are explored.
Equivalent systems {#s.equiv_systems}
------------------
Let $\mu$ be a positive integer. The homomorphism system $(A_2,G_2)$ with $A_2:G_2^{m_1}\to G_2^{k_2}$ is said to be *$\mu$-equivalent* to the homomorphism system $(A_1,G_1)$, $A_1:G_1^{m_1}\to G_1^{k_2}$, with $m_2\geq m_1$, if
- $\mu |S(A_1,G_1)|=|S(A_2,G_2)|$.
- There exist an injective map $\sigma: [1,m_1] \to [1,m_2]$ and affine homomorphisms $\phi_1,\ldots,\phi_{m_1}$, $\phi_i: G_2\to G_1$ such that the map $$\phi(x_1,\ldots,x_{m_2})=\left(\phi_1(x_{\sigma(1)}),\ldots,\phi_{m_1}(x_{\sigma(m_1)})\right)$$ induces a $\mu$-to-$1$ surjective map $\phi: S(A_2,G_2) \to S(A_1,G_1)$.
An affine homomorphism is a map $\phi_i:G_2\to G_1$ with $\phi_i(x)=b+\phi_i'(x)$, where $\phi_i'$ is a homomorphism and $b$ is a fixed element in $G_1$. Observe that, if necessary, we can restrict $\phi_i$ to map from the subgroup $S_i(A_2,G_2)$ (or a coset of the subgroup $S_i((A_2,\mathbf{0}),G_2)$) to $S_i(A_1,G_1)$. If $G_1=G_2$ and the $\{\phi_i\}_{i\in [1,m_1]}$ are affine automorphisms, then $\phi_i\left( S_i(A_2,G_2)\right)=S_i(A_1,G_1)$ and their sizes are the same. In this case the systems are said to be *auto-equivalent*.
Operations on equivalent systems and representability {#s.oper_between_representable}
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The propositions \[p.1-auto-equiv-rep\] through \[p.mu-equivalent\_2\] proved in this section expose how the property of equivalence between systems, as defined in Section \[s.equiv\_systems\], is related with their representability properties, Definition \[d.rep\_sys\]. For this section $G$, $G_1$ and $G_2$ are finite abelian groups and the systems are homomorphism systems.
$1$-auto-equivalent systems
---------------------------
\[p.1-auto-equiv-rep\] Let $((A_2,\mathbf{b}_2),G)$ be a $k_2\times m_2$ system $1$-auto-equivalent to $((A_1,\mathbf{b}_1),G)$, a $k_1\times m_1$ system. Assume $((A_2,b_2),G)$ is $\gamma'$-representable by $(K',H')$ with constants $\chi_1,\chi_2$. If the edges coloured by $\sigma(1),\ldots,\sigma(m_1)$ cover all the vertices of $H'$, then $(A_1,G)$ is $\gamma$-representable with the same constants $\chi_1,\chi_2$ and $\gamma_i=\gamma'_{\sigma(i)}$. If $((A_2,\mathbf{b}_2),G)$ is strongly representable, then so is $((A_1,\mathbf{b}_1),G)$.
Let $\phi$ be the map that defines the $1$-auto-equivalence $\phi: S(A_2,G) \to S(A_1,G)$ with $\phi(x_1,\ldots,x_{m_2})=\left(\phi_1(x_{\sigma(1)}),\ldots,\phi_{m_1}(x_{\sigma(m_1)})\right)$. Let $(K',H', \gamma',\linebreak[1] l', r', Q', p', c')$ be the vector defining the $\gamma'$-representation for $(A_2,G)$. Let $s$ be the uniformity of the edges of $H'$.
The vector $(K,H,\gamma,l,r,Q',p',c')$ defines the $\gamma$-representation of $((A_1,\mathbf{b}_1),G)$ as follows. $\gamma_i=\gamma'_{\sigma(i)}$ for $i\in[1,m_1]$. $H$ and $K$ are the hypergraph on the same vertex set of $H'$ and $K'$ respectively, and with the edges given by the colours $\sigma(1),\ldots,\sigma(m_1)$. Repaint the edge coloured $\sigma(i)$ with colour $i$. If $e=\{v_1,\ldots,v_s\}$ is an edge coloured $\sigma(i)$ in $K'$ and labelled $l'(e)$, then $e$ is an edge coloured $i$ in $K$ and labelled $l(e)=\phi_i(l'(e))$. $r_q(H_0)=r_q'(H_0')$ where $H_0'$ is the unique copy of $H'$ in $K'$ spanned by the vertices of $H_0$ seen as vertices of $K'$.
Each copy of $H$ in $K$ induces a unique copy of $H'$ in $K'$ and vice-versa. Moreover, $\phi$ is a bijection between the solution sets and $(K',H',\gamma',l',r',Q',p',c')$ is a $\gamma'$-representation for $((A_2,\mathbf{b}_2),G)$. Therefore, $(K,H,\gamma,l,r,Q,p,c)$ as defined above induces a $\gamma$-representation for $((A_1,\mathbf{b}_1,G)$ and have the same constants $\chi_1$ and $\chi_2$ as $(K',H',\gamma',l',r,Q',p',c')$. Since $\phi_i$ are affine automorphisms, if the representation for $((A_2,\mathbf{b}_2),G)$ is strong, the so is the representation for $((A_1,\mathbf{b}_1,G)$ here presented.
$\mu$-auto-equivalent systems
-----------------------------
\[p.mu-auto-equivalent\] Let $((A_2,\mathbf{b}_2),G)$ be a $k_2\times m_2$ system $\mu$-auto-equivalent to the $k_1\times m_1$ system $((A_1,\mathbf{b}_1),G)$, $m_2\geq m_1$. Let $$\begin{aligned}
\phi:S((A_2,\mathbf{b}_2),G)&\longrightarrow S((A_1,\mathbf{b}_1),G) \nonumber \\
(x_1,\ldots,x_{m_2}) &\longmapsto \left(x_{1},\ldots,x_{m_1}\right) \nonumber\end{aligned}$$ be the map that defines the $\mu$-auto-equivalence. Assume $((A_2,\mathbf{b}_2),G)$ is $(\gamma'_1,\ldots,\gamma'_{m_2})$-representable by $(K',H')$ with constants $\chi_1,\chi_2$. If the edges coloured by $1,\ldots,m_1$ cover all the vertices of $H'$, then $((A_1,\mathbf{b}_1),G)$ is $(\gamma'_{1},\ldots,\gamma'_{m_1})$-representable with $\chi_1,\chi_2$ as constants. If $((A_2,\mathbf{b}_2),G)$ is strongly representable, then so is $((A_1,\mathbf{b}_1),G)$.
Let $\iota$ be a map from $S((A_2,\mathbf{b}_2),G)$ to $[1,\mu]$ where, given $\mathbf{x}_1, \mathbf{x}_2\in S((A_2,\mathbf{b}_2),G)$ such that $\phi(\mathbf{x}_1)=\phi(\mathbf{x}_2)$ and $\mathbf{x}_1\neq \mathbf{x}_2$, then $\iota(\mathbf{x}_1)\neq \iota(\mathbf{x}_2)$. If $\phi$ is a $\mu$-to-$1$ map, such $\iota$ exist, is exhaustive and induces an equipartition in $S((A_2,\mathbf{b}_2),G)$.
Let $(K',H',\gamma',l',r',Q',p',c')$ be the vector defining the $\gamma'$-representation for $((A_2,\mathbf{b}_2),G)$. Let $s$ be the uniformity of the edges of $H'$. The candidate vector $(K,H,\gamma,l,r,Q,p,c)$ is defined as follows.
- $Q=Q'\times [1,\mu]$, $p=p'\prod_{i=m_1+1}^{m_2} \gamma'_i$, $c=c'$, $\gamma$ is such that $\gamma_i=\gamma'_i $ for $i\in[1,m_1]$.
- $H$ and $K$ are the hypergraphs on the vertex sets of $H'$ and $K'$ respectively. $e=\{v_1,\ldots,v_s\}$ is an edge in $K$ coloured $i\in[1,m_1]$ if and only $e$ is an edge coloured $i\in[1,m_1]$ in $K'$.
- $l$ is defined by $l(e)=l'(e)$ for $e$ an edge coloured $i\in[1,m_1]$.
- If $H_0\in C(H,K)$, then $r_q(H_0)=(r_q'(H_0'),\iota(r_0'(H_0')))$ where $H_0'$ is the unique copy of $H'$ in $K'$ spanned by the vertices of $H_0$, seen as vertices of $K'$.
Selecting $\mathbf{x}\in S((A_1,\mathbf{b}_1),G)$ and $q=(q',j)\in Q=Q'\times[1,\mu]$ is equivalent to select the $\mathbf{y}\in S((A_2,\mathbf{b}_2),G)$, with $\mathbf{y}\in \phi^{-1}(\mathbf{x})$ such that $\iota(\mathbf{y})=j$, and $q'\in Q'$, first coordinate of $q$. Moreover, each copy of $H$ in $K$ induces a unique copy of $H'$ in $K'$ and vice-versa. Therefore, the class of copies of $H$ related to $(\mathbf{x},q)$ is the same as the copies of $H'$ related to $(\mathbf{y},q')$.
Since each edge $e_i\in E(K')$, $i\in [1,m_1]$, is contained in $p'\frac{\prod_{j=1}^{m_2}\gamma'_j}{\gamma'_i}$ copies of $H'$ related to $(\mathbf{y},q')$, then it also contains, seen as an edge in $K$, $p'\frac{\prod_{j=1}^{m_2}\gamma'_j}{\gamma'_i}=p\frac{\prod_{j=1}^{m_1}\gamma'_j}{\gamma_i}$ copies of $H$ related to $(\mathbf{x},q)$. Therefore, $(K,H,\gamma,l,r,Q,p,c)$ as defined above induces a $\gamma$-representation for $(A_1,G)$ and have the same constants $\chi_1$ and $\chi_2$ as $(K',H',\gamma',l',r',Q',p',c')$. Moreover, since $\phi_i$ is the identity map for each $i$, if the representation for $((A_2,\mathbf{b}_2),G)$ is strong, then so is the presented representation for $((A_1,\mathbf{b}_1),G)$.
$\mu$-equivalent systems
------------------------
\[p.mu-equivalent\_1\] Let $((A_2,\mathbf{b}_2),G_2)$ be a $k_2\times m_1$ system $\mu$-equivalent to the $k_1\times m_1$ system $((A_1,\mathbf{b}_1),G_1)$ with $$\begin{aligned}
\phi:S((A_2,\mathbf{b}_2),G_2)&\longrightarrow S((A_1,\mathbf{b}_1),G_1) \nonumber \\
(x_1,\ldots,x_{m_1}) &\longmapsto \left(\phi_1(x_{1}),\ldots,\phi_1(x_{m_1})\right) \nonumber\end{aligned}$$ be the map that defines the $\mu$-equivalence. Assume $(A_2,G_2)$ is $(\gamma_1,\ldots,\gamma_{m_1})$-representable by $(K',H')$ with constants $\chi_1,\chi_2$. If $\phi_1:G_2\to G_1$ is surjective and $\phi^{-1}(\mathbf{x})=\prod_{i=1}^{m_1} \phi_1^{-1}((\mathbf{x})_i)$ then $((A_1,\mathbf{b}_1),G_1)$ is $(\gamma_{1},\ldots,\gamma_{m_1})$-representable with the same constants $\chi_1,\chi_2$. If $((A_2,\mathbf{b}_2),G)$ is strongly representable, then so is $((A_1,\mathbf{b}_1),G)$.
Observe that, for $i\in[1,m_1]$, $\phi_1(S_i((A_2,\mathbf{b}_2),G_2))=S_i((A_1,\mathbf{b}_1),G_1)$ as $\phi$ is surjective. Since $\phi_1$ is affine, $|\{y_i\in S_i((A_2,\mathbf{b}_2),G_2) : \phi_1(y_i)=x_i\}|$ is the same for each $x_i\in S_i((A_1,\mathbf{b}_1),G_1)$. Since $\phi^{-1}(\mathbf{x})=\prod_{i=1}^{m_1} \phi_1^{-1}((\mathbf{x})_i)$ and $\phi_1$ is affine, then we can let $\beta=|S_i((A_2,\mathbf{b}_2),G_2)|/|S_i((A_1,\mathbf{b}_1),G_1)|$, as its value is independent of $i\in[1,m_1]$. Therefore, $\mu=\beta^{m_1}$. Additionally, since $\phi_1$ is surjective, $\beta=|G_2|/|G_1|$.
Let $\iota$ be a map from $G_2$ to ${\mathbb Z}_{\beta}$ such that, if $y_1,y_2\in G_2$ with $\phi_1(y_1)=\phi_1(y_2)$ and $y_1\neq y_2$, then $\iota(y_1)\neq \iota(y_2)$. Since $\phi_1$ is a $\beta$-to-$1$ map between $G_2$ and $\phi_1(G_2)=G_1$, then such $\iota$ exist, is exhaustive and induces an equipartition of $G_2$ in $\beta$ classes. Moreover, $\iota$ induces the bijections $$\begin{array}{cl}
G_2 &\longrightarrow \phi_1(G_2)\times {\mathbb Z}_{\beta} \\ y &\longmapsto (\phi_1(y),\iota(y)) \end{array}
\text{ and }
\begin{array}{cl}
S((A_2,\mathbf{b}_2),G_2) &\longrightarrow S((A_1,\mathbf{b}_1),G_1)\times {\mathbb Z}_{\beta}^{m_1} \\ \mathbf{y} &\longmapsto (\phi(\mathbf{y}),\iota(\mathbf{y})) \end{array}$$ where $\iota((\mathbf{y}_1,\ldots,\mathbf{y}_{m_1}))=(\iota(\mathbf{y}_1),\ldots,\iota(\mathbf{y}_{m_1}))$. Let $\pi:{\mathbb Z}_{\beta}^{m_1}\to {\mathbb Z}_{\beta}^{m_1}/\langle 1,\ldots,1\rangle$ be the quotient map.
Let $(K',H',\gamma',l',r',Q',p',c')$ be the vector defining the $\gamma'$-representation for $((A_2,\mathbf{b}_2),G_2)$. Let $s$ be the uniformity of the edges of $H'$. The candidate vector $(K,H,\gamma,l,r,Q,p,c)$ is defined as follows.
- $Q=Q'\times \left[{\mathbb Z}_{\beta}^{m_1}/\langle 1,\ldots,1\rangle\right]$, $p=p'$, $c=c'$, $\gamma_i=\gamma'_i $ for $i\in[1,m_1]$.
- $H$ and $K$ are the hypergraphs on the same vertex sets and edge sets as $H'$ and $K'$ respectively. $e=\{v_1,\ldots,v_s\}$ is an edge in $K$ coloured $i\in[1,m_1]$ if and only $e$ is an edge coloured $i\in[1,m_1]$ in $K'$.
- $l(e)=\phi_1(l'(e))$ if $e$ is an edge coloured $i\in[1,m_1]$ as an edge in $K'$.
- Given $H_0\in C(H,K)$, let $H_0'$ be the unique copy of $H'$ in $K'$ spanned by the vertices of $H_0$ and let $\mathbf{y}=(\mathbf{y}_1,\ldots,\mathbf{y}_{m_1})=r_0'(H_0')\in S((A_2,\mathbf{b}_2),G_2)$ the solution spanned by $H_0'$. Then $r_q(H_0)=(r_q'(H_0'),\pi(\iota(\mathbf{y})))$.
Property RP\[prop\_rep1\] is fulfilled with the same parameters and each edge bears a label given by $l$. The function $r=(r_0,r_q)$ goes from $C(H,K)$ to $S((A_1,\mathbf{b}_1),G_1)\times Q$ by the definition of $r'=(r_0',r_q')$, $\iota$, $\phi_1$ and $\pi$. $r$ is surjective because $r'$ is surjective and $S((A_2,\mathbf{b}_2),G_2)$ is in bijection with $S((A_1,\mathbf{b}_1),G_1)\times {\mathbb Z}_{\beta}^{m_1}$. Observe that $r^{-1}(\mathbf{x},q)$ is the union of those $r'^{-1}(\mathbf{y},q')$, with $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$, such that $\phi(\mathbf{y})=\mathbf{x}$ and $q=(q',\pi(\iota(\mathbf{y})))$. This union has $\beta$ elements, as this is the size of each class in the quotient ${\mathbb Z}_\beta^{m_1}/\langle(1,\ldots,1)\rangle$. Therefore, $$\begin{aligned}
\left|r^{-1}(\mathbf{y},q)\right|=\beta \left|r'^{-1}(\mathbf{x},q')\right|=\beta p'c' \frac{|K'|^s}{|G_2|} \prod_{i=1}^{m_1} \gamma_i' =p c \frac{|K|^s}{|G_1|} \prod_{i=1}^{m_1} \gamma_i, \nonumber
\end{aligned}$$ which shows RP\[prop\_rep2\].
All the solutions $(\mathbf{y}_1,\ldots,\mathbf{y}_m)=\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ that conform the union just mentioned have the property that any component $\mathbf{y}_i$ takes all the possible $\beta$ values of $\phi_1^{-1} ((\mathbf{x})_i)$. Indeed, from all the solutions $(\mathbf{y}_1,\ldots,\mathbf{y}_m)=\mathbf{y}$ that, along with the $q'$, conform the sets of copies of $H$ given by $r^{-1}(\mathbf{x},q)$, there is only one solution $\mathbf{y}$ with $\mathbf{y}_i$ having a particular value in $\phi_1^{-1}((\mathbf{x})_i)$. Therefore, if two copies of $H$ in $K$ share an edge $e_i\in H_0\in r^{-1}(\mathbf{x},q)$, then they belong to the same set $r'^{-1}(\mathbf{y},q')$ if seen as copies of $H'$ in $K'$. Thus, there are $p'\frac{\prod_{j=1}^m \gamma'_j}{\gamma'_i}=p\frac{\prod_{j=1}^m \gamma_j}{\gamma_i}$ copies of $H$ in $K$ sharing $e_i$. This shows RP\[prop\_rep3\].
Let $\mathbf{x}\in S((A_1,\mathbf{b}_1),G_1)$ and $q=(q',j)\in Q$. Pick $e_i$ with $l(e_i)=(\mathbf{x})_i$ for some $i\in[1,m_1]$. Let $y_i=l'(e_i)$ by seen $e_i$ as an edge in $K'$. Let $\mathbf{y}$ be the unique solution to $S((A_2,\mathbf{b}_2),G_2)$ such that $\phi(\mathbf{y})=\mathbf{x}$, $(\mathbf{y})_i=y_i$, and $\pi(\iota(\mathbf{y}))=j$. If $((A_2,\mathbf{b}_2),G_2)$ is strongly representable, there exists a $H_0'$, $H_0'\in r'^{-1}(\mathbf{y},q')$, with $e_i\in H_0'$. If $H_0$ is the unique copy of $H$ in $K$ on the vertices of $H_0'$, then $e_i\in H_0$ and $H_0\in r^{-1}(\mathbf{x},q)$. This shows RP\[prop\_rep4\] for the system $((A_1,\mathbf{b}_1),G_1)$ when $((A_2,\mathbf{b}_2),G_2)$ is strongly representable and finishes the proof of the proposition.
\[p.mu-equivalent\_2\] Let $((A_2,\mathbf{b}_2),G_2)$ be a $k_2\times m_2$ system $\mu$-equivalent to the $k_1\times m_1$ system $((A_1,\mathbf{b}_1),G_1)$ with $m_2\geq m_1$. Let $$\begin{aligned}
\phi:S((A_2,\mathbf{b}_2),G_2)&\longrightarrow S((A_1,\mathbf{b}_1),G_1) \nonumber \\
(x_1,\ldots,x_{m_2}) &\longmapsto \left(\phi_1(x_{1}),\ldots,\phi_{m_1}(x_{m_1})\right) \nonumber
\end{aligned}$$ be the map that defines the $\mu$-equivalence. Assume $((A_2,\mathbf{b}_2),G_2)$ is $\gamma'$-strongly-representable by $(K',H')$ with constants $\chi_1,\chi_2$. Assume the following.
(i) \[h.1\] The edges coloured by $[1,m_1]$ cover all the vertices of $H'$.
(ii) \[h.2\] Given $\mathbf{x}\in S((A_1,\mathbf{b}_1),G_1)$ and $i\in[1,m_1]$, then $$\left\vert \{\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)\; : \; \phi(\mathbf{y})=\mathbf{x} \text{ and } (\mathbf{y})_i = y_i \}\right|$$ is constant for any $y_i\in S_i((A_2,\mathbf{b}_2),G_2)$ with $\phi_i(y_i)=(\mathbf{x})_i$.
Then $((A_1,\mathbf{b}_1),G)$ is strongly $\gamma$-representable with $\gamma_i=\gamma_i'\frac{|S_i((A_2,\mathbf{b}_2),G_2)|}{|S_i((A_1,\mathbf{b}_1),G_1)|}\frac{|G_1|}{|G_2|}$ for $i\in[1,m_1]$ and the same constants $\chi_1,\chi_2$.
The condition (\[h.2\]) is not superfluous. If $A_2=(1,1)$, $A_1=(1,2)$, $b_1=b_2=0$, $G_1=G_2={\mathbb Z}_2$ and $\phi(x_1,x_2)=(2x_1,x_2)$, then $\phi$ is one to one but $\phi_1$ does not satisfy (\[h.2\]) for the solution $(0,0)\in S(A_1,G_1)$. However, the number of solutions $\mathbf{y}\in S(A_2,G_2)$ with $\phi(\mathbf{y})=\mathbf{x}$ and $(\mathbf{y})_i=y_i$ is either zero (if there is no such solution $\mathbf{y}$ with $(\mathbf{y})_i=y_i$), or it is a positive fixed value for any $i\in[1,m_1]$ if there exist some solution $\mathbf{y}\in \phi^{-1}(\mathbf{x})$ with $(\mathbf{y})_i=y_i$. The reason being that $\phi$ and $\phi_i$ are affine homomorphisms and the preimage by $\phi$ and $\phi_i$ has a coset/subgroup-like structure. Therefore, the condition (\[h.2\]) can be rephrased as
1. \[h.2’\] Given $\mathbf{x}\in S((A_1,\mathbf{b}_1),G_1)$ and $i\in[1,m_1]$ then, for any $y_i\in S_i((A_2,\mathbf{b}_2),G_2)$ with $\phi_i(y_i)=(\mathbf{x})_i$, there exists a $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ with $\phi(\mathbf{y})=\mathbf{x}$ and $(\mathbf{y})_i=y_i$.
Since $\phi$ is surjective, so is $\phi_i:S_i((A_2,\mathbf{b}_2),G_2)\to S_i((A_1,\mathbf{b}_1),G_1)$ for $i\in[1,m_1]$. Since $\phi_i$ is affine, $|\{y_i\in S_i((A_2,\mathbf{b}_2),G_2) : \phi_i(y_i)=x_i\}|$ is the same for each $x_i\in S_i((A_1,\mathbf{b}_1),G_1)$. As $\phi$ and $\phi_i$ are affine homomorphisms, given $\mathbf{x}\in S((A_1,\mathbf{b}_1),G_1)$, the solutions $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ such that $\phi(\mathbf{y})=\mathbf{x}$ can be partitioned into $$\begin{gathered}
\label{e.union}
\left\{\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2) : \phi(\mathbf{y})=\mathbf{x}\right\}= \\ \bigcup_{\substack{y_i \in S_i((A_2,\mathbf{b}_2),G_2)\\ \phi_i(y_i)=(\mathbf{x})_i}} \left\{\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)\; :\; \phi(\mathbf{y})=\mathbf{x} \text{ and } (\mathbf{y})_i=y_i \right\}.\end{gathered}$$ By the assumptions, the size of the sets $\left\{\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)\; :\; \phi(\mathbf{y})=\mathbf{x} \text{ and } (\mathbf{y})_i=y_i \right\}$ is independent of each $y_i$ with $\phi_i(y_i)=(\mathbf{x})_i$ and we denote it by $\mu_i$. Therefore (\[e.union\]) is an equipartition. Let $\beta_i=|S_i((A_2,\mathbf{b}_2),G_2)|/|S_i((A_1,\mathbf{b}_1),G_1)|$ be the number of preimages by $\phi_i$ of each $x_i\in S_i((A_1,\mathbf{b}_1),G_1)$ in $S_i((A_2,\mathbf{b}_2),G_2)$. Then $\mu_i$ is such that $\mu_i\beta_i=\mu$.
Let $(K',H',\gamma',l',r',Q',p',c')$ be the vector defining the $\gamma'$-strong-representation for $((A_2,\mathbf{b}_2),G_2)$. Let $s$ be the uniformity of the edges of $H'$. The candidate vector $(K,H,\gamma,l,r,Q,p,c)$ is defined as follows.
- $Q=Q'$, $c=c'$,
- $\gamma_i=\gamma'_i \frac{|S_i((A_2,\mathbf{b}_2),G_2)|}{|S_i((A_1,\mathbf{b}_1),G_1)|}\frac{|G_1|}{|G_2|}$ for $i\in[1,m_1]$. $p=\mu p'\frac{|G_2|^{m_1-1}}{|G_1|^{m_1-1}}\left[\prod_{i=m_1+1}^{m_2}\gamma_i'\right] \left[\prod_{i=1}^{m_1} \frac{|S_i((A_1,\mathbf{b}_1),G_1)|}{|S_i((A_2,\mathbf{b}_2),G_2)|}\right]$.
- $H$ and $K$ are hypergraphs on the same vertex sets as $H'$ and $K'$ respectively. $e=\{v_1,\ldots,v_s\}$ is an edge in $K$ (respectively $H$) coloured $i\in[1,m_1]$ if and only if $e$ is an edge coloured $i\in[1,m_1]$ in $K'$ (respectively $H'$.)
- $l(e)=\phi_i(l'(e))$ if $e$ is an edge coloured $i\in[1,m_1]$ as an edge in $K'$.
- Given $H_0\in C(H,K)$, let $H_0'$ be the unique copy of $H'$ in $K'$ spanned by the vertices of $H_0$. Then $r_q(H_0)=r_q'(H_0')$.
RP\[prop\_rep1\] is satisfied for $(K,H)$ with the same bounds and the labelling function $l'$. By the hypothesis (\[h.1\]), each copy of $H'$ in $K'$ spans a unique copy of $H$ in $K$ and vice-versa. Since $$\label{e.345}
r^{-1}(\mathbf{x},q)=\bigcup_{\mathbf{y}\in \phi^{-1}(S((A_1,\mathbf{b}_1),G_1))\cap S((A_2,\mathbf{b}_2),G_2)} r'^{-1}(\mathbf{y},q)$$ and there are $\mu$ different $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ with $\phi(\mathbf{y})=\mathbf{x}$, then the union (\[e.345\]) is disjoint and $$|r^{-1}(\mathbf{x},q)|=\mu|r'^{-1}(\mathbf{y},q)|=\mu p' c\frac{|K'|^s}{|G_2|}\prod_{i=1}^{m_2} \gamma_i'.$$ as each set $r'^{-1}(\mathbf{y},q)$ contains $p' c\frac{|K'|^s}{|G_2|}\prod_{i=1}^{m_2} \gamma_i'$ copies of $H'$ in it.
By the definition of $\gamma$ and $p$ we have $$\begin{gathered}
|r^{-1}(\mathbf{x},q)|=\mu p' c\frac{|K'|^s}{|G_2|}\prod_{i=1}^{m_2} \gamma_i'= \\ \mu p' \frac{|K|^{s}}{|G_2|} \left[\prod_{i=m_1+1}^{m_2} \gamma_i' \right] \left[\prod_{i=1}^{m_1}\gamma_i\frac{|S_i((A_1,\mathbf{b}_1),G_1)|}{|S_i((A_2,\mathbf{b}_2),G_2)|} \right] \frac{|G_2|^{m_1}}{|G_1|^{m_1}} =p c\frac{|K|^s}{|G_1|}\prod_{i=1}^{m_1} \gamma_i.\end{gathered}$$ Since $r'$ is a $\gamma'$-representation function and $\phi=(\phi_1,\ldots,\phi_{m_1})$ defines the $\mu$-equivalence between systems (in particular, is surjective), RP\[prop\_rep2\] is satisfied for $r$.
Given $\mathbf{x}\in S((A_1,\mathbf{b}_1),G_1)$ and $q\in Q$, let $e_i$ be an edge coloured $i$ and with $l(e_i)=(\mathbf{x})_i$. $H_0$, a copy of $H$ in $K$, belongs to $r( \mathbf{x},q)$ and contains $e_i$ if and only if $H_0$, as a copy of $H'$ in $K'$, contains $e_i$ and belongs to one of the $r'^{-1}(\mathbf{y},q)$ with $\phi(\mathbf{y})=\mathbf{x}$. Since $((A_2,\mathbf{b}_2),G_2)$ is $\gamma'$-strongly-represented, each set $r'^{-1}(\mathbf{y},q)$ with $(\mathbf{y})_i=l'(e_i)$ contains an $H_0'$, a copy of $H'$ in $K'$, with $e_i\in H_0'$. By RP\[prop\_rep3\], there are $p' \frac{\prod_{j=1}^{m_2} \gamma_j'}{\gamma_i'}$ copies of $H'$ containing $e_i$ in any set $r'^{-1}(\mathbf{y},q)$ whenever $(\mathbf{y})_i=l'(e_i)$.
There are $\mu_i=\mu/\beta_i$ solutions $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ such that $\phi(\mathbf{y})=\mathbf{x}$ with $(\mathbf{y})_i=l'(e_i)$. Therefore, there is a total of $$\begin{aligned}
\mu_i &p' \frac{\prod_{j=1}^{m_2} \gamma_j'}{\gamma_i'}
=\frac{\mu}{\beta_i} p'
\left[\prod_{j=m_1+1}^{m_2} \gamma_j' \right]\frac{\prod_{j=1}^{m_1} \gamma_j'}{\gamma_i'}\nonumber \\
&= \mu {p'} \left[\prod_{j=m_1+1}^{m_2} \gamma_j'\right]
\frac{\left[\prod_{j=1}^{m_1} \gamma_j' \right] \left[\prod_{j=1}^{m_1}\frac{ |S_j((A_2,\mathbf{b}_2),G_2)|}{ |S_j((A_1,\mathbf{b}_1),G_1)|}\right] \frac{|G_1|^{m_1}}{|G_2|^{m_1}}}{\gamma_i' \beta_i \frac{|G_1|}{|G_2|} } \frac{|G_2|^{m_1-1}}{|G_1|^{m_1-1}} \prod_{j=1}^{m_1} \frac{ |S_j((A_1,\mathbf{b}_1),G_1)|}{ |S_j((A_2,\mathbf{b}_2),G_2)|} \nonumber \\
&= p \frac{\prod_{j=1}^{m_1} \gamma_j}{\gamma_i} \nonumber\end{aligned}$$ copies of $H'$ in $K'$ through $e_i$ that, seeing as copies of $H$ in $K$, belong to $r^{-1}(\mathbf{x},q)$. Hence, the vector $(K,H,\gamma,l,r,Q,p,c)$ fulfills RP\[prop\_rep3\].
To show RP\[prop\_rep4\], choose $q$ and let $e_i$ be an edge in $K$ and let $\mathbf{x}$ be a solution to $((A_1,\mathbf{b}_1),G_1)$ such that $(\mathbf{x})_i=l(e_i)$. By the surjectivity of $\phi$ there exists a $\mathbf{y}\in S((A_2,\mathbf{b}_2),G_2)$ with $\phi(\mathbf{y})=\mathbf{x}$. By the assumption (\[h.2\]), we can choose the solution $\mathbf{y}$ such that $(\mathbf{y})_i=l'(e_i)$. Since $r'^{-1}(\mathbf{y},q)$ contains a copy $H_0'$ of $H'$ with $e_i\in H_0'$, then $r^{-1}(\mathbf{x},q)$ contains $H_0$, the copy of $H$ over the vertices of $H_0'$, and satisfies $e_i\in H_0$. This shows RP\[prop\_rep4\] and finishes the proof of Proposition \[p.mu-equivalent\_2\].
#### Comment.
In Propositions \[p.mu-auto-equivalent\], \[p.mu-equivalent\_1\], and \[p.mu-equivalent\_2\], the permutation $\sigma$ has been omitted as the variables are assumed to be properly ordered so that $\sigma(i)=i$ for $i\in[1,m_1]$.
Proof of Theorem \[t.rem\_lem\_ab\_gr\]: from $G$ to ${\mathbb Z}_n^t$ {#s.proof_rl-lsg-1}
======================================================================
#### Sketch of the proof of Theorem \[t.rem\_lem\_ab\_gr\].
Let $((A,\mathbf{b}),G)$ be a homomorphism system with $A:G^m\to G^k$. We will see that each element of the family of the homomorphism systems on $m$ variables and $k$ equations, with $m\geq k+2$, admits a representation where the constants $\chi_1$ and $\chi_2$ involved only depend on $m$. For any given system $((A,\mathbf{b}),G)$, we find a sequence of $\mu$-equivalent systems $\{((A^{(i)},\mathbf{b}^{(i)},G^{(i)})\}$, $i\in[1,\kappa]$ for some $\kappa\in \mathbb{N}$, such that $((A^{(\kappa)},\mathbf{b}^{(\kappa)}),G^{(\kappa)})$ is strongly representable. Moreover, the sequence is equipped with affine morphisms $\phi:S((A^{(i+1)},\mathbf{b}^{(i+1)},G^{(i+1)})\to S((A^{(i)},\mathbf{b}^{(i)},G^{(i)})$ that fulfill the hypotheses of an appropriate proposition from Section \[s.oper\_between\_representable\]. By concatenating these propositions, we obtain the final result Proposition \[p.repr\_hom\]. The final argument of the construction is summarized in Section \[s.unwrap\_const\]. For the cases regarding $m< k+2$ and to show the second part of Theorem \[t.rem\_lem\_ab\_gr\], the additional argument from Section \[s.finish\_rem\_lem\_dkA1\] is used.
The sequence of systems $\{((A^{(i)},\mathbf{b}^{(i)},G^{(i)})\}_{i\in[1,\kappa]}$ deals with different features of the solution set $S((A,\mathbf{b}),G)$ so that, for the last element of the sequence, a $1$-strong-representation can be found using the methods from [@ksv13]. In Section \[s.rep\_indep\_vector\] the case of non-homogeneous systems is reduced to the homogeneous case $(A,G)$. In Section \[s.repr\_for\_Zt\_implies\_G\], we observe that the representation for any abelian group can be reduced to the homocyclic case ${\mathbb Z}_n^t$, for some appropriate $t$ and $n$. Section \[s.representability\_product\_cyclics\] is devoted to the $\gamma$-representation for any system with $G={\mathbb Z}_n^t$. In Section \[s.hom\_mat\_to\_integer\_mat\] we describe the interpretation of $A$ as an integer matrix in the case of $G={\mathbb Z}_n^t$. Once we have an integer matrix, we prepare the system for any determinantal in Section \[s.union\_of\_systems\] while Section \[s.gamma-effective\] prepares the systems to deal with the cases where $\gamma\neq 1$. Sections \[s.construction\_circular\_matrix\] and Section \[s.final\_composition\] are devoted to the representation by hypergraphs using the tools detailed in Section \[s.circ\_mat\_properties\].
Representation and the independent vector {#s.rep_indep_vector}
-----------------------------------------
Proposition \[p.hom\_to\_all\] below shows that we can restrict ourselves to consider homogeneous systems $A\mathbf{x}=0$.
\[p.hom\_to\_all\] Either there is no solution to $A\mathbf{x}=\mathbf{b}$, $x\in G^m$ or $((A,\mathbf{0}),G)$ is $1$-auto-equivalent to $((A,\mathbf{b}),G)$.
Assume that $A\mathbf{x}=\mathbf{b}$ has a solution $\mathbf{y}=(\mathbf{y}_1,\ldots,\mathbf{y}_m)$. The map $\phi: S((A,\mathbf{0}),G) \to S((A,\mathbf{b}),G)$ with $\phi((\mathbf{x}_1,\ldots,\mathbf{x}_m))=(\mathbf{x}_1+\mathbf{y}_1,\ldots,\mathbf{x}_m+\mathbf{y}_m)$ defines a $1$-auto-equivalence.
Representability for ${\mathbb Z}_n^t$ implies representability for $G$ {#s.repr_for_Zt_implies_G}
-----------------------------------------------------------------------
This section shows how to obtain, for some system $A'$ and integers $t$ and $n$, a system $(A',{\mathbb Z}_n^t)$ $\mu$-equivalent to the given homogeneous system $(A,G)$. Moreover, the map defining the $\mu$-equivalence fulfills the hypothesis of Proposition \[p.mu-equivalent\_1\]. Thus, a representation result for any system $(A,{\mathbb Z}_n^t)$ is enough.
By the Fundamental Theorem of Finite Abelian Groups, $G$ can be expressed, for some $n_1,\ldots,n_t> 1$ as the product of cyclic groups $
G={\mathbb Z}_{n_1}\times \cdots \times {\mathbb Z}_{n_t}, \text{ with } n_i|n_j \text{ for } i\geq j.
$ Let $G'={\mathbb Z}_{n_1}^t$. The group $G$ can be seen as a quotient of $G'$. Let us denote by $\tau:G'\to G$ the quotient map $$\tau(a_1,\ldots,a_t)= \left(\frac{n_1}{n_1}a_1,\ldots,\frac{n_1}{n_t}a_t\right)$$ and let $\beta=|G'|/|G|$. Let $\tau'$ denote the extension of $\tau$ from $G'$ to $G'^m$ using the diagonal action; if $(x_1,\ldots,x_m)=\mathbf{x}\in G'^m$ then $\tau'(\mathbf{x})=(\tau(x_1),\ldots,\tau(x_m))$.
Recall that the set of homomorphisms $A:G^m\to G^k$ are in bijection with $k\times m$ homomorphism matrices $(\vartheta_{i,j})$ for some homomorphisms $\vartheta_{i,j}:G\to G$ with $$\left(
\begin{array}{ccc}
\vartheta_{1,1} & \cdots & \vartheta_{1,m} \\
\vdots &\ddots & \vdots \\
\vartheta_{k,1} & \cdots & \vartheta_{k,m} \\
\end{array}\right)
\left(\begin{array}{c}
x_1 \\
\vdots \\
x_m \\
\end{array}\right)
= \left(\begin{array}{c}
b_1 \\
\vdots \\
b_k \\
\end{array}\right)
\iff
\sum_{i=1}^m \vartheta_{j,i}(x_i)=b_j, \; \forall j\in [1,k].\nonumber$$ See, for instance [@vandW91-2 Section 13.10, p. 66].
By considering the matrix of homomorphisms $(\vartheta_{i,j}')=(\vartheta_{i,j}\circ \tau)$, any homomorphism $A:G^m\to G^k$ induces a homomorphism $A':G'^m\to G'^k$. If we see $\mathbf{b}\in G'^k\supset G^k$, then the system $((A,\mathbf{b}),G)$ induces a system $((A',\mathbf{b}),G')$. Indeed if $\mathbf{y}\in S((A',\mathbf{b}),G')$ then $\tau'(\mathbf{y})\in S((A,\mathbf{b}),G)$ and for any $\mathbf{x}\in S((A,\mathbf{b}),G)$, then $\tau'^{-1}(\mathbf{x})\subset S((A',\mathbf{b}),G')$ and $\tau'^{-1}(\mathbf{x})\neq \emptyset$.
\[o.same\_gamma\] $\tau':S((A',\mathbf{b}),G')\to S((A,\mathbf{b}),G)$ is surjective. If $S_i((A,\mathbf{b}),G)$ is the translated subgroup obtained by projecting the solution set to the $i$-th coordinate of $G^m$, then $\tau^{-1}(S_i((A,\mathbf{b}),G))=S_i((A',\mathbf{b}),G')$ and $$\frac{|G|}{|S_i((A,\mathbf{b}),G)|}=\frac{|G'|}{|S_i((A',\mathbf{b}),G')|}.$$ Moreover, for any $\mathbf{x}\in S((A,\mathbf{b}),G)$, $\tau'^{-1}(\mathbf{x})=\prod_{i=1}^m \tau^{-1}((\mathbf{x})_i)$.
\[r.1\] Observe that $((A',\mathbf{b}),{\mathbb Z}_n^t)$ is $\mu$-equivalent to $((A,\mathbf{b}),G)$ with the surjective map $\tau':S(A',G')\to S(A,G)$ and that the hypotheses of Proposition \[p.mu-equivalent\_1\] regarding the map $\phi=\tau'$ hold by Observation \[o.same\_gamma\].
Therefore, using Proposition \[p.hom\_to\_all\], it is enough to find a $\gamma$-representation for $((A',0),{\mathbb Z}_n^t)$, alternatively denoted by $(A',{\mathbb Z}_n^t)$, with $\gamma_i=|{\mathbb Z}_n^t|/|S_i(A',{\mathbb Z}_n^t)|$.
Proof of Theorem \[t.rem\_lem\_ab\_gr\]: $\gamma$-representability of $(A,{\mathbb Z}_n^t)$ {#s.representability_product_cyclics}
===========================================================================================
In this section we prove the $\gamma$-representability of $(A,{\mathbb Z}_n^t)$ for homomorphism systems $A$ with $m\geq k+2$. The other cases with $m<k+2$ are treated in Section \[s.finish\_rem\_lem\_dkA1\]. Following Section \[s.repr\_for\_Zt\_implies\_G\], $A$ can be seen as a $k\times m$ matrix of homomorphisms. As previously mentioned, the construction involves creating a sequence of $\mu$-equivalent sequence, each element of the sequence being a modification of the pair matrix-group from the previous one.
From a homomorphism to an integer matrix {#s.hom_mat_to_integer_mat}
----------------------------------------
Let $g_i=(0,\ldots,0,\stackrel{i}{1},0,\ldots,0)$, $i\in [1,t]$, be the canonical generators of $G={\mathbb Z}_n^t$. Any variable $x_i$ in ${\mathbb Z}_n^t$ can be decomposed into $t$ variables $x_i=(x_{i,1},\ldots,x_{i,t})$, with $x_{i,j}\in {\mathbb Z}_n$. Therefore, any $k\times m$ homomorphism matrix in ${\mathbb Z}_n^t$ can be expressed as a $t k \times t m$ integer matrix by replacing each homomorphism $\psi:{\mathbb Z}_n^t\to{\mathbb Z}_n^t$ by a $t\times t$ integer matrix $\Psi=\left(
\psi_{i,j}\right)$; $\psi_{i,j}$ is the coefficient of $g_i$ in the image of $g_j$ by $\psi$ expressed as a linear combination of the generators $g_1,\ldots,g_t$. Indeed, the image of $g_j$ by $\psi$ is an element of ${\mathbb Z}_n^t$, hence it can be thought of as a tuple in $[0,n-1]^t$; $\psi_{i,j}$ is the $i$-th component of such tuple.
With these considerations, the system $A$ can be interpreted as an integer system of dimensions $t k\times t m$ with the variables in ${\mathbb Z}_n$: $A \left(x_{1,1},\cdots,x_{1,t},
\cdots,\linebreak[1] x_{m,1}\linebreak[1],\linebreak[1]\cdots\linebreak[1],\linebreak[1]x_{m,t} \right)^{\top}=0$ with $$A=\left(
\begin{array}{ccc}
\Psi_{1,1} & \cdots & \Psi_{1,m} \\
\vdots & \ddots & \vdots \\
\Psi_{k,1} & \cdots & \Psi_{k,m} \\
\end{array}\right), \text{ where $\Psi_{i,j}$ is a $t\times t$ block of integers.}$$
If $A^i$ is a column of zeros in $A$, we can exchange it with any column vector whose components are multiples of $n$. If the determinant of the $kt\times kt$ submatrix of $A$ formed by the first $kt$ columns is zero, as a matrix with coefficients in ${\mathbb Z}$, then we add appropriate multiples of $n$ to the main diagonal so that the modified matrix has non-zero determinant in ${\mathbb Z}$.[^14] The modified matrix and the original are equivalent in ${\mathbb Z}_n$.
Even though $A$ is treated as an integer matrix for most of Section \[s.representability\_product\_cyclics\], the arguments should take in consideration the origins of $A$ as a homomorphism matrix. In particular, the $t$ variables $x_{i,1},\ldots,x_{i,t}$ coming from $x_i$ are kept consecutive as they represent a unique variable $x_i$.
Union of systems: independent vectors simulation {#s.union_of_systems}
------------------------------------------------
Let $S(A)$ denote the Smith Normal Form of $A$. Recall that the $i$-th determinantal divisor of $A$, denoted by $D_i(A)$ and named $i$-th determinantal for short, is the greatest common divisor of the determinants of all the $i\times i$ submatrices of $A$ (choosing $i$ rows and $i$ columns). The product of the first $i$ elements in the diagonal of $S(A)$, $\prod_{j=1}^i d_j$, equals the $i$-th determinantal of $A$, so $D_i(A)=\prod_{j=1}^i d_j$. $A_i$ denotes the $i$-th row of $A$ while $A^i$ its $i$-th column.
\[p.union\_systems\_d\_k\] Let $A$ be a $k\times m$, $m\geq k$ integer matrix. Let $d_1,\ldots, d_k$ denote its diagonal elements of the Smith Normal Form of $A$. There is a matrix $A^{\text{\tiny{(1)}}}$, equivalent to $A$ (row reduced), such that the row $A_j^\text{\tiny{(1)}}$ satisfies $$\gcd\left(\left\{A_{j,i}^{\text{\tiny{(1)}}}\right\}_{i\in[1,m]}\right)=d_j.$$
Furthermore, assume that $d_i\neq 0$ for $i\in[1,k]$. The matrix $A^{\text{\tiny{(2)}}}$, obtained from $A^{\text{\tiny{(1)}}}$ by dividing the row $A_j^\text{\tiny{(1)}}$ by $d_j$, has $k$-th determinantal one.
Let $S=U^{-1}AV^{-1}$ be the Smith Normal Form of $A$, where $U$ and $V$ are integer unimodular matrices that convey, respectively, the row and column operations that transform $A$ into $S$. We have $S=\left(D|0\right)$, where $D$ is a $k\times k$ diagonal integer matrix with $\det(D)=D_k(A)$ and $0$ is an all–zero $k\times (m-k)$ matrix. $d_i$ is the $i$-th element in the main diagonal of $D$. Let $A^{\text{\tiny{(1)}}}=U^{-1}A=SV$. Notice that the system $A^{\text{\tiny{(1)}}}\textbf{x}=\mathbf{0}$ is equivalent to $A\textbf{x}=\mathbf{0}$.
As $A^{\text{\tiny{(1)}}}$ has been obtained from $S$ by column operations using integer coefficients, the $j$-th row $A_j^\text{\tiny{(1)}}$ is formed by integer multiples of $d_j$. Since $V$ is unimodular, then $$\gcd\left(\left\{A_{j,i}^{\text{\tiny{(1)}}}\right\}_{i\in[1,m]}\right)=d_j,$$ which proves the first part of the statement. Let $A^{\text{\tiny{(2)}}}$ be the matrix obtained by dividing each row $A_j^{\text{\tiny{(1)}}}$ by $d_j$. We have $A^{\text{\tiny{(2)}}}=S^{\text{\tiny{(2)}}}V$, where $S^{\text{\tiny{(2)}}}=\left(I_k|0\right)$ is the Smith Normal Form of $A^{\text{\tiny{(2)}}}$ and $I_k$ is the $k\times k$ identity matrix. This completes the proof.
The integer $d_i$ induces a homomorphism $d_i:G\to G$ with $d_i(x)=d_i x=\sum_{j=1}^{d_i}x$. Let $\mathcal{P}_{d_i}(G)$ denote the set $d_i^{-1}(0)\subset G$, this is, the subgroup of preimages of $0$ by the homomorphism induced by $d_i$ inside $G$.
\[o.sol\_set\] Using Proposition \[p.union\_systems\_d\_k\]: $$S(A,G)=\bigcup_{\mathbf{b}
} S((A^{\text{\tiny{(2)}}},\mathbf{b}),G), \; \text{for } \mathbf{b}\in\prod_{i=1}^k \mathcal{P}_{d_i}(G),$$ where $d_i$ is the greatest common divisor of the $i$-th row of $A^{\text{\tiny{(1)}}}$.
Let $\textbf{x}\in G^m$ be a solution to $A\mathbf{x}=\mathbf{0}$, or, equivalently, $A^{\text{\tiny{(1)}}}\mathbf{x}=\mathbf{0}$. Observe the $j$-th equation for $A^{\text{\tiny{(1)}}}$: $$A^{\text{\tiny{(1)}}}_{j,1}x_1 + \cdots + A^{\text{\tiny{(1)}}}_{j,m} x_m=0
\iff
d_j\left(A^{\text{\tiny{(2)}}}_{j,1}x_1+ \cdots + A^{\text{\tiny{(2)}}}_{j,m} x_m\right)=0.$$ Thus, $A^{\text{\tiny{(2)}}}_{j,1}x_1+ \cdots + A^{\text{\tiny{(2)}}}_{j,m} x_m$ is an element of $\mathcal{P}_{d_j}(G)$. Doing the same for all the rows (equations) of the system gives us that $A^{\text{\tiny{(2)}}}\mathbf{x}=\mathbf{b}$ for some independent vector $\mathbf{b}$ in $\prod_{i=1}^k \mathcal{P}_{d_i}(G) \subset G^k$. Also, any solution to $A^{\text{\tiny{(2)}}}\mathbf{x}=\mathbf{b}$ for some $\mathbf{b}\in\prod_{i=1}^k \mathcal{P}_{d_i}(G)$ is a solution to $A^{\text{\tiny{(1)}}}\mathbf{x}=\mathbf{0}$ by multiplying the $i$-th equation by $d_i$.
We introduce dummy variables $y_j\in G$ to account for those independent vectors that occur by Observation \[o.sol\_set\]. The variables $y_i\in G$ are called *simulating* variables.
\[o.simul\_small\_tor\_p-groups\] Assume $G={\mathbb Z}_n^{s}$. For each row $A_j^{\text{\tiny{(1)}}}$, the equation $$A_{j,1}^{\text{\tiny{(2)}}}x_1+\cdots+A_{j,m}^{\text{\tiny{(2)}}}x_m -\frac{n}{\gcd(n,d_j)}y_j=0,$$ where $y_j$ is a new variable with $y_j\in G$, is $|\mathcal{P}_{n/\gcd(n,d_j)}(G)|$-auto-equivalent to $$A_{j,1}^{\text{\tiny{(1)}}}x_1+\cdots+A_{j,m}^{\text{\tiny{(1)}}}x_m=d_j\left(A_{j,1}^{\text{\tiny{(2)}}}x_1+\cdots+A_{j,m}^{\text{\tiny{(2)}}}x_m\right)=0.$$ The application $(x_1,\ldots,x_m,y_i) \to (x_1,\ldots,x_m)$ gives the $|\mathcal{P}_{n/\gcd(n,d_j)}(G)|$-auto-equivalence.
Moreover, for each value of the $j$-th component of the independent vector $g\in \mathcal{P}_{d_j}(G)$, there are $|G|/|\mathcal{P}_{d_j}(G)|$ values for $y_j$ with $$\frac{n}{\gcd(n,d_j)}y_j=g.$$
Since $G=\mathbb{Z}_{n}^{s}$ then $\mathcal{P}_{d_j}(G)\cong \mathbb{Z}_{\gcd(n,d_j)}^{s}$. Observe that the introduction of $\overline{y}_j$ in $$A_{j,1}^{\text{\tiny{(2)}}}x_1+\cdots+A_{j,m}^{\text{\tiny{(2)}}}x_m -\overline{y}_j=0$$ with $\overline{y}_j\in \mathbb{Z}_{\gcd(n,d_j)}^{s}$ simulates the independent vector.
As $\frac{n}{\gcd(n,d_j)}: \mathbb{Z}_{n}^{s} \to \mathbb{Z}_{\gcd(n,d_j)}^{s}$ with $\frac{n}{\gcd(n,d_j)}(g)=\frac{n}{\gcd(n,d_j)}g$ is a $|\mathcal{P}_{n/\gcd(n,d_j)}(G)|$-to-$1$ surjective homomorphism, we can replace the variable $\overline{y}_j\in \mathbb{Z}_{\gcd(n,d_j)}^{s}$ by the variable $y_j\in \mathbb{Z}_{n}^{s}$ multiplied by $\frac{n}{\gcd(n,d_j)}$ and obtain the two parts of the observation.
Let $A^{\text{\tiny{(3)}}}$ denote the new matrix of the system with the simulating variables. This is, $A^{\text{\tiny{(3)}}}=(A^{\text{\tiny{(2)}}} \; Y)$ where $Y$\[page.Y\] is a collection of columns of a $k\times k$ diagonal integer matrix.
**Remark.** If $A$ is a $tk\times tm$ integer matrix coming from a homomorphism matrix, then we use Observation \[o.simul\_small\_tor\_p-groups\] on each row with $G={\mathbb Z}_n$ (or $s=1$). Additionally, Observation \[o.sol\_set\] should consider the matrices as $tk\times tm$ integer matrices and $\mathbf{b}\in {\mathbb Z}_n^{tk}$. Adding the simulating variables is only needed when $\gcd(d_{i},n)\neq 1$. To simplify the arguments, we may add some additional columns in the matrix $Y$, with its coefficients being multiples of $n$, so that the final matrix $A^{\text{\tiny{(3)}}}$ has dimensions $tk\times tm^{\text{\tiny{(3)}}}$, with $m^{\text{\tiny{(3)}}}=m+k$. Since $D_{tk}(A^{\text{\tiny{(2)}}})=1$, then $D_{tk}(A^{\text{\tiny{(3)}}})=1$.
\[r.2\] The system $(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$ is $\mu$-auto-equivalent to $(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ with $$\begin{aligned}
\phi: S(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t) &\longrightarrow S(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t) \nonumber \\
(x_1,\ldots,x_m,x_{m+1},\ldots,x_{m^{\text{\tiny{(3)}}}}) &\longmapsto (x_1,\ldots,x_m), \nonumber
\end{aligned}$$ where $\mu$ is the number of preimages by $\phi$ of each $\mathbf{x}\in S(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ in $S(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$. If $m^{\text{\tiny{(3)}}}=m+k$ then $\mu=\prod_{i=1}^{tk}\frac{n}{\gcd(d_i,n)}$.
From the determinantal to the determinant {#s.determinantal_to_determinant}
-----------------------------------------
\[lem:ext-mat\] Let $A$ be a $k\times m$ integer matrix, $m\geq k$. There is an $m\times m$ integer matrix $N$ that contains $A$ in its first $k$ rows and is such that $\det(N)=D_k(A)$.
Let us include a proof for completeness.
Let $S=UAV=(D|0)$ be the Smith Normal Form of $A$, where $U$ and $V$ are unimodular matrices and $D$ is a $k\times k$ diagonal matrix. Consider $$S'=\begin{pmatrix}
D & 0\\
0 & I_{m-k}\\
\end{pmatrix}\; \text{ and }\;
U'=\begin{pmatrix}
U & 0 \\
0 & I_{m-k} \\
\end{pmatrix}.$$ Then $N=U'^{-1} S' V^{-1}$ is an integer matrix as $U'$ is unimodular and satisfy the thesis of the lemma.
As $D_{tk}(A^{\text{\tiny{(2)}}})=1$, we use Lemma \[lem:ext-mat\] to extend the $tk\times tm$ integer matrix $A^{\text{\tiny{(2)}}}$ to a $tm\times tm$ determinant $1$ integer matrix $$N=\begin{pmatrix}
A^{\text{\tiny{(2)}}}\\
M\\
\end{pmatrix},\;
\det\begin{pmatrix}
A^{\text{\tiny{(2)}}}\\
M\\
\end{pmatrix}=\det(N)=1,$$ which is a part of the matrix $$A^{\text{\tiny{(4)}}}=\begin{pmatrix}
A^{\text{\tiny{(2)}}} & 0 & Y\\
M & I_{tm-tk} & 0\\
\end{pmatrix}.$$ Therefore, the matrix $A^{\text{\tiny{(4)}}}$ can be row reduced into a new matrix $A^{\text{\tiny{(5)}}}$ in such a way that $$A^{\text{\tiny{(5)}}}=\begin{pmatrix}I_{tm} &B\end{pmatrix}\sim A^{\text{\tiny{(4)}}}$$ for some $tm\times\left[(tm-tk)+(tm^{\text{\tiny{(3)}}}-tm) \right]= tm\times t\overline{m}^{\text{\tiny{(4)}}}$ integer matrix $B$. Moreover, we can assume that the columns of the matrix $I_{tm}$ from $A^{\text{\tiny{(5)}}}$ correspond to the ordered original variables $((x_{1,1},\ldots,x_{1,t}),\linebreak[1]\cdots,(x_{m,1},\ldots,x_{m,t}))$. Observe that $A^{\text{\tiny{(5)}}}$ has $tm$ rows and $tm^{\text{\tiny{(5)}}}$ columns, where $m^{\text{\tiny{(5)}}}=m+\overline{m}^{\text{\tiny{(4)}}}$.
\[r.3\] The system $((A^{\text{\tiny{(4)}}},\mathbf{0}),G)$, hence $((A^{\text{\tiny{(5)}}},\mathbf{0}),G)$, is $1$-auto-equivalent to $((A^{\text{\tiny{(3)}}},\mathbf{0}),G)$. Indeed, for any solution $\mathbf{y}\in S(A^{\text{\tiny{(4)}}},G)$ there exists one, and only one, solution $\mathbf{x}\in S(A^{ \text{\tiny{(3)}}},G)$ such that the projection $$\mathbf{y}=(\mathbf{y}_1,\ldots,\mathbf{y}_{tm^{\text{\tiny{(5)}}}}) \longmapsto (\mathbf{y}_1,\ldots,\mathbf{y}_{tm},\mathbf{y}_{tm+(tm-tk)+1},\ldots,\mathbf{y}_{tm^{\text{\tiny{(5)}}}} )$$ gives $\mathbf{x}$.[^15]
Let us show an observation that is helpful in Section \[s.final\_composition\].
\[o.number\_of\_solutions\] Let $A=\begin{pmatrix}A' &B\end{pmatrix}$, with $B$ being a $k\times m$, $m\geq k$ integer matrix and $A'$ denotes a square matrix of dimension $k$. Let $n$ be a positive integer and assume that $\gcd(D_{k}(B),n)=1$. Then, for any value of $x_1,\ldots,x_k$, $x_i\in {\mathbb Z}_n$, there are $n^{m-k}$ values for $(x_{k+1},\ldots,x_{k+m})\in {\mathbb Z}_n^{m}$ with $A\mathbf{x}=0$.
Extend the matrix $A$ with Lemma \[lem:ext-mat\] to a $1$-auto-equivalent system $$A'=
\begin{pmatrix}
A' & B & 0\\
0 & M & I_{m-k}\\
\end{pmatrix} \text{ with } \gcd\left(\det\begin{pmatrix} B \\ M \end{pmatrix},n\right)=1.$$ Select a value for $x_1,\ldots,x_k$ and any value for the last $m-k$ variables of $A'$. Then the value of the variables $x_{k+1},\ldots,x_{k+m}$ in ${\mathbb Z}_n$ is uniquely determined as the determinant is coprime with $n$.
Grouping the variables: on the matrix $B$ {#s.group_on_B}
-----------------------------------------
In Section \[s.hom\_mat\_to\_integer\_mat\] we have assigned an integer matrix in ${\mathbb Z}_n$ to a given homomorphism matrix. Let us partially reverse this transformation. Consider $A^{\text{\tiny{(5)}}}$ to be formed by $mm^{\text{\tiny{(5)}}}$ blocks of size $t\times t$, where $m^{\text{\tiny{(5)}}}=\overline{m}^{\text{\tiny{(4)}}}+m$. Let $\mathcal{A}^{\text{\tiny{(5)}}}_i$ be the matrix formed by the $i$-th row of blocks. Omitting the blocks of zeroes from the $I_{tm}$ part of $A^{\text{\tiny{(5)}}}$, $\mathcal{A}^{\text{\tiny{(5)}}}_i$ can be written as $$\mathcal{A}^{\text{\tiny{(5)}}}_i=\begin{pmatrix}I_t & \mathcal{B}_i\end{pmatrix},$$ where $\mathcal{B}_i$ corresponds to the rows $B_{(i-1)t+1},\ldots,B_{(i-1)t+t}$ from $A^{\text{\tiny{(5)}}}$. We can assume that $D_t(\mathcal{B}_i)\neq 0$ in ${\mathbb Z}$. Otherwise, we can add an appropriate multiple of $n$ to each of the elements of $\mathcal{B}_i$; the new matrix is equivalent in ${\mathbb Z}_n$ and has non-zero determinantal in ${\mathbb Z}$.
By Proposition \[p.union\_systems\_d\_k\], $\mathcal{B}_i$ has an equivalent, row reduced, matrix $\mathcal{B}_i^{\text{\tiny{(1)}}}$ where the greatest common divisors of the rows are the elements in the diagonal of the Smith Normal Form of $\mathcal{B}_i$. By performing such row reductions into $\mathcal{A}^{\text{\tiny{(5)}}}_i$, or in the whole $A^{\text{\tiny{(5)}}}$ using the corresponding rows, the matrix $I_t$ turns into a unimodular matrix $U_i$ related with the row operations conducted on $\mathcal{B}_i$ to obtain $\mathcal{B}_i^{\text{\tiny{(1)}}}$. Since $D_t(\mathcal{B}_i^{\text{\tiny{(1)}}})=D_t(\mathcal{B}_i)\neq 0$, $\mathcal{B}_i^{\text{\tiny{(1)}}}$ has no zero row in ${\mathbb Z}$.
As $U_i$ is unimodular, it induces an automorphism in $G={\mathbb Z}_n^t$ denoted by $\phi_i^{\text{\tiny{(1)}}}: G\to G$, with $$\phi_i^{\text{\tiny{(1)}}}(x)=\phi_i^{\text{\tiny{(1)}}}((x_1,\ldots,x_t))=\left[U_i^{-1}\begin{pmatrix}
x_1\\
\vdots\\
x_t\\
\end{pmatrix}\right]^{\top}.$$
Consider the matrix $A^{\text{\tiny{(6)}}}=\begin{pmatrix}I_{tm}& B^{\text{\tiny{(1)}}}\end{pmatrix}$ where $B^{\text{\tiny{(1)}}}$ is formed by collecting all the rows from $\mathcal{B}_i^{\text{\tiny{(1)}}}$, $i\in[1,m]$.
\[r.4\] $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ is $1$-auto-equivalent to $(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t)$ with $$\begin{aligned}
\phi:S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t) &\longrightarrow S(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t) \nonumber \\
\mathbf{x}=(\mathbf{x}_1,\ldots,\mathbf{x}_{m^{\text{\tiny{(5)}}}}) &\longmapsto (\phi_1^{\text{\tiny{(1)}}}(\mathbf{x}_1),\ldots,\phi_{m^{\text{\tiny{(5)}}}}^{\text{\tiny{(1)}}}(\mathbf{x}_{m^{\text{\tiny{(5)}}}})) \nonumber\end{aligned}$$ being the map between the solutions sets.
Towards $\gamma\neq 1$: constructing several systems {#s.gamma-effective}
----------------------------------------------------
We create several auxiliary systems to achieve an appropriate $\gamma\neq 1$ that are combined in Section \[s.final\_composition\]. The purpose of its combination is to create a strongly $1$-representable system $(A^{\text{\tiny{(7)}}},G^{\text{\tiny{(7)}}})$ with $S_i(A^{\text{\tiny{(7)}}},G^{\text{\tiny{(7)}}})=G^{\text{\tiny{(7)}}}$ for any $i$. $(A^{\text{\tiny{(7)}}},G^{\text{\tiny{(7)}}})$ is $\mu$-equivalent to $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ and the map of the $\mu$-equivalence fulfills the hypotheses of Proposition \[p.mu-equivalent\_2\]. See Remark \[r.last\].
Let $\mathcal{B}_{i}^{\text{\tiny{(2)}}}$ be the matrix obtained from $\mathcal{B}_{i}^{\text{\tiny{(1)}}}$ by dividing each row of $\mathcal{B}_i^{\text{\tiny{(1)}}}$, denoted by $\mathcal{B}_{i,[j]}^{\text{\tiny{(1)}}}$ with $j\in[1,t]$, by $d_{i,j}=\gcd(\mathcal{B}_{i,[j]}^{\text{\tiny{(1)}}})$. Therefore, the greatest common divisor of each row in $\mathcal{B}_{i}^{\text{\tiny{(2)}}}$ is one.[^16] Let $B^{\text{\tiny{(2)}}}$ be the matrix formed by collecting the rows in $\mathcal{B}_{i}^{\text{\tiny{(2)}}}$, $i\in[1,m]$. That is to say, for $i\in[1,m]$ and $j\in[1,t]$, the $(i-1)t+j$-th row of $B^{\text{\tiny{(2)}}}$ is the $j$-th row of $\mathcal{B}_{i}^{\text{\tiny{(2)}}}$.
Given $i\in[1,m]$ and $j\in[1,t]$, let $\mathcal{B}^{\text{\tiny{(2)}}}_{i(j)}$ denote the matrix $$\mathcal{B}^{\text{\tiny{(2)}}}_{i(j)}=
\begin{pmatrix}
B^{\text{\tiny{(2)}}}_{[t(i-1)+1,t(i-1)+j-1]} \\
\mathcal{B}_{i,[j]}^{\text{\tiny{(1)}}} \\
B^{\text{\tiny{(2)}}}_{[t(i-1)+j+1,ti]}\\
\end{pmatrix}$$ where $B^{\text{\tiny{(2)}}}_{[i_1,i_2]}$ denotes the set of rows with indices in $[i_1,i_2]$ from $B^{\text{\tiny{(2)}}}$ and $\mathcal{B}_{i,[j]}^{\text{\tiny{(1)}}}$ denotes the $j$-th row of $\mathcal{B}_{i}^{\text{\tiny{(1)}}}$. This is, all the rows of $\mathcal{B}^{\text{\tiny{(2)}}}_{i(j)}$ are the same as the rows of $\mathcal{B}^{\text{\tiny{(2)}}}_{i}$ except the $j$-th, which is the same as the $j$-th row in $\mathcal{B}_{i}^{\text{\tiny{(1)}}}$.
For $i\in [1,m]$ and $j\in[1,t]$ let $J_{(i,j)}$ be the matrix formed by $$\begin{aligned}
J_{(i,j)}'&=
\begin{pmatrix}
I_{t(i-1)} & 0 &0&0&0& B^{\text{\tiny{(2)}}}_{[1,t(i-1)]} & 0&0\\
0 & I_t & 0 & e_j &0& \mathcal{B}_{i(j)}^{\text{\tiny{(2)}}} & 0&0 \\
0 & 0 & I_{t(m-i)} &0 &0& B^{\text{\tiny{(2)}}}_{[ti+1,tm]} & 0&0\\
0 & 0 & 0 & 1& 0 & 0 &1&0 \\
0 & 0 & 0 & 0& I_{t-1} & 0 &0&I_{t-1} \\
\end{pmatrix}\nonumber \\
&\sim
\begin{pmatrix}
I_{t(i-1)} & 0 &0&0&0& B^{\text{\tiny{(2)}}}_{[1,t(i-1)]} & 0&0\\
0 & I_t & 0 & 0 &0& \mathcal{B}_{i(j)}^{\text{\tiny{(2)}}} & -e_j&0 \\
0 & 0 & I_{t(m-i)} &0 &0& B^{\text{\tiny{(2)}}}_{[ti+1,tm]} & 0&0\\
0 & 0 & 0 & 1& 0 & 0 &1&0 \\
0 & 0 & 0 & 0& I_{t-1} & 0 &0&I_{t-1} \\
\end{pmatrix} \nonumber\\ &=\begin{pmatrix} I_{t(m+1)} & B_{(i,j)}^{\text{\tiny{(3)}}}\end{pmatrix}=J_{(i,j)}\nonumber \end{aligned}$$ where $e_j=(0,\ldots,0,\stackrel{j}{1},0,\ldots,0)^{\top}\in{\mathbb Z}^t$.
The variables in the system associated to $J_{(i,j)}$ take values over $G_{(i,j)}=d_{i,j}^{-1}(0)\subset
{\mathbb Z}_n$, the subgroup of ${\mathbb Z}_n$ formed by the preimage of zero by the homomorphism induced by $d_{i,j}$ in ${\mathbb Z}_n$.[^17] The matrix $J_{(i,j)}$ can be considered as a homomorphisms system over $G_{(i,j)}$ or over $G_{(i,j)}^t$, by considering $J_{(i,j)}$ to be formed by blocks of size $t\times t$. In the first case the system is denoted by $(J_{(i,j)},G_{(i,j)})$ and, in the second, by $(J_{(i,j)},G_{(i,j)}^t)$.
With respect to $A^\text{\tiny{(6)}}$, $t$ equations and $2t$ variables in $G_{(i,j)}$ have been added to $J_{(i,j)}$. The added variables form the $(m+1)$-th block of $t$ variables in $J_{(i,j)}$ and the last block of $t$ variables over $G_{(i,j)}$. The added equation corresponds to the last one in $J_{(i,j)}$ and it involves the $2t$ variables added.
Let $J_{(1,0)}$ be the system induced by the matrix $$J_{(1,0)}=
\begin{pmatrix}
I_{tm} &0 & B^{\text{\tiny{(2)}}} & 0\\
0 & I_t & 0 &I_t \\
\end{pmatrix} =
\begin{pmatrix}
I_{t(m+1)} & B_{(1,0)}^{\text{\tiny{(3)}}} \\
\end{pmatrix}$$ that configures a system $(J_{(1,0)},{\mathbb Z}_n)$ or $(J_{(1,0)},{\mathbb Z}_n^t)$ if $J_{(1,0)}$ is seen as a block matrix. Let us denote $G_{(1,0)}={\mathbb Z}_n$.
Let $\Upsilon=(1,0)\cup \left\{[1,m]\times [1,t]\right\}$. The systems $J_{\kappa}$, $\kappa\in \Upsilon$, thought of as integer matrices, have some common properties.
(i) \[prop.G0\] $J_{\kappa}$ have $m^{\text{\tiny{(J)}}}=m^{\text{\tiny{(6)}}}+2=m^{\text{\tiny{(5)}}}+2$ variables and $k^{\text{\tiny{(J)}}}=k^{\text{\tiny{(6)}}}+1=m+1$ equations over $G_{\kappa}^t$.
(ii) \[prop.G1\] The groups $G_{\kappa}$ are cyclic: $G_{\kappa}=\frac{n}{\gcd(d_{\kappa},n)}\cdot {\mathbb Z}_n \subseteq {\mathbb Z}_n$. $J_{\kappa}$ can be seen as a homomorphism system $(J_{\kappa},G_{\kappa}^t)$.
(iii) \[prop.G2\] $J_{\kappa}$ can be displayed as $\begin{pmatrix}I_{t(m+1)} & B^{\text{\tiny{(3)}}}\end{pmatrix}$ for certain $B^{\text{\tiny{(3)}}}$ depending on $\kappa$ and with dimensions $t(m+1)\times t((m^{\text{\tiny{(5)}}}+2)-(m+1))$. All the rows $B^{\text{\tiny{(3)}}}_{[i]}$ from $B^{\text{\tiny{(3)}}}$ have $\gcd(B^{\text{\tiny{(3)}}}_{[i]},|G_{\kappa}|)=1$. Moreover, the block of $t$ consecutive rows $\mathcal{B}_i^\text{\tiny{(3)}}=B^{\text{\tiny{(3)}}}_{[(i-1)t+1,\ldots,(i-1)t+t]}$, $i\in [1,m+1]$, is such that $\gcd(D_{t}(\mathcal{B}_i^\text{\tiny{(3)}}),|G_{\kappa}|)=1$. Even more, $\gcd(D_{t}(\mathcal{B}_i^\text{\tiny{(3)}}),n)=1$
\[r.prop.G3\] For any $\kappa\in\Upsilon$, the homomorphism $$\begin{aligned}
f_{\kappa}:S(J_{\kappa},G_{\kappa}^t)&\longrightarrow S(A^{\text{\tiny{(6)}}},G_{\kappa}^t)\subset S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t) \nonumber \\
\begin{pmatrix}
\left(x_{(1,1)},\ldots,x_{(1,t)}\right) \\
\vdots \\
\left(x_{(m^{\text{\tiny{(J)}}},1)},\ldots,x_{(m^{\text{\tiny{(J)}}},t)}\right)
\end{pmatrix}
&\longmapsto
\begin{pmatrix}
\left(d_{1,1}\; x_{(1,1)},\ldots, d_{1,t}\; x_{(1,t)}\right) \\
\vdots \\
\left(d_{k^{\text{\tiny{(6)}}},1} \; x_{(k^{\text{\tiny{(6)}}},1)},\ldots, d_{k^{\text{\tiny{(6)}}},t} \; x_{(k^{\text{\tiny{(6)}}},t)}\right) \\
\left(x_{(k^{\text{\tiny{(J)}}}+1,1)},\ldots,x_{(k^{\text{\tiny{(J)}}}+1,t)}\right) \\
\vdots \\
\left(x_{(m^{\text{\tiny{(J)}}}-1,1)},\ldots,x_{(m^{\text{\tiny{(J)}}}-1,t)}\right) \\
\end{pmatrix}
\nonumber
\end{aligned}$$ is surjective and $|G_\kappa^t|$-to-$1$.
The variables with indices $[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$ from $(A^{\text{\tiny{(6)}}},G_{\kappa}^t)$ parameterize the solution set $(A^{\text{\tiny{(6)}}},G_{\kappa}^t)$. This is, any choice of $x_{(k^{\text{\tiny{(6)}}}+1,\cdot)},\ldots,x_{(m^{\text{\tiny{(6)}}},\cdot)}\in G_{\kappa}^t$ provide a unique solution to $(A^{\text{\tiny{(6)}}},G_{\kappa}^t)$. The same holds true for the variables indexed by $[k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}]$ in the system $(J_{\kappa},G_\kappa^t)$.
Assume $(i,j)\in \Upsilon \setminus \{(1,0)\}$. The $(i,j)$-th equation for $(A^{\text{\tiny{(6)}}},G_{\kappa}^t)$ can be written as $$x_{(i,j)}+d_{i,j} B_{[(i-1)t+j]}^{\text{\tiny{(2)}}}\cdot \left(x_{(k^{\text{\tiny{(6)}}}+1,j)},\ldots,x_{(m^{\text{\tiny{(6)}}},j)}\right)^{\top}=0$$ On the other hand, the $(i,j)$-th equation in $(J_{\kappa},G_{\kappa}^t)$ is $$\left\{\begin{array}{cr}
\left.\begin{array}{c}
y_{(i,j)} + B_{\kappa,[(i-1)t+j]}^{\text{\tiny{(3)}}}\cdot \left(y_{(k^{\text{\tiny{(J)}}}+1,j)},\ldots,y_{(m^{\text{\tiny{(J)}}},j)}\right)^{\top}= \\ =y_{(i,j)} + B_{[(i-1)t+j]}^{\text{\tiny{(2)}}}\cdot \left(y_{(k^{\text{\tiny{(J)}}}+1,j)},\ldots,y_{(m^{\text{\tiny{(J)}}}-1,j)}\right)^{\top} =0 \end{array}\right\} & \text{ if } (i,j)\neq \kappa \\
\left.\begin{array}{c}
y_{(i,j)} + B_{\kappa,[(i-1)t+j]}^{\text{\tiny{(3)}}}\cdot \left(y_{(k^{\text{\tiny{(J)}}}+1,j)},\ldots,y_{(m^{\text{\tiny{(J)}}},j)}\right)^{\top}= \\
y_{(i,j)} + d_{i,j} B_{[(i-1)t+j]}^{\text{\tiny{(2)}}}\cdot \left(y_{(k^{\text{\tiny{(J)}}}+1,j)},\ldots,y_{(m^{\text{\tiny{(J)}}}-1,j)}\right)^{\top}-y_{(m^{\text{\tiny{(J)}}},j)}=0
\end{array}\right\} & \text{ if } (i,j)= \kappa. \\
\end{array}\right.$$ Therefore, if we let $\left(x_{(k^{\text{\tiny{(6)}}}+1,\cdot)},\ldots,x_{(m^{\text{\tiny{(6)}}},\cdot)}\right)=\left(y_{(k^{\text{\tiny{(J)}}}+1,\cdot)},\ldots,y_{(m^{\text{\tiny{(J)}}}-1,\cdot)}\right)$, the variables $y_{(i,j)}$ and $x_{(i,j)}$ are such that $d_{i,j}y_{(i,j)}=x_{(i,j)}$ for $(i,j)\neq \kappa$. If $(i,j)=\kappa$, then $d_{i,j}G_{\kappa}=0$ and $(\mathbf{x})_{\kappa}=0=d_{i,j} (\mathbf{y})_{\kappa}$ for any pair of solutions $\mathbf{x}\in S(A^{\text{\tiny{(6)}}},G_{\kappa}^t)$ and $\mathbf{y}\in S(J_{\kappa},G_{\kappa}^t)$. This shows that the map $f_{\kappa}$ exists.
Since $f_{\kappa}$ maps the subset of parameterizing variables $(y_{(k^{\text{\tiny{(J)}}}+1,\cdot)},\ldots,y_{(m^{\text{\tiny{(J)}}}-1,\cdot)})$ to the parameterizing variables $(x_{(k^{\text{\tiny{(6)}}}+1,\cdot)},\ldots,x_{(m^{\text{\tiny{(6)}}},\cdot)})$ using the identity map, $f_{\kappa}$ is surjective. Moreover, since the image by $f_{\kappa}$ is independent of the variable $y_{(m^{\text{\tiny{(J)}}},\cdot)}$, the map is $|G_{\kappa}^t|$-to-$1$.
In the following part, Section \[s.circ\_mat\_properties\], we adapt to the case of homomorphisms matrices the properties of the $n$-circular matrices used to show $1$-strong-representations in [@ksv13] and [@can_tesis_09; @kraserven08].[^18] In particular, Proposition \[p.constr\_c\] in Section \[s.circ\_mat\_properties\] constructs, given an $n$-circular matrix, a matrix $C$ with good properties for the representation.
In Section \[s.construction\_circular\_matrix\] an $n$-circular matrix $\overline{J}_{\kappa}$ is constructed for each matrix $J_{\kappa}$, $\kappa\in \Upsilon$. The final construction of the $1$-strong-representation is conducted in Section \[s.final\_composition\]; it involves combining the matrices $\overline{J}_{\kappa}$, $\kappa\in \Upsilon$, in a single matrix $A^{\text{\tiny{(7)}}}$, as well as combining all the matrices $C_{\kappa}$, provided by Section \[s.circ\_mat\_properties\], in a single matrix $C$.
$n$-circular matrices and properties {#s.circ_mat_properties}
------------------------------------
An integer matrix $A$ formed by $k\times m$ square blocks, $m\geq k$, is said to be *block $n$–circular* if all the matrices formed by $k$ consecutive columns of blocks of $A$, $(A^i, \ldots, A^{i+k-1})$ (considering the indices modulo $m$) have determinant coprime with $n$. A matrix is called standard $n$–circular if it is $n$-circular and with the shape $\begin{pmatrix} I_k &B\end{pmatrix}$. When the size of the blocks is one this definition coincides with the one provided in [@ksv13 Definition 3]. The properties of the $n$-circular matrices described in Proposition \[p.constr\_c\] are used in the construction of the representation described in Section \[s.final\_composition\].
\[p.constr\_c\] Let $A$ be a $kt\times mt$ integer matrix, $m\geq k$, formed by $km$ blocks of size $t\times t$. Assume that $A$ is block $n$–circular. Then there exists a $m\times m$ block integer matrix $C=(\mathcal{C}_{i,j})$, each block of size $t\times t$ and $(i,j)\in[1,m]^2$, with the following properties.
i. \[p.p0\] $AC=0$
ii. \[p.p1\] The $i$-th row of $t\times t$ blocks is such that $\mathcal{C}_{i,j}=0$, for $j\in \{i+k+1,\ldots,i-1\}$ with indices modulo $m$. So, the matrix looks like $$C= \begin{pmatrix}
\ast & \ast & \ast & \ast & 0 & 0 \\
0& \ast & \ast & \ast & \ast & 0 \\
0&0& \ast & \ast & \ast & \ast \\
\ast&0&0& \ast & \ast & \ast \\
\ast&\ast&0&0& \ast & \ast \\
\ast&\ast&\ast&0&0& \ast \\
\end{pmatrix}$$
iii. \[p.p2\] $\gcd(\det(\mathcal{C}_{i,i}),n)=1$ and $\gcd(\det(\mathcal{C}_{i,k+i}),n)=1$ for all $i\in[1,m]$ with indices taken modulo $m$.
Consider the square matrix formed by the column blocks $A^{[i,i+k-1]}=(A^{[i]},\linebreak[1]\ldots,\linebreak[1]A^{[i+k-1]})$ with $i\in[1,m]$. By assumption $A^{[i,i+k-1]}$ is a square non-singular matrix as it has non-zero determianant. For the $j$-th vector in the column block $A^{[i+k]}$, $A^{[i+k],j}$, with $j\in[1,t]$, we can find rational coefficients $b_{(i-1)t+w,(i+k-1)t+j}$, $w\in[1,kt]$, with $$A^{[i+k],j}=\sum_{w\in[1,kt]} b_{(i-1)t+w,(i+k-1)t+j}\; A^{[i,i+k-1],w}$$ where $A^{[i,i+k-1],w}$ stands for the $w$-th column in $A^{[i,i+k-1]}$ and corresponds to the $((i-1)t+w)$-th column in $A$. Moreover, since the determinant is coprime with $n$, there exists an integer $c_{(i+k-1)t+j,(i+k-1)t+j}$, coprime with $n$, such that $$\label{e.coeff_C}
-c_{(i+k-1)t+j,(i+k-1)t+j}\; A^{[i+k],j}=\sum_{w\in[1,kt]} c_{(i-1)t+w,(i+k-1)t+j}\; A^{[i,i+k-1],w}$$ where, for $w\in[1,kt]$, $c_{(i-1)t+w,(i+k-1)t+j}=-c_{(i+k-1)t+j,(i+k-1)t+j} \; b_{(i-1)t+w,(i+k-1)t+j}$ are integers.
The coefficients of the matrix $C$ are
(a) $c_{w_1,w_2}$ whenever the subscripts $(w_1,w_2)$ coincide with one the $c$’s found in the relations given by (\[e.coeff\_C\]) for the $mk$ column vectors of $A$.
(b) \[c.matrix\_c.2\] $0$ otherwise.
Consider the matrix $C$ as divided into $t\times t$ blocks $\mathcal{C}_{\cdot,\cdot}\;$. $C$ satisfy property \[p.p1\]. Indeed, given a column of $C$ indexed by $j=j_1t+j_2$, with $j_1\in[0,m-1]$ and $j_2\in[1,t]$, the indices $i$ of the rows involved in the relations given by (\[e.coeff\_C\]) satisfy $i\in[(j_1-k)t,(j_1+1)t]$.
The relation (\[e.coeff\_C\]) can be rearranged as $$0=c_{(i+k-1)t+j,(i+k-1)t+j} A^{[i+k],j}+\sum_{w\in[1,kt]} c_{(i-1)t+w,(i+k-1)t+j} A^{[i,i+k-1],w}$$ and can be extended to $\sum_{w\in[1,tm]} c_{w,(i+k-1)t+j} A^l$ considering that all the other $c$’s that appear in the sum are zero by (\[c.matrix\_c.2\]). Thus \[p.p0\] is satisfied.
Observe that $\mathcal{C}_{i,i}$ is a diagonal matrix where all the elements in the diagonal are coprime with $n$. Hence the first part of property \[p.p2\] is satisfied. To show the second part observe that, for each $i\in[1,m]$, indices modulo $m$, $$\begin{pmatrix}
A^{[i-k]} & \cdots & A^{[i-1]}
\end{pmatrix}
\begin{pmatrix}
0 & 0 & \cdots & 0 & \mathcal{C}_{i-k,i} \\
& & I_{t(k-1)} & & \vdots \\
& & & & \mathcal{C}_{i-1,i} \\
\end{pmatrix}
=
\begin{pmatrix}
A^{[i-k+1]} & \cdots & A^{[i-1]} & \overline{A}^{[i]}
\end{pmatrix}$$ where the columns of $\overline{A}^{[i]}$ are multiples of the columns of $A^{[i]}$ by (\[e.coeff\_C\]). Indeed, $\overline{A}^{[i],j}= -c_{(i-1)t+j,(i-1)t+j} A^{[i],j}$. Therefore $$\det\begin{pmatrix}
A^{[i-k+1]} & \cdots & A^{[i-1]} & \overline{A}^{[i]}
\end{pmatrix}
=
\det\begin{pmatrix}A^{[i-k+1]} & \cdots & A^{[i-1]} & A^{[i]}\end{pmatrix} \prod_{j\in[1,t]}c_{(i-1)t+j,(i-1)t+j}$$ which is a product of integers coprime with $n$. Since $$\det \begin{pmatrix}
0 & 0 & \cdots & 0 & \mathcal{C}_{i-k,i} \\
& & I_{t(k-1)} & & \vdots \\
& & & & \mathcal{C}_{i-1,i} \\
\end{pmatrix}= \pm\det(\mathcal{C}_{i-k,i})$$ and $(\pm\det(\mathcal{C}_{i-k,i}))\cdot \det\begin{pmatrix}
A^{[i-k]} & \cdots & A^{[i-1]}
\end{pmatrix} =\det\begin{pmatrix}
A^{[i-k+1]} & \cdots & A^{[i-1]} & \overline{A}^{[i]}
\end{pmatrix}$, then $\det(\mathcal{C}_{i-k,i})$ is an integer coprime with $n$. This proves the second part of \[p.p2\] and finalizes the proof of the proposition.
Construction of the $n$-circular matrix {#s.construction_circular_matrix}
---------------------------------------
Let $n$ be a positive integer and let $G$ be an abelian group of order $n$. For our purposes, we can assume $G={\mathbb Z}_n$. Let $A$ be a $kt\times mt$ matrix $A=\begin{pmatrix}I_{tk}& B \end{pmatrix}$ though of as built with $km$ blocks of size $t\times t$. Moreover, we shall assume that $\gcd\left(D_t(\mathcal{B}_i),n\right)=1$, where $\mathcal{B}_i$ is the $i$-th block of $t$ rows of the submatrix $B$, $i\in[1,k]$. In this section we build a $tk^{\text{\tiny{(9)}}}\times tm^{\text{\tiny{(9)}}}$ integer matrix $A^{\text{\tiny{(9)}}}=\begin{pmatrix}I_{tk^{\text{\tiny{(9)}}}}& B \end{pmatrix}$ such that:
- $(A^{\text{\tiny{(9)}}},G^t)$ is $1$-auto-equivalent to $(A,G^t)$.
- $A^{\text{\tiny{(9)}}}$ is $n$-circular with blocks of length $1$, hence $n$-circular with blocks of size $t$.
We enlarge the $t\times (m-k)t$ matrix $\mathcal{B}_i$ using Lemma \[lem:ext-mat\] to the $(m-k)t\times (m-k)t$ matrix $$\mathcal{B}_i^{\text{\tiny{(4)}}}=\begin{pmatrix}
\mathcal{B}_i\\
M_i\\
\end{pmatrix}
\text{ with }
\det\begin{pmatrix}
\mathcal{B}_i\\
M_i\\
\end{pmatrix}=1.$$ By adding some new variables taking values in $G$, $\mathcal{A}_i=\begin{pmatrix}I_t & \mathcal{B}_i\end{pmatrix}$ turns into the matrix denoted by $\mathcal{A}^{\text{\tiny{(8)}}}_i$ with $$\mathcal{A}^{\text{\tiny{(8)}}}_i=
\begin{pmatrix}
I_t & 0 & \mathcal{B}_i \\
0 & I_{(m-k-1)t} & M_i \\
\end{pmatrix}
=\begin{pmatrix}I_{(m-k)t} & \mathcal{B}_i^{\text{\tiny{(4)}}}\end{pmatrix}$$
Let us denote by $B^{\text{\tiny{(4)}}}$ the matrix formed by attaching together all the rows in $\{\mathcal{B}_i^{\text{\tiny{(4)}}}\}_{i\in[1,k]}$ $$B^{\text{\tiny{(4)}}}=
\begin{pmatrix}
\mathcal{B}_1^{\text{\tiny{(4)}}}\\
\vdots \\
\mathcal{B}_{k}^{\text{\tiny{(4)}}}
\end{pmatrix}.$$ Denote by $ A^{\text{\tiny{(8)}}}$ the matrix $A^{\text{\tiny{(8)}}}=\begin{pmatrix}I_{k(m-k)t} & B^{\text{\tiny{(4)}}}\end{pmatrix}$. The variables added with respect to $A$ take values over the whole $G$. The system $(A^{\text{\tiny{(8)}}},G)$ and $(A^{\text{\tiny{(8)}}},G^t)$ are $1$-auto-equivalent to $(A,G)$ and $(A,G^t)$ respectively.
#### A Lemma for the building blocks. {#s.lemma_building_blocks}
Lemma \[lem:ext3\] improves [@ksv13 Lemma 11] so that each block can be constructed by adding a linear number of rows with respect to the original number of columns.
\[lem:ext3\] Let $n$ and $r$ be positive integers and let $M$ be an $r\times r$ integer matrix with determinant coprime with $n$. There are $r\times r$ integer matrices $S$ and $T$ such that $$M'=
\begin{pmatrix}
I_r \\
S \\
M \\
T\\
I_r\\
\end{pmatrix}$$ is a $5r\times r$ integer matrix with the property that each $r\times r$ submatrix of $M'$, consisting of $r$ consecutive rows, has a determinant coprime with $n$.
We detail the construction of $T$. Let us define the matrices $r-i\times r$ matrices $$M^i=\begin{pmatrix} M^i_{i+1} \\ \vdots \\ M^i_r \\
\end{pmatrix},\; i\in [0,r-1],$$ together with the rows $T_{i+1}$ inductively. Let $M^0=M$. Let $d_i=\gcd(M^{i-1}_{i,i},\ldots,M^{i-1}_{r,i})$, $i\in[1,r]$, be the greatest common divisor of the column $M_{\cdot,i}^{i-1}$. Let $$T_{i}=\lambda_{i}^{i} M^{i-1}_{i}+ \cdots + \lambda_r^{i} M^{i-1}_{r}$$ where $\lambda_{i}^{i}, \ldots,\lambda_r^{i}$ are such that $$\label{eq.6}
\lambda_{i}^{i} M^{i-1}_{i,i}+ \cdots + \lambda_r^{i} M^{i-1}_{r,i}=d_{i}$$ and where $\lambda_{i}^{i}$ is some prime, $p_{i}$, larger than $n$. This $p_{i}$ exists, subjected to the constrain (\[eq.6\]), by the Dirichlet theorem regarding the containment of infinitely many primes in the arithmetic progressions $a+b{\mathbb Z}$ with $\gcd(a,b)=1$. Observe that $$\det
\begin{pmatrix}
M^{i-1}_{i} \\
\vdots \\
M^{i-1}_r\\
T_1\\
\vdots\\
T_{i-1}
\end{pmatrix}
=p_{i}
\det
\begin{pmatrix}
M^{i-1}_{i+1} \\
\vdots \\
M^{i-1}_r\\
T_1\\
\vdots\\
T_{i}
\end{pmatrix}$$
The rows of the matrix $M^i$, denoted by $M^i_j$, are $M^i_{j}=M^{i-1}_j - (M^{i-1}_{j,i}/d_i) T_i$, for $j\in [i+1,r]$ and with $T_0=\mathbf{0}$. The first $i$ columns of $M^i$ are the zero columns.
Observe that $\gcd(d_i,n)=1$ as $$\det\begin{pmatrix}
M^{i-1}_{i+1}\\
\vdots\\
M^{i-1}_{r}\\
T_1\\
\vdots \\
T_i\\
\end{pmatrix}=\det\begin{pmatrix}
M^i_{i+1} \\
\vdots \\
M^i_r\\
T_1\\
\vdots\\
T_i\\
\end{pmatrix}$$ is coprime with $n$ and the original matrix $M$ has determinant coprime with $n$. Therefore the equivalent matrix $$\begin{pmatrix}
M^i_{i+1} \\
\vdots \\
M^i_r\\
T_1\\
\vdots\\
T_i\\
\end{pmatrix}
\sim
\begin{pmatrix}
M_{i+1} \\
\vdots \\
M_r\\
T_1\\
\vdots\\
T_i\\
\end{pmatrix}$$ also has a determinant coprime with $n$. This shows the property regarding the coprimality of the determinant of consecutive rows for the first $r$ rows constructed in this way, $T_1,\ldots,T_r$. Observe that $$T_i=(\overbrace{0 \; \cdots \; 0}^{i-1}\; d_i \; \ast \; \cdots \; \ast).$$ Since each $d_i$ is coprime with $n$, we can add the identity matrix after the matrix $T$ and the claimed properties are satisfied. The matrix $S$ is built similarly but we start from the last column and we construct a lower diagonal matrix $S$.
#### Attaching building blocks. {#page.attaching_building}
We use Lemma \[lem:ext3\] on each matrix $\mathcal{B}_i^{\text{\tiny{(4)}}}$ to obtain matrices $$\begin{pmatrix}
I_{t(m-k)}\\
S_i\\
\mathcal{B}_i^{\text{\tiny{(4)}}}\\
T_i\\
I_{t(m-k)}\\
\end{pmatrix}=
\begin{pmatrix}
I_{t(m-k)}\\
\mathcal{B}_i^\text{\tiny{(5)}}\\
I_{t(m-k)}\\
\end{pmatrix}$$ that are put together into a large matrix $$A^{\text{\tiny{(9)}}}=
\begin{pmatrix}
I_{(4k+1)t(m-k)}
\begin{array}{c}
I_{t(m-k)}\\
\mathcal{B}_1^\text{\tiny{(5)}}\\
I_{t(m-k)}\\
\vdots\\
I_{t(m-k)}\\
\mathcal{B}_k^\text{\tiny{(5)}}\\
I_{t(m-k)}\\
\end{array}
\end{pmatrix}=\begin{pmatrix}I_{(4k+1)t(m-k)} & B^{\text{\tiny{(9)}}}\end{pmatrix}$$ that is $n$-circular. This is, any $r=(4k+1)t(m-k)$ consecutive columns form a matrix with determinant coprime with $n$. Indeed, the matrix formed by the first $r$ columns is the identity matrix. On the other cases, some columns of the left most identity matrix $I_{r}$ are selected, along with some other columns from the $B^{\text{\tiny{(9)}}}$ part. Therefore the determinant is, up to a sign, the determinant of the submatrix formed by the columns selected in $B^{\text{\tiny{(9)}}}$ and the rows corresponding to the indices of the columns not picked from $I_{r}$. If the set of columns selected are consecutive and contains all the columns of $B^{\text{\tiny{(9)}}}$, the determinant is coprime with $n$ as so is the determinant formed with $t(m-k)$ consecutive rows from $B^{\text{\tiny{(9)}}}$. Since the first and the last square blocks of $B^{\text{\tiny{(9)}}}$ are identity matrices, the remaining cases are shown.
For $G^t={\mathbb Z}_n^t$, the equations induced by the new rows in $B^{\text{\tiny{(9)}}}$ with respect to $B^{\text{\tiny{(4)}}}$ are $$x_i+\sum_{j=1}^{t(m-k)} B^{\text{\tiny{(9)}}}_{i,j}\; x_{r+j}=0, \; x_i\in {\mathbb Z}_n.$$ Therefore, $(A^{\text{\tiny{(9)}}},G^t)$ is a $k^{\text{\tiny{(9)}}}\times m^{\text{\tiny{(9)}}}$ homomorphism system $1$-auto-equivalent to $(A^{\text{\tiny{(8)}}},G^t)$.
\[r.ext\_matrix\_auto\] The system $(A^{\text{\tiny{(9)}}},{\mathbb Z}_n^t)$ is $1$-auto-equivalent to $(A,{\mathbb Z}_n^t)$ by projecting onto the original coordinates using maps $\phi_i$ equal to the identity map. Indeed, any solution to $(A,{\mathbb Z}_n^t)$ can be extended uniquely to a solution in $(A^{\text{\tiny{(9)}}},{\mathbb Z}_n^t)$ as the last $m-k$ variables in both systems parameterize the solutions in both cases.
Final composition for $\gamma\neq 1$ and representation for $(A^{\text{(7)}},G)$ {#s.final_composition}
--------------------------------------------------------------------------------
#### Joining the matrices and groups.
Let $\left\{\overline{J}_{\kappa}\right\}_{\kappa\in \Upsilon}$, be the $n$–circular integer matrices obtained from $\left\{ J_{\kappa}=\begin{pmatrix}I_{t(m+1)} & B_{\kappa}^{\text{\tiny{(3)}}}\end{pmatrix} \right\} _{\kappa\in \Upsilon}$ using the procedure in Section \[s.construction\_circular\_matrix\]. This applies by (\[prop.G2\]) in Section \[s.gamma-effective\] regarding $\gcd(D_{t}(\mathcal{B}_{\kappa,i}^\text{\tiny{(3)}}),n)=1$. All the matrices $\overline{J}_{\kappa}$ have the same dimensions $tk^{\text{\tiny{(J')}}}=(4k^{\text{\tiny{(J)}}}+1)t(m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}})$, $tm^{\text{\tiny{(J')}}}=(4k^{\text{\tiny{(J)}}}+2)t(m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}})$ over $G_{\kappa}$.
Consider $\Upsilon$ to be ordered lexicographically; given $(\kappa_1,\kappa_2),(\kappa_3,\kappa_4)\in \Upsilon$, $(\kappa_1,\kappa_2)<(\kappa_3,\kappa_4)$ if and only if $\kappa_1 m+\kappa_2<\kappa_3 m + \kappa_4$. The columns of the matrix $A^{\text{\tiny{(7)}}}$ correspond to the columns $\left(\overline{J}_{\kappa}\right)^v$ using the lexicographical order for the ordered set $[1,t m^{\text{\tiny{(J')}}}] \times \Upsilon \owns (v,\kappa)$. The rows $A^{\text{\tiny{(7)}}}$ correspond to the rows $\left(\overline{J}_{\kappa}\right)_w$ using the lexicographically ordered set $[1,t k^{\text{\tiny{(J')}}}]\times \Upsilon\owns (w,\kappa)$. The coefficients of $A^{\text{\tiny{(7)}}}$ are zero wherever the intersection of a column and a row does not appear in any of the matrices $\overline{J}_{\kappa}$. This is, the $(i,j)$ element of $A^{\text{\tiny{(7)}}}$ is given by $$(A^{\text{\tiny{(7)}}})_{i,j}=
\left\{\begin{array}{cl}
\left(\overline{J}_{\kappa}\right)_{\lambda_1,\mu_1} &\begin{array}{ll}
\text{if} &\left\{
\begin{array}{c}
i=(\lambda_1-1)(1+mt)+1+(\kappa_1-1)t+\kappa_2\\
j=(\mu_1-1)(1+mt)+1+(\kappa_1-1)t+\kappa_2\\
\end{array}\right.\\
\phantom{.}&\text{for some }
\left\{\begin{array}{c}
\lambda_1\in[1,tk^{\text{\tiny{(J')}}}],\mu_1\in[1,tm^\text{\tiny{(J')}}]\\
\kappa=(\kappa_1,\kappa_2)\in \Upsilon\\
\end{array}\right.\\
\end{array}\\
0 & \text{otherwise.}
\end{array}\right.$$
Consider $\overline{A^{\text{\tiny{(7)}}}}$ to be the block-diagonal matrix containing the matrices $\{\overline{J}_{\kappa}\}_{\kappa\in\Upsilon}$ as the blocks in the diagonal. The matrix $A^{\text{\tiny{(7)}}}$ can be seen as an appropriate permutation of rows and columns of the block matrix $\overline{A^{\text{\tiny{(7)}}}}$. Let $P_1$ and $P_2$ denote, respectively, the row and column permutations so that $A^{\text{\tiny{(7)}}}=P_1 \overline{A^{\text{\tiny{(7)}}}} P_2$.
$A^{\text{\tiny{(7)}}}$ can be considered as formed by $t^2 k^{\text{\tiny{(J')}}} m^{\text{\tiny{(J')}}}$ blocks of size $(1+tm)\times (1+tm)$ over the groups $G=\prod_{\kappa \in \Upsilon} G_{\kappa}$. Furthermore, $t^2$ of the $(1+tm)\times (1+tm)$ blocks can be grouped in a single block of size $t(1+tm)\times t(1+tm)$. This allows us to interpret $A^{\text{\tiny{(7)}}}$ as formed by $k^{\text{\tiny{(9)}}} m^{\text{\tiny{(9)}}}$ blocks of size $t(1+tm)\times t(1+tm)$ over the groups $G^t=\left(\prod_{\kappa \in \Upsilon} G_{\kappa}\right)^t$. Therefore, if we denote $k^{\text{\tiny{(7)}}}=k^{\text{\tiny{(J')}}}$ and $m^{\text{\tiny{(7)}}}=m^{\text{\tiny{(J')}}}$, $A^{\text{\tiny{(7)}}}$ can be considered as a $k^{\text{\tiny{(7)}}}\times m^{\text{\tiny{(7)}}}$ homomorphism system over $G^t$ denoted by $(A^{\text{\tiny{(7)}}},G^t)$. $(A^{\text{\tiny{(7)}}},G^t)$ has the particularity that the solution set of the system $A^{\text{\tiny{(7)}}}$, $S(A^{\text{\tiny{(7)}}},G^t)$, is the cartesian product of the solution sets $\left\{S\left(\overline{J}_{\kappa},G_{\kappa}^t\right)\right\}_{\kappa\in \Upsilon}$.
#### Matrix $C$ for $A^{\text{\tiny{(7)}}}$.
Since each of the matrices $\{\overline{J}_{\kappa}\}_{\kappa\in \Upsilon}$ is $n$–circular with block size $1$, we use Proposition \[p.constr\_c\] to find band-shape matrices $C_{\kappa}$ related to $\overline{J}_{\kappa}$ for $\kappa\in\Upsilon$.
All the $\{C_{\kappa}\}_{\kappa\in \Upsilon}$ are joined into a single $C$ fulfilling the properties stated in Proposition \[p.constr\_c\] for $A=A^{\text{\tiny{(7)}}}$. Indeed, let $\overline{C}$ be the block matrix with the matrices $\{C_{\kappa}\}_{\kappa\in \Upsilon}$ in the diagonal and zeros everywhere else. Observe that $\overline{A^{\text{\tiny{(7)}}}} \; \overline{C}=\mathbf{0}$. Let $C=P_2^{-1}\overline{C}P_2$, where $P_2$ is the column permutation from $\overline{A^{\text{\tiny{(7)}}}}$ into $A^{\text{\tiny{(7)}}}$. Then the equality $A^{\text{\tiny{(7)}}}C=\mathbf{0}$ follows.
If we group the consecutive rows and columns of $C$ by blocks of size $t(1+mt)\times t(1+mt)$, then $C$ can be considered as a $(t(1+mt))^2$-sized-block matrix $C=\left(\mathcal{C}_{i,j}\right)$ with $(i,j)\in [1,m^{\text{\tiny{(7)}}}]\times [1,m^{\text{\tiny{(7)}}}]$. Moreover, $C$ has the band-shape inherited from $\{C_{\kappa}\}_{\kappa\in \Upsilon}$. In particular.
- $\mathcal{C}_{i,i}$, $i\in[1,m^{\text{\tiny{(7)}}}]$, is a $t(1+mt)\times t(1+mt)$ upper triangular matrix where each coefficient in the diagonal is coprime with $n$.
- $\mathcal{C}_{i,i+k^{\text{\tiny{(7)}}}}$, $i\in[1,m^{\text{\tiny{(7)}}}]$, is a $t(1+mt)\times t(1+mt)$ lower triangular matrix where each coefficient in the diagonal is coprime with the order of the group on which it is acting.[^19]
- $\mathcal{C}_{i,j}=0$ for $i\in[1,m^{\text{\tiny{(7)}}}]$ and $j\in [i+k^{\text{\tiny{(7)}}}+1,\ldots,i-1]$, indices modulo $m^{\text{\tiny{(7)}}}$.
#### Strongly $1$-representation for $(A^{\text{\tiny{(7)}}},G^t)$. {#s.strongly_1_rep}
We use a similar machinery as in [@ksv13 Lemma 4] to construct, assuming $m^{\text{\tiny{(7)}}}\geq k^{\text{\tiny{(7)}}}+2$, a strongly $1$-representation for $(A^{\text{\tiny{(7)}}},G)$ denoted by $(K,H)$.
Let the hypergraph $K$ have the vertex set $V(K)=\left(\prod_{\kappa\in \Upsilon} G_{\kappa}\right)^t\times[1,m^{\text{\tiny{(7)}}}]$. $H$ has $[1,m^\text{\tiny{(7)}}]$ as its vertex set and $m^\text{\tiny{(7)}}$ edges $e_i=\{i,\ldots,i+k^\text{\tiny{(7)}}\} \mod m^{\text{\tiny{(7)}}}$ for $i\in[1,m^\text{\tiny{(7)}}]$ with $e_i$ coloured $i$. Since $m^{\text{\tiny{(7)}}}\geq k^{\text{\tiny{(7)}}}+2$, $H$ has $m^{\text{\tiny{(7)}}}$ pair-wise distinct edges.
The edges in $K^{\text{\tiny{(7)}}}$ form a $m^{\text{\tiny{(7)}}}$-partite $(k^{\text{\tiny{(7)}}}+1)$-uniform hypergraph. The edge $e_i=\{g_i,\ldots,g_{i+k^{\text{\tiny{(7)}}}}\}$, with $g_j\in G^t\times \{j\}\subset V(K)$, is coloured $i$ and is labelled $g$ if and only $$\sum_{j=i}^{i+k^{\text{\tiny{(7)}}} } \mathcal{C}_{i,j}(g_j)=g.$$ These labels on the edges define a labelling function $l:E(K)\to G^t$. Furthermore, the uniformity of the edges is $k^{\text{\tiny{(7)}}}+1$ which is bounded by $m^{\text{\tiny{(7)}}}=|V(H)|$. This shows RP\[prop\_rep1\] from Definition \[d.rep\_sys\] with $\chi_1=m^{\text{\tiny{(7)}}}$.
For RP\[prop\_rep2\], observe that all the copies of $H$ in $K$ should contain one, and exactly one, vertex in each of the vertex clusters $G^t\times \{j\}$ in $V(K)$. Given $H_0\in C(H,K)$ with $V(H_0)=\{g_1,\ldots,g_{m^{\text{\tiny{(7)}}}}\}$, the labels on the edges $e_1,\ldots,e_{m^{\text{\tiny{(7)}}}}$ of $H_0$ are given by $(l(e_1),\ldots,l(e_{m^{\text{\tiny{(7)}}}}))^{\top}=C(g_1,\ldots,g_{m^{\text{\tiny{(7)}}}})^{\top}$. Since $A^{\text{\tiny{(7)}}}C=0$, then $(l(e_1),\ldots,l(e_{m^{\text{\tiny{(7)}}}}))\in S(A^{\text{\tiny{(7)}}},G^t)$. Let $Q=\{1\}$, $r_q(H_0)=1$ for each $H_0\in C(H,K)$ and $p=1$. Then we define $$\begin{aligned}
r:C(H,K) &\longrightarrow S(A^{\text{\tiny{(7)}}},G^t)\times Q\nonumber \\
H_0=\{e_1,\ldots,e_{m^{\text{\tiny{(7)}}}}\}&\longmapsto (r_0(H_0),r_q(H_0))=\left((l(e_1),\ldots,l(e_{m^{\text{\tiny{(7)}}}})),1\right). \nonumber \end{aligned}$$
We claim that $r$ induces a strong $1$-representation for some $c$ bounded by a function of $m^{\text{\tiny{(7)}}}$. Let $(\mathbf{x}_1,\ldots,\mathbf{x}_{m^{\text{\tiny{(7)}}}})=\mathbf{x}\in S(A^{\text{\tiny{(7)}}},G^t)$ be a solution. For each $i\in[1,m^{\text{\tiny{(7)}}}]$, there are $|G^t|^{k^{\text{\tiny{(7)}}}}$ edges labelled $\mathbf{x}_i$. Indeed, as $\mathcal{C}_{i,i}$ is a block with determinant coprime with the order of the group $G^t$, for any choice of the vertices $\{g_{i+1},\ldots,g_{i+k}\}$ there is a unique vertex $g_{i}\in G\times \{i\}$ such that $\sum_{j=i}^{i+k^{\text{\tiny{(7)}}} } \mathcal{C}_{i,j}(g_j)=\mathbf{x}_i$, namely $g_i=\mathcal{C}^{-1}_{i,i} (\mathbf{x}_i-\sum_{j=i+1}^{i+k^{\text{\tiny{(7)}}} } \mathcal{C}_{i,j}(g_j))$.[^20]
Given the solution $\mathbf{x}$ and an edge $e_i$ coloured $i$ and labelled $\mathbf{x}_i$ in $K$, there is a unique $H_0$, copy of $H$ in $K$, with $r_0(H_0)=\mathbf{x}$ and $e_i\in H_0$. Indeed, if $e_i$ has the vertices $\{g_i,\ldots,g_{i+k^{\text{\tiny{(7)}}} }\}$ as its support, then there exists a unique $g_{i-1}\in G^t\times \{i-1\}\subset V(K)$ such that $l(\{g_{i-1},\ldots,g_{i+k^{\text{\tiny{(7)}}} -1}\})=\mathbf{x}_{i-1}$. This process can be repeated a total of $m^{\text{\tiny{(7)}}}-k^{\text{\tiny{(7)}}}-1$ times. Indeed, the vertices $g_{i-1},g_{i-2},\ldots,g_{i+k^{\text{\tiny{(7)}}}+1 }$ that complete, together with $g_i,\ldots,g_{i+k^{\text{\tiny{(7)}}}}$, a copy of $H$ in $K$ can be uniquely determined using the coordinate values $\mathbf{x}_{i-1},\ldots,\mathbf{x}_{i+k^{\text{\tiny{(7)}}}+1}$ of the solution $\mathbf{x}$ and a subset of the previously determined vertices.
Let $H_0$ denoted this copy of $H$ in $K$. By the existence of $r$, $r_0(H_0)\in S(A^{\text{\tiny{(7)}}},G^t)$. Even more, we claim that $r_0(H_0)=\mathbf{x}$. Indeed, by the $n$–circularity of $A^{\text{\tiny{(7)}}}$, any $m^{\text{\tiny{(7)}}}-k^{\text{\tiny{(7)}}}$ consecutive values of $\{\mathbf{x}_j\}$ determines the solution. The copy $H_0$ has been constructed so that it contains edges labelled with the $m^{\text{\tiny{(7)}}}-k^{\text{\tiny{(7)}}}$ consecutive values $\mathbf{x}_i,\mathbf{x}_{i-1},\ldots,\mathbf{x}_{i+k^{\text{\tiny{(7)}}}+1}$ and coloured appropriately with $\{i,i-1,\ldots,i+k^{\text{\tiny{(7)}}}+1\}$. Since the only solution $\mathbf{y}\in S(A^{\text{\tiny{(7)}}},G^t)$ that satisfies $(\mathbf{y})_j=\mathbf{x}_j$ for $j\in\{i,i-1,\ldots, i+k^{\text{\tiny{(7)}}}+1\}$ is $\mathbf{y}=\mathbf{x}$ and $r_0(H_0)\in S(A^{\text{\tiny{(7)}}},G^t)$, then $r_0(H_0)=\mathbf{x}$.
Hence, given $\mathbf{x}\in S(A^{\text{\tiny{(7)}}},G^t)$ and $i\in[1,m^{\text{\tiny{(7)}}}]$, each copy of $e_i$ coloured $i$ and labelled $\mathbf{x}_i$ can be extended to a unique copy of $H$ in $K$ related to $\mathbf{x}$. This shows RP\[prop\_rep3\] and RP\[prop\_rep4\]. Moreover, the number of copies of $H$ related to $\mathbf{x}$ in $K$ is $$|r^{-1}(\mathbf{x},1)|=|G^t|^{k^{\text{\tiny{(7)}}}}=\frac{|G^t|^{k^{\text{\tiny{(7)}}}+1}}{|G^t|}=\left(\frac{1}{m^{\text{\tiny{(7)}}}}\right)^{k^{\text{\tiny{(7)}}}+1} \frac{|K|^{k^{\text{\tiny{(7)}}}+1}}{|G^t|}=c \frac{|K|^{k^{\text{\tiny{(7)}}}+1}}{|G^t|}$$ as there are $|G^t|^{k^{\text{\tiny{(7)}}}}$ edges labelled $(\mathbf{x})_i$ for any $i\in[1,m^{\text{\tiny{(7)}}}]$. This shows RP\[prop\_rep2\] and finishes the strong $1$-representation for $(A^{\text{\tiny{(7)}}},G^t)$ by $(K,H)$.
#### Relation with previous systems.
Observe that each of the systems $\{(J_\kappa,G_\kappa^t)\}_{\kappa\in \Upsilon}$ have one more equation and two more variables than $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ and that $(A^{\text{\tiny{(7)}}},G^t)$ have the same number of equations and variable as any $(\overline{J}_\kappa,G_{\kappa}^t)$. Thus, $A^{\text{\tiny{(7)}}}$ has dimensions $k^{\text{\tiny{(7)}}}=(4k^{\text{\tiny{(6)}}}+5)(m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}+1)$ and $m^{\text{\tiny{(7)}}}=(4k^{\text{\tiny{(6)}}}+6)(m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}+1)$ over $G^t$.
If $x_i$ is a variable in $(A^{\text{\tiny{(7)}}},G^t)$, then it can be decomposed into $t$ variables $x_{(i,j)}\in G$ with $j\in [1,t]$. Furthermore, each $x_{(i,j)}$ can be understood as formed by $mt+1$ variables $x_{(i,j),\kappa}\in G_{\kappa}$, where $\kappa\in\Upsilon$.
\[r.last\] The system $(A^{\text{\tiny{(7)}}},G^t)$ is $\mu$-equivalent to $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ with the injection $$\begin{aligned}
\sigma: [1,m^{\text{\tiny{(6)}}}] &\longrightarrow [1,m^{\text{\tiny{(7)}}}] \nonumber \\
i & \longmapsto
\left\{
\begin{array}{ll}
4(i-1) (m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}+1)+ 3(m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}+1)+1 & \text{ if }\; i\in[1,k^{\text{\tiny{(6)}}}] \\
(4 k^{\text{\tiny{(6)}}}+5)(m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}+1)+i-k^{\text{\tiny{(6)}}} & \text{ if }\; i\in[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]\\
\end{array} \right. \nonumber\end{aligned}$$ and $$\begin{aligned}
\phi:S(A^{\text{\tiny{(7)}}},G^t)
&\longrightarrow S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t) \nonumber \\
(x_1,\ldots,x_{m^{\text{\tiny{(6)}}}}) &\longmapsto (\phi_1(x_{\sigma(1)}),\ldots,\phi_{m^{\text{\tiny{(6)}}}}(x_{\sigma(m^{\text{\tiny{(6)}}})})) \nonumber\end{aligned}$$ with $$\phi_i(x_{(\sigma(i),j)})= \left\{
\begin{array}{cl}
d_{i,j} \sum_{\kappa\in \Upsilon} x_{(\sigma(i),j),\kappa} & \text{ for }\; i\in[1,k^{\text{\tiny{(6)}}}]\\
\sum_{\kappa\in \Upsilon} x_{(\sigma(i),j),\kappa} & \text{ for }\; i\in[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}] \\
\end{array}\right. ,$$ where $d_{i,j}$ is the greatest common divisor of the $j$-th row of the block $\mathcal{B}_i^{\text{\tiny{(1)}}}$ from $A^{\text{\tiny{(6)}}}=\begin{pmatrix} I_{tm} & B^{\text{\tiny{(1)}}} \end{pmatrix}$.[^21]
Additionally, for any $\mathbf{x}\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ and $i\in[1,m^{\text{\tiny{(6)}}}]$, the number of $\mathbf{y}\in S(A^{\text{\tiny{(7)}}},G^t)$ with $\phi(\mathbf{y})=\mathbf{x}$ and fixed $(\mathbf{y})_{\sigma(i)}$ is independent of the $(\mathbf{y})_{\sigma(i)}\in \phi_i^{-1}((\mathbf{x})_i)$.
The system $(A^{\text{\tiny{(7)}}},G^t)$ is formed by joining the systems $\{(\overline{J}_\kappa,G_{\kappa}^t)\}_{\kappa\in \Upsilon}$ together. The $k^{\text{\tiny{(J')}}}\times m^{\text{\tiny{(J')}}}$ system $(\overline{J}_\kappa,G_{\kappa}^t)$ is $1$-auto-equivalent to $(J_\kappa,G_{\kappa}^t)$ by Remark \[r.ext\_matrix\_auto\] for any $\kappa\in \Upsilon$.[^22] Let $\phi'_{\kappa}$ be the map defining the $1$-auto-equivalence from $S(\overline{J}_{\kappa},G_{\kappa}^t)$ to $S(J_{\kappa},G_{\kappa}^t)$
Any solution $\mathbf{x}\in S(A^{\text{\tiny{(7)}}},G^t)$ induces $(mt+1)$ solutions $\overline{\mathbf{x}}_\kappa\in S(\overline{J}_{\kappa},G_{\kappa}^t)$ and vice-versa. By the $1$-auto-equivalence, these solutions can be seen in $(J_{\kappa},G_{\kappa}^t)$ considering $\mathbf{x}_\kappa=\phi_{\kappa}'(\overline{\mathbf{x}}_\kappa)\in S(J_{\kappa},G_{\kappa}^t)$. We use the maps $f_{\kappa}$ from Remark \[r.prop.G3\] to conclude that $$\phi(\mathbf{x})=\sum_{\kappa\in \Upsilon} f_{\kappa}(\mathbf{x}_\kappa)\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$$ as $\phi$ is the sum over $\Upsilon$ of the compositions of the homomorphisms $f_{\kappa}$ with the $\phi_{\kappa}'$.
Since the homomorphism $f_{(1,0)}$ associated to $(J_{(1,0)},{\mathbb Z}_n^t)=(J_{(1,0)},G_{(1,0)}^t)$ is surjective, so is $\phi$.[^23] As $\phi$ is a homomorphism, $\phi$ is $\mu$-to-$1$ for $\mu=|S(A^{\text{\tiny{(7)}}},G^t)|/|S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)|$.
Let us show the second part of the result for $\mathbf{x}\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$. Given $\mathbf{y}_{\kappa}\in S(J_{\kappa},G_{\kappa}^t)$, $(\mathbf{y}_{\kappa})_{i}\in G_{\kappa}^t$, $i\in[1,m^{\text{\tiny{(J)}}}]$, denotes the $i$-th coordinate of $\mathbf{y}_{\kappa}$ and $(\mathbf{y}_{\kappa})_{i,j}\in G_{\kappa}$ denotes the $(i,j)$-th coordinate of the solution with $(i,j)\in[1,m^{\text{\tiny{(J)}}}]\times[1,t]$.
Any collection of solutions $\mathbf{y}_{\kappa}\in S(J_{\kappa},G_{\kappa}^t)$ induce a unique $\mathbf{y}\in(A^{\text{\tiny{(7)}}},G^t)$. The variables indexed in $[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$ parameterize the solutions in $S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$. Therefore, if we have $$\label{eq.x_as_sum_y}
(\mathbf{x})_i=\sum_{\kappa\in \Upsilon} (\mathbf{y}_{\kappa})_{i+1}, \text{ for all } i\in[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$$ for our selected $\mathbf{x}$, then $\phi(\mathbf{y})=\mathbf{x}$. The condition (\[eq.x\_as\_sum\_y\]) is also necessary; if the collection of solutions $\{\mathbf{y}_{\kappa}\}_{\kappa\in \Upsilon}$ inducing $\mathbf{y}$ does not satisfy (\[eq.x\_as\_sum\_y\]) for some index $i\in [k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$, then $\phi(\mathbf{y})\neq \mathbf{x}$. The variables that parameterize the solutions for any system $(J_{\kappa},G_{\kappa}^t)$ are those indexed in $[k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}]$; once the value of $(\mathbf{y}_{\kappa})_i$ is selected for $i\in[k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}]$, the solution $\mathbf{y}_\kappa \in S(J_{\kappa},G_{\kappa}^t)$ exists and is unique. As the proof of Remark \[r.prop.G3\] highlights, the variable $m^{\text{\tiny{(J)}}}$ does not appear in (\[eq.x\_as\_sum\_y\]); the value of $\phi(\mathbf{y})$ is independent of the values $(\mathbf{y}_{\kappa})_{m^{\text{\tiny{(J)}}}}$ for $\kappa\in \Upsilon$.
Pick an $i\in[k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$ and a value for $(\mathbf{y})_{i+1}\in G^t$ such that $\phi_i((\mathbf{y})_{i+1})=(\mathbf{x})_i$. All the solutions $\mathbf{y}\in S(A^{\text{\tiny{(7)}}},G^t)$ with $\phi(\mathbf{y})=\mathbf{x}$ can be found by selecting a value for the remaining parameterizing variables $(\mathbf{y})_{i_1}$ with $i_1\in [k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}]\setminus\{i+1\}$ appropriately to configure a solution in $S(A^{\text{\tiny{(7)}}},G^t)$ with $\phi(\mathbf{y})=\mathbf{x}$. For $i_1\in [k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}-1]\setminus\{i+1\}$, we can select any $(\mathbf{y})_{i_1}\in G^t$ as long as $$(\mathbf{x})_{i_1-1}=\sum_{\kappa \in\Upsilon} (\mathbf{y}_{\kappa})_{i_1}=\phi_{i_1-1}((\mathbf{y})_{i_1}).$$ Additionally, we can select any value for $(\mathbf{y})_{m^{\text{\tiny{(J)}}}}$. Observe that the number of choices is independent on the particular value $(\mathbf{y})_{l+1}$ with $\phi_l((\mathbf{y})_{l+1})=(\mathbf{x})_l$ we have picked.
Given $\mathbf{x}\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$, $i\in[1,k^{\text{\tiny{(6)}}}]$ and any $y^{(i)}\in G^t$ with $\phi_i(y^{(i)})=(\mathbf{x})_i$ we shall find all the solutions $\mathbf{y}\in S(A^{\text{\tiny{(7)}}},G^t)$ with $\phi(\mathbf{y})=\mathbf{x}$ and $(\mathbf{y})_i=y^{(i)}$. For $\kappa\in[1,m]\times[1,t]$, select any solution $\mathbf{y}_{\kappa}'''\in S(J_{\kappa},G_{\kappa}^t)$ such that $$\label{eq.23}
(\mathbf{y}_{\kappa}''')_{i}=y^{(i)}_{\kappa}.$$ Pick an auxiliary solution $\mathbf{y}_{(1,0)}'\in S(J_{(1,0)},G_{(1,0)}^t)$, defined by the last $m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}}$ variables,
- choosing a value for $(\mathbf{y}_{(1,0)}')_{m^{\text{\tiny{(J)}}}}$ in ${\mathbb Z}_n^t$ (any value.)
- $(\mathbf{y}_{(1,0)}')_{j+1}=(\mathbf{x})_{j}-\sum_{\kappa\in\Upsilon\setminus \{(1,0)\}}(\mathbf{y}_{\kappa}''')_{j+1}$ for $j\in [k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$.
Let $\mathbf{y}'$ be the solution in $S(A^{\text{\tiny{(7)}}},G^t)$ defined by the $mt+1$ solutions $\{\mathbf{y}_{(1,0)}',\{\mathbf{y}_{\kappa}'''\}_{\kappa\in \Upsilon \setminus \{(1,0)\}}\}$. Observe that $\phi(\mathbf{y}')=\mathbf{x}$. Indeed, for $j\in [k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$, $$\phi_{j}(\mathbf{y}')=(\mathbf{y}_{(1,0)}')_{j+1}+\sum_{\kappa\in \Upsilon\setminus \{(1,0)\} } (\mathbf{y}_{\kappa}''')_{j+1}=(\mathbf{x})_{j} -\sum_{\kappa\in \Upsilon\setminus \{(1,0)\} } (\mathbf{y}_{\kappa}''')_{j+1}+\sum_{\kappa\in \Upsilon\setminus \{(1,0)\} } (\mathbf{y}_{\kappa}''')_{j+1}= (\mathbf{x})_{j}.$$ Since $\phi(\mathbf{y}')\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ and $\mathbf{x}$ is unique given the value of its last $m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}$ variables, the claim follows and we have a solution $\mathbf{y}'\in S(A^{\text{\tiny{(7)}}},G^t)$ such that $\phi(\mathbf{y}')=\mathbf{x}$.
However, it is not yet clear that $(\mathbf{y})_i=y^{(i)}$; by (\[eq.23\]), we know this equality holds for all coordinates in $\Upsilon\setminus (1,0)$. If $\mathbf{y}_{(1,0)}'$ would be such that $(\mathbf{y}_{(1,0)}')_j=y^{(i)}_{(1,0)}$, then the solution $\mathbf{y}'\in S(A^{\text{\tiny{(7)}}},G^t)$ defined before satisfies the claims $\phi(\mathbf{y}')=\mathbf{x}$ and $(\mathbf{y}')_i=y^{(i)}$. Let us define $\epsilon_{i,j}=\epsilon_{i,j}(\mathbf{x},\{\mathbf{y}_{\kappa}'''\}_{\kappa\in \Upsilon \setminus\{(1,0)\}},y^{(i)})$, for $j\in[1,t]$, as the difference between the $j$-th components of the $i$-th coordinate of $\mathbf{y}_{(1,0)}'$ and its aimed value $(y^{(i)}_{(1,0)})_j$: $$(\mathbf{y}_{(1,0)}')_{i,j}-(y^{(i)}_{(1,0)})_j=\epsilon_{i,j}(\mathbf{x},\{\mathbf{y}_{\kappa}'''\}_{\kappa\in \Upsilon \setminus\{(1,0)\}},y^{(i)}).$$ Observe that $\epsilon_{i,j}\in G_{i,j}$. Indeed, $$(\phi_i(y^{(i)}))_j=d_{i,j} \left(\sum_{\kappa\in \Upsilon} (y^{(i)}_{\kappa})_{j}\right)=(\mathbf{x})_{i,j}\stackrel{\phi(\mathbf{y}')=\mathbf{x}}{=}(\phi_i(\mathbf{y}'))_{j}=d_{i,j}\left((\mathbf{y}'_{(1,0)})_{i,j}+\sum_{\kappa\in \Upsilon \setminus \{(1,0)\}} (y_{\kappa}^{(i)})_j\right).$$ Therefore $d_{i,j} \epsilon_{i,j}=d_{i,j} \left((\mathbf{y}_{(1,0)}')_{i,j}-(y^{(i)}_{(1,0)})_j\right)=0$ as claimed.
Using $\epsilon_{i,j}$ we pick, one for each $j\in[1,t]$, a total of $t$ auxiliary solutions $\mathbf{y}^{(j)}_{(1,0)}\in S(J_{(1,0)},G_{(i,j)}^t)\subset S(J_{(1,0)},G_{(1,0)}^t)$, such that
- $(\mathbf{y}_{(1,0)}^{(j)})_{m^{\text{\tiny{(J)}}}}=0$.
- $(\mathbf{y}_{(1,0)}^{(j)})_{i,j}=\epsilon_{i,j}$.
- $ (\mathbf{y}_{(1,0)}^{(j)})_{i,r}=0$ for $r\in[1,t]\neq j$.[^24]
These $\{\mathbf{y}^{(j)}_{(1,0)}\}_{j\in[1,t]}$ exist because the greatest common divisor of the coefficients of the $((i-1)t+j)$-th row of $B^{\text{\tiny{(3)}}}_{(1,0)}$ from $J_{(1,0)}$, that define the $(i,j)$-th variable, is $1$.
Consider the collection of $t$ solutions $\{\mathbf{y}_{(i,j)}''\}_{j\in[1,t]}$, $\mathbf{y}_{(i,j)}''\in S(J_{(i,j)},G_{(i,j)}^t)$, that are determined by sharing the last $m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}}$ variables with $\mathbf{y}_{(1,0)}^{(j)}$: $(\mathbf{y}_{(i,j)}'')_r=(\mathbf{y}_{(1,0)}^{(j)})_{r}$ for $r\in[k^{\text{\tiny{(J)}}}+1,m^{\text{\tiny{(J)}}}]$. Observe that $(\mathbf{y}_{(i,j)}'')_i=0$ since
- $(\mathbf{y}_{(i,j)}'')_{i,j}=0$ as $(\mathbf{y}_{(i,j)}'')_{i,j}=(\mathbf{y}_{(i,j)}'')_{m^{\text{\tiny{(J)}}},1}=(\mathbf{y}_{(1,0)}^{j})_{m^{\text{\tiny{(J)}}},1}=0$.
- For $r\neq j$, the equations defining $(\mathbf{y}_{(i,j)}'')_{i,r}$ and $(\mathbf{y}_{(1,0)}^{(j)})_{i,r}$ from the last $m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}}$ variables are the same. Since $(\mathbf{y}_{(1,0)}^{(j)})_{i,r}=0$, then so is $(\mathbf{y}_{(i,j)}'')_{i,r}$.
Let $\mathbf{y}$ be the solution in $S(A^{\text{\tiny{(7)}}},G^t)$ formed by
- $\mathbf{y}_{\kappa}=\mathbf{y}_{\kappa}'''$ for $\kappa\in([1,m]\setminus\{i\})\times [1,t]$.
- $\mathbf{y}_{\kappa}=\mathbf{y}_{\kappa}''+\mathbf{y}_{\kappa}'''$ for $\kappa\in\{i\}\times [1,t]$.
- $\mathbf{y}_{(1,0)}=\mathbf{y}'_{(1,0)}-\sum_{j=1}^t \mathbf{y}_{(1,0)}^{(j)}$.
Observe that
- for $\kappa\in \Upsilon$, $\mathbf{y}_{\kappa}\in S(J_{\kappa},G_{\kappa}^t)$ as $\mathbf{y}_{(1,0)}^{(j)}\in S(J_{(1,0)},G_{(i,j)}^t)\subset S(J_{(1,0)},G_{(1,0)}^t)$ and $\mathbf{y}_{(i,j)}''\in S(J_{(i,j)},G_{(i,j)}^t)$ for all $j\in[1,t]$.
- $\phi(\mathbf{y})=\mathbf{x}$ as $\phi(\mathbf{y})\in S(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ and, for $r\in [k^{\text{\tiny{(6)}}}+1,m^{\text{\tiny{(6)}}}]$, $$\begin{aligned}
\phi_r(\mathbf{y})&=(\mathbf{y}_{(1,0)})_{r+1}+\sum_{\kappa\in([1,m]\setminus\{i\})\times [1,t]} (\mathbf{y}_{\kappa})_{r+1}+\sum_{j\in[1,t]}(\mathbf{y}_{(i,j)})_{r+1} \nonumber \\
&=
(\mathbf{y}_{(1,0)}')_{r+1}-\sum_{j=1}^t (\mathbf{y}_{(1,0)}^{(j)})_{r+1}
+\nonumber \\
&\qquad
+\sum_{\kappa\in([1,m]\setminus\{i\})\times [1,t]} (\mathbf{y}_{\kappa}''')_{r+1}+\sum_{j\in[1,t]}\left((\mathbf{y}_{(i,j)}'')_{r+1} + (\mathbf{y}_{(i,j)}''')_{r+1}\right) \nonumber \\
&=(\mathbf{y}_{(1,0)}')_{r+1}+\sum_{\kappa\in\Upsilon \setminus\{(1,0)\}}(\mathbf{y}_{\kappa}''')_{r+1}=(\mathbf{x})_{r} \nonumber\end{aligned}$$
- $(\mathbf{y})_i=y^{(i)}$. Indeed, for $\kappa\in ([1,m]\setminus\{i\})\times [1,t]$, $(\mathbf{y}_{\kappa})_i=(\mathbf{y}_{\kappa}''')_i=y^{(i)}_{\kappa}$ by hypothesis. Since $(\mathbf{y}_{(i,j)}'')_i=0$, $(\mathbf{y}_{(i,j)})_i=(\mathbf{y}_{(i,j)}'')_i+(\mathbf{y}_{(i,j)}''')_i=y^{(i)}_{(i,j)}$ for $j\in[1,t]$. Additionally, for each $j\in[1,t]$, $$(\mathbf{y}_{(1,0)})_{i,j}=(\mathbf{y}'_{(1,0)})_{i,j}-\sum_{r=1}^t (\mathbf{y}_{(1,0)}^{(r)})_{i,j}=(\mathbf{y}'_{(1,0)})_{i,j}-(\mathbf{y}_{(1,0)}^{(j)})_{i,j}=(\mathbf{y}'_{(1,0)})_{i,j}-\epsilon_{i,j}=
(y^{(i)}_{(1,0)})_j.$$
Therefore, the set of solutions $\{\mathbf{y}_{\kappa}\}_{\kappa\in\Upsilon}$ provides a unique solution $\mathbf{y}$ with $\phi(\mathbf{y})=\mathbf{x}$ and $(\mathbf{y})_i=y^{(i)}$ as desired. Using Observation \[o.number\_of\_solutions\] on the matrices $J_{\kappa}$, the number of choices made to find $\mathbf{y}$ is independent of the particular $y^{(i)}$. Even more, the excesses $\epsilon_{i,j}$ define a quotient structure among the possible choices of $\{\mathbf{y}_\kappa'''\}_{\kappa\in\Upsilon\setminus \{(1,0)\}}$ and value for $(\mathbf{y}_{(1,0)}')_{m^{\text{\tiny{(J)}}}}$ (the other choices for the solutions $\mathbf{y}^{(j)}_{(1,0)}$ can be thought to be fixed depending on $\epsilon_{i,j}$.) This finishes the proof of the remark.
Since the system is $n$–circular and $|G|$ is a divisor of $n^\alpha$, for some positive integer $\alpha$, $S_i(A^{\text{\tiny{(7)}}},G^t)=G^t$ for all $i\in[1,m^{\text{\tiny{(7)}}}]$.
Observation on adding variables {#s.adding_variables}
-------------------------------
The number of variables in $A^{\text{\tiny{(7)}}}$, as well as its relative order, is the same as for matrix $\overline{J}_{\kappa}$. This is, the $i$-th variable in $\overline{J}_{\kappa}$, seen as a system $(\overline{J}_{\kappa},G_{\kappa}^t)$, is one of the coordinates that configure the $i$-th variable in the system $(A^{\text{\tiny{(7)}}},G^t)$.
The hypergraph $H$ used in the representation of $(A^{\text{\tiny{(7)}}},G^t)$ can be obtained in the following way. Consider the original $k\times m$ system $(A,G_0)$. Let $H_0$ be a $(k+1)$-uniform hypergraph over the vertex set $V=\{v_1,\ldots,v_{m}\}$ with edges $e_i=\{v_{i},\ldots,v_{i+k} \}$, indices modulo $m$. Let us pair the edge $e_i$ with the $i$-th variable of $A$, $x_i$. Let $e_i=\{v_i,\ldots,v_{i+k} \}$ and $e_j=\{v_j,\ldots,v_{j+k}\}$ be the edges associated with $x_i$ and $x_j$ respectively. The variable $x_i$ is said to be *before* $x_j$ if $v_j\in e_i$. If $x_i$ is before $x_j$ then the variables $x_{i+1},\ldots,x_{j-1}$ are said to be *in-between* $x_i$ and $x_j$.
Assume that $(A',G')$ is built from $(A,G_0)$ by adding a new variable and a new equation. Let $H_1$ denote the hypergraph associated with $(A',G')$ as described in the procedure above. Alternatively, $H_1$ can be constructed from $H_0$ as follows. When a new variable $x_l$ is added to the system $(A,G)$, a new vertex $v_{m+1}$ is added to $V(H)$. If a new equation is added, the uniformity of the edges increases by $1$. The edges $e_1,\ldots,e_{l-1}$ in $H_1$ start at the same point as the corresponding edges in $H_0$ and have one more vertex in them. The edges $e_{l+1},\ldots,e_{m+1}$ in $H_1$ start one vertex later and have one additional vertex than the corresponding ones in $H_0$ (they finish two vertices later than $e_{l},\ldots,e_{m}$ in $H_0$.) In particular, if two edges $e_i$ and $e_j$, with associated variables $x_i$ and $x_j$, share a vertex in $H_0$ and $x_i$ is before $x_j$, then the corresponding edges in $H_1$ share the same number of vertices if the new variable $x_l$ is between $x_i$ and $x_j$.[^25] If the added variable $x_l$ is not between them, then the corresponding edges to $e_i$ and $e_j$ in $H_1$ share an additional vertex. Moreover, some edges that did not share any vertex in $H_0$, may share a vertex in $H_1$.
If $(A',G')$ is built from $(A,G_0)$ by adding only a new variable, then $H_1$ is built from $H_0$ by adding a new vertex $v_{m+1}$ but not increasing the uniformity of the edges. If $x_i$ is before $x_j$ in $H_0$ and the new variable $x_l$ is between $x_i$ and $x_j$, then $e_i$ and $e_{j+1}$, the edges corresponding to $x_i$ and $x_j$ in $(A',G')$ intersect in one vertex less. Otherwise, the intersection between the new edges does not change with respect to the intersection of the corresponding edges in $H_0$.
Observe that the hypergraph $H$ provided by the procedures from Section \[s.final\_composition\] to represent $(A^{\text{\tiny{(7)}}},G^t)$ can be thought of as coming from an embryonal cycle $H_0$ as described above. Let $H_2$ be the hypergraph obtained from $H$ by removing the edges related to the added variables from $(A,G_0)$ to $(J_{\kappa},G_{\kappa}^t)$. Observe that most of the added variables involve the addition of equations. In these cases, removing the edges from $H$ represents no problems in terms of connectedness of $H_2$. However, some variables were added without the addition of any equation. Those free variables correspond to the parts of the construction dealing with $\gamma\neq 1$, Section \[s.gamma-effective\], and to simulate the independent vectors in Section \[s.union\_of\_systems\]. Since the number of variables used in the simulation of the independent vectors is less than $k$, the $k+1$-uniformity of the hypergraph embryo $H_0$ allows $H_2$ to be connected.
The remaining cases involve the additional variable added in Section \[s.gamma-effective\] to each of the systems $(J_{\kappa},G_{\kappa}^t)$ with respect to $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$. Since $m^{\text{\tiny{(6)}}}-k^{\text{\tiny{(6)}}}\geq 2$, the procedure from Section \[s.construction\_circular\_matrix\] ads at least four variables between any pair of the first $k^{\text{\tiny{(J)}}}$ variables from $(J_{\kappa},G_\kappa^t)$. Notice that the first $k^{\text{\tiny{(6)}}}$ variables of $J_{\kappa}$ contains the original set of $m$ variables of $(A,G_0)$. Hence, the set of added variables is non-empty and well distributed throughout the original set of variables, with several variables placed in-between original ones and others before, increasing the connectedness. This justifies that $H_2$ is connected and supported over the same set of vertices as $H$.
From the representation of $(A^{\text{(7)}},G^t)$ to $(A,{\mathbb Z}_n^t)$ {#s.unwrap_const}
--------------------------------------------------------------------------
Let us summarize the steps followed to construct $(A^{\text{\tiny{(7)}}},G^t)$ and its relation with the previous systems. Recall that the original homomorphism system $(A',\prod_{i=1}^t{\mathbb Z}_{n_i})=(A_0,G_0)$ has dimensions $k\times m$. Moreover, we assume that $m\geq k+2$.
\[figure.summary\]
**From** **To** **Relation** **Dimensions** **Description** **In**
----------------------------------------------------------------------------- ------------------------------------------------------------------------------ -------------------- -------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------- --------------------------------------
$(A_0,G_0)$ $(A,{\mathbb Z}_n^t)$ $\mu$-equivalent 1 $k$ eq., $m$ var. From $G_0=\prod_{i=1}^t {\mathbb Z}_{n_i}$ to ${\mathbb Z}_{n_1}^t$ \[s.repr\_for\_Zt\_implies\_G\]
$(A,{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ equivalent $k$ eq., $m$ var. row reduction \[s.union\_of\_systems\]
$(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$ $\mu$-auto-equiv. $k$ eq., $m+k=m^{\text{\tiny{(3)}}}$ var. from determinantal $n$ to $1$, independent vector simulation \[s.union\_of\_systems\]
$(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(4)}}},{\mathbb Z}_n^t)$ $1$-auto-equiv. $m^{\text{\tiny{(3)}}}$ eq., $2m^{\text{\tiny{(3)}}}- k^{\text{\tiny{(3)}}}$ var. determinantal 1 to determinant 1 \[s.determinantal\_to\_determinant\]
$(A^{\text{\tiny{(4)}}},{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t)$ equivalent $m^{\text{\tiny{(3)}}}\times 2m^{\text{\tiny{(3)}}}- k^{\text{\tiny{(3)}}}$ row reduction to $\left( I_{m^{\text{\tiny{(3)}}}} \;B \right)$ \[s.determinantal\_to\_determinant\]
$(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ $1$-auto-equiv. $k^{\text{\tiny{(5)}}}$ eq., $m^{\text{\tiny{(5)}}}$ var. row-reduce $t$-row blocks in $B$; product of $\gcd$ of the rows is the determinantal of the block \[s.group\_on\_B\]
$(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ $(J_{\kappa},G_{\kappa}^t)$, with $\kappa \in\Upsilon$ splitting $k^{\text{\tiny{(6)}}}+1$ eq., $m^{\text{\tiny{(6)}}}+2$ var. each find systems $J_{\kappa}=\left( I_{k^{\text{\tiny{(6)}}}} \; B \right)$ with $D_{t}(B_{\text{\tiny{[ti+1,ti+t]}}})=1$ \[s.gamma-effective\]
$(J_{\kappa},G_{\kappa}^t)$, with $\kappa\in\Upsilon$ $\left(\overline{J}_{\kappa},G_{\kappa}^t\right)$, with $\kappa\in \Upsilon$ $1$-auto-equiv. $k^{\text{\tiny{(J')}}}$ eq.,$m^{\text{\tiny{(J')}}}$ var. each find $n$-circular systems for $(J_{\kappa},G_{\kappa}^t)$ \[s.construction\_circular\_matrix\]
$\left(\overline{J}_{\kappa},G_{\kappa}^t\right)$ with $\kappa\in \Upsilon$ $(A^{\text{\tiny{(7)}}},G^t)$ joining $k^{\text{\tiny{(J')}}}=k^{\text{\tiny{(7)}}}$ eq.,$m^{\text{\tiny{(J')}}}=k^{\text{\tiny{(7)}}}$ var. group the systems $(\overline{J}_{\kappa},G_{\kappa}^t)$ in a single one \[s.final\_composition\]
$(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ $(A^{\text{\tiny{(7)}}},G^t)$ $\mu$-equivalent 2 $k^{\text{\tiny{(7)}}}$ eq.,$m^{\text{\tiny{(7)}}}$ var. conclusion from joining the systems \[s.final\_composition\]
Where $k^{\text{\tiny{(J')}}}=(4 k^{\text{\tiny{(J)}}}+1)(m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}})$, $m^{\text{\tiny{(J')}}}=(4 k^{\text{\tiny{(J)}}}+2)(m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}})$, and $G=\prod_{\kappa\in \Upsilon} G_{\kappa}$.
We prove the first part of Theorem \[t.rem\_lem\_ab\_gr\] under the conditions $m\geq k+2$ by concatenating Proposition \[p.mu-equivalent\_2\], \[p.mu-equivalent\_1\], \[p.mu-auto-equivalent\] and \[p.1-auto-equiv-rep\] between the different pairs of $\mu$-equivalent systems appropriately. For instance we shall use
- Proposition \[p.mu-equivalent\_2\] from $(A^{\text{\tiny{(7)}}},G^t)$ to $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ by Remark \[r.last\].
- Proposition \[p.1-auto-equiv-rep\] from $(A^{\text{\tiny{(6)}}},{\mathbb Z}_n^t)$ to $(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t)$ by Remark \[r.4\].
- No proposition is needed from $(A^{\text{\tiny{(5)}}},{\mathbb Z}_n^t)$ to $(A^{\text{\tiny{(4)}}},{\mathbb Z}_n^t)$ as they are equivalent.
- Proposition \[p.1-auto-equiv-rep\] from $(A^{\text{\tiny{(4)}}},{\mathbb Z}_n^t)$ to $(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$ by Remark \[r.3\].
- Proposition \[p.mu-auto-equivalent\] from $(A^{\text{\tiny{(3)}}},{\mathbb Z}_n^t)$ to $(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ by Remark \[r.2\].
- Proposition \[p.mu-equivalent\_1\] from $(A^{\text{\tiny{(1)}}},{\mathbb Z}_n^t)$ to $(A_0,G_0)$ by Remark \[r.1\] (with $A^{\text{\tiny{(1)}}}=A'$ and $(A_0,G_0)=(A,G)$.)
These propositions can be concatenated by Section \[s.adding\_variables\]. Indeed, the edges related to the remaining variables, after the composition of the maps defining the $\mu$-equivalences, still cover all the vertices of the initial hypergraph $H$. Thus this shows the following proposition.
\[p.repr\_hom\] Let $G$ be a finite abelian group and let $m,k$ be two positive integers with $m\geq k+2$. Let $A$ be a homomorphism $A:G^m\to G^k$ and let $\mathbf{b}\in G^k$ be given. Then the system of linear configurations $((A,\mathbf{b}),G)$ is $\gamma$-strongly-representable with $\gamma_i=|G|/|S_i(A,G)|$ and where $\chi_1,\chi_2$ depend only on $m$.
By means of Theorem \[t.rep\_sys\_rem\_lem\], Proposition \[p.repr\_hom\] proves the first part of Theorem \[t.rem\_lem\_ab\_gr\] when $m\geq k+2$. The second part of Theorem \[t.rem\_lem\_ab\_gr\] and the treatment of the cases when $m<k+2$ are proved in Section \[s.finish\_rem\_lem\_dkA1\].
Proof of Theorem \[t.rem\_lem\_ab\_gr\]: second part and the cases $m<k+2$ {#s.finish_rem_lem_dkA1}
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The cases where $k>m$ can be reduced to $m=k$ by eliminating the redundant equations (for instance, thinking of them as equations in ${\mathbb Z}_n$ using Section \[s.repr\_for\_Zt\_implies\_G\] and Section \[s.hom\_mat\_to\_integer\_mat\] and performing Gaussian elimination on the matrices only allowing integer operations.)
Let $G_0$ be a finite abelian group. If the system $A\mathbf{x}=0$, $\mathbf{x}\in \prod_{i=1}^m X_i$, with $\left|S\left(A,G_0,\prod_{i=1}^m X_i\right)\right| < \delta |S(A,G_0)|$ has $k$ equations and $m$ variables, then $A'\mathbf{x}=0$, $\mathbf{x}\in \prod_{i=1}^m X_i \times G_0^2$, with $$A'=\left(A
\left|\begin{array}{cc}
|G_0| &|G_0| \\
0 & 0 \\
\vdots & \vdots \\
0 & 0 \\
\end{array}\right.\right)$$ has $k$ equations in $m+2$ variables and $\left|S\left(A',G_0,\prod_{i=1}^m X_i\times G_0^2\right)\right| < \delta |S(A',G_0)|$. Therefore, the cases $m<k+2$ can be proved using the second part of Theorem \[t.rem\_lem\_ab\_gr\] restricted to the case $m\geq k+2$. This is encapsulated in Observation \[lem:red2\_ext\_3\].
\[lem:red2\_ext\_3\] Assume that the $\gamma$-representation of the $k\times m$ homomorphism system $(A,G_0)$, with $m\geq k+2$, is constructed using a system with an $n$–circular matrix $(A^{\text{\tiny{(7)}}},G)$ by the methods exposed in sections \[s.rep\_indep\_vector\] through \[s.unwrap\_const\]. Let $I\subset[1,m]$ be a set of indices such that $X_i=G_0$, $i\in I$. Then Theorem \[t.rem\_lem\_ab\_gr\] holds with $X_i'=\emptyset$.
Suppose $m\geq k+2$. By reordering the variables, assume that $I=[j,m]$ for some $j$. Represent the system $(A,G_0)$ by the $s$-uniform hypergraph pair $(K_0,H)$. By the construction in Section \[s.final\_composition\] and Section \[s.unwrap\_const\] and the comments in Section \[s.adding\_variables\], $H$ is connected and $K_0$ is $|H|$-partite. Let $V_1,\ldots,V_{|H|}$ denote the stable sets in $K_0$ and recall that $|V_i|=|V_j|$ for $i,j\in[1,|H|]$. Let $K=K_0\left(\prod_{i=1}^m X_i\right)$ represent the hypergraph where only the edges labelled $x_i\in X_i$ and colored $i$ appear. Let $K'$ and $H'$ be the hypergraphs obtained by removing the edges colored $i\in[j,m]$ from $K$ and $H$ respectively. Assume $e_1=\{1,\ldots,s\}\subset H$ is the edge related to $x_1$.
Let $d$ be the number of vertices in $H'$ with no edges. By the construction of the representation from the $n$–circular matrix provided in the third part of Section \[s.final\_composition\] and knowing that the deleted edges have the largest indices, the stable sets $V_i$ (clusters of vertices) of $K'$ with no edge incident with them are the consecutive ones $V_{|H|-d+1},\ldots,V_{|H|}$. Since every subset of $s$ clusters from $K'$ span, at most, one color class of edges, the connected component of $H'$ has one vertex in each of the first $|H|-d$ clusters. The isolated vertices of $H'$ can be placed in any cluster $V_i, i\in[1,|H|]$. Thus, each copy of $H$ in $K\left(\prod_{i=1}^m X_i\right)$ generates ${|K|-(|H|-d) \choose d}\approx c'|K|^d$ copies of $H'$ in $K'$, for some constant $c'$.
Let $H''$ be the hypergraph built from $H'$ by removing its isolated vertices. Let $K''$ be built from $K'$ by removing the isolated stables $V_{|H|-d+1},\ldots,V_{|H|}$. Then $(K'',H'')$ is a representation of the system $(A,G_0)$ where $(x_1,\ldots,x_{j-1})$ is a solution if and only if there is some $(x_j,\ldots,x_m)\in G_0^{m-j+1}$ for which $(x_1,\ldots,x_{j-1},x_j,\ldots,x_m)$ is a solution. Indeed, we can extend any copy $H''$ in $K''$ to $\left|\prod_{i=|H|-d+1}^{|H|} V_i\right|=|V_i|^d=c|K|^d$ copies of $H$ in $K$ by selecting one additional vertex in each of the last $d$ clusters. Since $X_i=G_0$, $K$ has all the possible edges coloured $i\in I$ in the vertices $\{(\cdot,j)\}_{j\in \{i_1,\ldots,i_s\}}$, where $\{i_1,\ldots,i_s\}$ is the support of the edge coloured $i$ in $H$. Therefore, any choice in $\prod_{i=|H|-d+1}^{|H|} V_i$ completes a copy of $H''$ in $K''$ to a copy of $H$ in $K$. On the other hand, any copy of $H$ in $K$ generates one copy of $H''$ in $K''$. Thus, $$|C(H, K)|=|C(H'', K'')| \;|V_i|^d=|C(H'', K'')|\; c|K|^d$$ and the proportions $|K|^{|H|}/|C(H, K)|$ and $|K''|^{|H''|}/|C(H'', K'')|$ are such that $$\frac{|K|^{|H|}}{|C(H, K)|}=\frac{|K|^{|H|}}{c|K|^d |C(H'', K'')|}=c''\frac{|K''|^{|H''|}}{|C(H'', K'')|}$$ where $c''$ only depends on $m$.
Therefore, if $|S(A,G_0,\prod_{i=1}^{j-1} X_i\times G_0^{m-j+1})|<\delta |S(A,G_0)|$, then, by the representability for $(A,G_0)$ by $(K,H)$, $|C(H,K)|<\delta' |K|^{|H|}$. Hence $|C(H'',K'')|<\delta'' |K''|^{|H''|}$. By applying the same procedure of Theorem \[t.rep\_sys\_rem\_lem\] to $(K'',H'')$, we show a removal lemma by obtaining sets $X_i'\subset X_i$, $i\in[1,j-1]$, as those are the represented variables, such that $\prod_{i=1}^{j-1} X_i\setminus X_i'$ has no solution (for any value of the last variables.) Therefore, we obtain the additional property that $X_i'=\emptyset$ for $i\in [j,m]$.
Let us observe that Observation \[lem:red2\_ext\_3\] can be used to obtain a similar additional conclusion for [@ksv13 Theorem 1].
Conclusions and final comments
==============================
In this paper we have presented Definition \[d.rep\_sys\], a notion of representation of a system of configurations using a pair of hypergraphs that generalizes previous definitions. Additionally, the notion is strong enough to translate the combinatorial removal lemma, Theorem \[t.rem\_lem\_edge\_color\_hyper\], to the context of system of configurations, Theorem \[t.rep\_sys\_rem\_lem\]. We observe that the systems of configurations induced by “copies of a hypergraph H in K” is representable. Additionally, we present a representation for the systems induced by “patterns of the permutation $\tau$ in the permutation $\sigma$”.
The extra flexibility given by Definition \[d.rep\_sys\] with respect to previous notions allows us to show that the configuration systems defined by homomorphisms between finite abelian groups are representable (see Proposition \[p.repr\_hom\]). The combination of Proposition \[p.repr\_hom\] and Theorem \[t.rep\_sys\_rem\_lem\] is reflected in the removal lemma for homomorphism systems of finite abelian groups, Theorem \[t.rem\_lem\_ab\_gr\], which is the main result of the paper. Several applications of Theorem \[t.rem\_lem\_ab\_gr\] are given in the introduction.
In [@canszeven14+], a more algebraic definition of representation tied with the infinite aspect of the compact abelian groups is presented as [@canszeven14+ Definition 3.7], which follows the lines of [@sze10]. Indeed, the strong version of Definition \[d.rep\_sys\] can be seen as the discrete and combinatorial analogue of [@canszeven14+ Definition 3.7]. Let us mention that the construction of the representation for the homomorphisms of finite abelian groups presented in sections \[s.proof\_rl-lsg-1\]-\[s.finish\_rem\_lem\_dkA1\] can be adapted to fit [@canszeven14+ Definition 3.7]. It is natural to ask if a removal lemma result holds for homomorphisms in compact abelian groups. However, such result presents technical difficulties involving some aspects of the construction presented here that arise when generic compact abelian groups are considered.[^26]
Acknowledgements {#acknowledgements .unnumbered}
================
The author wants to thank Oriol Serra, Balázs Szegedy and Pablo Candela for fruitful discussions, improvement suggestions as well as carefully reading the manuscript.
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[^1]: Department of Mathematics, University of Toronto, Canada. Supported by a University of Toronto Graduate Student Fellowship. Charles University, Czech Republic. Supported by ERC-CZ project LL1201 CORES. E-mail: [[email protected]]{}
[^2]: Recall that a (hyper)graph is composed by an ordered pair of sets; the vertex set and the set of edges, which are subsets of vertices. A hypergraph is said to be $k$-uniform if all the edges are subsets of $k$ vertices. A graph is a $2$-uniform hypergraph. A copy of a triangle is an injective map from $\{1,2,3\}$ to the vertex set of the graph that sends an edge of the triangle to an edge in the graph. To be precise, Ruzsa and Szemerédi showed the $(6,3)$–Theorem, which states the following: in any $3$-uniform hypergraph with $\delta n^2$ edges, there are $6$ vertices spanning (or inducing) $3$ edges if $n>n_0(\delta)$.
[^3]: The $k$-term arithmetic progressions are the configurations $\{a,a+d,\ldots,a+(k-1)d\}$ with $a\in{\mathbb Z}$ and $d\in {\mathbb Z}_{\geq 0}$
[^4]: Szemerédi’s Regularity Lemma states that the vertex set of any graph can be partitioned into finitely many parts, such that between most of the pairs, we have a quasi-random bipartite graph.
[^5]: The Removal Lemma for Hypergraphs states that if a given $k$-uniform hypergraph $K$ has few copies of a fixed $k$-uniform hypergraph $H$, then it can be made free of copies of $H$ by removing few edges. See Theorem \[t.rem\_lem\_edge\_color\_hyper\].
[^6]: In this paper, the argument of Ruzsa and Szemerédi can be found in the proof of Theorem \[t.multi\_szem\_ab\_gr\] but the construction of the hypergraph and the use of the Removal Lemma for Hypergraphs has been substituted by the removal lemma for homomorphisms Theorem \[t.rem\_lem\_ab\_gr\].
[^7]: The determinantal of order $i$ of an integer matrix $A$ is the greatest common divisor of all the $i\times i$ submatrices of $A$. In this paper, the term *determinantal* is used to refer to the determinantal of maximal order. See [@new72 Chapter II, Section 13] or Section \[s.union\_of\_systems\] for additional details and more of its properties.
[^8]: \[f.footnote\_7\]For instance, the projection onto the first coordinate of the solution set of the equation $x_1+2(x_2+x_3)=0$, with $x_i\in {\mathbb Z}_6$, is isomorphic to ${\mathbb Z}_3$.
[^9]: See [@vandW91-2 Section 13.10]
[^10]: The determinantal can be assumed to be $1$. This restriction implies that $\left|S((A,\mathbf{b}),G)\right|= |G^{m-k}|$, which is not true in the general case, as Observation \[o.number\_of\_solutions\], or footnote \[f.footnote\_7\] shows.
[^11]: The argument of Section \[s.adding\_variables\] and of Observation \[lem:red2\_ext\_3\] could be applied to [@ksv13 Theorem 1] to add this extra property.
[^12]: See Definition \[d.rep\_sys\] for the additional conditions of the strong-representability.
[^13]: For a detailed argument of how to obtain a removal lemma for directed and coloured graphs, the reader may refer to [@vena_master].
[^14]: This can be done as, for instance, $n^{i kt}$ grows faster than $(kt)! n^i$ when $i$ increases and $n\geq2$.
[^15]: The coordinates to be omitted correspond to the columns of $\begin{pmatrix}0 & I_{tm-tk}\end{pmatrix}^{\top}$ for $A^{\text{\tiny{(4)}}}$. The value of these variables is determined by the values on the first $m$ coordinates.
[^16]: We could have chosen to divide the coefficients of the row $\mathcal{B}_{i,j}^{\text{\tiny{(1)}}}$ by the the minimum $\overline{d}_{i,j}$ such that $\gcd(d_{i,j}/\overline{d}_{i,j},n)=1$.
[^17]: Observe that, if $\gcd(d_{i,j},n)=1$, then $G_{(i,j)}=\{0\}$.
[^18]: Although the final representation to prove [@ksv13 Theorem 1] is not strong, the one established in [@ksv13 Lemma 4] has the strong property.
[^19]: The matrix is lower triangular and not only block lower triangular (with the blocks in the diagonal having determinant coprime with $n$.) Indeed, the matrices $\overline{J}_{\kappa}$ built using Lemma \[lem:ext3\] are $n$-circular for blocks of size $1$.
[^20]: All the rational numbers appearing in $\mathcal{C}^{-1}_{i,i}$ have denominators co-prime with $|G^t|$, hence inducing automorphisms in $G$.
[^21]: See Section \[s.group\_on\_B\].
[^22]: Indeed, $(\overline{J}_{\kappa},G_{\kappa}^t)$ is built from $(J_{\kappa},G_{\kappa}^t)$ by adding some variables and the same number of equations in a way that the new equations are: new variable equal linear equation involving old variables. Therefore, a projection onto the right variables using the identity as maps $\phi_i$ configure the application $\phi$ of the $1$-auto-equivalence.
[^23]: Observe that $S(J_{\kappa},G_{\kappa}^t)$ contains the trivial solution $\mathbf{0}$ and $\{f_{\kappa}\}_{\kappa\in\Upsilon}$ are homomorphisms.
[^24]: There are $|G_{(i,j)}^t|^{m^{\text{\tiny{(J)}}}-k^{\text{\tiny{(J)}}}-2}$ such solutions for each $j\in[1,t]$ by Observation \[o.number\_of\_solutions\]. We only select one per each $j$.
[^25]: Here $x_l$ is assumed to be between the variables corresponding to $x_i$ and $x_j$ in $H_1$.
[^26]: These difficulties are related with the extension of matrices performed in Section \[s.construction\_circular\_matrix\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper we prove that no absolutely maximally entangled, AME, state with minimal support exists with 7 sites and 5 levels.
General AME states are pure multipartite states that, when reduced to half or less of the sites, the maximum entropy mixed state is obtained. They have found applications in teleportation and quantum secret sharing, and finding conditions for their existence is a well known open problem. We consider the version of this problem for minimally supported AME states. We single out known both sufficient and necessary conditions in that case. From our negative result, we show that the necessary condition is not sufficient. The proof uses a recent result on the theory of general, nonlinear, classical codes.
author:
- 'Antonio Bernal [^1]'
title: On the Existence of Absolutely Maximally Entangled States of Minimal Support
---
[2]{}
Introduction
============
In this paper we consider pure states of $n$ qudrits, $|\Psi\rangle\in ({{\mathbb C}}^d)^{\otimes n}$, such that, when tracing out half or more of the sites, the mixed state of maximum confusion is obtained. Those states have been called absolutely maximally entangled, AME, or $AME(n,d)$, in [@qss] in the context of quantum secret sharing schemes. The same concept had already appeared in [@scott] in the context of quantum error correcting codes, under the term $\floor{n/2}$-uniform”.
AME states have found applications in fields like teleportation or quantum secret sharing, and provide links between different areas of mathematics, like coding theory, orthogonal arrays, quantum error correcting codes or combinatorial designs, see [@amecombinatorial], [@qss] and [@ameexistenceandapplications].
A well known open problem is to determine conditions for the existence of AME states. This paper deals with the problem of existence of AME states that are supported on a minimal set of kets from the computational basis.
For AME states of minimal support, a necessary condition is that $d\ge\ceil{n/2}+1$ if $n\ge4$ and $d$ is any integer [@amecombinatorial], and a sufficient condition is that $d\ge n-1$, when $d$ is a prime power, [@ameorthogonal; @amecombinatorial; @ameexistenceandapplications].
We prove that there is no $AME(7,5)$ state with minimal support. The result is proved using the standard theory of linear codes, along with a recent result that relates linear and nonlinear codes, see [@kokkalaetal]. Since the case where $n=7$ and $d=5$ is not forbidden by the above necessary condition, we see that the condition is not sufficient.
The organization of the paper is a follows. Sections \[amegeneral\] and \[linearcodes\] are devoted to review the general definitions and both necessary and sufficient conditions. In section \[negativeexample\], it is proved that no $AME(7,5)$ states of minimal support exist. Section \[conclusions\] contains concluding remarks and some open questions.
Absolutely maximally entangled states {#amegeneral}
=====================================
Let $n$ and $d$ be integers $n,d\ge 2$. Let $\P$ be a pure multipartite state on $n$ sites, where the local Hilbert space is $d$-dimensional. That is, $\P\in ({{\mathbb C}}^d)^{\otimes n}$.
We say that $\P$ is absolutely maximally entangled with $n$ sites and local dimension $d$, $AME(n,d)$, if for any partition of $\{1,\ldots,n\}$ into two disjoint subsets $A$ and $B$, with $|B|=m\le|A|=n-m$, the density obtained from $\P\langle\Psi|$ tracing out the sites on the entries in $A$ is multiple of the identity, $$\operatorname{Tr}_A\P\langle\Psi| = \frac{1}{d^{m}}Id_{{{\mathbb C}}^{\otimes m}}.$$
If $V$ is a vector space $v\in V$ and ${\cal B}\subset V$ is a basis of $V$, the support of $v$ in the basis $\cal B$ is the number of nonzero coordinates of $v$ in the basis $\cal B$.
A linear algebra argument shows that any $AME(n,d)$ state has support on the computational basis of at least $d^{\floor{n/2}}$.
Given two integers $n$, $d$, with $n,d\ge 2$, we will say that an $AME(n,d)$ state $\P$ is of minimal support if the support of $\P$ in the computational basis is $d^{\floor{n/2}}$.
There is a characterization of $AME(n,d)$ states of minimal support in terms of classical codes.
We consider the set ${{\mathbb Z}}_d=\{0,\ldots,d-1\}$. A code over the alphabet ${{\mathbb Z}}_d$ of wordlength $n$ is a subset ${{\cal C}}\subset{{\mathbb Z}}_d^n$. On ${{\cal C}}$ we consider the Hamming distance. Given two words $w,w'\in{{\cal C}}$, the Hamming distance between $w$ and $w'$, $D_H(w,w')$ is the number of coordinates on which the words $w$ and $w'$ differ. The minimum distance $\delta$ of the code ${{\cal C}}$ is the minimum of the distances $D_H(w,w')$ between different words $w,w'\in{{\cal C}}$. The well known Singleton bound establishes that $|{{\cal C}}|\le d^{n-\delta + 1}$. A code is called maximum distance separable, MDS, if the singleton bound is an equality. See [@roth] for general properties of codes.
\[mdsequivalence\] The existence of $AME(n,d)$ of minimal support is equivalent to the existence of MDS codes of wordlength $n$, alphabet size $d$ and minimum distance $\ceil{n/2}+1$. The words of the code and the kets of the state are in one onto one correspondence.
The following property follows by a combinatorial argument involving the associated MDS code.
\[nimpliesnminusone\] Let $n\ge 3$ be an integer. If there is an $AME(n,d)$ state of minimal support, then there is an $AME(n-1,d)$ state of minimal support.
So, given $d$, the set of all $n$ such that $AME(n,d)$ states of minimal support exist is an interval.
For any integer $d\ge 2$, there is an integer ${{\cal N}}(d)$ such that, an $AME(n,d)$ state of minimal support exists if, and only if, $n\le {{\cal N}}(d)$.
We finally mention the necessary condition for the existence of AME states of minimal support:
\[necessarycondition\] If $n\ge 4$ and an $AME(n,d)$ state of minimal support exists, then $d\ge\ceil{\frac{n}{2}}+1$.
This condition forbids many combinations $(n,d)$ for possible $AME(n,d)$ states of minimal support. For example, although $AME(6,2)$ states exist, none of them can be of minimal support, [@amecombinatorial].
Theorem \[necessarycondition\] can be read as an upper bound for ${{\cal N}}(d)$.
For any integer $d\ge 3$, ${{\cal N}}(d)\le 2d-2$, if ${{\cal N}}(d)$ is even, and ${{\cal N}}(d)\le 2d-3$, if ${{\cal N}}(d)$ is odd.
We observe that theorem \[necessarycondition\] is true when $d\ge 3$, for any $n\ge 2$, the cases not covered in theorem \[necessarycondition\] being trivial. Since $AME({{\cal N}}(d),d)$ states of minimal support exist, the statement is another way to write the inequality $\ceil{{{\cal N}}(d)/2}+1\le d$.
The results discused so far are true for general integer values of the local dimension $d$.
Using linear MDS codes {#linearcodes}
======================
In the case where $d$ is a prime power, the alphabet $\{0,\ldots,d-1\}$ can be given a unique field structure, $GF(d)$. In this case, there is more detailed information on certain cases.
In the case of linear $MDS[n,k]$ codes over the field $GF(d)$, where $n$ stands for the code lenght and $k$ is the code dimension, the Singleton identity reads $$k=n-\delta + 1,$$ where $\delta$ is the minimum distance. The linear MDS codes that give rise to $AME(n,d)$ states of minimal support have, according to the Singleton identity and theorem \[mdsequivalence\], dimension $k=\floor{n/2}$.
When $d$ is the power of a prime number, we have the theory of generalized Red Solomon, GRS, codes and their extensions, that are known to be MDS. If $4\le n\le d+1$ and $2\le k\le n-2$, there is linear MDS code of lenght $n$ and dimension $k$ over $GF(d)$, see [@roth] for details.
The following result gives many examples of $AME$ states of minimal support. It has been stated in [@ameexistenceandapplications] resorting to the theory of linear MDS codes, as referred to above, and in [@ameorthogonal] using the theory of orthogonal arrays[^2].
\[someame\] There are $AME(n,d)$ states of minimal support, whenever $n\ge 4$ and $d\ge n-1$ is a power of a prime number.
\[corollary\] If $d$ is a prime power, $d\ge 3$, then ${{\cal N}}(d)\ge d+1$.
A negative example {#negativeexample}
==================
\[noamecase\] There is no $AME(7,5)$ state of minimal support, ${{\cal N}}(5)=6$.
As in [@roth], define $L_d(k)$ as the maximum wordlength of any linear MDS code of dimension $k$ over $GF(d)$, $d$ being a prime power. Several bounds an equalities are known about $L_d(k)$, see [@roth]. We will use that $L_d(3)=d+1$ if $d$ is an odd prime power. In particular, we use that $L_5(3)=6$.
This shows that no linear MDS code over $GF(5)$ exists with wordlength 7 and dimension 3.
Now suppose that an $AME(7,5)$ state of minimal support exists. By theorem \[mdsequivalence\], there is a MDS code over $GF(5)$ with wordlength 7 and minimum distance $5$.
The code given in theorem \[mdsequivalence\] however, is not guaranteed to be linear, so this bound $L_5(3)=6$ on the theory of linear codes does not suffice to prove the statement.
To end the proof, we note a result of [@kokkalaetal], that any MDS code, not necessarily linear, over an alphabet of size 5, code size $5^k$, $k\ge 3$, and minimum distance $\delta\ge 3$, can be transformed to a linear MDS code with the same parameters and dimension $k$ with a permutation of coordinates, followed by a permutation of the symbols at each coordinate separately.
This proves that no $AME(7,5)$ state of minimal support exists and ${{\cal N}}(5)\le 6$, corollary \[corollary\] gives the reverse inequality.
The necessary condition given in theorem \[necessarycondition\] does not forbid the existence of $AME(7,5)$ states of minimal support. This necessary condition, therefore, is no sufficient.
Conclusions
===========
The existence problem for $AME(n,d)$ states is a non trivial one, even for states minimally supported.
$AME(n,d)$ states of minimal support exist if, and only if $n\le{{\cal N}}(d)$, and the necessary and sufficient conditions reviewed in this paper can be read as: $$d+1\le {{\cal N}}(d) \le 2d-2,\text{ or } 2d-3,$$ for $d\ge 3$, the inequality on the right being valid for any integer $d$ and the one on the left being valid for all $d$ power of a prime number. We have seen that the upper bound for ${{\cal N}}(d)$ is not tight, since ${{\cal N}}(5)=6$.
The theory of linear codes is restricted to the case where the local dimension is a prime power. To investigate other local dimensions, further consideration of general (nonlinear) codes and of combinatorial structures, like orthogonal arrays, seems needed. Sharper estimates on the maximum number of sites ${{\cal N}}(d)$ for which there are $AME$ states of minimal support for a given local dimension $d$ are desirable too.
[1]{}
D.Goyeneche, K. Życzkowski, *Genuinely multipartite entangled states and orthogonal arrays*, Phys. Rev. A **90**, 022316 (2014)
D.Goyeneche, D. Alsina, J.I. Latorre, A. Riera, K. Życzkowski, *Absolutely maximally entangled states and combinatorial designs*, Phys. Rev. A **92**, 032316 (2015)
W. Helwig, W. Cui, J.I. Latorre, A. Riera, H. Lo, *Absolute Maximal Entanglement and Quantum Secret Sharing,* Phys. Rev. A **86**, 052335 (20129
W. Helwig, W. Cui, *Absolutely Maximally Entangled States: Existence and Applications*, arXiv:1306.2536 \[quant-ph\]
J. I. Kokkala, D. S. Krotov and P. R. J. Östegard, *On the classification of MDS codes*, IEEE Trans. Inf. Theory **61**(12), 6485-6492 (2015)
A.J. Scott, *Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions*, Phys. Rev. A **69**, 052330 (2004)
R.M. Roth, *Intruduction to Coding Theory,* Cambridge University Press, (2006)
[^1]: Electronic address: `[email protected]`\
Supported by project FIS2013-41757-P
[^2]: Due to a typographical error, the result is stated in [@ameorthogonal] for a general integer dimension $d$. The authors ment to state it in the case where $d$ is a prime power.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system. We find explicit conditions on the initial data, with ADM mass $M,$ such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius $r$ along each geodesic is bounded by $2M,$ the timelike lines $r=c\in [0,2M]$ are incomplete, and for $r>2M$ the metric converges asymptotically to the Schwarzschild metric with mass $M$. The initial data that we construct guarantee the formation of a black hole in the evolution. We also give examples of such initial data with the additional property that the solutions exist for all $r\geq 0$ and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities.'
author:
- |
H[å]{}kan Andréasson\
Mathematical Sciences\
Chalmers University of Technology\
Göteborg University\
S-41296 Göteborg, Sweden\
email: [email protected]\
\
Markus Kunze\
Fachbereich Mathematik\
Universität Duisburg-Essen\
D-45117 Essen, Germany\
email: [email protected]\
\
Gerhard Rein\
Mathematisches Institut der Universität Bayreuth\
D-95440 Bayreuth, Germany\
email: [email protected]
title: The formation of black holes in spherically symmetric gravitational collapse
---
\[section\]
\[theorem\][Definition]{} \[theorem\][Proposition]{} \[theorem\][Example]{} \[theorem\][Remark]{} \[theorem\][Corollary]{} \[theorem\][Lemma]{}
Introduction
============
One of the many striking predictions of General Relativity is the assertion that under appropriate conditions astrophysical objects like stars or galaxies undergo a gravitational collapse resulting in a spacetime singularity. This was first proven by Oppenheimer and Snyder [@OS] who constructed a semi-explicit example of a homogeneous spherically symmetric ball of dust, i.e., of a pressure-less fluid, which under its self-consistent, general relativistic gravitational interaction collapses. During this collapse the scalar curvature of spacetime blows up at the centre of symmetry, and the geometry of spacetime breaks down there. This is referred to as the formation of a spacetime singularity. An important feature of the Oppenheimer-Snyder solution is that during the collapse a two-dimensional spacelike sphere evolves which encloses the singularity and through which no causal curve, i.e., no light ray or particle trajectory, can pass outward. In this way the spacetime singularity is isolated from the outside part of spacetime by a so-called event horizon, and the singularity cannot be seen or in any other way be experienced by observers outside the event horizon. This configuration was later termed a black hole.
In the 1960s Penrose [@pen] proved that the formation of spacetime singularities from regular initial data is not restricted to spherically symmetric, especially constructed or isolated examples but is a genuine, stable feature of spacetimes. However, this result gives little information about the geometric structure of a spacetime with such a singularity. In particular, it is in general not known if every spacetime singularity which arises from the gravitational collapse of regular initial data is covered by an event horizon. Since the existence of so-called naked singularities (for which, by definition, the latter is not true) would violate predictability (it would not be possible to predict from the initial data what an observer would see if he could observe a singularity), the cosmic censorship conjecture was formulated which demands that any singularity which arises from the gravitational collapse of *generic* regular initial data is indeed hidden behind an event horizon. The restriction to generic data means that naked singularities are allowed to occur for a “null set” of the initial data. An important example where naked singularities do form for a null set, but for which cosmic censorship holds true, is the spherically symmetric Einstein-scalar field system, cf. [@Chr94; @Chr99a]. Actually the above is an informal statement of the so-called weak cosmic censorship conjecture [@wald 12.1]; we will not be concerned with the strong version in the present paper. For a mathematical discussion and the definition of the weak cosmic censorship conjecture we refer to [@Chr99].
To deal with this conjecture in full generality is out of reach of the present level of mathematics, but under the assumption of spherical symmetry progress has been made in recent years. One important outcome of these investigations is that the answer is sensitive to which model is chosen to describe the matter. Christodoulou [@Chr84] showed that for dust, i.e., the matter model used by Oppenheimer and Snyder, cosmic censorship is violated. On the other hand, in a series of papers Christodoulou investigated a massless scalar field as matter model and showed in 1999 that weak and strong cosmic censorship hold true for this matter model; see [@Chr99a] and the references therein.
In the present investigation the main example considered as a matter model is the so-called collisionless gas as described by the Vlasov equation. It is used extensively in astrophysics, cf. [@BT], to describe galaxies or globular clusters which are viewed as large ensembles of mass points which interact only through the gravitational field that the ensemble creates collectively. In a relativistic context this leads to the Einstein-Vlasov system. All results available for this system support the following
[**Conjecture:**]{} [*Weak cosmic censorship holds for the Einstein-Vlasov system.*]{}
We mention explicitly that, in contrast to dust, small, spherically symmetric initial data launch global solutions, i.e., the solutions are geodesically complete and hence satisfy cosmic censorship, cf. [@RR1]. Also, the numerical simulations [@AR1; @OC; @RRS2] which treat large initial data support the hypothesis that naked singularities do not form in the evolution. We point out a further interesting feature of Vlasov matter observed in these numerical studies: In a one-parameter family of solutions which for large parameters, i.e., large amplitudes of the initial data, collapse to a black hole the smallest black hole always has a strictly positive ADM mass, i.e., there is a mass gap. For some other models, e.g. a scalar field, the mass of the black hole as a function of the parameter is continuous and arbitrarily small black holes can form, cf. [@CG] for a review.
The aim of the present paper is to find explicit conditions on the initial data which ensure the formation of black holes. This class of initial data has the important property that, except for “boundary cases”, properly restricted small perturbations of the data lead to solutions with the same properties. In this sense the established behaviour of the solutions is stable and not restricted to especially constructed solutions or initial data, respectively. It turns out that some of our results can be formulated for a general matter model which satisfies certain specific assumptions, and in order to give a broader impact to our results we shall do so. At the same time we emphasize that the Vlasov matter model is the only one which is presently known to actually satisfy all the assumptions needed for our arguments to go through.
As an interesting corallary to our main result we show that it is in fact possible to choose initial data for the Einstein-Vlasov system, which lead to formation of black holes, such that the solutions exist for all Schwarzschild time and all $r\geq 0.$ We thus obtain global existence in Schwarzschild coordinates for initial data which are not small, and to the best of our knowledge this is the first global existence result in Schwarzschild coordinates for initial data which lead to gravitational collapse and formation of black holes.
One aspect of our result is that there is a set of initial data which leads to gravitational collapse such that weak cosmic censorship holds. This point should be related to an earlier result by Rendall [@Rend92], where it is shown that there exist initial data for the spherically symmetric Einstein-Vlasov system such that a trapped surface forms in the evolution. The occurrence of a trapped surface signals the formation of an event horizon. Indeed, Dafermos [@D05] has proved that if a spherically symmetric spacetime contains a trapped surface and the matter model satisfies certain hypotheses then weak cosmic censorship holds true. In [@DR05] it was then shown that Vlasov matter does satisfy the required hypotheses. Hence, by combining these results it follows that initial data exist which lead to gravitational collapse and for which weak cosmic censorship holds. However, the proof in [@Rend92] rests on a continuity argument, and it is not possible to tell whether or not a given initial data set will give rise to a black hole. This is in contrast to the explicit conditions on the initial data, together with the detailed asymptotic structure, that we obtain in the present work. In this regard it is natural to relate our results to those of Christodoulou on the spherically symmetric Einstein-scalar field system [@Chr87] and [@Chr91]. In [@Chr87] it is shown that if the final Bondi mass $M$ is different from zero, the region exterior to the sphere $r=2M$ tends to the Schwarzschild metric with mass $M$. In Theorem \[bh\] below we show that solutions of the spherically Einstein-Vlasov system, under certain conditions on the initial data, also converge to the Schwarzschild metric asymptotically. Furthermore, in [@Chr91] explicit conditions on the initial data are specified which guarantee the formation of trapped surfaces. This paper played a crucial role in Christodoulou’s proof [@Chr99a] of the weak and strong cosmic censorship conjectures mentioned above. The conditions on the initial data in [@Chr91] allow the ratio of the Hawking mass and the area radius to cover the full range, i.e., $2m/r\in (0,1),$ whereas our conditions always require $2m/r$ to be quite close to one. However, we believe that to understand gravitational collapse in the case of Vlasov matter the essential situation is when $2m/r$ is large. We thus hope that the results in the present paper will lead to progress on the general understanding of gravitational collapse and the weak cosmic censorship conjecture in the case of Vlasov matter.
The Vlasov matter model has a further property to recommend it when compared to other matter models. For the Vlasov-Poisson system, which arises as the Newtonian limit of the Einstein-Vlasov system in a rigorous sense [@RR2; @Rl0], and which is used extensively in astrophysics, there is a global existence and uniqueness result for general, smooth initial data [@LP; @Pf]. This means in particular that any breakdown of a solution of the Einstein-Vlasov system can be expected to be a genuine, general relativistic effect such as a spacetime singularity and not only remainder of some bad behaviour which the matter model exhibits already on the Newtonian level.
To be more specific, consider now a smooth spacetime manifold $M$ equipped with a spacetime metric $g_{\alpha \beta}$; Greek indices run from $0$ to $3$. Then the Einstein equations read $$\label{feqgen}
G_{\alpha \beta} = 8 \pi T_{\alpha \beta},$$ where $G_{\alpha \beta}$ is the Einstein tensor, a non-linear second order differential expression in the metric $g_{\alpha \beta}$, and $T_{\alpha \beta}$ is the energy-momentum tensor given by the matter content (or other fields) of the spacetime. To obtain a closed system, the field equations (\[feqgen\]) have to be supplemented by $$\label{matevol}
\mbox{evolution equation(s) for the matter}$$ and $$\label{emtdef}
\mbox{the definition of $T_{\alpha \beta}$ in terms of the matter and the metric}.$$ It is often possible to specify conditions on (\[matevol\]) and (\[emtdef\]) under which one can establish geometric properties of a spacetime described by the Einstein-matter system (\[feqgen\]), (\[matevol\]), (\[emtdef\]). The Penrose singularity theorem mentioned above is of this nature, and part of our arguments will also be presented in this form.
However, in order to verify such general conditions, in particular with respect to the existence of local or global solutions to the corresponding initial value problem, a specific matter model must be chosen, and in the present paper this is a collisionless gas. All the particles in the gas are assumed to have the same rest mass, normalized to unity, and to move forward in time. Hence, their number density $f$ is a non-negative function supported on the mass shell $$PM := \left\{ g_{\alpha \beta} p^\alpha p^\beta = -1,\ p^0 >0 \right\},$$ a submanifold of the tangent bundle $TM$ of the spacetime manifold $M$; $p^\alpha$ are the canonical momenta corresponding to general coordinates $x^\alpha=(t,x^a)$ on $M$. We use coordinates $(t,x^a)$ with zero shift, and Latin indices run from $1$ to $3$. On the mass shell $PM$ the variable $p^0$ becomes a function of the remaining variables $(t,x^a,p^b)$: $$p^0 = \sqrt{-g^{00}} \sqrt{1+g_{ab}p^a p^b}.$$ The number density $f=f(t,x^a,p^b)$ satisfies a continuity equation, the so-called Vlasov equation, which says that $f$ is constant along the geodesics of the spacetime metric, $$\label{vlgen}
\partial_t f + \frac{p^a}{p^0}\,\partial_{x^a} f
-\frac{1}{p^0}\,\Gamma^a_{\beta \gamma} p^\beta p^\gamma\,\partial_{p^a} f = 0,$$ where $\Gamma^\alpha_{\beta \gamma}$ are the Christoffel symbols induced by the metric $g_{\alpha \beta}$. The energy-momentum tensor is given by $$\label{emtvlgen}
T_{\alpha \beta}
=\int p_\alpha p_\beta f \,|g|^{1/2} \,\frac{dp^1 dp^2 dp^3}{-p_0},$$ where $|g|$ denotes the modulus of the determinant of the metric. The system (\[feqgen\]), (\[vlgen\]), (\[emtvlgen\]) is the Einstein-Vlasov system in general coordinates. For an introduction to relativistic kinetic theory and the Einstein-Vlasov system we refer to [@And05] and [@Rend05].
If, for comparison, the matter is to be described as a perfect fluid with density $\Rho$, four-velocity field $U^\alpha$, and pressure $P$, then the matter evolution equations are the Euler equations $$U^\alpha \nabla_\alpha \Rho + (\Rho +P) \nabla^\alpha U_\alpha = 0,$$ $$(\Rho +P)U^\alpha \nabla_\alpha U_\beta +
(g_{\alpha \beta} + U_\alpha U_\beta) \nabla^\alpha P = 0,$$ where $\nabla_\alpha$ is the covariant derivative corresponding to the metric $g_{\alpha \beta}$. The energy-momentum tensor in this case is $$T_{\alpha \beta}
= \Rho U_\alpha U_\beta + P (g_{\alpha \beta} + U_\alpha U_\beta ).$$ To close the Einstein-Euler system it has to be supplemented by an equation of state $P=P(\Rho)$. The choice $P=0$ yields the dust matter model referred to above.
Due to the complexity of the field equations (\[feqgen\]) very little can be said about the questions at hand for these equations in their general form. Since on the other hand these questions are of considerable interest also in spacetimes satisfying simplifying symmetry assumptions, we from now on focus on asymptotically flat, spherically symmetric spacetimes and write down the metric $$ds^2=-e^{2\mu(t,r)}dt^2+e^{2\lambda(t,r)}dr^2+r^2(d\theta^2+\sin^2\theta\,d\varphi^2)$$ in Schwarzschild coordinates. Here $t\in{\mathbb R}$ is the time coordinate, $r\in [0, \infty[$ is the area radius, i.e., $4 \pi r^2$ is the area of the orbit of the symmetry group $\mathrm{SO}(3)$ labeled by $r$, and the angles $\theta\in[0, \pi]$ and $\varphi\in[0, 2\pi]$ parameterize these orbits. Asymptotic flatness means that the metric quantities $\lambda$ and $\mu$ have to satisfy the boundary conditions $$\label{boundc}
\lim_{r\to\infty}\lambda(t, r)=\lim_{r\to\infty}\mu(t, r)=0.$$ For a metric of this form the $00$, $11$, and $01$ components of the Einstein equations are found to be $$\label{ein1}
e^{-2\lambda}(2r\lambda_r-1)+1=8\pi r^2 e^{-2\mu} T_{00},$$ $$\label{ein2}
e^{-2\lambda}(2r\mu_r+1)-1 = 8\pi r^2 e^{-2\lambda} T_{11},$$ $$\label{ein3}
\lambda_t = 4 \pi r T_{01},$$ where subscripts indicate partial derivatives. The $22$ and $33$ components are also nontrivial, but they are not needed for our analysis, and the remaining components vanish identically due to the symmetry assumption.
Our aim is to find explicit conditions on the initial data such that the corresponding solutions of the spherically symmetric, asymptotically flat version of the system (\[feqgen\]), (\[matevol\]), (\[emtdef\]) have the following property: There is an outgoing radial null geodesic $\gamma^+$ originating from $r=r_0>0$, i.e., $$\label{gamma+}
\frac{d \gamma^+}{ds}(s)=e^{(\mu-\lambda)(s,\gamma^+(s))},\;\gamma^+(0)=r_0,$$ such that the solution exists on the outer region $$\label{ddef}
D:=\{(t,r) \in [0,\infty[^2 \mid r \geq \gamma^+(t)\},$$ and $\gamma^+$ has the property that $$\label{limgamma}
\lim_{s\to \infty}\gamma^+(s) < \infty.$$ This indicates that the matter distribution undergoes a gravitational collapse, and a black hole forms. In the case of Vlasov matter we obtain a more detailed picture which supports this interpretation: There exists an extremal, radially outgoing null geodesic $\gamma^\ast$ in the outer domain $D$ such that $\lim_{s\to \infty}\gamma^\ast (s) = 2 M$ where $M$ is the ADM mass of the solution, and as $t\to\infty$ the metric converges for $r>2 M$ to the Schwarzschild metric representing a black hole of mass $M$.
In the next section we state our main results for the Einstein-Vlasov system, where we specify classes of spherically symmetric initial data which lead to solutions showing the above behaviour. The Vlasov equation and the corresponding energy-momentum tensor components in the case of spherical symmetry are stated there. In Section \[secgenmat\] we give a general formulation of one of our results where no particular matter model is considered. The reason for this is that most steps in the proof of Theorem \[vlasov2\] below are of a general character and—besides the fact that for the Einstein-Vlasov system there is an existence theory for the initial value problem which guarantees the existence of solutions on $D$—the specific properties of Vlasov matter are used only in one key lemma. Hence it is natural to precisely single out the required conditions on the level of the macroscopic matter quantities. This clarifies the main mechanism in our method, and it may lead to applications of our method to other matter models. Using an additional feature of Vlasov matter we construct an alternative, and in some respects larger, class of initial data which ensure the formation of black holes, cf. Theorem \[vlasov1\].
The proofs of our results then proceed as follows. After stating some general auxiliary results in Section \[prelim\] we prove Theorem \[genmat\], which is the general-matter version of Theorem \[vlasov2\], in Section \[secgenmatproof\]. The latter result is then established in Section \[secvlasov2\] by showing that Vlasov matter satisfies the required general conditions on the matter for a suitable class of initial data. Theorem \[vlasov1\] is established in Section \[secvlasov1\] together with Corallary \[ssinthemiddle\] on global existence in Schwarzschild coordinates. For all these results it is essential to make sure that in the outer region $D$ all the matter moves inward. In the case of general matter this is a condition which we have to impose on the solution, whereas in the case of Vlasov matter we can specify conditions on the initial data such that this is true. In Section \[bhproof\] we prove the convergence of our solution to a Schwarzschild black hole of the corresponding ADM mass in the case of Vlasov matter.
Main results for Vlasov matter {#secvlasres}
==============================
In this section Eqns. (\[boundc\])–(\[ein3\]) will be supplemented by the spherically symmetric version of the Vlasov equation together with expressions for the relevant components of the energy-momentum tensor so that a closed system is obtained, known as the spherically symmetric, asymptotically flat Einstein-Vlasov system.
In order to exploit the symmetry it is useful to introduce non-canonical variables on momentum space and write $f=f(t,r,w,L)$. For a detailed derivation of the corresponding equations we refer to [@Rein95]; here we just state the result. The Vlasov equation is $$\label{vlasov}
\partial_{t}f+e^{\mu-\lambda}\frac{w}{E}\partial_{r}f
-\left(\lambda_{t}w+e^{\mu-\lambda}\mu_{r}E-
e^{\mu-\lambda}\frac{L}{r^3E}\right)\,\partial_{w}f=0,$$ where $$E=E(r,w,L):=\sqrt{1+w^{2}+L/r^{2}} = e^\mu p^0.$$ The variables $w\in ]-\infty,\infty[$ and $L\in [0,\infty[$ can be thought of as the radial component of the momentum and the square of the angular momentum respectively. Notice that the latter is conserved along characteristics of the Vlasov equation. The matter quantities are given by $$\begin{aligned}
\rho(t,r)
&=&
e^{-2 \mu} T_{00}(t,r) =
\frac{\pi}{r^{2}}
\int_{-\infty}^{\infty}\int_{0}^{\infty}Ef(t,r,w,L)\,dL\,dw,
\label{rho}\\
p(t,r)
&=&
e^{-2 \lambda} T_{11}(t,r) =
\frac{\pi}{r^{2}}\int_{-\infty}^{\infty}\int_{0}^{\infty}
\frac{w^{2}}{E}f(t,r,w,L)\,dL\,dw,
\label{p}\\
j(t,r)
&=&
-e^{-(\lambda+\mu)} T_{01}(t,r) =
\frac{\pi}{r^{2}}
\int_{-\infty}^{\infty}\int_{0}^{\infty}w\,f(t,r,w,L)\,dL\,dw.
\label{j}\end{aligned}$$ Notice that the quantities $\rho,\, p,\, j$ appear on the right hand sides of the field equations (\[ein1\])–(\[ein3\]), and they are given in terms of $f$ alone, which is the main reason for using the non-canonical variables $w$ and $L$. The system (\[boundc\])–(\[ein3\]), (\[vlasov\])–(\[j\]) is the spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates. As initial data we need to prescribe an initial distribution function $\open{f}=\open{f}(r,w,L)\geq 0$, which should be compactly supported in $]0,\infty[ \times ]-\infty,\infty[\times ]0,\infty[$, and such that $$\label{notsinit}
\int_0^r 4\pi\eta^2\open{\rho}(\eta)\,d\eta
=4\pi^2 \int_0^r\int_{-\infty}^{\infty}\int_0^{\infty}
E\open{f}(\eta,w,L)\,dL\,dw\,d\eta < \frac{r}{2}.$$ The origin $r=0$ is excluded from the support for technical reasons, but this could be avoided by using Cartesian coordinates. The condition (\[notsinit\]) implies that the equations (\[ein1\]) and (\[ein2\]) have solutions $\lambda$ and $\mu$, cf. Section \[prelim\], and since $\open{f}$ has compact support, a property which is inherited by $f(t)$, the matter terms are well defined. If in addition $\open{f}$ is $C^1$ we say that the initial data is *regular*. As is shown in [@RR1] or [@Rein95], regular initial data launch a unique local solution for which all the derivatives which appear in the system exist classically. In Section \[secvlasov2\] we discuss in more detail that this local solution extends to the whole outer region $D$ defined in (\[ddef\]).
To state our main results let $0<r_0<r_1$ be given, put $M=r_1/2$ (this is going to be the ADM mass of the solution), and fix $0<M_\mathrm{out}<M$ such that $$\label{icnts}
\frac{2(M-M_\mathrm{out})}{r_0}<\frac{8}{9}.$$ **Remark.** The value $8/9$ is chosen for definiteness, and any number less than one would do, effecting the values of some of the constants below.
Two different theorems will be stated below, corresponding to the following two situations.
- Let $R_1>r_1$ be such that $$\label{mediumstrip}
R_1-r_1<\frac{r_1-r_0}{6},$$ or
- let $R_1>r_1$ be such that $$\label{smallstrip}
\sqrt{\frac{R_1-r_1}{R_1}}<\min\left\{\frac{1}{6},\frac{r_0^2}{12\kappa R_1M},
\frac{r_1-r_0}{8\kappa R_1}\right\},$$ where the (explicit) constant $\kappa>0$ will be specified in Theorems \[vlasov2\] and \[genmat\] below.
Finally, we define $$R_0:=\frac{1}{2}(r_1+R_1).$$ Denote by $\open{\rho}\,$ the energy density induced by the initial distribution function $\open{f}$. We require that all the matter in the outer region $[r_0, \infty[$ is initially located in the strip $[R_0,R_1]$, with $M_\mathrm{out}$ being the corresponding fraction of the ADM mass $M$, i.e., $$\label{checkM}
\int_{r_0}^{\infty}4\pi r^2\open{\rho}(r)dr = \int_{R_0}^{R_1}4\pi r^2\open{\rho}(r)dr=M_\mathrm{out}.$$ Furthermore, the remaining fraction $M-M_\mathrm{out}$ should be initially located within the ball of area radius $r_0$, i.e., $$\label{M-checkM}
\int_{0}^{r_0}4\pi r^2\open{\rho}(r)dr=M-M_\mathrm{out}.$$ **Remark.** The set up described above is quite similar to the set up in [@Chr91] for a scalar field. In [@Chr91] it is not required to have matter in an “inner” strip $[0,r_0]$, as is the case here in view of (\[M-checkM\]) and the condition $M_\mathrm{out}<M.$ The reason why we need some matter in the region $r\le r_0$ is to ensure that initially ingoing matter continues to be ingoing for all times, cf. Lemma \[ingoinglemma\] below. If one only considers purely radially ingoing particles, i.e., with no angular momentum (which results in a non-smooth distribution function $f$), then we could allow for $M_\mathrm{out}=M.$ It is interesting to note that $p=\rho$ holds for Vlasov matter, if the particles have no angular momentum and their rest mass is zero, which is the case for the scalar field considered in [@Chr91].
Now we are in the position to formulate our main results for Vlasov matter. Corresponding to Case (i) above, we prove
\[vlasov1\] Let $r_0,\, r_1, M$, and $M_\mathrm{out}$ be given as above, and let $R_1$ satisfy (\[mediumstrip\]). Then there exists a set ${\cal{I}}_1$ of regular initial data for the spherically symmetric Einstein-Vlasov system such that if $\open{f}\in{\cal{I}}_1$, then (\[checkM\]) and (\[M-checkM\]) hold, the corresponding solution exists on $D$, and $$\lim_{s\to \infty}\gamma^+(s) < \infty,
\quad \lim_{s\to \infty} \int_{\gamma^+(s)}^\infty 4 \pi r^2 \rho(s,r)\,dr > 0,$$ where $\gamma^+$ satisfies (\[gamma+\]).
By abuse of notation we denote by $D$ both the outer region in spacetime defined by (\[ddef\]) and the part of the mass shell with $(t,r) \in D$.
The next theorem addresses Case (ii) above, assuming the stronger condition (\[smallstrip\]). This allows for a more straightforward proof, and the constraints on the momentum variables of the initial distribution function $\open{f}$ which are used to specify the set ${\cal{I}}_1$ will be slightly relaxed. Hence, the initial data set ${\cal{I}}_1$ does not contain ${\cal{I}}_2$ in Theorem \[vlasov2\] below, but it is larger in the sense that data in ${\cal{I}}_2$ are quite close to containing a trapped surface, which is not necessarily the case for data in ${\cal{I}}_1$. The precise form of ${\cal{I}}_1$ and ${\cal{I}}_2$ is specified in the proofs.
\[vlasov2\] Let $r_0,\, r_1, M$, and $M_\mathrm{out}$ be given as above and let $R_1$ satisfy (\[smallstrip\]) with $\kappa=6$. Then there exists a set ${\cal{I}}_2$ of regular initial data for the spherically symmetric Einstein-Vlasov system such that if $\open{f}\in{\cal{I}}_2$, then (\[checkM\]) and (\[M-checkM\]) hold, the corresponding solution exists on $D$, and $$\lim_{s\to \infty}\gamma^+(s) < \infty,
\quad \lim_{s\to \infty} \int_{\gamma^+(s)}^\infty 4 \pi r^2 \rho(s,r)\,dr > 0,$$ where $\gamma^+$ satisfies (\[gamma+\]).
The Einstein-Vlasov system has a wide variety of static, spherically symmetric solutions with finite ADM mass and finite radius, i.e., compact support of the matter, cf. [@Rss1; @RRss1; @RRss2]. Particularly interesting examples of initial data for which our results apply are obtained if the matter for $r \leq r_0$ is represented by such a static solution, more precisely:
\[ssinthemiddle\] Let $f_s$ be a static solution of the spherically symmetric Einstein-Vlasov system with finite ADM mass $M_s>0$ and finite radius $r_s >0$. Define $r_0=r_s$, let $r_1 > r_0$ be arbitrary, $M=r_1/2$, and $M_\mathrm{out} = M -M_s$; the latter quantity is positive. Then the initial data sets ${\cal{I}}_1$ and ${\cal{I}}_2$ both contain data $\open{f}$ which coincide with the given static solution for $0\leq r\leq r_0$. The corresponding solution $f$ of the Einstein-Vlasov system exists for all $r\geq 0,\ t\geq 0$ and coincides with the static solution $f_s$ for all $r\leq \gamma^+(t)$ and $t\geq 0$.
We prove this result at the end of Section \[secvlasov1\]. It represents a global existence result for the Einstein-Vlasov system in Schwarzschild time for data which are not small.
In the next section we formulate a version of Theorem \[vlasov2\] for quite general matter models. One reason for this is that the main mechanism behind our method becomes very transparent by posing sufficient conditions on the macroscopic matter terms rather than conditions on the initial distribution function $\open{f}$ as we did in the theorems above. Theorem \[vlasov2\] will then be a consequence of this generalization, cf. Section \[secvlasov2\], whereas Theorem \[vlasov1\] is established in Section \[secvlasov1\].
In these proofs it turns out that for the classes of initial data that we specify we can obtain somewhat sharper asymptotic information on $\gamma^+$ and the mass in the outer region; see (\[precise\]) below. More importantly, we can establish the following additional information which shows that the solution evolves towards a Schwarzschild black hole of mass $M$.
\[bh\] In the situation of Theorem \[vlasov1\] or Theorem \[vlasov2\] the following holds:
- There exist constants $\alpha, \beta > 0$ depending only on the initial data set ${\cal I}_1$ or ${\cal I}_2$ respectively such that if $$t \geq 0 \ \mbox{and}\ r \geq 2 M + \alpha e^{-\beta t}$$ then $f(t,r,\cdot,\cdot) = 0$, i.e., we have vacuum, and the metric equals the Schwarzschild metric $$ds^2 = -\left(1-\frac{2 M}{r}\right)\, dt^2 + \left(1-\frac{2 M}{r}\right)^{-1}dr^2\,
+ r^2(d\theta^2 + \sin^2 \theta d\varphi^2),$$ representing a black hole of mass $M$.
- For all $t\geq 0$ and $\gamma^+ (t) \leq r \leq 2 M + \alpha e^{-\beta t}$, $$\mu(t,r)
\leq \ln \left(\frac{\alpha e^{-\beta t}}{2 M + \alpha e^{-\beta t}}\right)^{1/2}$$ so that in the outer region $D$, $$\lim_{t\to \infty} \mu(t,r) = -\infty\ \mbox{for}\
r \leq 2 M,$$ and the timelike lines $r=c,$ where $c\in [0,2M],$ are incomplete and their proper lengths are uniformly bounded by a constant depending on $\alpha,\, \beta$ and $M.$
- Let r\^:= { rr\_0 &|& \
&& (0)=r \_[s]{}(s) < }, and let $\gamma^\ast$ be the radially outgoing null geodesic with $\gamma^\ast (0) = r^\ast$. Then $$\lim_{s\to\infty}\gamma^\ast (s) = 2 M,$$ and every radially outgoing null geodesic $\gamma$ with $\gamma(0) > r^\ast$ is future complete with $\lim_{s\to\infty}\gamma (s) = \infty$.
The result for general matter models {#secgenmat}
====================================
In this section we specify the general assumptions on a matter model sufficient for our method to be applied. In order to keep the discussion consistent with the Vlasov part of our arguments, and in view of the right hand sides of the field equations (\[ein1\]), (\[ein2\]), (\[ein3\]), it is convenient to use the notation $$\label{genmatquant}
\rho := e^{-2\mu} T_{00},\ p := e^{-2\lambda} T_{11},\
j := - e^{-\mu-\lambda}T_{01}.$$ Firstly, we assume that the following two conditions are satisfied.
- The dominant energy condition holds. (DEC)
- The radial pressure $p$ is non-negative. (NNP)
The dominant energy condition (DEC) plays a central role in general relativity and is the main criterion that a matter model should satisfy to be considered realistic. We refer to [@HE] for its definition. The non-negative pressure condition (NNP) is restrictive in the sense that it rules out, for example, a Maxwell field as matter model. However, for most astrophysical models it is a standard assumption, with e.g. fluid models satisfying this condition. For the purpose of this paper we only need to focus on two consequences of these two criteria, cf. [@HE] and [@P]. The (DEC) condition implies, together with the (NNP) condition, that $$\label{nnrho}
0\leq p\leq \rho\ \mbox{and}\ |j| \leq \rho.$$ Furthermore, by (DEC) any geodesic $(s,R(s))$ of a material particle or a light ray satisfies $$\label{mattergeodesic}
\left|\frac{dR(s)}{ds}\right| \leq e^{(\mu-\lambda)(s,R(s))}.$$ The meaning of the latter condition is that locally the speed of energy flow is less than or equal to the speed of light.
Let $\lambda,\, \mu,\, \rho,\, p,\, j$ correspond to a solution of the spherically symmetric Einstein-matter equations (\[boundc\])–(\[ein3\]), (\[matevol\]), (\[emtdef\]) in Schwarzschild coordinates, launched by initial data from a class $\cal{I}$. In order to investigate the global structure of the solutions it is necessary that they exist globally in an appropriate sense. In the situation at hand they need to exist on the outer region $D$ defined in (\[ddef\]). In the spherical symmetric case the main obstruction for obtaining global solutions arises from the difficulties related to the centre of symmetry $r=0$. For example, for a massless scalar field or a collisionless gas as matter model it has been shown that solutions remain regular away from $r=0$ for general initial data, cf. [@Chr99; @And1; @RRS]. On the other hand, for dust a singularity of shell crossing type can also occur at some $r>0$. Although in that case there are no true geometric spacetime singularities, such behaviour has to be ruled out in order not to interfere with the analysis of the solution on $D$. This can be achieved by proper assumptions on the initial data, cf. [@Chr84]. In view of (\[mattergeodesic\]) a possible break down of solutions at $r=0$ will have no influence on the outer domain $D$. Hence we formulate a third condition, concerning global existence of solutions in the outer domain, as follows.
- For solutions launched by data from the set $\cal{I}$, $\gamma^+$ defined by (\[gamma+\]) exists on $[0,\infty[$, and $\lambda,\, \mu,\, \rho,\, p,\, j\in C^1(D).$ (GLO)
The three conditions above are of a quite general nature. The fourth and final condition however, is tightly connected to our method of proof.
- There exists a constant $c_1>0$ such that $\rho \leq -c_1 j$ in $D$. (GCC)
The acronym (GCC) stands for “gravitational collapse condition”, and this condition plays a crucial role for our method of proof. We emphasize that our main results show that for Vlasov matter there are initial data sets such that (GCC) holds. As a first consequence of (GCC) and (\[nnrho\]), note that $j\le 0$ in $D$, i.e., the matter is ingoing for all times. In this respect our present results complement [@AKR1], where purely outgoing matter is considered.
Let us now assume that our matter model satisfies (DEC) and (NNP), and that there exists an initial data set $\cal{I}$ such that (GLO) and (GCC) hold as well. Then we have the following result, which should be viewed as a version of Theorem \[vlasov2\] for general matter.
\[genmat\] Let $r_0,\, r_1,\, M$, and $M_\mathrm{out}$ be given as above and let $R_1$ satisfy (\[smallstrip\]) with $\kappa=2c_1$. Assume that there exists an initial data set ${\cal{I}}_3\subset\cal{I}$ such that (\[checkM\]) and (\[M-checkM\]) hold for all initial data in ${\cal{I}}_3$. Then for any solution launched by initial data in ${\cal{I}}_3$, $$\lim_{s\to \infty}\gamma^+(s) < \infty,
\quad \lim_{s\to \infty} \int_{\gamma^+(s)}^\infty 4 \pi r^2 \rho(s,r)\,dr > 0,$$ where $\gamma^+$ satisfies (\[gamma+\]).
The detailed information on the gravitational collapse which for Vlasov matter is provided in Theorem \[bh\] is not available in the present situation, but the following still holds.
[**Remark**]{}. In the situation of Theorem \[genmat\], $$\lim_{t\to \infty} \mu(t,r) = -\infty\ \mbox{for}\
\lim_{s\to\infty}\gamma^+(s) \leq r \leq r_1$$ for some $r_1 > \lim_{s\to\infty}\gamma^+(s)$. If $r^\ast$ and $\gamma^\ast$ are defined as in Theorem \[bh\] then $$\lim_{s\to\infty}\gamma^\ast (s) < \infty,$$ and every radially outgoing null geodesic $\gamma$ with $\gamma(0) > r^\ast$ is future complete with $\lim_{s\to\infty}\gamma (s) = \infty$.
These assertions will be established in Section \[bhproof\]. Concerning the question which matter models besides Vlasov matter satisfy our conditions above we note the following:
[**Remark**]{}. For a spherically symmetric perfect fluid with density ${\cal R}$, pressure $P=P({\cal R})$, and radial velocity field $u$, the (DEC) and (NNP) conditions and Eqn. (\[nnrho\]) respectively are satisfied provided that $0\leq P({\cal R}) \leq {\cal R}$, which restricts the equation of state. The (GCC) condition holds for example with $c_1=\sqrt{2}$ if $-e^{\lambda} u \geq 1$ on $D$. In the kinetic context of the Vlasov model we derive analogous estimates on the particle level from conditions on the initial data.
Preliminaries {#prelim}
=============
In this section we collect some general facts concerning the spherically symmetric Einstein-matter equations under the assumptions (DEC) and (NNP) that have been specified in the previous section.
A quantity which plays an important role is the quasi-local mass $m(t,r)$. Typically, the spherically symmetric Einstein-matter system is supplemented by the requirement of a regular centre, i.e., $\lambda (t,0)=0$. Using this boundary condition the field equation (\[ein1\]) implies that $$\label{e2lamb}
e^{-2\lambda}=1-\frac{2m}{r},$$ where the quasi-local mass would be given by $m(t,r) := \int_0^r 4\pi\eta^2\rho(t,\eta)\,d\eta$. Then $m(t,\infty)$ is a conserved quantity, the ADM mass. However, in the present context we want to investigate the system on the outer domain $D$, regardless of whether or not the solution remains regular in the region $r<\gamma^+(t)$. Hence we do not use the usual boundary condition at $r=0$. Instead, we assume that the ADM mass $M>0$ is given and redefine the quasi-local mass by $$\label{m-def}
m(t,r)=M-\int_r^\infty 4\pi\eta^2\rho(t,\eta)\,d\eta.$$ Then $\lim_{r\to\infty} m(t, r)=M$, $0\le m\le M$, and $m_r=4\pi r^2\rho$ holds. Defining $\lambda$ by (\[e2lamb\]), (\[genmatquant\]) shows that (\[ein1\]) and the boundary condition in (\[boundc\]) are satisfied. In addition, we need to modify (\[notsinit\]) to $$\label{notsinit2}
\open{m}(r)<\frac{r}{2},\quad r\in ]0, \infty[,$$ a condition that once again will be included in the notion of regular initial data.
By (\[ein1\]) and (\[ein2\]), $$\label{mur}
\lambda_r=\Big(4\pi r\rho-\frac{m}{r^2}\Big)e^{2\lambda},
\quad\mu_r=\Big(\frac{m}{r^2}+4\pi r p\Big)e^{2\lambda}.$$ In view of (\[boundc\]), $\mu=\hat{\mu}+\check{\mu}$, where we define $$\begin{aligned}
\hat{\mu}(t, r)
&:=&
-\int_r^\infty\frac{m(t, \eta)}{\eta^2}\,e^{2\lambda(t,\,\eta)}\,d\eta,\label{hatmu}
\\
\check{\mu}(t, r)
&:=&
-\int_r^\infty 4\pi\eta\,p(t, \eta)\,e^{2\lambda(t,\,\eta)}\,d\eta.
\label{checkmu}\end{aligned}$$
\[hatmu-lem\] The following assertions hold.
- $2\hat{\mu}\le\mu-\lambda\le\hat{\mu}\le\hat{\mu}+\lambda$.
- $\mu+\lambda\le\hat{\mu}+\lambda$.
- $(\mu-\lambda)(t, r)=2\hat{\mu}(t, r)
+\int_r^\infty 4\pi\eta\,(\rho-p)(t, \eta)\,e^{2\lambda(t,\,\eta)}\,d\eta$.
- $\hat{\mu}_t(t, r)=\int_r^\infty 4\pi j(t, \eta)
\,e^{(\mu+\lambda)(t,\,\eta)}e^{2\lambda(t,\,\eta)}\,d\eta$. In particular, if $j\le 0$, then also $\hat{\mu}_t\le 0$.
[**Proof:**]{} In view of (\[boundc\]), $$\lambda(t, r)=-\int_r^\infty\Big(4\pi\eta\,\rho(t, \eta)
-\frac{m(t, \eta)}{\eta^2}\Big)\,e^{2\lambda}\,d\eta
= - \int_r^\infty 4\pi\eta\,\rho(t, \eta)\,e^{2\lambda}\,d\eta -\hat{\mu} ,$$ and by (\[nnrho\]) the relation $\mu-\lambda\ge 2\hat{\mu}$ follows. On the other hand, by (\[e2lamb\]), $\lambda\ge 0$. Thus $\check{\mu}\le 0$ leads to $\mu-\lambda\le\mu\le\hat{\mu}\le\hat{\mu}+\lambda$, and part (a) is established. Part (b) follows from $\check{\mu}\le 0$. As to (c), we observe that $$\hat{\mu}+\lambda+\int_r^\infty
4\pi\eta\,(\rho-p)\,e^{2\lambda}\,d\eta=\check{\mu},$$ which gives the claim. By (\[e2lamb\]) and (\[ein3\]), ${(e^{2\lambda}\frac{m}{r^2})}_t
=\frac{1}{2r}{(e^{2\lambda}-1)}_t=-4\pi\,e^{\mu+\lambda}e^{2\lambda}j$. Hence (d) follows from (\[hatmu\]).
$\Box$
\[int0infty\] For $r\in [0, \infty[$ the following holds: $$\begin{aligned}
& & \int_r^\infty 4\pi\eta\,(\rho+p)(t, \eta)
\,e^{(\mu+\lambda)(t,\,\eta)}e^{2\lambda(t,\,\eta)}\,d\eta
=1-e^{(\mu+\lambda)(t,\,r)}\le 1,
\\ & & \int_r^\infty 4\pi\eta\,\rho(t, \eta)
\,e^{(\hat{\mu}+\lambda)(t,\,\eta)}e^{2\lambda(t,\,\eta)}\,d\eta
=1-e^{(\hat{\mu}+\lambda)(t,\,r)}\le 1.\end{aligned}$$
[**Proof:**]{} It suffices to integrate $$\begin{aligned}
\label{gsto}
\partial_r(e^{\mu+\lambda})
& = &
e^{\mu+\lambda}(\mu_r+\lambda_r) = e^{\mu+\lambda} 4\pi r\,(p+\rho),
\nonumber \\
\partial_r(e^{\hat{\mu}+\lambda})
& = &
e^{\hat{\mu}+\lambda}(\hat{\mu}_r+\lambda_r)
=e^{\hat{\mu}+\lambda}\Big(e^{2\lambda}\frac{m}{r^2}
+\Big(4\pi r\rho-\frac{m}{r^2}\Big)e^{2\lambda}\Big)
\nonumber \\
& = &
4\pi r\rho\,e^{\hat{\mu}+\lambda}e^{2\lambda},\end{aligned}$$ observing that $\lim_{r\to\infty}\hat{\mu}(t, r)
=\lim_{r\to\infty}\lambda(t, r)=\lim_{r\to\infty}\mu(t, r)=0$. For Vlasov matter, the first relation has been used in [@And1 Lemma 1].
$\Box$
Next we consider outgoing and ingoing radial null geodesics $\gamma^+$ and $\gamma^-$, respectively.
\[gammapm\] Let $\gamma^{\pm}$ be the solutions to $$\frac{d\gamma^{\pm}}{ds}(s)=\pm\,e^{(\mu-\lambda)(s,\,\gamma^{\pm}(s))},
\quad\gamma^+(0)=r_0<r_1=\gamma^-(0).$$ Then
- $\gamma^+$ is strictly increasing, $s\mapsto m(s, \gamma^+(s))$ is increasing, and the limits $\lim_{s\to\infty}\gamma^+(s)\in ]r_0, \infty]$ and $\lim_{s\to\infty} m(s, \gamma^+(s))\in [m(0, r_0), M]$ exist.
- $\gamma^-$ is strictly decreasing, $s\mapsto m(s, \gamma^-(s))$ is decreasing, and the limits $\lim_{s\to\infty}\gamma^-(s)\in [0, r_1[$ and $\lim_{s\to\infty} m(s, \gamma^-(s))\in [0, m(0, r_1)]$ exist.
- The relation $$\frac{d}{ds}(\hat{\mu}+\lambda)(s, \gamma^{\pm}(s))
=\Big(\hat{\mu}_t-4\pi r\,e^{\mu+\lambda}(j\mp\rho)\Big)
\bigg|_{(t,\,r)=(s, \gamma^{\pm}(s))}$$ holds. In particular, if $j\le 0$ and $\rho=j=0$ along $\gamma^{\pm}$, then also $\frac{d}{ds}(\hat{\mu}+\lambda)(s, \gamma^{\pm}(s))\le 0$.
[**Proof:**]{} Differentiating (\[e2lamb\]) w.r.t. $t$ and using (\[ein3\]) implies that $m_t=-4\pi r^2 e^{\mu-\lambda} j$. Since $\rho\ge j$ according to (\[nnrho\]), this yields $$\begin{aligned}
\frac{d}{ds}\,m(s, \gamma^+(s)) & = & m_t(s, \gamma^+(s))
+m_r(s, \gamma^+(s))\frac{d\gamma^+}{ds}(s)
\\ & = & (-4\pi r^2 e^{\mu-\lambda} j
+4\pi r^2\rho\,e^{\mu-\lambda})\big|_{(t,\,r)=(s, \gamma^+(s))}\ge 0.\end{aligned}$$ Thus part (a) is obtained from $m\le M$. Since $\rho\ge -j$, the proof of (b) is analogous to (a). As to (c), note that by definition of $\hat{\mu}$, (\[ein1\]), and (\[ein3\]), $$\begin{aligned}
\lefteqn{\frac{d}{ds}(\hat{\mu}+\lambda)(s, \gamma^{\pm}(s))}
\\ & = & \Big(\hat{\mu}_t+\hat{\mu}_r\frac{d\gamma^{\pm}}{ds}
+\lambda_t+\lambda_r\frac{d\gamma^{\pm}}{ds}\Big)\bigg|_{(t,\,r)=(s, \gamma^{\pm}(s))}
\\ & = & \Big(\hat{\mu}_t\pm\frac{m}{r^2}\,e^{2\lambda}e^{\mu-\lambda}
-4\pi r\,e^{\mu+\lambda}j\pm\Big(4\pi r\rho-\frac{m}{r^2}\Big)e^{2\lambda}
e^{\mu-\lambda}\Big)\bigg|_{(t,\,r)=(s, \gamma^{\pm}(s))}
\\ & = & \Big(\hat{\mu}_t-4\pi r\,e^{\mu+\lambda}(j\mp\rho)\Big)
\bigg|_{(t,\,r)=(s, \gamma^{\pm}(s))},\end{aligned}$$ as desired. The last claim follows from Lemma \[hatmu-lem\](d).
$\Box$
Proof of Theorem \[genmat\] {#secgenmatproof}
===========================
In this section we use the hypotheses stated in Section \[secgenmat\] to prove Theorem \[genmat\]. The proof is short and emphasizes that the crucial mechanism is captured in the (GCC) condition. Our main results which show in particular that the (GCC) condition holds for Vlasov matter are established in the next sections.
Consider the out- and ingoing null geodesics $\gamma^+$ and $\gamma^-$ defined in Lemma \[gammapm\]. The claims follow if we can show that these geodesics never intersect. By continuity and monotonicity there exists $T \in ]0,\infty]$ such that $$\label{fg-esti}
r_0\le\gamma^+(t) < \gamma^-(t)\le r_1,\quad t\in [0, T[;$$ it will be shown that actually $T=\infty$ holds. In view of (\[checkM\]) we have initially that $\rho=p=j=0$ for $r\geq R_1.$ The (GCC) condition implies that $j\leq 0$ in $D$, meaning that the flow of matter is ingoing. Therefore $$\label{ingoing}
\rho=p=j=0\quad\mbox{and}\quad m=M
\quad\mbox{for}\quad (t,r)\in [0, T[\times [R_1, \infty[.$$ By Lemma \[int0infty\], (\[nnrho\]), the (GCC) condition, and Lemma \[hatmu-lem\](d) for $s\in [0,T[$ and $r\in [\gamma^+(s), \infty[$, $$\begin{aligned}
1-e^{(\mu+\lambda)(s,\,r)} & = & \int_r^\infty 4\pi\eta\,(\rho+p)(s, \eta)
\,e^{(\mu+\lambda)(s,\,\eta)}e^{2\lambda(s,\,\eta)}\,d\eta
\\ & \le & 2c_1\int_r^\infty 4\pi\eta\,|j(s, \eta)|
\,e^{(\mu+\lambda)(s,\,\eta)}e^{2\lambda(s,\,\eta)}\,d\eta
\\ & \le & -2c_1 R_1\int_r^\infty 4\pi j(s, \eta)
\,e^{(\mu+\lambda)(s,\,\eta)}e^{2\lambda(s,\,\eta)}\,d\eta
\\ & = & -2c_1 R_1\hat{\mu}_t(s, r),\end{aligned}$$ since $j(s, \eta)\neq 0$ implies $\eta\le R_1$. Thus $$\label{mut}
\hat{\mu}_t(s, r)\le
-\frac{1}{2c_1 R_1}\Big(1-e^{(\mu+\lambda)(s,\,r)}\Big).$$ This in turn implies that $$\begin{aligned}
\label{est}
\lefteqn{\hat{\mu}(t, \gamma^{\pm}(t))-\hat{\mu}(0, \gamma^{\pm}(0))}\nonumber
\\ & = & \int_{0}^t\frac{d}{ds}\,\hat{\mu}(s, \gamma^{\pm}(s))\,ds\nonumber
\\ & = & \int_{0}^t\Big(\hat{\mu}_t(s, \gamma^{\pm}(s))
\pm\hat{\mu}_r(s, \gamma^{\pm}(s))e^{(\mu-\lambda)(s,\,\gamma^{\pm}(s))}\Big)\,ds\nonumber
\\ & \le & \int_{0}^t\Big(-\frac{1}{2c_1 R_1}\Big(1-e^{(\mu+\lambda)(s,\,\gamma^{\pm}(s))}\Big)
\pm\frac{m(s,
\gamma^{\pm}(s))}{\gamma^{\pm}(s)^2}\,e^{(\mu+\lambda)(s,\,\gamma^{\pm}(s))}\Big)\,ds
\nonumber
\\ & \le & -\frac{t}{2c_1 R_1}
+\int_{0}^t\Big(\frac{1}{2c_1 R_1}
+\frac{m(s, \gamma^{\pm}(s))}{\gamma^{\pm}(s)^2}\Big)\,e^{(\mu+\lambda)(s,\,\gamma^{\pm}(s))}\,ds.\end{aligned}$$ Now for any $r\in [r_0,r_1]$ and $t\in [0,T[$ it follows from $\hat{\mu}_r\ge 0$ and (\[e2lamb\]) that $$\hat{\mu}(t,r)\leq \hat{\mu}(t,R_1)
=-\int_{R_1}^{\infty}\frac{M\,d\eta}{\eta^2(1-2M/\eta)}.$$ Using $M=r_1/2$ we get $$\hat{\mu}(t,R_1)=\frac{1}{2}\log\Big(\frac{R_1-r_1}{R_1}\Big),$$ so that for $r\in [r_0,r_1],$ $$\label{emuhat}
e^{\hat{\mu}(t,\,r)}\leq e^{\hat{\mu}(t,R_1)}=\sqrt{\frac{R_1-r_1}{R_1}}.$$ By (\[mattergeodesic\]) and the properties of the initial matter distribution there is vacuum in the region $\gamma^+(t) \leq r\leq\gamma^-(t)$. Hence $m(t, r)=M-M_\mathrm{out}$ and (\[icnts\]) imply that $$\label{explambdaless3}
e^{\lambda(t,r)}\leq\frac{1}{\sqrt{1-2(M-M_\mathrm{out})/r_0}}<3$$ for $\gamma^+(t) \leq r\leq\gamma^-(t)$. From Lemma \[hatmu-lem\](b) and (\[smallstrip\]), recalling $\kappa=2c_1$, we obtain in particular that $$e^{(\mu+\lambda)(s,\gamma^{\pm}(s))}\leq
e^{(\hat{\mu}+\lambda)(s,\gamma^{\pm}(s))} <
\min\left\{\frac12,\frac{r_0^2}{8c_1R_1M}\right\} =: d.$$ Thus (\[est\]) yields $$\begin{aligned}
\hat{\mu}(t, \gamma^{\pm}(t))-\hat{\mu}(0, \gamma^{\pm}(0))
& \le & -\frac{t}{2c_1 R_1}
+d\int_{0}^t\Big(\frac{1}{2c_1 R_1}+\frac{M}{r_0^2}\Big)\,ds
\\ & = & -\bigg(\frac{1-d}{2c_1 R_1}-d\,\frac{M}{r_0^2}\bigg)t
\\ & \le & -\bigg(\frac{1}{4c_1 R_1}-d\,\frac{M}{r_0^2}\bigg)t
\\ & \le & -\frac{t}{8c_1 R_1},\quad t\in [0, T[.\end{aligned}$$ Hence Lemma \[hatmu-lem\](a) leads to the estimate $$\begin{aligned}
\label{2step}
|\gamma^{\pm}(t)-\gamma^{\pm}(0)|
& = & \bigg|\int_{0}^t\,e^{(\mu-\lambda)(s,\,\gamma^{\pm}(s))}\,ds\bigg|
\le\int_{0}^t\,e^{\hat{\mu}(s,\,\gamma^{\pm}(s))}\,ds
\nonumber\\ & \le & e^{\hat{\mu}(0,\,\gamma^{\pm}(0))}
\int_{0}^t\,e^{-\frac{s}{8c_1 R_1}}\,ds\le 8c_1 R_1 \sqrt{\frac{R_1-r_1}{R_1}},
\nonumber\end{aligned}$$ where we used (\[emuhat\]) in the last inequality. By the third condition in (\[smallstrip\]), $$\sqrt{\frac{R_1-r_1}{R_1}}<\frac{r_1-r_0}{16 c_1R_1},$$ so that $$|\gamma^{\pm}(t)-\gamma^{\pm}(0)|<\frac{r_1-r_0}{2},\quad t\in [0, T[.$$ Since $\gamma^-(0)-\gamma^+(0)=r_1-r_0$, this implies that $\gamma^-(T)-\gamma^+(T) > 0$. Hence, if we choose $T$ in (\[fg-esti\]) to be maximal, then $T=\infty$, i.e., $\gamma^+$ and $\gamma^-$ do never intersect. This completes the proof of Theorem \[genmat\].
$\Box$
[**Remark.**]{} In the above proof we have obtained the more explicit information that $$\label{precise}
\lim_{s\to \infty}\gamma^+(s) < \frac{r_0+r_1}{2},
\quad m(s,\gamma^+(s))= M-M_\mathrm{out},\; s\geq 0,$$ the latter since all the matter originally to the right of $\gamma^- (s) > \gamma^+(s)$ necessarily stays there.
Proof of Theorem \[vlasov2\] {#secvlasov2}
============================
We first check that the (DEC), (NNP), and (GLO) conditions hold for Vlasov matter. Then we show that there exists a class of initial data such that the corresponding solutions satisfy the (GCC) condition with $c_1=3$. Hence Theorem \[vlasov2\] will follow from Theorem \[genmat\].
The characteristic system associated to the Vlasov equation (\[vlasov\]) is $$\begin{aligned}
\frac{dR}{ds} & = & e^{(\mu-\lambda)(s,\,R)}\,\frac{W}{E}, \label{char1}
\\[1ex] \frac{dW}{ds} & = & -\lambda_t(s, R)W-e^{(\mu-\lambda)(s,\,R)}\mu_r(s, R)E
+e^{(\mu-\lambda)(s,\,R)}\frac{L}{R^3 E}, \label{char2}
\\[1ex] \frac{dL}{ds} & = & 0. \label{char3}\end{aligned}$$ If $s\mapsto (R, W, L)(s)$ is a solution with data $(R, W, L)(0)=(r, w, L)$, then $$f(s, R(s), W(s), L)=\open{f}(r,w, L)$$ is constant in $s$. Hence $(R(s), W(s), L)\in \supp f(s)$ iff $(r,w,L) \in \supp \open{f}$. Such characteristics will be addressed as characteristics in $\supp f$.
Direct inspection of the definition in (\[p\]) shows that (NNP) holds for Vlasov matter. It is moreover well-known that the (DEC) condition is satisfied for Vlasov matter; see [@And05 Sec. 1.4]. Alternatively, we can check (\[nnrho\]) and (\[mattergeodesic\]) directly. The latter follows from (\[char1\]) above, whereas the former is a consequence of the expressions for the matter terms given in (\[rho\]), (\[p\]), and (\[j\]).
To see that the (GLO) condition holds for any regular initial data set we argue as follows. First of all, a regular initial data launches a local-in-time solution on some time interval $[0,T[$, and the corresponding theorems in [@RR1] or [@Rein95] also give a condition under which this local solution can be extended to a global one. In order to see that the local solution can always be extended to the whole outer domain $D$ we first observe that the spherically symmetric Einstein-Vlasov system on $D$, with (\[e2lamb\]) and (\[m-def\]) replacing the usual boundary condition of a regular centre and with (\[gamma+\]) included, has again a well-posed initial value problem for regular data supported in $]r_0,\infty[$. This can be shown in the same way as for the system on the whole space, the essential point being that no characteristic of the Vlasov equation can enter region $D$ at the boundary $r=\gamma^+(t)$. To the local solution on $D$ we can now apply the arguments from [@RRS] and conclude that the solution exists on all of $D$. This is possible due to the fact that the estimates in [@RRS] address a situation where matter is bounded away from the centre or is controlled in a neighborhood of the centre so that these estimates can be applied on $D$. We emphasize that for our present analysis only the behaviour of the solution on $D$ plays a role. We have chosen to present our results in the form that we have Vlasov matter also inside $r<\gamma^+(t)$, and this part of the solution may or may not break down, but this is irrelevant for our arguments.
Hence it remains to show that the (GCC) condition holds. To this end we let $0<r_0<r_1<R_1$, $R_0=(r_1+R_1)/2$, and $M=r_1/2$. For a parameter $W_-<0$ to be specified below and regular data $\fn$ with ADM mass $M$ we formulate the following
[**General support condition:**]{} For all $(r,w,L) \in \supp \fn\,$ the following holds: $$r \in ]0,r_0] \cup [R_0,R_1],$$ and if $r\in [R_0,R_1]$ then $$w \leq W_-$$ and also $$\label{hypoL}
0< L <\frac{3L}{\eta}\,\open{m}(\eta) +\eta\,\open{m}(\eta),\ \eta\in [r_0,R_1].$$ We use the notation $\open{m}$ when $\rho=\open{\rho}\,$ in (\[m-def\]). Furthermore, we abbreviate $$\label{Gammadef}
\Gamma = \Gamma(r_1,R_1) := \sqrt{\frac{R_1-r_1}{R_1+r_1}}.$$ The following lemma shows that if the support condition holds, then the particles in the outer domain $D$ keep moving inward in a controlled way.
\[ingoinglemma\] Let $\fn$ be regular and satisfy the general support condition for some $W_-<0$. Then for all $(r,w,L) \in \supp f(t)$ such that $(t, r)\in D$, $$w \leq \Gamma(r_1,R_1) W_-.$$ In particular, $j\leq 0$ on $D$.
**Proof:** Let $[0, T[$ denote the maximal time interval such that for $t < T$ $$\label{bootcd}
w < 0 \ \mbox{for}\ (r,w,L)\in\supp f(t)\ \mbox{with}\
(t,r) \in D.$$ Since $W_-<0$, $T>0$ by continuity. By the definition of $j$, $$\label{jle0}
j(t, r)\le 0 \ \mbox{for}\ (t,r)\in D_T:=D \cap ([0,T[\times [0,\infty[).$$ Let $(R, W, L)(s)$ be a characteristic in $\supp f$. Then $$\begin{aligned}
\frac{d}{ds}(e^{-\lambda}W) & = &
-\,e^{-\lambda}\Big(W\lambda_t+W\lambda_r\frac{dR}{ds}-\frac{dW}{ds}\Big)
\\ & = & \frac{4\pi R}{E}\,e^{\mu}(2WEj-W^2\rho-E^2 p)
+e^{\mu}\Big(1-\frac{2m}{R}\Big)\frac{L}{R^3 E}
\\ & & +\,e^{\mu}\frac{m}{R^2}\,\Big(\frac{w^2}{E}-E\Big)
\\ & = & -\,\frac{4\pi^2}{R}\,e^{\mu}\,\int_{-\infty}^\infty\int_0^\infty
\bigg[\sqrt{\frac{\tilde{E}}{E}}\,w
-\sqrt{\frac{E}{\tilde{E}}}\,\tilde{w}\bigg]^2\,f\,d\tilde{L}\,d\tilde{w}
\\ & & -\,e^{\mu}\frac{m}{R^2}\bigg(\frac{1+L/R^2}{E}+\frac{2L}{R^2 E}\bigg)
+e^{\mu}\frac{L}{R^3 E}\,,\end{aligned}$$ where $E=E(R, W, L)$ and $\tilde{E}=\tilde{E}(R, \tilde{w},
\tilde{L})$. Therefore $$\frac{d}{ds}(e^{-\lambda}W)\le -e^{\mu}\frac{m}{R^2}\bigg(\frac{1+L/R^2}{E}
+\frac{2L}{R^2 E}\bigg)+e^{\mu}\frac{L}{R^3 E}.$$ Differentiating (\[e2lamb\]) w.r.t. $t$ and using (\[ein3\]) leads to $m_t=-4\pi r^2 e^{\mu-\lambda} j$, which by (\[jle0\]) is non-negative on $D_T$. It follows that $m(s, r)\ge m(0, r)=\open{m}\,(r)$. Thus as long as the characteristic remains in $D_T$, $$\begin{aligned}
\frac{d}{ds}(e^{-\lambda}W)
& \le & -e^{\mu}\frac{\open{m}\,(R)}{R^2}\bigg(\frac{1+L/R^2}{E}
+\frac{2L}{R^2 E}\bigg)+e^{\mu}\frac{L}{R^3 E}
\\ & = & e^{\mu}\,\frac{1}{R^3 E}\bigg(L-\frac{3L}{R}\,\open{m}\,(R)
-R\,\open{m}\,(R)\bigg).\end{aligned}$$ Now $R(0)\in [R_0, R_1]$ and $\dot{R}(s)\le 0$ by (\[char1\]) and (\[bootcd\]) yields $R_1\ge R(0)\ge R(s)\geq \gamma^+(s) \geq r_0$. Hence condition (\[hypoL\]) implies that, as long as the characteristic remains in $D_T$, $\frac{d}{ds}(e^{-\lambda}W)<0$, so that $$W(s)\le e^{\lambda(s,\,R(s))-\lambda(0,\,R(0))}\,W_- .$$ But $\lambda\ge 0$, so $W_-<0$ leads to $$W(s)\le\Big(\min_{r\in [R_0, R_1]} e^{-\lambda(0,\,r)}\Big)\,W_-.$$ In view of (\[e2lamb\]), $$e^{-\lambda(0,\,r)}\geq \sqrt{1-\frac{2M}{R_0}}=\sqrt{\frac{R_1-r_1}{R_1+r_1}},
\quad r\in [R_0,R_1],$$ and recalling (\[Gammadef\]) it follows that $$W(s)\leq \Gamma(r_1,R_1) W_-<0$$ as long as the characteristic remains in $D_T$. By the maximality of $T$ in (\[bootcd\]), $T=\infty$, and the proof is complete.
$\Box$
In order to specify the initial data set ${\cal{I}}_2$, let $r_0,\, r_1,\, M$, and $M_\mathrm{out}$ be given as in Section \[secvlasres\] and let $R_1$ be such that (\[smallstrip\]) holds for $\kappa=6$. We require that $W_-<0$ satisfies the estimate $$\label{condw}
\Gamma(r_1,R_1)\, |W_-|\geq 1.$$ Then $$\begin{aligned}
\label{I2def}
{\cal{I}}_2 := \Bigl\{ \fn
&\mid&
\fn \ \mbox{is regular, satisfies (\ref{checkM}), (\ref{M-checkM}),
the general support condition,}\nonumber \\
&&
\mbox{and for}\ (r,w,L)\in \supp \fn\ \mbox{with}\ r\in [R_0,R_1],
\sqrt{L}/r_0 \leq \Gamma\, |W_-|\Bigr\}.\nonumber \\
&&
\ \label{condL1}\end{aligned}$$ Consider now a solution $f$ launched by initial data from this set. Condition (\[condw\]) and Lemma \[ingoinglemma\] imply that $$\label{condw2}
|w|\geq\Gamma(r_1,R_1)\,|W_-|\geq 1\quad\mbox{on}\quad\supp f \cap D,$$ and since $L$ is conserved along characteristics, (\[condL1\]) leads to $\sqrt{L}/r \leq \sqrt{L}/r_0 \leq |w|$ for all particles in $\supp f \cap D$. Hence the definition (\[rho\]) of $\rho$ implies that on $D$, $$\begin{aligned}
\rho(t,r)
&\leq&
\frac{\pi}{r^2}\int_{-\infty}^{\infty}\int_0^{\infty} f\,dL\,dw
+ \frac{\pi}{r^2}\int_{-\infty}^{\infty}\int_0^{\infty} |w|f\,dL\,dw\nonumber\\
&&
{}+
\frac{\pi}{r^2}\int_{-\infty}^{\infty}\int_0^{\infty} \sqrt{L}/r f\,dL\,dw\nonumber\\
&\leq&
3\,\frac{\pi}{r^2}\int_{-\infty}^{\infty}\int_0^{\infty} |w|f\,dL\,dw =3\,|j(t,r)|.\end{aligned}$$ Accordingly, ${\cal{I}}_2$ satisfies the (GCC) condition with $c_1=3$, and Theorem \[vlasov2\] follows from Theorem \[genmat\].
$\Box$
We briefly show that the set ${\cal{I}}_2$ is far from empty. Therefore fix $0<r_0<r_1<R_0<R_1$, $M=r_1/2$, and $0<M_\mathrm{out}<M$ such that $R_0=(r_1+R_1)/2$, (\[icnts\]), and (\[smallstrip\]) are satisfied. Let $0\leq f_1\in C^1$ have $r$-support in $[r_0-\delta,r_0]$ for some $0<\delta<r_0/9$, and let $0\leq f_2\in C^1$ have $r$-support in $[R_0, R_1]$. Fix the compact $w$-support of $f_2$ in $]-\infty,W_{-}]$ with $W_-<0$ such that (\[condw\]) holds, and fix its $L$-support in $[0, L_2]$ so that $$\frac{\sqrt{L_2}}{r_0} \leq \Gamma(r_1,R_1)\,|W_-|$$ and $$L <(M-M_\mathrm{out})\Big(\frac{3L}{\eta}+\eta\Big),\quad L\in[0, L_2],
\quad\eta\in [r_0,R_1].$$ Now take $\open{f}=A f_1+B f_2,$ where $A>0$ and $B>0$ are chosen such that (\[checkM\]) and (\[M-checkM\]) are satisfied. Note that $\open{m}(\eta)\ge M-M_\mathrm{out}$ for $\eta\in [r_0, R_1]$, whence (\[hypoL\]) holds as well; thus the general support condition if verified. It remains to check (\[notsinit2\]). If $r\in ]0, r_0-\delta]$, then $\open{m}(r)=0$. If $r\in [r_0-\delta, R_0]$, then $\open{m}(r)\le M-M_\mathrm{out}$ yields in view of (\[icnts\]), $$\frac{2\open{m}}{r}\leq \frac{2(M-M_\mathrm{out})}{r_0-\delta}<1.$$ If $r\in [R_0, \infty[$, then $$\frac{2\open{m}}{r}\leq \frac{2M}{R_0}<1,$$ since $2M=r_1<R_0.$ Hence $\open{f}$ is regular and has all the properties that are required in the definition of ${\cal{I}}_2$.
[**Remark.**]{} The set ${\cal{I}}_2$ has “non-empty interior”, in the sense that sufficiently small perturbations of initial data in the “interior” of this set belong to ${\cal I}_2$ as well, provided that the support is changed very little and $M$ is left invariant. This is due to the fact that the various parameters entering into the definition of ${\cal{I}}_2$ are defined in terms of inequalities and hence can be varied.
Proof of Theorem \[vlasov1\] {#secvlasov1}
============================
The set up is closely related to the set up in the proof of Theorem \[vlasov2\]. As we saw above, the (DEC), (NNP), and (GLO) conditions are satisfied for Vlasov matter, and we will again construct an initial data set such that the (GCC) condition holds with $c_1=3$. However, since this result relies on condition (\[mediumstrip\]) instead of (\[smallstrip\]), we cannot simply invoke Theorem \[genmat\] after the (GCC) condition has been verified; instead an additional step needs to be added to the proof. For this new argument a slightly stronger condition on the momentum variable $w$ needs to be imposed on $\supp \fn$. We now require that $W_-<0$ satisfies $$\label{condwt1}
\Gamma(r_1,R_1)^2 |W_-|^2\geq \frac{10}{d},$$ where $$d:=\min\left\{\frac12,\frac{r_0}{12 R_1},\frac{r_1-r_0}{300 R_1}\right\}.$$ Then $$\begin{aligned}
\label{I1def}
{\cal{I}}_1 := \Bigl\{ \fn
&\mid&
\fn \ \mbox{is regular, satisfies (\ref{checkM}), (\ref{M-checkM}),
the general support condition,}\nonumber \\
&&
\mbox{and for}\ (r,w,L)\in \supp \fn\ \mbox{with}\ r\in [R_0,R_1],
\sqrt{L}/r_0 \leq 1.
\Bigr\}
\ \label{condL1t1}\end{aligned}$$ The same construction as at the end of the previous section shows that this set is not empty, and the same remark as at the end of the previous section applies.
Let $f$ be a solution launched by initial data from ${\cal{I}}_1$. It is clear from these conditions that Lemma $\ref{ingoinglemma}$ applies, and since $10/d\geq 1$, it follows that (\[condw2\]) holds as well. Thus the argument leading to $\rho\leq 3|j|$ on $D$ in the proof of Theorem \[vlasov2\] applies again. Hence, the (GCC) condition is satisfied with $c_1=3$.
Consider the expression $$\rho(s, r)-p(s, r)=
\frac{\pi}{r^2}\int_{-\infty}^\infty \int_0^\infty
\Big(E-\frac{w^2}{E}\Big) \,f(s, r, w,L)\,dL\,dw.$$ Since $E^2\ge w^2\ge\Gamma^2(r_1,R_1)\,W_-^2$ by Lemma \[ingoinglemma\], we get for $r\in [\gamma^+(s), R_1]$ from $\sqrt{L}/r_0\le 1$, $$\label{c0-def}
E-\frac{w^2}{E}=\frac{1}{E}\,(E^2-w^2)=\frac{1}{E}\,\Big(1+\frac{L}{r^2}\Big)
\le\frac{2}{E}\le\frac{2}{\Gamma^2\,W_-^2}\,E=:c_0 E,$$ so that $$\label{c0}
\rho(s, r)-p(s, r)\le c_0\rho(s, r).$$ After this preparation, we again show that the out- and ingoing null geodesics $\gamma^+$ and $\gamma^-$ do not intersect. We choose $T\in ]0,\infty[$ such that (\[fg-esti\]) holds. In this case we cannot rely on the smallness of $e^{\hat{\mu}}$ as in the proof of Theorem \[genmat\], so we need to control the evolution also when $e^{\hat{\mu}}$ is not small. For this part the estimate (\[c0\]) is essential.
We fix $t_\ast^{\pm}\in [0, T[$ by requiring that $$e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))} > d\,\,\mbox{for}\,\,
s\in [0, t_\ast^{\pm}[,\quad
e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))}\le d\,\,\mbox{for}\,\,
s\in [t_\ast^{\pm}, T[.$$ First we note that $t_\ast^{\pm}$ is well-defined, since by Lemma \[gammapm\](c), $$\label{monotone}
\frac{d}{ds}\,e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))}\le 0.$$ [*Step 1:*]{} Consider $s\in [0, t_\ast^{\pm}]$; if $t_\ast^{\pm}=0$, then this step is omitted. For $\eta\ge\gamma^{\pm}(s)$, $$d\le e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))}
\le e^{(\hat{\mu}+\lambda)(s,\,\eta)},$$ since $(\hat{\mu}+\lambda)_r=4\pi r\rho\,e^{2\lambda}\ge 0$ by (\[gsto\]). Hence Lemma \[hatmu-lem\](c) and (\[c0\]) yield $$\begin{aligned}
(\mu-\lambda)(s, \gamma^{\pm}(s)) & = & 2\hat{\mu}(s, \gamma^{\pm}(s))
+\int_{\gamma^{\pm}(s)}^\infty 4\pi\eta\,(\rho-p)(s, \eta)\,e^{2\lambda(s,\,\eta)}\,d\eta
\\ & \le & 2\hat{\mu}(s, \gamma^{\pm}(s))
+\frac{c_0}{d}\int_{\gamma^{\pm}(s)}^\infty 4\pi\eta\,\rho(s, \eta)
\,e^{(\hat{\mu}+\lambda)(s,\,\eta)}e^{2\lambda(s,\,\eta)}\,d\eta
\\ & \le & 2\hat{\mu}(s, \gamma^{\pm}(s))+\frac{c_0}{d},\end{aligned}$$ where for the last estimate Lemma \[int0infty\] has been used.
Now we make the following observation: There is at least one characteristic $(\bar{R},\bar{W},\bar{L})(s)$ with $\bar{R}(0)\in [R_0,R_1],$ which does not leave the strip $[r_1,R_1]$ during the finite time interval $[0, T]$. In fact, if at time $t=T$ all characteristics had left the strip $[r_1, R_1]$ (and thus had entered the region $r<r_1$), then $m(T, r_1)=M$. From (\[e2lamb\]) and $2M=r_1$ it would follow that $\lambda(T,r_1)=\infty$. However, this contradicts the (GLO) condition which holds for Vlasov matter.
Since $\gamma^{\pm}(s)\le r_1\le \bar{R}(s)$, and since $\hat{\mu}_r\geq 0$, we thus obtain in view of Lemma \[hatmu-lem\](a) that $$\begin{aligned}
(\mu-\lambda)(s, \gamma^{\pm}(s))
& \le & 2\hat{\mu}(s,\gamma^{\pm}(s))
+\frac{c_0}{d}\le 2\hat{\mu}(s, \bar{R}(s))+\frac{c_0}{d}
\\ & \le & (\mu-\lambda)(s, \bar{R}(s))+\frac{c_0}{d},
\quad s\in [0, t_\ast^{\pm}].\end{aligned}$$ Next note that $|W|\ge 1$ by (\[condw2\]), and hence due to (\[char1\]) and observing $\bar{R}^2\ge r_0^2\ge L$, $$|\dot{\bar R}|=\frac{|W|}{E}\,e^{\mu-\lambda}
\geq\frac{|W|}{\sqrt{2+W^2}}\,e^{\mu -\lambda}
\geq\frac{1}{2}\,e^{\mu-\lambda}.$$ Therefore for all $t\in [0, t_\ast^{\pm}]$ the estimate $$\begin{aligned}
\label{1step}
|\gamma^{\pm}(t)-\gamma^{\pm}(0)|
& = &
\bigg|\int_0^t\pm\,e^{(\mu-\lambda)(s,\,\gamma^{\pm}(s))}\,ds\bigg|
\le e^{\frac{c_0}{d}}\int_0^t e^{(\mu-\lambda)(s,\,\bar{R}(s))}\,ds\nonumber\\
& \le &
-2e^{\frac{c_0}{d}}\int_0^t\dot{\bar{R}}(s)\,ds
=2e^{\frac{c_0}{d}}(\bar{R}(0)-\bar{R}(t))\nonumber\\
& \le & 2e^{\frac{c_0}{d}}(R_1-r_1)\end{aligned}$$ is obtained.\
[*Step 2:*]{} Let $t\in [t_\ast^{\pm}, T[$; if $t_\ast^{\pm}=T$, then this step is omitted. The arguments here are basically the ones presented in Section \[secgenmatproof\]. The computation leading to (\[est\]) is almost identical, and $$\begin{aligned}
\label{est2}
&&
\hat{\mu}(t, \gamma^{\pm}(t))-\hat{\mu}(t_{\ast}^{\pm},
\gamma^{\pm}(t_{\ast}^{\pm}))
\nonumber\\
&&
\qquad \le -\frac{t-t_{\ast}^{\pm}}{2c_1 R_1}
+\int_{t_{\ast}^{\pm}}^t\Big(\frac{1}{2c_1 R_1}
+\frac{m(s, \gamma^{\pm}(s))}{\gamma^{\pm}(s)^2}\Big)\,
e^{(\mu+\lambda)(s,\,\gamma^{\pm}(s))}\,ds\end{aligned}$$ for $c_1=3$. By Lemma \[hatmu-lem\](b), $e^{(\mu+\lambda)(s,\,\gamma^{\pm}(s))}
\le e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))}\le d$. Next we use the facts that $m/r<1/2,\; \gamma^{\pm}(s)\ge r_0$, and the definition of $d$ to obtain the estimate $$\begin{aligned}
\hat{\mu}(t, \gamma^{\pm}(t))-\hat{\mu}(t_\ast^{\pm}, \gamma^{\pm}(t_\ast^{\pm}))
& \le & -\frac{1}{2c_1 R_1}(t-t_\ast^{\pm})
+d\int_{t_\ast^{\pm}}^t\Big(\frac{1}{2c_1 R_1}+\frac{1}{2r_0}\Big)\,ds
\\ & = & -\bigg(\frac{1-d}{2c_1 R_1}-d\,\frac{1}{2r_0}\bigg)(t-t_\ast^{\pm})
\\ & \le & -\bigg(\frac{1}{4c_1 R_1}-d\,\frac{1}{2r_0}\bigg)(t-t_\ast^{\pm})
\\ & \le & -\frac{1}{8c_1 R_1}(t-t_\ast^{\pm}),\quad t\in [t_\ast^{\pm}, T[.\end{aligned}$$ Hence by Lemma \[hatmu-lem\](a), $$\begin{aligned}
\label{22step}
|\gamma^{\pm}(t)-\gamma^{\pm}(t_\ast^{\pm})|
& = & \bigg|\int_{t_\ast^{\pm}}^t\,e^{(\mu-\lambda)(s,\,\gamma^{\pm}(s))}\,ds\bigg|
\le\int_{t_\ast^{\pm}}^t\,e^{\hat{\mu}(s,\,\gamma^{\pm}(s))}\,ds
\nonumber\\ & \le & e^{\hat{\mu}(t_\ast^{\pm},\,\gamma^{\pm}(t_\ast^{\pm}))}
\int_{t_\ast^{\pm}}^t\,e^{-\frac{(s-t_\ast^{\pm})}{8c_1 R_1}}\,ds
\nonumber\\ & \le & e^{(\hat{\mu}+\lambda)(t_\ast^{\pm},\,\gamma^{\pm}(t_\ast^{\pm}))}
\int_{t_\ast^{\pm}}^\infty\,e^{-\frac{(s-t_\ast^{\pm})}{8c_1 R_1}}\,ds\le 8c_1 R_1 d.\end{aligned}$$ Adding the contributions (\[1step\]) from Step 1 and (\[22step\]) from Step 2, the final estimate $$|\gamma^{\pm}(t)-\gamma^{\pm}(0)|\le 2e^{c_0/d}(R_1-r_1)
+8c_1 R_1 d$$ is obtained for all $t\in [0, T[$. From (\[c0-def\]) and (\[condwt1\]) we have $c_0/d\leq 1/5$. The third condition on $d$ together with (\[mediumstrip\]) thus imply that $$|\gamma^{\pm}(t)-\gamma^{\pm}(0)|<\frac{r_1-r_0}{2}.$$ As in the proof of Theorem \[genmat\] we conclude that $\gamma^+$ and $\gamma^-$ do not intersect, completing the proof of Theorem \[vlasov1\].
$\Box$
[**Remarks.**]{} (a) The sharper estimates stated in (\[precise\]) clearly hold also in this case.\
(b) The solution must necessarily enter the regime of Step 2, more precisely, $$\lim_{s\to \infty} e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))} = 0$$ for both null geodesics. Otherwise, the monotonicity implied by Eqn. (\[monotone\]) yields a positive constant $c>0$ such that $e^{(\hat{\mu}+\lambda)(s,\,\gamma^{\pm}(s))}>c$ for all time, and hence, $$|\dot \gamma^\pm| = e^{\mu - \lambda} =
e^{\hat{\mu} + \lambda}e^{\check{\mu} - 2\lambda}
> c e^{\check{\mu} -2\lambda}.$$ Since no matter can cross the two null geodesics, $$\begin{aligned}
(\check{\mu} -2\lambda)(s,r)
&=&
\int_r^\infty 4 \pi \eta(2 \rho - p)e^{2\lambda} d\eta + 2\hat{\mu}(s,r)\\
&\geq&
2\hat{\mu}(s,r) = -2 \int_r^\infty \frac{\open{m}(r_0)}{\eta^2}
\frac{1}{1-2\open{m}(r_0)/\eta}\,d\eta\\
&=&
\ln \frac{r-2\open{m}(r_0)}{r}\end{aligned}$$ for $r=\gamma^\pm(s)$. If we insert this into the estimate for $\dot \gamma^\pm$ it follows that this quantity is bounded from below by a positive constant which contradicts the finite limits of $\gamma^\pm(s)$ as $s\to \infty$.
It remains to prove Cor. \[ssinthemiddle\].
[**Proof of Corollary \[ssinthemiddle\]:**]{} Let $f_s$ be a static solution. By [@And2], $2m_s(r)/r < 8/9$ for $r>0$ where $m_s$ is the local ADM mass induced by $f_s$. In particular, $M_s < r_s/2 < r_1/2 = M$, and (\[icnts\]) holds. As described above we can now specify the matter distribution for $r\geq r_0$, and we obtain initial data $\open{f}$ in ${\cal{I}}_1$ or in ${\cal{I}}_2$ which coincide with the given static solution for $0\leq r\leq r_0$.
Since no matter travels from the outer domain $D$ to the inner one where $r\leq \gamma^+(t)$, the only way the matter in the outer domain can affect the static solution is through the metric. Consider the time-independent version of the Vlasov equation (\[vlasov\]). Dropping all the time derivatives we see that in the remaining equation the factor $e^{\lambda - \mu}$ can be canceled. Therefore, the static Einstein-Vlasov system is formulated in terms the quantities $f, \lambda$, and $\mu_r$, but not $\mu$ itself. By (\[e2lamb\]) and (\[mur\]), $\lambda$ and $\mu_r$ are on $r\leq \gamma^+(t)$ not affected by the matter in the outer domain $D$, and therefore $f=f_s, \lambda, \mu_r$ remain time-independent for $r\leq \gamma^+(t)$.
$\Box$
Notice that the metric coefficient $\mu$ of course does change on the interior region, cf. Thm \[bh\] (b).
Proof of Theorem \[bh\] {#bhproof}
=======================
As a first step we estimate $\mu -\lambda$ from below for $r > 2 M$, using Lemma \[hatmu-lem\] (a): (-)(t,r) && 2(t,r) = -2 \_r\^ e\^[2(t,)]{}d\
&=& -2 \_r\^d-2 \_r\^d\
&=& , r> 2 M. Now consider any characteristic in the matter support, and let $R(t)$ denote its radial coordinate. Then by Lemma \[ingoinglemma\] and as long as $R(t) > 2 M$, $$\frac{dR}{ds} = e^{(\mu-\lambda)(s,R)}\frac{W}{E} \leq - C e^{(\mu-\lambda)(s,R)}
\leq - C \frac{R-2M}{R};$$ for initial data from the set ${\cal I}_1$ respectively ${\cal I}_2$ one can take $C:=\Gamma |W_-|/\sqrt{2+\Gamma^2 W_-^2}$ respectively $C:=1/\sqrt{3}$. Integrating this differential inequality we find that as long as $R(t) > 2 M$ the estimate -C t && \_[R(0)]{}\^[R(t)]{} dr = R(t) - R(0) + 2 M\
&& 2 M - R\_1 + 2 M holds, and hence $$R(t) \leq 2 M + (R_1 - 2 M)e^{\frac{1}{2 M}(R_1 - 2 M - C t)},$$ which proves the support estimate in part (a). Since all the matter, which has ADM mass $M$, is contained in the region where $r \leq 2 M + \alpha e^{-\beta t}=: \sigma(t)$, the assertion on the metric follows. Moreover, for any $r\leq \sigma(t)$ the monotonicity of $\mu$ with respect to $r$ implies that $$\mu(t,r) \leq \ \mu(t,\sigma(t)) = \hat\mu(t,\sigma(t)) =
\ln \left(\frac{\sigma(t) -2 M}{\sigma(t)}\right)^{1/2},$$ which is the first assertion of part (b). The second follows immediately since the integral $\int_0^\infty e^{\mu(t,r)}dt$ is the proper length of a coordinate line of constant $r,\theta$, and $\varphi$ in the outer region $D$. This completes the proof of part (b). As to (c) we first observe that any radially outgoing null geodesic which enters the region $r > 2 M$ escapes to $r=\infty$ and is future complete, since by part (a) the metric on $r>2 M+ \epsilon$ where $\epsilon >0$ is arbitrary eventually equals the Schwarzschild one for which the asserted properties of the geodesics hold. Now consider the extremal geodesic $\gamma^\ast$. If there existed some time $t>0$ such that $\gamma^\ast(t) > 2 M$, then by continuous dependence on the initial data the same would be true for all radially outgoing null geodesics with $\gamma(0)$ sufficiently close to but less than $r^*$. Hence such geodesics would escape to $r=\infty$ in contradiction to the definition of $r^\ast$. This shows that the extremal, radially outgoing null geodesic $\gamma^\ast$ has the property that $\lim_{t\to\infty} \gamma^\ast(t) \leq 2 M$.
It remains to show that the limit above cannot be strictly less than $2 M$. To this end we consider a radially outgoing null geodesic as long as $\gamma(t) < \sigma(t) = 2 M + \alpha e^{-\beta t}$. Then $$\frac{d\gamma}{ds} =
e^{(\mu-\lambda)(s,\gamma(s))} \leq e^{\mu(s,\sigma(s))}
=
\left(\frac{\sigma(s) -2 M}{\sigma(s)}\right)^{1/2}
\leq
C e^{-\beta s/2},$$ and hence for any $0\leq t_0 \leq t$ and as long as $\gamma(t) < \sigma(t)$, $$\gamma(t) \leq \gamma(t_0) + C e^{-\beta t_0/2},$$ where the constant $C>0$ again depends only on the initial data set. Now assume that $R^\ast := \lim_{t\to \infty}\gamma^\ast(t) < 2 M$, choose $t_0>0$ such that $R^\ast + C e^{-\beta t_0/2} < 2 M$, and consider the radially outgoing null geodesic $\gamma^{\ast\ast}$ with $\gamma^{\ast\ast}(t_0) = R^\ast$. Then by construction, $\gamma^{\ast\ast}(t) < 2 M < \sigma(t)$ for all $t\geq t_0$, and since $\gamma^{\ast\ast}(t_0) = R^\ast > \gamma^{\ast}(t_0)$ it follows that $\gamma^{\ast\ast}(0) > \gamma^{\ast}(0)=r^\ast$. Hence $\gamma^{\ast\ast}$ is a radially outgoing null geodesic which at time $t=0$ starts to the right of $r^\ast$ and does not escape to $r=\infty$. This is in contradiction to the definition of $r^\ast$.
$\Box$
We conclude this section by proving the remark after Theorem \[genmat\]. Under our general matter conditions the matter is ingoing in the region $D$, in particular, the matter is for all time restricted to the region where $r\leq R_1$. Hence for $r\geq R_1$ the metric is again equal to the Schwarzschild one with mass $M$, and if we replace $2 M$ by $R_1$ in the above argument for part (c) we obtain the assertions on $\gamma^\ast$ in the general matter context. As to the divergence of $\mu$ we observe that $$\frac{d}{ds} \hat \mu (s,\gamma^-(s)) =
\hat \mu_t (s,\gamma^-(s)) + \hat \mu_r(s,\gamma^-(s))\frac{d\gamma^-}{ds} (s) \leq 0$$ so that the limit $
\hat \mu_\infty := \lim_{s\to \infty}\hat \mu (s,\gamma^-(s))
$ exists. The fact that $\gamma^-(s)>0$ is decreasing with $$\left| \frac{d\gamma^-}{ds} (s)\right| = e^{(\mu-\lambda)(s,\gamma^-(s))}
\geq e^{2 \hat\mu (s,\gamma^-(s))} \geq e^{2 \hat\mu_\infty},$$ implies that $\hat\mu_\infty = -\infty$. Since $\mu \leq \hat\mu$ we conclude that $$\lim_{s\to \infty}\mu (s,\gamma^-(s)) = -\infty,$$ from which the assertion follows by the monotonicity of $\mu$ with respect to $r$. [**Acknowledgement:**]{} The authors are grateful for discussions with A. Rendall.
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| {
"pile_set_name": "ArXiv"
} |
{width="2"}
[ [**Center for Sustainable Engineering of Geological and Infrastructure Materials (SEGIM)**]{}\
\[0.1in\] Department of Civil and Environmental Engineering\
\[0.1in\] McCormick School of Engineering and Applied Science\
\[0.1in\] Evanston, Illinois 60208, USA ]{}
\
0.5in
\
\
[**SEGIM INTERNAL REPORT No. 15-09/047A**]{}\
\
[ **Roozbeh Rezakhani [^1] and Gianluca Cusatis [^2]** ]{}
[[**Abstract**]{}: Discrete fine-scale models, in the form of either particle or lattice models, have been formulated successfully to simulate the behavior of quasi-brittle materials whose mechanical behavior is inherently connected to fracture processes occurring in the internal heterogeneous structure. These models tend to be intensive from the computational point of view as they adopt an “a priori” discretization anchored to the major material heterogeneities (e.g. grains in particulate materials and aggregate pieces in cementitious composites) and this hampers their use in the numerical simulations of large systems. In this work, this problem is addressed by formulating a general multiple scale computational framework based on classical asymptotic analysis and that (1) is applicable to any discrete model with rotational degrees of freedom; and (2) gives rise to an equivalent Cosserat continuum. The developed theory is applied to the upscaling of the Lattice Discrete Particle Model (LDPM), a recently formulated discrete model for concrete and other quasi-brittle materials, and the properties of the homogenized model are analyzed thoroughly in both the elastic and inelastic regime. The analysis shows that the homogenized micropolar elastic properties are size-dependent, and they are functions of the RVE size and the size of the material heterogeneity. Furthermore, the analysis of the homogenized inelastic behavior highlights issues associated with the homogenization of fine-scale models featuring strain-softening and the related damage localization. Finally, nonlinear simulations of the RVE behavior subject to curvature components causing bending and torsional effects demonstrates, contrarily to typical Cosserat formulations, a significant coupling between the homogenized stress-strain and couple-curvature constitutive equations.]{}
*Keywords:* Asymptotic Expansion Homogenization; Asymptotic Expansion; Discrete Models; Lattice Models; Cosserat Continuum; Strain Softening; Size Effect.\
Introduction
============
Discrete fine-scale models, in the form of either particle or lattice models, have been formulated successfully in the literature to simulate the behavior of a variety of different materials. Their use has become more and more popular in the last few decades due to a number of appealing properties that make them advantageous compared to continuum based formulations.
The geometry of discrete models is built with reference to the actual internal structure of the material of interest and it consists of “particles” connected through either “contact points” or “connecting struts” (also called “lattice elements”). This “a priori” discretization allows simulating material heterogeneity efficiently in the case of materials - such as concrete, rock, sea-ice, and toughened ceramics - characterized by hard and stiff inclusions embedded in a more compliant, weak, and brittle, matrix. In addition, the intrinsic particle/lattice spacing automatically provides the formulation with an internal characteristic length which can be made randomly variable if the discrete model is constructed according to the actual random distribution of material heterogeneity.
The degrees of freedom (displacements and rotations) are defined only at a finite number of points – referred also as “nodes” thereinafter – which, depending on the formulation, may or may not correspond to the partice center of mass or particle centroid. Strain and stress measures are defined at a finite number of points coinciding with the contact points or with some specified points along the connecting struts. The constitutive behavior is formulated through vectorial, as opposed to tensorial, stress versus strain relationships and stress tractions are supposed to be distributed over either a “contact area” or the cross sectional area of the connecting struts (in this paper, this area will be generically referred to as “facet”). Finally, the classical concepts of equilibrium and compatibility are formulated through algebraic equations, instead of partial differential equations typical of continuum mechanics. One of the main advantages of discrete models is that the discreteness of the formulation permits handling naturally displacement discontinuities arising during damage localization and fracture processes.
Rigid particle models, under the name of Discrete Element Method (DEM), were first formulated to simulate both natural materials, such as geomaterials [@Cundall-1; @Serrano-1; @Cundall-3; @Kawai-1], as well as man-made materials like concrete [@Zubelewicz-3; @Plesha-1; @Zubelewicz-4]. A somewhat similar model is the rigid-body-spring model (RBSM), which subdivides the material domain into rigid polyhedral elements interconnected by zero-size springs [@Kawai-1; @Bolander-1; @Bolander-2; @Bolander-3].
Lattice models, pioneered by Hrennikoff [@Hrennikoff-1] to solve elastic problems in the pre-computers era, were later developed by many authors to model fracture in quasi-brittle materials in both 2D [@Schlangen-1], and 3D [@Cusatis-1; @Cusatis-2; @Lilliu-1; @Berton-1].
More recently, various discrete models, in the form of either lattice or particle models, have been quite successful in simulating concrete materials [@Lilliu-1; @cusatis-ldpm-1; @cusatis-ldpm-2; @Leite-1; @Kim-1]. For an extensive review of the currently available models for concrete the reader is directed to a recent special issue [@CementComposite] collecting several papers covering a wide variety of concrete mechanics phenomena spanning several length scales, from the scale of cement particles to that of reinforced concrete structural members.
In most applications of interest in practice, fine-scale models lead to fairly large computational systems characterized by a huge computational cost making their practical use rather limited. For example, the full-scale computational analysis of an average concrete bridge would require millions of degrees of freedom or the simulation of a rock formation would require billions of degrees of freedom. The solution of such large problems, although possible in principle with large super computer clusters, is unimaginable in everyday engineering practice. For this reason, many studies have been devoted to finding optimal and rigorous approaches for multiscale computation.
Among different multiscale techniques available in the literature [@Galvanetto-1], the ones based on homogenization theory have been widely used over the past decades. The homogenization theory relies on two main assumptions. The first is the existence of a certain volume of material, the so called Representative Volume Element (RVE) or Unit Cell (UC), carrying a complete description of the internal material structure [@Gitman-1; @Kouznetsova-1]. The second is that the size of such a volume is much smaller than the size of the overall solid volume under consideration. The latter is also known as the “scale separation” assumption.
Hill [@Hill-1], Eshelby [@Eshelby-1], Hashin and Strikman [@Hashin-1] pioneered analytical homogenization techniques which were developed later by other authors [@Christensen-1; @Nemat-Nasser-1]. Analytical homogenization is able to reasonably approximate material properties when the exact solution of the boundary value problem associated with the RVE problem can be obtained. However, in this approach, elastic behavior, small strains, and relatively simple internal structure are the limiting assumptions typically adopted. When complicated heterogeneous structures are considered, or constitutive behavior of constituents are nonlinear, other homogenization techniques [@Lopez-Pamies-1; @Lopez-Pamies-2] needs to be considered.
To overcome these difficulties, computational homogenization is often used in the literature [@Smith-1; @Feye-1; @Kouznetsova-1; @Miehe-1]. In this approach, a single RVE is assigned to each calculation point (e.g. gauss point in a Finite Element mesh) in the macro domain and at each step of the nonlinear analysis, macro-strain increments are imposed as essential boundary conditions to the RVE. The solution of the RVE boundary value problem is then averaged for the calculation of the associated macroscopic stress tensor. Since no assumption is made for the macroscopic constitutive law, this method can be used for materials featuring extremely nonlinear behavior.
A somewhat similar but more mathematically rigorous homogenization technique is the so-called Asymptotic Expansion Homogenization (AEH) that uses the asymptotic expansion of the displacement field based on a length parameter representing the ratio between the length scale of material heterogeneity and the macroscopic length scale. Starting from this expansion hierarchical boundary value problems are obtained at different scales. This approach can easily handle problems with multiple (more than 2) scales in both space and time [@Fish-3]; it does not make assumptions on the character of the macroscopic constitutive equations; and its implementation in computer codes is relatively simple.
Within the extensive literature on AEH, remarkable is the work of the following authors. Hassani [@Hassani-1; @Hassani-2] investigated formulation of homogenization theory and topology optimization and its numerical application to materials with periodic microstructure. Chung [@Chung-1] presented detailed derivation of multiple scale formulation for elastic solids. Fish employed this approach to study elastic as well as elasto-plastic composites [@Fish-2]. Ghosh [@Ghosh-1] adopted MH along with Voronoi Cell Finite Element Method (VCFEM) to study the behavior of composites with random meso-structure [@Ghosh-2]. More recently, Fish [@Fish-3] introduced the Generalized Mathematical Homogenization (GMH) to derive continuum constitutive equations starting from Molecular Dynamics (MD).
All the aforementioned work is relevant to Cauchy continuum formulations. However, homogenization schemes were also used for the multiscale analysis of Cosserat continuum models, in which an independent rotation field appears in addition to the displacement field. Feyel [@Feye-1] built a homogenization scheme to couple a Cauchy continuum formulation at the micro-scale giving rise to a Cosserat continuum formulation at the macro-scale. Asymptotic homogenization technique was employed by Forest [@Forest-1] for upscaling elastic Cosserat solids. In this work, the author studied various types of asymptotic expansions for the displacement and rotation fields and investigated their effect on the resulting macroscopic continuum behavior. Results of this investigation, showed that the nature of the homogenized continuum depends on the ratio of the Cosserat characteristic length of constituents, size of heterogeneity and typical size of the structure.
Chan et al. [@Chan-1] derived the governing constitutive equations for strain gradient elasticity for both homogeneous and functionally graded materials using the strain energy density function and the related definitions of the stress fields. They showed that additional terms appear in the equations that are related to the strain gradient nonlocality and the interaction between material nonhomogeneity. Bardenhagen et al. [@Bardenhagen-1] obtained a nonlinear higher order gradient continuum representation of discrete periodic micro-structures by means of an energy approach. The developed model was then employed to investigate the existence and stability of localization bands and their relationship to the model loss of ellipticity. Finally, homogenization of discrete atomic models into equivalent continuum can be found in publications where the authors exploited asymptotic analysis techniques [@Caillerie-1] and the mathematical $\Gamma$-convergence method [@Braides-1].
The present study derives a general multiscale homogenization scheme suitable for upscaling materials whose fine-scale behavior can be successfully approximated through the use of discrete models featuring both translational and rotational degrees of freedom.
The Fine-Scale Problem
======================
With reference to Figure \[TwoScaleAnalysis\]a, let us consider the interaction of two adjacent particles, $I$ and $J$, sharing a generic facet. If one limits the analysis to the case of small strains and displacements – which is a reasonable assumption in absence of large plastic deformation prior to fracture as observed in brittle and quasi-brittle materials – meaningful measures of deformation [@cusatis-ldpm-1] can be defined as
$$\label{eps}
\epsilon^{IJ}_{\alpha}=\frac{1}{r} \left(\mathbf{U}^J + \mb \Theta^{J} \times \mathbf{c}^{J} - \mathbf{U}^I - \mb{\Theta}^I \times \mathbf{c}^I \right) \cdot \mathbf{e}^{IJ}_{\alpha}$$
and
$$\label{curvature}
\chi^{IJ}_{\alpha}=\frac{1}{r} \left( \mb{\Theta}^{J} - \mb{\Theta}^I \right) \cdot \mathbf{e}^{IJ}_{\alpha}$$
where $\epsilon^{IJ}_{\alpha}=$ facet strains; $\chi^{IJ}_{\alpha}=$ facet curvatures; $r=|\mathbf{x}^{IJ}|$; $\mathbf{x}^{IJ}=\mathbf{x}^J-\mathbf{x}^I $ is the vector connecting the particle nodes $P_I$ and $P_J$; $\mathbf{e}^{IJ}_{\alpha}$ ($\alpha=N,M,L$) are unit vectors defining a facet Cartesian system of reference such that $\mathbf{e}^{IJ}_{N}=$ is orthogonal to the facet and $\mathbf{e}^{IJ}_{N} \cdot \mathbf{x}^{IJ} >0$; $\mathbf{U}^I$, $\mathbf{U}^J$ = displacement vectors of node $P_I$ and $P_J$; $\mb{\Theta}^{I}$, $\mb{\Theta}^{J}$ = rotation vectors of node $P_I$ and $P_J$; and $\mathbf{c}^{I}$, $\mathbf{c}^{J}$ = vectors connecting nodes $P_I$ and $P_J$ to the facet centroid, see Fig. \[TwoScaleAnalysis\]a. It must be observed here that displacements and rotations are assumed to be independent variables.
For given strain and curvature vectors, a vectorial constitutive equation provide stress, $\mathbf{t}^{IJ}$, and couple, $\mathbf{m}^{IJ}$, tractions on each facet. Formally one can write $\mathbf{t}^{IJ} = t_{{\alpha}}(\epsilon_{N}, ...) \mathbf{e}^{IJ}_{\alpha}$ and $\mathbf{m}^{IJ} = m_{{\alpha}}(\chi_{N}, ...) \mathbf{e}^{IJ}_{\alpha}$ where, in general, summation rule applies over $\alpha$. As an example, the elastic behavior can be formulated through the following equations
$$\label{elastic}
t_{\alpha} = E_\alpha \epsilon_{\alpha}; \hspace{0.25 in} m_{\alpha} = W_{\alpha} \chi_{\alpha}= E_\alpha \ell_\alpha^2 \chi_{\alpha}; \hspace{0.25 in} (\alpha=N,M,L)$$
in which each traction component is proportional to the associated strain or curvature (summation rule does not apply); and $E_\alpha$, $W_\alpha$ are fine-scale elastic constants which are related by a characteristic length $\ell_\alpha$. An example of nonlinear facet constitutive equations is reported in Appendix \[LDPM\], Section \[LDPM-Constitutive\], with reference to the so-called Lattice Discrete Particle Model (LDPM) that will be considered in the numerical examples. Finally, the computational discrete fine-scale framework is completed by imposing the equilibrium of each single particle subject to the effect of all surrounding particles. Translational and rotational dynamic equilibrium equations read
$$\label{motion-1}
M_u^I\ddot{\mathbf U}^I +\mathbf{M}_{u \theta}^I\ddot{\mb \Theta}^I - V^I \mathbf{b}^0=\sum_{\mathcal{F}_I} A \mathbf{t}^{IJ}$$
and
$$\label{motion-2}
\mathbf{M}_{\theta}^I\ddot{\mb \Theta}^I = \sum_{\mathcal{F}_I} A ( \mathbf{w}^{IJ} +\mathbf{m}^{IJ})$$
where $\mathbf{w}^{IJ} = \mathbf c^I \times \mathbf {t}^{IJ}$ is the moment of the traction $\mathbf {t}^{IJ}$ with respect to the particle node $P_I$; $\mathcal{F}_I$ is the set of facets surrounding node P$_I$ and obtained by collecting all the facets associated with each node pair $(I,J)$; $A$ = facet area; superimposed dots represent time derivatives; $V^I$ is the particle volume; $\mathbf{b}^0$ is the body force vector; $M_u^I$ = mass of node $P_I$; and $\mathbf{M}_{u \theta}^I$, $\mathbf{M}_{\theta}^I$ = moment of inertia tensors. It is worth observing that $\mathbf{M}_{u \theta}^I = \mathbf{0}$ and $\mathbf{M}_{\theta}^I = M_{\theta}^I \mathbf{I}$ if the particle node is the particle center of mass; the axes of the system of reference are parallel to the particle principal axes of inertia; and the principal moments of inertia are the same in all directions. These conditions, although applicable only to a limited number of cases (e.g. spherical particles), do not reduce the conceptual generality of the derivation that will be presented in this paper and will be assumed thereinafter for simplicity.
Asymptotic Expansion Homogenization {#HomogTheory}
=====================================
In this section, the two-scale homogenization of the general fine-scale problem introduced in the previous section is pursued by means of the approach proposed in Ref. [@Fish-3]. In the original formulation only central forces were assumed to act on the particles and, consequently, the rotational equilibrium equation was not considered.
Two Scale Approximation and Asymptotic Expansions
-------------------------------------------------
In order to perform a two-scale asymptotic expansion homogenization, a periodic discrete system, composed by a number of adjacent RVEs, is considered in this section. In Figure \[TwoScaleAnalysis\]b, the generic macroscopic material domain and the corresponding global coordinate system $\bf X$ are shown. At any point in the macroscopic domain, two separate length scales and the corresponding local coordinate systems, $\bf x$ and $\bf y$, are introduced to represent (1) the macroscopic domain, in which the problem is defined as homogeneous continuum with no detail of material heterogeneity, and (2) the meso-scale domain, in which heterogeneity is modeled by the discrete meso-scale model. Vector ${\mathbf}X$, as shown in the figure, is the vector connecting the origin of the global macroscopic coordinate system to the mass center of a generic RVE. In Figure \[TwoScaleAnalysis\]c, a zoomed view of the macroscopic material point is shown in the local meso-scale coordinate system $\bf{y}$, in which a representative volume of heterogeneous material is depicted. One should consider that in Figure \[TwoScaleAnalysis\]a, particles $I$ and $J$ are shown in the local macroscopic coordinate system $\bf x$. Therefore, they should be plotted in smaller size compared to Figure \[TwoScaleAnalysis\]c, but this was not done for the sake of clarity. If the separation of scales exists, one can write the following relationship linking macro and meso local coordinate systems $$\begin{aligned}
\label{scale-link-1}
\mathbf{x}=\eta \mathbf{y}; \hspace{0.25 in} 0< \eta <<1\end{aligned}$$
where $\eta$ is a very small positive scalar. In addition, the displacement of a generic node P$_I$, $\mathbf{U}^I = \mathbf{u}(\mathbf{x}^I, \mathbf{y}^I)$, can be approximated by means of the following asymptotic expansion
$$\mathbf{u}(\mathbf x, \mathbf y) \approx \mathbf u^0(\mathbf x, \mathbf y)+\eta \mathbf u^1(\mathbf x, \mathbf y)
\label{disp-expansion}$$
where only terms up to order $\mathcal{O}(\eta)$ are considered. Functions $\mathbf u^0(\mathbf x, \mathbf y)$, and $\mathbf u^1(\mathbf x, \mathbf y)$ are continuous with respect to $\mathbf{x}$ and discrete (i.e. defined only at finite number of points) with respect to $\mathbf{y}$.
In order to define the asymptotic expansion for rotations, it is convenient first to postulate the existence of a continuous displacement-like field ${\mathbf}{d}^\eta (\mathbf x)$ such that $2 {\mathbf}\Theta^I= \mb \nabla \times {\mathbf}{d}^\eta|_{\mathbf x =\mathbf x^I}$. If ${\mathbf}{d}^\eta (\mathbf x)$ is replaced by a two-scale approximation similar to the one in Equation \[disp-expansion\], one can write, ${\boldsymbol \Theta}^I = \mb{\theta}(\mathbf{x}^I, \mathbf{y}^I)$, and
$$\mb \theta (\mathbf x, \mathbf y) \approx \eta^{-1} \mb \omega^{0}(\mathbf x, \mathbf y) + \mb \varphi^0(\mathbf x, \mathbf y)+ \mb \omega^{1}(\mathbf x,\mathbf y)+\eta \mb\varphi^{1}(\mathbf x, \mathbf y)
\label{rot-expansion}$$
where $2 \boldsymbol\omega^{0} = \boldsymbol \nabla_y \times {\mathbf}{d}^0$; $2 \boldsymbol\varphi^{0} = \boldsymbol \nabla_x \times {\mathbf}{d}^0$; $2 \boldsymbol \omega^{1} = \boldsymbol \nabla_y \times {\mathbf}{d}^1$; $2 \boldsymbol \varphi^{1} = \boldsymbol \nabla_x \times {\mathbf}{d}^1$; and subscripts $x$ and $y$ identify the nabla operator in the coarse- and fine-scale, respectively. Thus, $\mb\omega^{0}$, $\mb\omega^{1}$ should be interpreted as rotations in the fine-scale whereas $\mb\varphi^{0}$, $\mb\varphi^{1}$ as the corresponding coarse-scale rotations. It is worth observing here that, contrarily to the expansion of displacements, the asymptotic expansion for rotations features a term of order $\mathcal{O}(\eta^{-1})$ and two distinct terms of order $\mathcal{O}(1)$.
In the macroscopic coordinate $\mathbf{x}$, the difference in position between nodes $P_I$ and $P_J$ can be considered as infinitesimal. Hence, in order to obtain the asymptotic expansion of strains and curvatures, it is convenient first to obtain the Taylor series expansion of displacement and rotation at nodes $P_J$ around point $P_I$ of coordinate $\mathbf{x}^I$ in the local coordinate system $\bf x$. By assuming that the displacement and rotation fields in Equations \[disp-expansion\] and \[rot-expansion\], are continuous and differentiable with respect to $\mathbf{x}$, one can write
$$\begin{aligned}
\label{taylor-1-J}
\begin{aligned}
U_i^J=u_i(\mathbf{x}^J,\mathbf{y}^J) = u_i^J+u^J_{i,j} \,x^{IJ}_j + \frac{1}{2}u^J_{i,jk}\, x^{IJ}_j x^{IJ}_k +\cdots
\end{aligned}\end{aligned}$$
$$\begin{aligned}
\label{taylor-2-J}
\begin{aligned}
\Theta_i^J=\theta_i(\mathbf{x}^J,\mathbf{y}^J)= \theta_i^J+\theta_{i,j}^J x^{IJ}_j +\frac{1}{2} \theta^J_{i,jk}\, x^{IJ}_j x^{IJ}_k +\cdots
\end{aligned}\end{aligned}$$
where $u_i^J=u_i(\mathbf{x}^I,\mathbf{y}^J)$; $ u^J_{i,j} = \partial u_i/\partial x_j (\mathbf{x}^I,\mathbf{y}^J)$; $u^J_{i,jk} =\partial^2 u_i/\partial x_j \partial x_k (\mathbf{x}^I,\mathbf{y}^J)$; $\theta_i^J= \theta_i(\mathbf{x}^I,\mathbf{y}^J)$; $ \theta^J_{i,j} = \partial \theta_i/\partial x_j (\mathbf{x}^I,\mathbf{y}^J)$; $\theta^J_{i,jk} =\partial^2 \theta_i/\partial x_j \partial x_k (\mathbf{x}^I,\mathbf{y}^J)$; $x^{IJ}_j$ is a vector connecting node $P_I$ to node $P_J$ in the $\mathbf{x}$ space. By substituting Equations \[disp-expansion\] and \[rot-expansion\] into Equation \[eps\], and using the Taylor expansion of displacement and rotation of node $P_J$ around node $P_I$ (Equations \[taylor-1-J\] and \[taylor-2-J\]) one obtains the multiple scale definition of facet strains (see Appendix \[exp-strains-details\] for details)
$$\label{eps-expansion}
\epsilon_{\alpha}=\eta^{-1} \epsilon_{\alpha}^{-1} + \epsilon_{\alpha}^0 + \eta \epsilon_{\alpha}^1$$
where
$$\label{eps-expansion-minus}
\epsilon_{\alpha}^{-1} = \bar{r}^{-1} \bigg[ u_i^{0J} - u^{0I}_i + \varepsilon_{ijk} \omega_j^{0J} \bar c_{k}^{J} - \varepsilon_{ijk} \omega_j^{0I} \bar c_{k}^{I} \bigg]e^{IJ}_{\alpha i}$$
$$\label{eps-expansion-zero}
\epsilon_{\alpha}^0 =\bar{r}^{-1} \bigg[ u_i^{1J} + u^{0J}_{i,j} y^{IJ}_j - u^{1I}_i + \varepsilon_{ijk} \bigg( \varphi_j^{0J} + \omega_j^{1J} + \omega_{j,m}^{0J} y^{IJ}_m \bigg) \bar c_{k}^{J} - \varepsilon_{ijk} \bigg( \varphi_j^{0I} + \omega_j^{1I} \bigg) \bar c_{k}^{I} \bigg] e^{IJ}_{\alpha i}$$
$$\label{eps-expansion-plus}
\begin{split}
\epsilon_{\alpha}^1 = \bar{r}^{-1} \bigg[ u^{1J}_{i,j} y^{IJ}_j + \frac{1}{2}u^{0J}_{i,jk} y^{IJ}_j y^{IJ}_k + \varepsilon_{ijk} \bigg( \varphi_j^{1J} + \varphi_{j,m}^{0J} y^{IJ}_m + \omega_{j,m}^{1J} y^{IJ}_m + \frac{1}{2} \omega^{0J}_{j,mn} y^{IJ}_m y^{IJ}_n \bigg) \bar c_{k}^{J} - \varepsilon_{ijk} \varphi_j^{1I} \bar c_{k}^{I} \bigg] e^{IJ}_{\alpha i}
\end{split}$$
In the previous equations, $\varepsilon_{ijk}$ is the Levi-Civita permutation symbol and length type variables have been changed into their $\mathcal{O}(1)$ counterparts by using Equation \[scale-link-1\]: $r=\eta \bar{r}$, $c_{k}^I=\eta \bar{c}_{k}^I$, $c_{k}^J=\eta \bar{c}_{k}^J$.
Similarly, multiple scale definition of facet curvature can be calculated as (see Appendix \[exp-strains-details\] for details)
$$\label{curv-expansion}
\eta \chi_{\alpha}=\eta^{-1} \psi_{\alpha}^{-1} + \psi_{\alpha}^{0} + \eta \psi_{\alpha}^1$$
where
$$\label{curv-expansion-minus2}
\psi_{\alpha}^{-1} =\bar{r}^{-1} \bigg[ \omega_i^{0J} - \omega_i^{0I} \bigg] {e}^{IJ}_{\alpha i}$$
$$\label{curv-expansion-minus1}
\psi_{\alpha}^{0} = \bar{r}^{-1} \bigg[ \varphi_i^{0J}+ \omega_i^{1J} + \omega_{i,j}^{0J} y^{IJ}_j - \varphi_i^{0I} - \omega_i^{1I} \bigg] {e}^{IJ}_{\alpha i}$$
$$\label{curv-expansion-zero}
\psi_{\alpha}^1 = \bar{r}^{-1} \bigg[ \varphi_i^{1J} + \omega_{i,j}^{1J} y^{IJ}_j + \varphi_{i,j}^{0J} y^{IJ}_j + \frac{1}{2} \omega_{i,jk}^{0J} y^{IJ}_j y^{IJ}_k - \varphi_i^{1I} \bigg] {e}^{IJ}_{\alpha i}$$
It is worth noting that in this section as well as in the rest of the paper superscript $IJ$ has been dropped when the permutation of $I$ and $J$ is not associated with a sign change.
Multiple-Scale Equilibrium Equations
------------------------------------
In order to obtain the correct scale separation of the governing equations, a rescaling of the discrete equilibrium equations needs to be performed. For the sake of simplicity, and since only quasi-static problems are concerned in the current research, it is assumed $\mathbf M^I_{u \theta} = 0$ and $\mathbf{M}_{\theta}^I = M_{\theta}^I \mathbf{I}$ on the left hand side of Equation \[motion-1\]. Rescaling is pursued by assuming that the material density, mass per unit volume, is of order zero: $\rho \sim \mathcal{O}(1)$, which along with the displacement asymptotic expansion implies that the left-hand-side of Equation \[motion-1\] is $\sim \mathcal{O}(\eta^3)$. By dividing both sides of Equation \[motion-1\] by $\eta^3$, and considering that all length variables should be considered $\sim \mathcal{O}(\eta^1)$, one obtains
$$\label{motion-1-rescaled}
\bar{M}_u^I\ddot { {\mathbf u}}^I - \bar{V}^I \mathbf{b}^0=\eta^{-1}\sum_{\mathcal{F}_I}{\bar{A}\, t_{\alpha} {\mathbf e}_\alpha^{IJ}}$$
where $\bar{M}_u^I = M_u^I/\eta^3$, $\bar{V}^I=V^I/\eta^3$ , $\bar{A} = A/\eta^2$ are all quantities $\sim \mathcal{O}(1)$. For reason of dimensionality, body forces $b^{0}_i$ can be always assumed to be proportional to gravity $\rho g$ and, consequently, they can be considered $\mathcal{O}(1)$ quantities as well.
One can rescale the rotational equation in a similar fashion by recognizing that, according to the previous discussion, the rotational moment of inertia is $\sim \mathcal{O}(\eta^5)$. Dividing both sides of Equation \[motion-2\] by $\eta^4$ one obtains
$$\label{motion-2-rescaled}
\eta \bar {M_\theta^I}\ddot{\mb \theta}^I=\eta^{-1} \sum_{\mathcal{F}_I} \bar{A}\, (\eta^{-1} w_{\alpha}{\mathbf e}_\alpha^{IJ} + \eta^{-1} m_{\alpha}{\mathbf e}_\alpha^{IJ})$$
where $\bar{M}_\theta^I = M_\theta^I/\eta^5$ is $\sim \mathcal{O}(1)$.
In the elastic regime one can write: $t_{\alpha} = \eta^{-1} t^{-1}_{\alpha} + t^{0}_{\alpha} + \eta t^{1}_{\alpha}$; where $t^{(\cdot)}_{\alpha} = E_\alpha \epsilon^{(\cdot)}_\alpha$, and $E_\alpha$ is assumed to be $\sim \mathcal{O}(1)$. In addition, $q_\alpha = \eta^{-1} m_{\alpha} = \eta^{-1} q^{-1}_{\alpha} + q^{0}_{\alpha} + \eta q^{1}_{\alpha}$ in which $q^{(\cdot)}_{\alpha} =\bar{W}_\alpha \psi^{(\cdot)}_\alpha$; $\bar{W}_\alpha=E_\alpha {\bar{\ell}}^{2}_\alpha$; and $\bar{\ell} = \ell / \eta$. Finally, $p_{\alpha} = \eta^{-1}w_{\alpha} = \eta^{-1} p^{-1}_{\alpha} + p^{0}_{\alpha} + \eta p^{1}_{\alpha}$ where $p^{(\cdot)}_{\alpha} {\mathbf e}_\alpha^{IJ} = \bar{\mathbf{c}}^I \times t^{(\cdot)}_{\alpha} {\mathbf e}_\alpha^{IJ}$. Since $w_\alpha$ and $m_\alpha$ are moments, it is reasonable that the asymptotic expansion of those variables divided by $\eta$ is similar to the one for tractions $t_\alpha$, considering that length type variables are considered to be $\sim \mathcal{O}(\eta)$.
Introducing these traction expressions along with the asymptotic definition of displacement and rotation fields Equations \[disp-expansion\] and \[rot-expansion\] (which also imply $\eta \ddot{\mb \theta}^{0I} =\ddot{\mb \omega}^{0I}+\mathcal{O}(\eta)$), into the rescaled equilibrium equations leads to
$$\label{motion-1-sep}
\eta^{-2} \sum_{\mathcal{F}_I}{\bar{A}\, t^{-1}_{\alpha} {\mathbf e}_\alpha^{IJ}}+ \eta^{-1} \sum_{\mathcal{F}_I}{\bar{A}\, t^{0}_{\alpha} {\mathbf e}_\alpha^{IJ}} + \sum_{\mathcal{F}_I}{\bar{A}\, t^1_{\alpha} {\mathbf e}_\alpha^{IJ}} - \bar{M}_u^I\ddot { {\mathbf u}}^{0I} + \bar{V}^I \mathbf{b}^0+\mathcal{O}(\eta) = \mathbf{0}$$
and
$$\begin{aligned}
\label{motion-2-sep}
\begin{aligned}
\eta^{-2}\sum_{\mathcal{F}_I} \bar{A}\, (p^{-1}_{\alpha}{\mathbf e}_\alpha^{IJ} + q^{-1}_{\alpha}{\mathbf e}_\alpha^{IJ}) + \eta^{-1}\sum_{\mathcal{F}_I} \bar{A}\, (p^{0}_{\alpha}{\mathbf e}_\alpha^{IJ} + q^{0}_{\alpha}{\mathbf e}_\alpha^{IJ}) \\ - \bar {M_\theta^I}\ddot{\mb \omega}^{0I} + \sum_{\mathcal{F}_I} \bar{A}\, (p^{1}_{\alpha}{\mathbf e}_\alpha^{IJ} + q^{1}_{\alpha}{\mathbf e}_\alpha^{IJ}) + \mathcal{O}(\eta) = \mathbf{0}
\end{aligned}\end{aligned}$$
in which terms of different orders are gathered together. The multiple scale equations reported above can also be used for nonlinear constitutive equations provided that facet tractions and facet moments can be expressed through the multiple scale decomposition exploited above. It will be shown later in the paper that this can be indeed achieved under some reasonable assumptions.
The RVE Problem {#RVEproblem}
---------------
Let’s first consider the equilibrium equations at the $\mathcal{O}(\eta^{-2})$ scale. From Equations \[motion-1-sep\] and \[motion-2-sep\], it is evident that the $\mathcal{O}(\eta^{-2})$ equilibrium equations represent the equilibrium of all particles in the RVE subjected to the stress tractions $t_\alpha^{-1}$ and the moment tractions $q_\alpha^{-1}$ and without any applied external load. Consequently, solution of the $\mathcal{O}(\eta^{-2})$ problem implies $t_\alpha^{-1}=0$ and $q_\alpha^{-1}=0$, which in turn, leads to $\epsilon_{\alpha}^{-1}=0$ and $\psi_{\alpha}^{-1}=0$. By taking into consideration the definitions of $\epsilon^{-1}_{\alpha}$ and $\psi^{-1}_{\alpha}$ (Equations \[eps-expansion-minus\] and \[curv-expansion-minus2\]) such result indicates that the $\mathcal{O}(\eta^{-2})$ problem represents a rigid body rototranslation of the RVE. This can be expressed as
$$\label{U0}
u_i^0(\mathbf{X}, \mathbf{y}) = v_i^0(\mathbf{X}) + \varepsilon_{ijk} y_k \omega_j^{0}(\mathbf{X})$$
in which the fields $\mathbf{v}^0$ and $\mb \omega^{0}$ are only dependent on macroscopic coordinate system $\bf X$, i.e. these quantities varies smoothly in the macro-scale material domain; they do not change within the RVE domain; and they can be calculated when kinematic boundary conditions are specified for the $\mathcal{O}(\eta^{-2})$ problem. These boundary conditions must describe the physical fact that the RVE is attached to a point in the macroscopic continuum. Hence, $\mathbf{v}^0$ must correspond to the macroscopic displacement field, and $\mb \omega^{0}$ must be equal to the macroscopic rotation field: $\mb \varphi^{0} = \mb \omega^{0}$. Since $\mb \omega^{0}$ is constant over the RVE, then $\mb \varphi^{0}$ is also constant in the RVE.
On the basis of Equation \[U0\] and the discussion above, one can rewrite the $\mathcal{O}(1)$ strains and curvatures as (See Appendix \[Revised-strain-curvature\] for details) $$\label{eps-expansion-zero'}
\epsilon_{\alpha}^0 =\bar{r}^{-1} \left( u_i^{1J} - u^{1I}_i + \varepsilon_{ijk} \omega_j^{1J} \bar c_{k}^{J} - \varepsilon_{ijk} \omega_j^{1I} \bar c_{k}^{I}\right) {e}^{IJ}_{\alpha i} + P^\alpha_{ij} \left(\gamma_{ij}
+ \varepsilon_{jmn} \kappa_{im} y_{n}^{c} \right) $$ $$\label{curv-expansion-minus1'}
\psi_{\alpha}^{0} = \bar{r}^{-1} \left( \omega_i^{1J}- \omega_i^{1I} \right) {e}^{IJ}_{\alpha i}+ P^\alpha_{ij} \kappa_{ij}$$
where $\gamma_{ij} = v^{0}_{j,i} - \varepsilon_{ijk} \omega_k^{0}$, $\kappa_{ij}=\omega_{j,i}^{0}$ are the macroscopic Cosserat strain and curvature tensors, respectively. The vector ${\mathbf}{y}^c$ is the position vector of the centroid of the common facet between particle $I$ and $J$ and $P^\alpha_{ij} = n^{IJ}_i e^{IJ}_{\alpha j}$ is a projection operator. Comparing the first term of Equation \[eps-expansion-zero’\] with Equation \[eps\], it can be concluded that this term is the lower scale definition of the three components of the facet strains (one normal and two tangential) written in terms of fine-scale displacements and rotations $u^1$ and $\omega^1$. The second term of Equation \[eps-expansion-zero’\], $P^\alpha_{ij} \left(\gamma_{ij} + \varepsilon_{jmn} \kappa_{im} y_{n}^{c} \right)$, is the projection of macroscopic Cosserat strain and curvature tensors on each facet. Similarly, Equation \[curv-expansion-minus1’\] shows that the $\mathcal{O}(1)$ curvature includes a fine-scale term (see Equation \[curvature\]), which depends on fine-scale rotation term $\omega^1$, and a coarse-scale term corresponding to the projection of macroscopic curvature tensor on each facet. Therefore, Equations \[eps-expansion-zero’\] and \[curv-expansion-minus1’\] express the $\mathcal{O}(1)$ facet strains and curvatures as the sum of their fine-scale counterparts and the projection of macroscopic strain and curvature tensors onto the facet level. It is worth nothing that the projection operator $P^\alpha_{ij}$ corresponds exactly to the one used in the microplane model [@Bazant-2; @xinwei-1] if $ e^{IJ}_{N i}\equiv n^{IJ}_i $, i.e. the discrete model is formulated in such a way the facets are orthogonal to the associated lattice struts. In addition, it must be noted that the term $\varepsilon_{jmn} \kappa_{im} y_{n}^{c}$ transforms the macroscopic curvature tensor, which is constant over the RVE, to different strain values at different positions $y^c_n$ inside the RVE, which is then projected on the facets through the operator $P^\alpha_{ij}$. Expanding this term for different components of curvature tensor, it can be shown that it perfectly corresponds to the strain field generated by curvatures in classical beam theories.
Strains and curvatures of order $\mathcal{O}(\eta)$ can also be rewritten by taking into account Equation \[U0\]. One gets
$$\label{eps-expansion-plus'}
\begin{aligned}
\epsilon_{\alpha}^1 = \bar{r}^{-1} \bigg[& u^{1J}_{i,j} y^{IJ}_j + \varepsilon_{ijk} \varphi_j^{1J} \bar c_{k}^{J} + \varepsilon_{ijk} \omega_{j,m}^{1J} y^{IJ}_m \bar c_{k}^{J} - \varepsilon_{ijk} \varphi_j^{1I} \bar c_{k}^{I} \\
&+ \frac{1}{2}v^{0}_{i,jk} y^{IJ}_j y^{IJ}_k + \frac{1}{2} \varepsilon_{ijk} \omega^{0}_{j,mn} y^{IJ}_m y^{IJ}_n y_{k}^{c} + \varepsilon_{ijk} \omega_{j,m}^{0} y^{IJ}_m \bar c_{k}^{J} \bigg] e^{IJ}_{\alpha i}
\end{aligned}$$
$$\label{curv-expansion-zero'}
\psi_{\alpha}^1 = \bar{r}^{-1} \left[ \varphi_i^{1J} + \omega_{i,j}^{1J} y^{IJ}_j - \varphi_i^{1I} + \omega_{i,j}^{0} y^{IJ}_j + \frac{1}{2} \omega_{i,jk}^{0J} y^{IJ}_j y^{IJ}_k \right] {e}^{IJ}_{\alpha i}$$
Detailed mathematical derivation of Equations \[eps-expansion-zero’\] through \[curv-expansion-zero’\] is provided in Appendix \[Revised-strain-curvature\].
In the previous derivation, where linear elastic behavior was assumed, the equilibrium equations at the $\mathcal{O}(\eta^{-2})$ scale were shown to represent the rigid body motion conditions for the RVE and, consequently, they led to zero strains, $\epsilon_{\alpha}^{-1}$, curvatures, $\psi_{\alpha}^{-1}$, tractions, $t_\alpha^{-1}$, and moments, $p_\alpha^{-1}$, and $q_\alpha^{-1}$, at the $\mathcal{O}(\eta^{-1})$ scale. These conditions can be reasonably assumed *a priori* in the case of nonlinear material behavior. In this case case one may write $t_\alpha = t_\alpha(\epsilon_{\beta}^{0}+\eta \epsilon_{\beta}^{1})$; $p_\alpha = p_\alpha(\epsilon_{\beta}^{0}+\eta \epsilon_{\beta}^{1})$, and $q_\alpha = q_\alpha(\eta^{-1}\psi^0_\beta+\psi^1_\beta)$ in which $\alpha,~\beta = N,M,L$. Since $\eta$ is a small quantity, one can also write the Taylor expansion of $t_\alpha$ and $p_\alpha$ around the $\mathcal{O}(1)$ component of strain and the Taylor expansion of $q_\alpha$ around the $\mathcal{O}(\eta^{-1})$ component of curvature:
$$\label{TaylorExpansions}
\begin{split}
& t_\alpha = t_\alpha(\epsilon_{\beta}^{0}+\eta \epsilon_{\beta}^{1}) = t_\alpha(\epsilon_{\beta}^{0}) + \eta \frac{\partial t_\alpha(\epsilon_{\beta}^{0})}{ \partial \epsilon^0_\gamma } \epsilon_\gamma^1 \\
& p_\alpha = p_\alpha(\epsilon_{\beta}^{0}+\eta \epsilon_{\beta}^{1}) = p_\alpha(\epsilon_{\beta}^{0}) + \eta \frac{\partial p_\alpha(\epsilon_{\beta}^{0})}{ \partial \epsilon^0_\gamma } \epsilon_\gamma^1 \\
& q_\alpha = q_\alpha(\eta^{-1}\psi^0_\beta+\psi^1_\beta) = q_\alpha(\eta^{-1}\psi^0_\beta) + \eta \frac{\partial q_\alpha(\eta^{-1}\psi^0_\beta)}{ \partial \psi^0_\gamma}\psi_\gamma^1
\end{split}$$
which can be rewritten as $t_{\alpha} = t^{0}_{\alpha} + \eta t^{1}_{\alpha}$; $p_{\alpha} = p^{0}_{\alpha} + \eta p^{1}_{\alpha}$; $q_{\alpha} = q^{0}_{\alpha} + \eta q^{1}_{\alpha}$, with the following conditions
$$\label{ZeroOne-Terms-Def}
\begin{gathered}
t_\alpha^0=t_\alpha(\epsilon^0_\beta); ~~~ p_\alpha^0=p_\alpha(\epsilon^0_\beta); ~~~ q_\alpha^0=q_\alpha(\eta^{-1} \psi^0_\beta); \\
t_\alpha^1=\frac{\partial t^0_\alpha}{ \partial \epsilon^0_\gamma } \epsilon_\gamma^1; ~~~ p_\alpha^1=\frac{\partial p^0_\alpha}{ \partial \epsilon^0_\gamma } \epsilon_\gamma^1; ~~~ q_\alpha^1=\frac{\partial q^0_\alpha}{ \partial \psi^0_\gamma }\psi_\gamma^1
\end{gathered}$$
This demonstrates that Equations \[motion-1-sep\], and \[motion-2-sep\] are valid also in the case of nonlinear material behavior under the assumption that traction and moments at the $\mathcal{O}(\eta^{-1})$ scale are zero as required, in the linear case, by the rigid body motion of the RVE.
The RVE problem is governed by the $\mathcal{O}(\eta^{-1})$ terms in Equations \[motion-1-sep\] and \[motion-2-sep\]. Considering those terms and scaling back all the variables, one can write the $\mathcal{O}(\eta^{-1})$ equations as
$$\label{RVE-1}
\sum_{\mathcal{F}_I}{{A}\, t^{0}_{\alpha} {{\mathbf}e}_\alpha^{IJ}} = 0; \hspace{0.25 in} \sum_{\mathcal{F}_I} {A}\, ({\mathbf}{c}^I \times t^{0}_{\alpha} {\mathbf}{e}_\alpha^{IJ} + m^{0}_{\alpha}{{\mathbf}e}_\alpha^{IJ}) = 0$$
Equations \[RVE-1\] are force and moment equilibrium equations of each single particle inside the RVE subjected to $\mathcal{O}(1)$ facet traction $t_\alpha^0$ and moment $m_\alpha^0$ vectors, which, in turn, are functions of $\epsilon^0_\alpha$ and $\psi^0_\alpha$, consisting of a coarse-scale and a fine-scale term (see Equations \[eps-expansion-zero’\] and \[curv-expansion-minus1’\]). In other words, Equations \[RVE-1\] state that the macroscopic strain, $\gamma_{ij} = v^{0}_{j,i} - \varepsilon_{ijk} \omega_k^{0}$, and curvature, $\kappa_{ij}=\omega_{j,i}^{0}$, tensors should be applied on all RVE facets as negative eigenstrains, and the fine-scale solution, in terms of displacements $u^1_i$ and rotations $\omega^1_i$ of each particle, must be calculated satisfying its force and moment equilibrium equations, while periodic boundary conditions are enforced on the RVE. The solution of the equilibrium equations also provides facet traction $t_\alpha^0$ and moment $m_\alpha^0$ vectors that are later used to compute the macroscopic stress and couple tensors.
The Macroscopic Problem {#macro-derivation}
-----------------------
Finally, let us consider the $\mathcal{O}(1)$ equilibrium equations in Equations \[motion-1-sep\] and \[motion-2-sep\]. The $\mathcal{O}(1)$ translational equilibrium equation for each particle in the RVE reads
$$\label{macro-1}
{M}_u^I \ddot {u}_i^{0I} = \eta \sum_{\mathcal{F}_I}{{A} \frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + {V}^I {b}^{0}_i$$
where all the variables have been scaled back in the original system of reference, and $t_i^{IJ}=t^0_\beta e_{\beta i}^{IJ}$. By using Equation \[U0\] and by averaging the contribution of all particles in the RVE, one can write (see Appendix \[MacroEquil-Derivation\] for details)
$$\label{macro-1-1-averaged}
\rho_u \ddot {v}_i^{0} = \frac{1}{V_0}\sum_I \sum_{\mathcal{F}_I}{\eta A \frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + b_i$$
where $V_0$ is the volume of the RVE; $\rho_u=\sum_I {M}^I_u/V_0$ is the mass density of the macroscopic continuum; $b_i=b^0_i (1 - \phi)$; and $\phi= 1 -\sum_I {V}^I/V_0$ is the porosity of the macroscopic continuum. Equation \[macro-1-1-averaged\] was derived under the assumption that $\sum_I {M}_u^I y^{I}_i=0$, which corresponds to the assumption that the local system of reference is the mass centroid of the particle system within the RVE.
Before proceeding with the derivations, let’s take a closer look at the definition of $\epsilon^{1}_{\alpha}$ and the term $(\partial {t}^{IJ}_{i}/\partial \epsilon^0_{\alpha}) \epsilon^1_{\alpha}$ on the RHS of Equation \[macro-1-1-averaged\]. Each facet in the material domain is shared between two particles, say $I$ and $J$. Therefore, by summing up the contributions of two adjacent particles, one obtains
$$\begin{aligned}
\label{higher-strain-1}
\begin{aligned}
\frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} + \frac {\partial {t}^{JI}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} = & \frac{1}{\bar{r}} \frac {\partial {t}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \bigg[ \bigg( u^{1J}_{n,j} y^{IJ}_j + \varepsilon_{njk} \varphi_j^{1J} \bar c_{k}^{J} + \varepsilon_{njk} \omega_{j,m}^{1J} y^{IJ}_m \bar c_{k}^{J} - \varepsilon_{njk} \varphi_j^{1I} \bar c_{k}^{I} \\
& \hspace{0.5 in} + \frac{1}{2} v^{0}_{n,jk} y^{IJ}_j y^{IJ}_k + \frac{1}{2} \varepsilon_{njk} \omega^{0}_{j,mo} y^{IJ}_m y^{IJ}_o y_{k}^{c} + \varepsilon_{njk} \omega_{j,m}^{0} y^{IJ}_m \bar c_{k}^{J} \bigg) e^{IJ}_{\alpha n} \bigg] \\
& + \frac{1}{\bar{r}} \frac {\partial {t}_{i}^{JI}}{\partial \epsilon^0_{\alpha}} \bigg[ \bigg( u^{1I}_{n,j} y^{JI}_j + \varepsilon_{njk} \varphi_j^{1I} \bar c_{k}^{I} + \varepsilon_{njk} \omega_{j,m}^{1I} y^{JI}_m \bar c_{k}^{I} - \varepsilon_{njk} \varphi_j^{1J} \bar c_{k}^{J} \\
& \hspace{0.5 in} + \frac{1}{2} v^{0}_{n,jk} y^{JI}_j y^{JI}_k + \frac{1}{2} \varepsilon_{njk} \omega^{0}_{j,mo} y^{JI}_m y^{JI}_o y_{k}^{c} + \varepsilon_{njk} \omega_{j,m}^{0} y^{JI}_m \bar c_{k}^{I} \bigg) e^{JI}_{\alpha n} \bigg]
\end{aligned}\end{aligned}$$
Considering the definition of the vector $\mathbf{y} ^{IJ} = \mathbf{y} ^{J} - \mathbf{y} ^{I}$, one can write $y_m^{IJ} = - y_m^{JI}$ and $\bar c^I_k-\bar c^J_k=y_k^{IJ}$. In addition, $e^{IJ}_{\alpha i} = - e^{JI}_{\alpha i}$ and $t_{i}^{IJ} = -t_{i}^{JI}$ hold for each facet. Finally, the sign of $\epsilon^{0}_\alpha$ does not change by interchanging $I$ and $J$ in its definition. This leads to $\partial t_{i}^{IJ}/\partial \epsilon^{0}_\alpha = - \partial t_{i}^{JI}/\partial \epsilon^{0}_\alpha$. Taking all above facts into account, Equation \[higher-strain-1\] can be written as
$$\begin{aligned}
\label{higher-strain-2}
\begin{aligned}
\frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} + \frac {\partial {t}^{JI}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} = \frac{1}{\bar{r}}\frac {\partial {t}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \bigg[ y^{IJ}_m \bigg( & u^{1J}_{n,m} - u^{1I}_{n,m} + \varepsilon_{njk} \omega_{j,m}^{1J} \bar c_{k}^{J} - \varepsilon_{njk} \omega_{j,m}^{1I} \bar c_{k}^{I} \\
& + v^{0}_{n,jm} y^{IJ}_j - \varepsilon_{njk} \omega_{j,m}^{0} + \varepsilon_{njk} \omega^{0}_{j,mo} y^{IJ}_o y_{k}^{c} \bigg) e^{IJ}_{\alpha n} \bigg] \\
\end{aligned}\end{aligned}$$
Comparing the expression inside the bracket on the RHS of Equation \[higher-strain-2\] to the definition of $\epsilon^0_{\alpha}$ in Equation \[eps-expansion-zero’\], it can be concluded that
$$\begin{aligned}
\label{higher-strain-3}
\begin{aligned}
\frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} + \frac {\partial {t}^{JI}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha} = \frac {\partial {t}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \frac {\partial \epsilon^0_{\alpha}}{\partial x_m} y_m^{IJ} = \frac {\partial {t}_{i}^{IJ}}{\partial x_m} y_m^{IJ}
\end{aligned}\end{aligned}$$
Therefore, one can average the term $(\partial {t}^{IJ}_{i}/\partial \epsilon^0_{\alpha}) \epsilon^1_{\alpha}$ on each facet and replace it with $1/2(\partial {t}_{i}^{IJ}/\partial x_m) y_m^{IJ}$ in the equilibrium Equations \[macro-1-1-averaged\], which can be rewritten as
$$\label{macro-1-2-averaged}
\rho_u \ddot {v}_i^{0} = \frac{1}{2V_0} \sum_I \sum_{\mathcal{F}_I}{A} r \frac{\partial {t}^{IJ}_{i}}{\partial x_j} n_j^{IJ} + b_i$$
Finally, by considering that (1) $\partial (t^{IJ}_i n^{IJ}_j )/ \partial x_j = \partial t^{IJ}_i/ \partial x_j n^{IJ}_j + t^{IJ}_i \partial n^{IJ}_j/ \partial x_j$ and (2) $\partial n^{IJ}_j/ \partial x_j=0$ for the periodicity of the problem; and by recalling that $t_i^{IJ}=t^0_\alpha e_{\alpha i}^{IJ}$, one obtains
$$\label{macro-eq-cont}
\rho_u \ddot{v}^0_i = \sigma^0_{ji,j} + b_i$$
and
$$\label{macro-stress-formula}
\sigma^0_{ij} = \frac{1}{2V_0} \sum_I \sum_{\mathcal{F}_I}{A} r t^0_\alpha P_{i j}^{\alpha}$$
Equation \[macro-eq-cont\] is the classical partial differential equation governing the equilibrium of continua whereas Equation \[macro-stress-formula\] provides the macroscopic stress tensor by averaging the solution of the RVE problem. It is worth mentioning that Equation \[macro-stress-formula\] coincides with the virial stress formula for atomistic systems derived in Ref. [@Fish-3], but it is also equivalent to the averaging formula used in the classical microplane model [@Bazant-2] formulation and derived through an energetic equivalence.
The $\mathcal{O}(1)$ moment equilibrium equation is considered next. Since the purpose in this section is to average the equation of motion of all particles inside the RVE and derive the macroscopic equilibrium equation governing the entire RVE, to have a consistent formulation for all particles and RVEs, one must consider the moment of all forces with respect to a fixed point in space.
For the generic particle $I$, by taking the moment of all forces with respect to the origin of a global macroscopic coordinate system as shown in Figure \[TwoScaleAnalysis\]b and by considering the results of the $\mathcal{O}(\eta^{-2})$ problem, one can write (see Appendix \[MacroEquil-Derivation\] for details)
$$\label{macro-2}
{M}_u^I \varepsilon_{ijk} X^{I}_j \left( \ddot {v}_k^{0} + \varepsilon_{kmn} \eta^{-1} \ddot {\omega}_m^{0} x^{I}_n \right) + \eta^{-1} {M_\theta^I} \ddot{\omega}_i^{0} = \eta \sum_{\mathcal{F}_I} A \left( {\frac {\partial {w}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} \right) + {V}^I \varepsilon_{ijk} X^{I}_j {b}_k^{0}$$
where $X^I_j$ is the position vector of particle $I$ in global coordinate system; $w^{IJ}_i = \varepsilon_{ijk} X^C_j t^{IJ}_k$ is the moment of facet traction with respect to the point $O$; $X^C_j$ is the position vector of the contact point $C$ between the particles $I$ and $J$ in the global coordinate system, and $m_i^{IJ}=m^0_\beta e_{\beta i}^{IJ}$. Also, $x_j^I$ and $x_j^C$ are the position vectors of the particle $I$ and the contact point $C$ with respect to the mass center of the RVE, respectively.
By summing up the moment equilibrium equations of all particles inside the RVE and dividing by the volume of the RVE, and considering that $X^I_j = X_j + x^I_j$, one obtains (see Appendix \[MacroEquil-Derivation\] for details)
$$\label{macro-2-averaged-init}
\frac{1}{V_0} \sum_I {M}_u^I \varepsilon_{ijk} X_j \ddot {v}_k^{0} + \rho_{im}^{\theta} (\eta^{-1} \ddot{\omega}_m^{0}) = \frac{\eta}{V_0} \sum_I \sum_{\mathcal{F}_I} A \left( {\frac {\partial {w}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} \right) + \frac{1}{V_0} \sum_I {V}^I \varepsilon_{ijk} X_j {b}_k^{0}$$
where $\rho_{im}^{\theta} =\sum_I \left[ M_\theta^I \delta_{im} + M_u^I \varepsilon_{ijk} \varepsilon_{kmn} x_j^I x_n^I \right]/V_0$ is the inertia tensor of the RVE. In deriving Equation \[macro-2-averaged-init\], the particle density $M^I/V^I$ was assumed to be constant for all particles; and the local system of reference at the center of the RVE was chosen such that $\sum_I M_u^I x^I_i x^I_j =0$ for any $i \neq j$, i.e. as mentioned earlier in this paper, the axes of the system of reference are principal axes of inertia for the system of particles within the RVE.
Before moving forward with the derivation, let’s first consider the second term on the RHS of Equation \[macro-2-averaged-init\]. For a facet in the material domain which is shared between particles $I$ and $J$, by summing the contribution of two particles $I$ and $J$ on the term $(\partial {m}_{i}^{IJ}/\partial \psi^{0}_{\alpha}) \psi^1_{\alpha}$ and by considering the definition of $\psi^1_{\alpha}$ (see Equation \[curv-expansion-zero’\]), one gets
$$\begin{aligned}
\label{higher-curv-1}
\begin{aligned}
\frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} + \frac {\partial {m}_{i}^{JI}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} = & \frac{1}{\bar{r}} \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \left[ \varphi_i^{1J} + \omega_{i,j}^{1J} y^{IJ}_j - \varphi_i^{1I} + \omega_{i,j}^{0} y^{IJ}_j + \frac{1}{2} \omega_{i,jk}^{0J} y^{IJ}_j y^{IJ}_k \right] {e}^{IJ}_{\alpha i} \\
& + \frac{1}{\bar{r}} \frac {\partial {m}_{i}^{JI}}{\partial \psi^{0}_{\alpha}} \left[ \varphi_i^{1I} + \omega_{i,j}^{1I} y^{JI}_j - \varphi_i^{1J} + \omega_{i,j}^{0} y^{JI}_j + \frac{1}{2} \omega_{i,jk}^{0I} y^{JI}_j y^{JI}_k \right] {e}^{JI}_{\alpha i}
\end{aligned}\end{aligned}$$
Since the moment stress vector applied on a single facet belonging to two particles $I$ and $J$ are the same in magnitude but opposite in direction, one can write $m_{i}^{IJ} = -m_{i}^{JI}$; and consequently, $\partial m_{i}^{IJ}/\psi^{0}_\alpha = - \partial m_{i}^{JI}/\psi^{0}_\alpha$. In addition, the sign of $\psi^{0}_\alpha$ does not change by interchanging $I$ and $J$ in its definition, and that $y_m^{IJ} = - y_m^{JI}$, $e^{IJ}_{\alpha i} = - e^{JI}_{\alpha i}$, Equation \[higher-curv-1\] can be written as
$$\begin{aligned}
\label{higher-curv-2}
\begin{aligned}
\frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} + \frac {\partial {m}_{i}^{JI}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} = & \frac{1}{\bar{r}} \frac{\partial {m}_{i}^{IJ}}{\partial \psi^{0}_\alpha} \bigg[ y^{IJ}_j \left(\omega_{n,j}^{1J} + \omega_{n,jk}^{0} y^{IJ}_k - \omega_{n,j}^{1I} \right) \bigg] e^{IJ}_{\alpha n}
\end{aligned}\end{aligned}$$
If one compares the definition of $\psi^0_{\alpha}$ (see Equation \[curv-expansion-minus1’\]) to the expression in the bracket on the RHS of Equation \[higher-curv-2\], it yields
$$\begin{aligned}
\label{higher-curv-3}
\begin{aligned}
\frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} + \frac {\partial {m}_{i}^{JI}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} = \frac{\partial {m}_{i}^{IJ}}{\partial \psi^{0}_\alpha} \frac{\partial \psi^{0}_\alpha}{\partial x_j} y^{IJ}_j = \frac{\partial {m}_{i}^{IJ}}{\partial x_j} y^{IJ}_j
\end{aligned}\end{aligned}$$
As a result, one can replace the term $(\partial {m}_{i}^{IJ}/\partial \psi^{0}_{\alpha}) \psi^1_{\alpha}$ in Equation \[macro-2-averaged-init\], with the averaged expression derived in the above Equation \[higher-curv-3\]. Similarly to the derivation relevant to the translational equation of motion, one can replace the term $(\partial {w}_{i}^{IJ}/{\partial \epsilon^0_{\alpha}}) \epsilon^1_{\alpha} $ on the RHS of Equation \[macro-2-averaged-init\], by the average value $1/2(\partial {w}_{i}^{IJ}/{\partial x_m}) y_m^{IJ}$ for each facet. Equation \[macro-2-averaged-init\] can be then rewritten as
$$\begin{aligned}
\label{macro-2-3-averaged}
\begin{aligned}
\rho_{im}^{\theta} (\eta^{-1} \ddot{\omega}_m^{0}) = \frac{\eta}{2V_0}\sum_I \sum_{\mathcal{F}_I} {{A} \bigg( \frac{\partial w^{IJ}_i}{\partial x_j} y^{IJ}_j} + \frac{\partial m^{IJ}_i}{\partial x_j} y^{IJ}_j \bigg) + \frac{1}{V_0} \sum_I \bigg( {V}^I \varepsilon_{ijk} X_j {b}_k^{0} - {M}_u^I \varepsilon_{ijk} X_j \ddot {v}_k^{0} \bigg)
\end{aligned}\end{aligned}$$
Using ${X}^C_j = {X}_j + {x}^C_j$ in the definition of $w^{IJ}_i$ along with the identity equations $\partial (m^{IJ}_i n^{IJ}_j )/ \partial x_j = \partial m^{IJ}_i/ \partial x_j n^{IJ}_j$ and $\partial (w^{IJ}_i n^{IJ}_j )/ \partial x_j = \partial w^{IJ}_i/ \partial x_j n^{IJ}_j$, Equation \[macro-2-3-averaged\] can be written as
$$\begin{aligned}
\label{macro-2-4-averaged}
\begin{aligned}
\rho_{im}^{\theta} (\eta^{-1} \ddot{\omega}_m^{0}) = & \frac{\eta}{2V_0}\sum_I \sum_{\mathcal{F}_I} {A} (y^{IJ}_j \varepsilon_{imk} X_m t^{IJ}_k)_{,j} + \frac{\eta}{2V_0}\sum_I \sum_{\mathcal{F}_I} {A} (y^{IJ}_j \varepsilon_{imk} x^C_m t^{IJ}_k + y^{IJ}_j m^{IJ}_i)_{,j} \\
& + ( \varepsilon_{ijk} X_j {b}_k - \rho_u \varepsilon_{ijk} X_j \ddot {v}_k^{0} )
\end{aligned}\end{aligned}$$
The last term on the RHS of Equation \[macro-2-4-averaged\] is written considering the fact that $b_k^{0}$ and $\ddot {v}_k^{0}$ are equal for all particles inside the RVE. Furthermore, the first term on the RHS of Equation \[macro-2-4-averaged\] can be expanded as $(y^{IJ}_j \varepsilon_{imk} X_m t^{IJ}_k)_{,j} = \varepsilon_{ijk} y^{IJ}_j t^{IJ}_k + \varepsilon_{imk} X_m (y^{IJ}_j t^{IJ}_{k})_{,j}$, in which $\partial y^{IJ}_j/ \partial x_j=0$ is used. Therefore, Equation \[macro-2-4-averaged\] becomes
$$\begin{aligned}
\label{macro-2-5-averaged}
\begin{aligned}
\rho_{im}^{\theta} (\eta^{-1} \ddot{\omega}_m^{0}) = & \frac{1}{2V_0}\sum_I \sum_{\mathcal{F}_I} {A} \varepsilon_{ijk} x^{IJ}_j t^{IJ}_k+ \frac{1}{2V_0}\sum_I \sum_{\mathcal{F}_I} {A} (x^{IJ}_j \varepsilon_{imk} x^C_m t^{IJ}_k + x^{IJ}_j m^{IJ}_i)_{,j} \\
& + \varepsilon_{ijk} X_j \bigg( \frac{1}{2V_0}\sum_I \sum_{\mathcal{F}_I} A (x^{IJ}_m t^{IJ}_{k})_{,m} + {b}_k^{0} - \rho_u \ddot {v}_k^{0} \bigg)
\end{aligned}\end{aligned}$$
The last term on the RHS of Equation \[macro-2-5-averaged\] is the moment of the translational equilibrium equation of the RVE (see Equation \[macro-1-2-averaged\]) around the origin of the macroscopic global coordinate system; therefore, it is equal to zero. Comparing the first term on RHS of Equation \[macro-2-5-averaged\] with definition of macroscopic stress tensor of the RVE in Equation \[macro-stress-formula\], one can replace it with $\varepsilon_{ijk} \sigma^0_{jk}$. The second term on the RHS of Equation \[macro-2-5-averaged\] is the divergence of the averaged moment stress tensor of the RVE. The macro-scale rotational equation of motion can be then written as follows
$$\label{macro-rotational-final}
\begin{gathered}
\rho_{\theta ij} (\eta^{-1} \ddot {\omega}_j ^{0}) = {\varepsilon}_{ijk} {\sigma}_{ij}^0 + \frac{\partial \mu^0_{ji}}{\partial x_j}
\end{gathered}$$
where
$$\label{macro-momentstress-formula}
\begin{gathered}
\mu^0_{ij} = \frac{1}{2V_0}\sum_I \sum_{\mathcal{F}_I} {A}r (m_{\alpha}^0 P_{ij}^{\alpha} + t_{\alpha}^0 Q_{ij}^{\alpha})
\end{gathered}$$
and the matrix $Q_{ij}^{\alpha}$ is defined as $Q_{ij}^{\alpha} = n_i^{IJ} \varepsilon_{jkl} x^C_k e_{\alpha l}^{IJ}$. $\mu^0_{ij}$ is the macroscopic moment stress tensor calculated using the results of RVE analysis, and Equation \[macro-rotational-final\] corresponds to the classical rotational equilibrium equation of Cosserat continua [@Chan-1; @Cosserat-1]. According to Equation \[macro-momentstress-formula\] for macroscopic moment stress tensor and considering that ${x}^C_k = {x}^{I}_k + {c}^I_k$, one can conclude that $\mu^0_{ij}$ consists of three terms: (1) the effect of the facet couple traction ${\mathbf}{m}$; (2) the effect of the moment of the facet stress traction ${\mathbf}{t}$ around the particle node which the facet belongs to, and (3) the effect of the moment of the facet stress traction ${\mathbf}{t}$, transferred to the particle node, around the centroid of the RVE. As result, the moment stress tensor is characterized by three length scales: (1) the facet size, associated to ${\mathbf}{m}$; (2) the particle size or facet spacing; and (3) the size of the RVE.
Numerical Results {#NumRes}
=================
The homogenization theory formulated and discussed in the previous sections was implemented in the MARS computational software [@mars-1] with the objective of upscaling the Lattice Discrete Particle Model (LDPM). LDPM, formulated, calibrated, and validated by Cusatis and coworkers [@cusatis-ldpm-1; @cusatis-ldpm-2], is a meso-scale discrete model which simulates the mechanical interaction of concrete coarse aggregate pieces. LDPM has shown superior capabilities in modeling concrete behavior under dynamic loading [@cusatis-Jovanca; @cusatis-alonerate], Alkali Silica Reaction (ASR) deterioration [@cusatis-mohammed], as well as failure and fracture of fiber-reinforced concrete [@cusatis-Ed1; @cusatis-Ed2].
The complete LDPM formulation is summarized in Appendix \[LDPM\]. It is worth mentioning here that the LDPM computational units are polyhedral cells whose construction is anchored to the Delaunay triangulation of the simulated concrete aggregate pieces that are assumed to be spherical and size-graded according to the Fuller size distribution. In the LDPM formulation, each polyhedral cell represents one concrete spherical aggregate piece embedded in the surrounding mortar and the interfaces among the cells represent potential mortar cracks. Figure \[PeriodicLDPM\]a shows a typical LDPM system of polyhedral cells and Figure \[PeriodicLDPM\]b its periodic approximation.
[0.45]{} ![Polyhedral particle distribution in a LDPM prism: (a) generic LDPM system, (b) Periodic LDPM system.[]{data-label="PeriodicLDPM"}](CellPrism.pdf "fig:")
[0.45]{} ![Polyhedral particle distribution in a LDPM prism: (a) generic LDPM system, (b) Periodic LDPM system.[]{data-label="PeriodicLDPM"}](CellPrismPeriodic.pdf "fig:")
The generic RVE shown in Figure \[PeriodicLDPM\]b is constructed as follows. Eight nodes are created at the vertexes of a cube (Figure \[RVEgen\]a). Then nodes are randomly placed on a RVE edge parallel to $x$ axis, see node $a$ in Figure \[RVEgen\]b. Then, these nodes are duplicated on the other three parallel edges along the $x$ axis, see nodes $b, c$, and $d$ in Figure \[RVEgen\]b. Similar procedure is carried out over the edges parallel to $y$ and $z$ axes. Next, the node generation on the RVE surfaces is performed by randomly placing nodes on a cube face with $z$ axis as normal vector, see node $e$ in Figure \[RVEgen\]c. The same nodes are then duplicated on the opposite RVE faces, see node $f$ in Figure \[RVEgen\]c. Nodes on parallel cube faces with $x$ and $y$ axes as normal vectors are constructed with the same algorithm. Finally, nodes are placed inside the RVE based on the general LDPM procedure (see Appendix \[LDPM\] and relevant publications [@cusatis-ldpm-1] for details).
As mentioned earlier in this paper the RVE analysis is conducted by imposing periodic boundary conditions. This is obtained by setting the displacements and rotations of the RVE vertexes to be zero and by imposing, through a master-slave constraint, that the periodic edge nodes and face nodes have the same rotations and displacements.
The overall multiscale numerical procedure adopted in this paper can be summarized as follows.
- The finite element method is employed to solve the macro-scale homogeneous problem in which external loads and essential BCs are applied incrementally. During each numerical step, strain increments $\Delta{\gamma}_{ij} = \Delta{v}^{0}_{j,i} - \varepsilon_{ijk} \Delta{\varphi}_k^{0}$ and curvature increments $\Delta{\kappa}_{ij}=\Delta{\omega}_{j,i}^{0}$ tensors are calculated at each integration point based on the nodal displacement and rotation increments of the corresponding finite element.
- The macroscopic strain and curvature increments are projected into the RVE facets through the proper projection operators: $\Delta{\epsilon}^c_{\alpha} = P^\alpha_{ij} \left(\Delta{\gamma}_{ij} + \varepsilon_{jmn} \Delta{\kappa}_{im} y_{n}^{c} \right)$ and $\Delta{\psi}^c_{\alpha} = P^\alpha_{ij} \Delta{\kappa_{ij}}$. These projected strains and curvatures are imposed, upon sign change, as eigen-strains and eigen-curvatures, $\Delta{\epsilon}^0_{\alpha} =\Delta{\epsilon}^c_{\alpha}+\Delta{\epsilon}^f_{\alpha}=\Delta{\epsilon}^c_{\alpha}- (-\Delta{\epsilon}^f_{\alpha})$ and $\Delta{\psi}^0_{\alpha} =\Delta{\psi}^c_{\alpha}+\Delta{\psi}^f_{\alpha}=\Delta{\psi}^c_{\alpha}- (-\Delta{\psi}^f_{\alpha})$ (See section \[RVEproblem\]), to the RVE allowing the calculation of the fine-scale solution governed by the fine-scale constitutive equations.
- Finally, the fine-scale facet tractions and moments are used to compute, through Equations \[macro-stress-formula\] and \[macro-momentstress-formula\], the macroscopic stresses, $\sigma^0_{ij}$, and couple stresses, $\mu_{ij}^0$, for each Gauss point in the FE mesh.
Elastic RVE Analysis {#Elastic Analysis}
--------------------
This section presents the analysis of the elastic macroscopic behavior of one LDPM RVE. The macroscopic homogenized behavior is analyzed with reference to the classical constitutive equation for Cosserat elasticity, which, in non-dimensional variables, can be written as:
$$\label{Cauchy-Constitutive}
\hat{\sigma}_{ij} = p_0\hat{\gamma}_{kk} \delta_{ij} + p_1\hat{\gamma}_{(ij)} + p_2 \hat{\gamma}_{[ij]}~; ~~~\hat{\mu}_{ij} = q_0 \hat{\kappa}_{kk} \delta_{ij} + q_1 \hat{\kappa}_{(ij)} + q_2 \hat{\kappa}_{[ij]}$$
where $\hat{\sigma}_{ij} = \sigma_{ij} / (2 \mu + \chi)$ and $\hat{\mu}_{ij}= L\mu_{ij} /[(2 \mu+\chi)D^2]$ are the normalized stress and couple tensors, $L$ = characteristic size of the structure of interest, $D$ = size of the RVE; $\hat{\gamma}_{ij}=\gamma_{ij}$, $\hat{\kappa}_{ij} =L \kappa_{ij}$ normalized strains and curvatures; $p_0=\lambda/ (2 \mu + \chi) $, $p_1=1$; $p_2 = \chi / (2 \mu+\chi)$; $q_0=\pi_1/[(2 \mu + \chi)D^2]$, $q_1=(\pi_2+\pi_3)/[(2 \mu + \chi)D^2]$; $q_2=(\pi_2-\pi_3)/[(2 \mu + \chi)D^2]$; $\delta_{ij}$ = kronecker delta; $\mu$, $\lambda$, $\chi$, $\pi_1$, $\pi_2$, and $\pi_3$ are the elastic constants; and the subscript parentheses and brackets represent extraction of the symmetric and antisymmetric, respectively, part of the tensors.
In this section, eight different LDPM RVE sizes $D$= 15, 20, 25, 35, 50, 75, 100, and 150 mm are considered and 5 RVEs, characterized by different placement of the aggregate pieces, is studied for each case. It is worth mentioning that, in LDPM, different spherical aggregate placement inside the RVE yield to different RVE polyhedral particle configurations. The numerical calculations were performed by assuming the concrete mix design and model parameters reported in Appendix \[LDPM\]. Figure \[young-poisson-norm\]a shows the homogenized values of $p_0$, $p_2$, and of the normalized Young’s modulus defined as $e=E/(2\mu +\chi)=(3 \lambda +2\mu +\chi)/(2 \lambda +2\mu +\chi)$, as function of the RVE size normalized by the maximum spherical aggregate size, $d=D/d_a$. The error bars represent the scatter in the results obtained by simulating 5 different RVEs of the same size but with different realization of spherical aggregate positions inside the RVE. As one can see, the calculated values of the parameters tend to converge to a constant value as the size of the RVE increases and, at the same time, the results become independent of the spherical aggregate distribution inside the RVE. The value of $p_2$ is very close to zero for all RVE sizes and decreases rapidly with respect to the RVE size; this suggests that, for the analyzed fine-scale model, the homogenized stress tensor is symmetric. This result is due to the fact that in the LDPM formulation facet moments are zero, and this leads to facet traction distributions around each particle that have zero moment resultant around the particle node. In Figure \[young-poisson-norm\]b the homogenized Poisson’s ratio is reported based on the equation $\nu = \lambda / (2 \lambda +2\mu +\chi)$ and the calculated asymptotic value, 0.18, corresponds well with the value of 0.175 calculated by exploiting the equivalence between particle models and microplane models [@cusatis-ldpm-1]. Finally, Figure \[young-poisson-norm\]c shows the homogenized parameters, $q_0$, $q_1$, and $q_2$, as a function of the RVE size. These quantities also converge to an asymptotic value and become independent of the RVE spherical aggregate distribution for large enough value of $D/d_a$. By virtue of these results and by recalling the definitions of $q_0$, $q_1$, and $q_2$, it is interesting to note that the macroscopic Cosserat elastic parameters of the homogenized continuum depend quadratically on the RVE size.
[0.32]{} ![Variation of elastic normalized effective material properties: (a) $p_0$, $p_2$ and normalized Young modulus $E$. (b) $\nu$ Poisson’s ratio. (c) $q_0$, $q_1$ and $q_2$, with respect to the ratio of RVE size to maximum spherical aggregate size.[]{data-label="young-poisson-norm"}](p-Constants.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Variation of elastic normalized effective material properties: (a) $p_0$, $p_2$ and normalized Young modulus $E$. (b) $\nu$ Poisson’s ratio. (c) $q_0$, $q_1$ and $q_2$, with respect to the ratio of RVE size to maximum spherical aggregate size.[]{data-label="young-poisson-norm"}](Poisson-Norm.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Variation of elastic normalized effective material properties: (a) $p_0$, $p_2$ and normalized Young modulus $E$. (b) $\nu$ Poisson’s ratio. (c) $q_0$, $q_1$ and $q_2$, with respect to the ratio of RVE size to maximum spherical aggregate size.[]{data-label="young-poisson-norm"}](q-constants.pdf "fig:"){width="\textwidth"}
Nonlinear RVE Analysis
----------------------
In this section, the nonlinear response of the RVE is investigated under different strain and curvature loading conditions. Three different RVE sizes, $D=$25, 50, and 100 mm, and 7 different spherical aggregate placement inside the RVE are considered for each case. Typical polyhedral particle systems and geometry of each RVE size are shown in Figure \[RVEgeom\]. The nonlinear homogenized behavior of the RVE is studied under the effect of uniaxial strain tension and compression, hydrostatic compression, bending and torsional curvatures. In the following numerical examples, concrete mix design and model parameters are the same as the ones used in the elastic analysis.
[0.1]{} ![RVE geometry and polyhedral particle distribution: (a) 25 mm (b) 50 mm (c) 100 mm[]{data-label="RVEgeom"}](UC-Cell-25.pdf "fig:"){width="90.00000%"}
[0.3]{} ![RVE geometry and polyhedral particle distribution: (a) 25 mm (b) 50 mm (c) 100 mm[]{data-label="RVEgeom"}](UC-Cell-50.pdf "fig:"){width="55.00000%"}
[0.3]{} ![RVE geometry and polyhedral particle distribution: (a) 25 mm (b) 50 mm (c) 100 mm[]{data-label="RVEgeom"}](UC-Cell-100.pdf "fig:"){width="100.00000%"}
### Nonlinear Analysis of RVE subject to components of the strain tensor {#non-ten}
Figure \[stress-strainNonlinear-tens\] shows the homogenized stress-strain curves for different RVE sizes and polyhedral particle realizations relevant to RVEs subjected to uniaxial tensile strain. The results illustrate that the different polyhedral particle realizations do not affect the linear elastic and nonlinear pre-peak responses, but on the other hand, it clearly influences on the post-peak softening response. One can notice that the post-peak response of smaller RVE sizes is more scattered, while fine-scale randomness effect on the homogenized response diminishes for the larger RVEs [@Gitman-1; @Nguyen-1]. Therefore, one can conclude that the mesh realization is a more influential factor on the post-peak softening response of the RVEs of smaller sizes. Average of peak stress and strain values of different mesh realizations are calculated for each RVE size, and its variation with respect to the RVE size is plotted in Figures \[PeakStress\] and \[PeakStrain\]. As one can see these quantities as well as mesh realization effect decrease as the size of the RVE increases.
[0.3]{} ![Macroscopic stress-strain curve for three different RVE size: 25mm, 50mm, 100mm under uni-axial tension[]{data-label="stress-strainNonlinear-tens"}](25_sig_str.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Macroscopic stress-strain curve for three different RVE size: 25mm, 50mm, 100mm under uni-axial tension[]{data-label="stress-strainNonlinear-tens"}](50_sig_str.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Macroscopic stress-strain curve for three different RVE size: 25mm, 50mm, 100mm under uni-axial tension[]{data-label="stress-strainNonlinear-tens"}](100_sig_str.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Variation of (a) average peak stress and (b) average peak strain, with respect to the RVE size.[]{data-label="PeakStat"}](sig_peak.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Variation of (a) average peak stress and (b) average peak strain, with respect to the RVE size.[]{data-label="PeakStat"}](str_peak.pdf "fig:"){width="\textwidth"}
Furthermore, the average stress-strain curves of different polyhedral particle configurations for each RVE size are calculated and plotted in Figure \[AverageSigmaTens\]. As one can see clearly, increasing size of the RVE affects the post-peak behavior and increases the brittleness of the response. This is consistent with the well-known size effect associated to damage localization in quasi-brittle materials [@Bazant-Book].
[0.3]{} ![(a) Average tensile stress-strain curves for three different RVE sizes. (b) Coarse- and fine-scale strain energy density for different RVE sizes.[]{data-label="TenSigEneAvg"}](Avg_sig_str.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Average tensile stress-strain curves for three different RVE sizes. (b) Coarse- and fine-scale strain energy density for different RVE sizes.[]{data-label="TenSigEneAvg"}](TensileEne.pdf "fig:"){width="\textwidth"}
This phenomenon is depicted in Figure \[rve-damaged-tens\], which shows damaged RVEs of different sizes at the end of the tensile loading process. The contour plots present meso-scale crack opening distributions corresponding to macroscopic imposed uniaxial strain equal to $10^{-3}$. One can easily notice that the damaged area does not scale with the RVE size leading to the post peak size dependency on the RVE size.
Evolution of damage for a 100 mm RVE is also shown in Figure \[Curve100\] at five different macroscopic strain levels. Strain levels (1) and (2) are in pre-peak regime, in which damage is distributed throughout the RVE, which corresponds to the fact that homogenized response is not size dependent in the pre-peak regime. At strain level (3) which corresponds to the peak of the stress-strain curve, damage is still distributed over the RVE; However, as the material undergoes softening, damage localization initiates. Strain levels (4) and (5) are relevant to the softening branch of the response, in which damage localization is clearly visible. The size dependence of the homogenized softening RVE response leads to mesh-dependence of the macroscopic response. This issue has been investigated by some authors [@Gitman-1; @Nguyen-1; @Gitman-2] with reference to continuum-based fine scale models. The complete analysis of this aspect with reference to the current LDPM-based homogenization scheme will be pursed in future work by the writers.
Finally, in Figure \[TensileEne\], the Hill-Mandel condition is verified by comparing the RVE strain energy density calculated through fine-scale and macroscopic quantities.
Next, the nonlinear homogenized behavior of the RVE is studied under confined (uniaxial strain) and hydrostatic compression. For the confined compression test, a strain tensor with a longitudinal component up to -0.03 is considered, whereas for the hydrostatic compression case, all normal components of the strain tensor are set equal and with value up to -0.03. Figure \[stress-strainNonlinear-comp\] shows the nonlinear response of RVEs of different sizes and 7 different polyhedral particle configurations.
[0.3]{} ![Volumetric stress-strain curve for three different RVE sizes under confined compression and hydrostatic compression[]{data-label="stress-strainNonlinear-comp"}](25NonComp.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Volumetric stress-strain curve for three different RVE sizes under confined compression and hydrostatic compression[]{data-label="stress-strainNonlinear-comp"}](50NonComp.pdf "fig:"){width="\textwidth"}
[0.3]{} ![Volumetric stress-strain curve for three different RVE sizes under confined compression and hydrostatic compression[]{data-label="stress-strainNonlinear-comp"}](100NonComp.pdf "fig:"){width="\textwidth"}
In this case, due to the confinement, the stress-strain response is strain-hardening, and as one can see the different polyhedral particle realizations do not affect significantly the homogenized response in both the elastic and inelastic regime. In addition, the average of different mesh realization stress-strain responses is calculated and plotted for each RVE size in Figure \[AverageSigmaComp\]. The nonlinear compressive response does not depend on the RVE size, which is consistent with the fact that plastic deformations are distributed through out the specimen, and strain localization does not take place. Finally, the Hill-Mandel condition is verified with reference to the confined compression test, and the fine- and coarse-scale strain energy density of different RVE sizes are plotted in Figure \[compEne\].
[0.3]{} ![(a) Average compressive volumetric stress-strain curves for three different RVE sizes. (b) Coarse- and fine-scale strain energy density for different RVE sizes.[]{data-label="AverageSigmaComp-compEne"}](AverageCompNon.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Average compressive volumetric stress-strain curves for three different RVE sizes. (b) Coarse- and fine-scale strain energy density for different RVE sizes.[]{data-label="AverageSigmaComp-compEne"}](CompEne.pdf "fig:"){width="\textwidth"}
### Nonlinear Analysis of RVE subject to components of the curvature tensor {#non-cur}
In this section, the nonlinear homogenized behavior of RVEs of 3 different sizes, 50, 75, and 100 mm and 5 five different mesh configurations for each size, is studied under the effect of components of macroscopic curvature tensor. Bending and torsional behavior of the RVEs are investigated by applying macroscopic curvature tensors with the only non-zero components of $\kappa_{12} = 1$ and $\kappa_{11} = 1$, respectively. Figure \[rve-damaged-curv\] shows crack opening contour of damaged RVEs at the macroscopic curvature for $\kappa_{12} = 0.5$. The resulting crack pattern conforms with the fracture mode that one may expect from bending theories. Multiple crack lines are generated in the tensile strain domain, which is the top half of the RVEs, while half bottom part in under compression. As more strain is applied in compressive part, splitting cracks take place in the latter region due to transverse tensile stress. Typical crack pattern of RVEs under torsion are plotted in Figure \[rve-damaged-tor\] for $\kappa_{11} = 0.5$. Crack opening contours show that the amount of damage close to the RVE center is negligible, while it increases as the facets are placed at further positions. This corresponds to the deformation mechanism and strain distribution in solids subject to torsion.
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Sig_12_all_n.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Sig_all_avg_n.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Sig_scaled_11-12.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Ene_all_n.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Ene_12_scaled.pdf "fig:"){width="\textwidth"}
[0.3]{} ![(a) Homogenized couple stresses $\mu_{11}$ and $\mu_{12}$ versus curvature $\kappa_{11}$ and $\kappa_{12}$ for five different polyhedral particle configurations for each RVE size. (b) Average of the homogenized couple stress of different polyhedral particle configurations for each RVE size for $\kappa_{11}$ and $\kappa_{12}$ cases. (c) Scaled couple stress versus curvature curves. (d) Macro and Fine-Scale strain energy density evolution. (e) Scaled strain energy density evolution for the case $\kappa_{12}$. (f) Trace of stress tensor due to elastic and nonlinear analysis of RVE under macroscopic $\kappa_{12}$.[]{data-label="NC"}](Trace_n.pdf "fig:"){width="\textwidth"}
Homogenized moment stress components $\mu_{12}$ and $\mu_{11}$ versus macroscopic curvature tensor components $\kappa_{12}$ and $\kappa_{11}$ of RVEs of different sizes and polyhedral particle configurations are plotted in Figure \[mu\_all\]. One can see that effect of different polyhedral particle realizations on the homogenized response is negligible, which is due to the occurrence of distributed damage inside the RVE. Homogenized response of RVEs with different polyhedral particle realizations are averaged for each size and plotted in Figure \[mu\_all\_avg\]. It can be seen that the homogenized response consists of an initial elastic part and a hardening branch, which is related to the confinement due to the fact that all components of the strain tensor are zero, and the RVE cannot expand laterally. It is illustrated that, at any level of macroscopic curvature, magnitude of the moment stress for larger RVE sizes is bigger compared to the smaller ones. Size dependency of moment stress was discussed in Section \[Elastic Analysis\], and it was shown that the elastic Cosserat coefficients are proportional to the RVE size squared. In order to study size dependency in the nonlinear regime, the normalized quantities $\hat{\mu}_{ij} = {\mu}_{ij}/D$ and $\hat{k}_{ij} = k_{ij} \times D$ are plotted in Figure \[mu\_avg\_scaled\]. One can see that the normalized curves of three different RVE sizes are unique for both bending and torsion. This implies that the proportionality of the homogenized micropolar properties to the RVE size squared is still valid in the nonlinear regime. In Figure \[Ene\_all\], the Hill-Mandel condition is verified, and coarse- and fine-scale strain energy density are plotted for each RVE size for both the aforementioned cases.
Finally, the existence of coupling effect characterized by the dependency of the homogenized stress tensor on the curvature tensor is investigated in both elastic and nonlinear regimes. The macroscopic curvature $\kappa_{12}=1$ is applied on the RVEs, and the trace of the homogenized stress tensor is calculated and plotted in Figure \[trace\] for different RVE sizes. One can observe that the trace of the stress tensor for the case of elastic RVE behavior is zero throughout the analysis. On the other hand, for the case of nonlinear behavior, it increases monotonically with the curvature. This implies that in elastic regime, stresses and strains are totally uncoupled from couple stresses and curvatures, whereas these quantities are strongly coupled in the nonlinear case. This aspect has been investigated very little in the literature where fully uncoupled behavior has been always postulated.
Tension Test on a Concrete Prism with Parallel Elastic Bars {#prismExample}
-----------------------------------------------------------
In this section, the behavior of a reinforced concrete prism under tension is studied in a full fine-scale simulation, and the obtained results are compared to the solution of the same problem through a two-scale homogenization algorithm, in which the concrete prism is modeled as a homogeneous continuum with a meso-scale material RVE assigned to every macroscopic integration point. Figure \[fullldpm-Cbar\] shows the concrete prism and the two elastic bars attached to it, which are simulated by LDPM and solid finite elements, respectively. The same specimen in a two-scale homogenization problem is depicted in Figure \[homogenization-Cbar\], in which concrete prism is modeled by tetrahedral finite elements. Cross section of the concrete prism is 100 mm $\times$ 100 mm, and its height is 500 mm. Two rigid loading plates are attached at the top and the bottom of the whole specimen cross section to apply the boundary condition. Young modulus and Poisson’s ratio of the elastic bars are 28 GPa and 0.18, respectively. The same LDPM parameters used in the previous sections are adopted here. The specimen is pulled in the longitudinal direction up to a displacement equal to 0.7 mm. The RVE size is chosen to be 30 mm which approximately corresponds to the volume of each tetrahedral FE in the coarse mesh. This is done to mitigate the mesh-dependence due to the softening behavior of the RVE. The concrete prism and the elastic bars are connected through a master-slave algorithm. The numerical simulations of the coarse scale are performed by neglecting the couple stresses which are expected to be negligible for this particular application.
[0.2]{} 
[0.3]{} 
[0.4]{} 
The global force-displacement response of the full fine-scale and homogenization problems are plotted in Figure \[stress-strain-curve\]. Since concrete prism and elastic bars are tied and deform together during the loading process, distributed damage takes place through the whole specimen during the initial stages of the loading process, see Figure \[prism\]a. This damage state represents the linear elastic and the first hardening segment of the stress-strain response of the structure. The same damage state is captured through the homogenization procedure. Figure \[prism\]e shows finite elements normal strain distribution along the loading direction through the specimen. One can see that the strain values are all in the same range, and no localization has occurred. The response of the full fine-scale and homogenization problems show excellent agreement in the elastic and the first hardening segment. As further deformation is applied on the structure, damage localizes in one section of the concrete bar and this causes a sudden drop in the global force-displacement curve. Subsequently, since the elastic bars and the concrete prism are forced to deform in parallel, the overall system can carry more load leading to a rehardening of the global response. Analysis of Figure \[stress-strain-curve\] shows that five damage localization events occur during the deformation process which corresponds to five sudden drops in the load-displacement curve. Crack pattern of the specimen is plotted after the formation of two, four, and five damage localization in Figure \[prism\]b, c, and d. It is interesting to show that the homogenization framework is able to generate the same damage distribution pattern. Figures \[prism\]f, g, and h show that two, four and five strain localization band appear in the specimen, which corresponds to the damage configuration obtained from full fine-scale problem. The global load-displacement curve of the homogenization problem also shows five sudden drops which conforms to the full fine-scale response, see Figure \[stress-strain-curve\]. The homogenized response captures well the displacement at which the first three localization events occur, while it underestimates its value for later events. This is likely due to the relatively coarse mesh adopted at the macroscopic scale.
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](cntr1.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](cntr2.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](cntr3.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](cntr4.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](Homog1.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](Homog2.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](Homog3.pdf "fig:")
[0.2]{} ![(Top row) Crack opening contour at different loading from full fine-scale simulation (Bottom row) Strain distribution contour at different loading steps from homogenization algorithm[]{data-label="prism"}](Homog4.pdf "fig:")
Conclusions
===========
This paper presents the asymptotic expansion homogenization of fine-scale periodic discrete systems featuring independent translational and rotational degrees of freedom. Employing consistent asymptotic expansion of displacement and rotation fields, a rigorous analytical derivation was performed for elastic behavior, and it was extended to the nonlinear case upon making reasonable assumptions on the rigid body motions of a RVE. Based on this work, the following general conclusions can be drawn.
- The equivalent homogenized continuum is of Cosserat-type characterized by nonsymmetric stress and couple tensors energetically conjugate to nonsymmetric strain and curvature tensors, respectively. The classical linear and rotational momentum balance equations can be derived from the homogenization of the fine-scale equilibrium equations.
- The fine-scale kinematic quantities, namely facet strains and curvatures, are demonstrated to be related to the projection of the coarse-scale strains and curvatures into the local facet system of reference. This allows a straightforward implementation of the RVE problem into any computational framework.
- Similarly to previous research, the derived formula linking the fine-scale response to the coarse-scale stress tensor corresponds to the virial stress formulation commonly used for atomistic systems.
- The derived formula linking the fine-scale response to the coarse-scale couple tensor is shown to consist of three terms with clear physical meaning. The first term is associated with the fine-scale couple tractions and it can be related to the facet size, which, in turn can be associated with the size of weak spots in the material internal structure. The second term arises from the moment of the fine-scale stress tractions with respect to the particle node. As such, it depends on the size of the fine-scale particles and it can be related to the spacing or characteristic distance of the weak spots in the material internal structure. Finally, the third term is the effect of the moment of fine-scale stress tractions with respect to the center of the RVE and, consequently, it depends on the RVE size.
The developed framework was then implemented in a computational software and applied to the upscaling of LDPM. Specific to this fine-scale model, the numerical results demonstrate the following interesting features of the equivalent homogenized continuum.
- The macro-scale elastic parameters relating the stress tensor to the strain tensor become independent on RVE size and on the random position of the polyhedral particles inside the RVE for RVE sizes larger than about 5 times the maximum spherical aggregate size. On the contrary, the macro-scale parameters relevant to the relationship between curvature and couple tensors are shown to depend on the RVE size squared and they become independent on the random position of the polyhedral particles inside the RVE for RVE sizes larger than about 5 times the maximum spherical aggregate size.
- The non-symmetric part of the macro-scale stress tensor is negligible since the relevant parameter is at least one order of magnitude smaller than the one governing the symmetric part. As a consequence, the linear and rotational momentum balance equations are decoupled.
- In the elastic regime the stress-strain and couple-curvature constitutive equations are completely uncoupled.
- In the non linear regime, for tensile loading and because the fine-scale behavior is strain-softening, the response is RVE-size-dependent. This is an expected result, although very often not acknowledged by most authors in the literature, associated with strain localization induced by softening. On the contrary, such dependence is not observed for compressive dominated loading conditions because the LDPM fine-scale behavior in compression is strain-hardening.
- The coarse-scale couple-curvature constitutive equations scale with the square of the RVE size in the nonlinear range also but, contrarily to the elastic case, they show a strong coupling with the stress-strain constitutive equations. Such coupling, never considered in the current literature of Cosserat media, will be studied in future work by the authors.
ACKNOWLEDGEMENTS
================
This material is based upon work supported by the National Science Foundation under grant no. CMMI-1201087.
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Short Review of the Lattice Discrete Particle Model (LDPM) Geometrical Construction and Constitutive Equations {#LDPM}
==============================================================================================================
LDPM model generation procedure and governing constitutive equations are explained in the following two sections.
LDPM model construction {#LDPM-Construction}
-----------------------
Concrete meso-scale structure is modeled by LDPM through the following steps:
- Spherical aggregate generation is the first step which is carried out assuming that each aggregate piece can be approximated as a sphere. Under this assumption, the following spherical aggregate size distribution function proposed by Stroeven [@Stroeven-1] is considered
$$\label{psd}
f(d) = \frac{qd_0^q}{[1-(d_0/d_a)^q]d^{q+1}}$$
in which $d_a$ and $d_0$ are the maximum and minimum spherical aggregate size, respectively, and $q$ is a material parameter. It can be shown [@Stroeven-1] that Equation \[psd\] is associated with a sieve curve (percentage of spherical aggregate by weight retained by a sieve of characteristic size $d$) in the form
$$\label{sieve}
f(d) = \bigg(\frac{d}{d_a}\bigg)^{n_f}$$
where $n_f = 3-q$. For $n_f=0.5$ Equation \[sieve\] corresponds to the classical Fuller curve which for its optimal packing properties, is extensively used in concrete technology. Considering concrete cement content $c$, water-to-cement ratio $w/c$, specimen volume, maximum $d_a$ and minimum $d_0$ spherical aggregate size along with the considered distribution function Equation \[sieve\], the spherical aggregate system can be generated using a random number generator.
- By using a try-and-error random procedure, spherical aggregate pieces are introduced into the concrete volume from the largest to the smallest size. Figure \[DogbonePRTC\] shows the spherical aggregate system generated for a typical dogbone specimen.
- Delaunay tetrahedralization of the spherical aggregate piece centers is employed to define the interactions of the spherical aggregate system (Figure \[PolyCellsGeom\]).
- Finally, a three-dimensional domain tessellation anchored to the Delaunay tetrahedralization is carried out to create a system of polyhedral particles interacting through triangular facets, and a lattice system composed of the line segments connecting the spherical aggregate centers. Figure \[DogboneCells\] shows the final polyhedral particle discretization of a typical dogbone specimen.
[0.3]{} ![(a) Spherical aggregate system for a typical dogbone specimen. (b) LDPM polyhedral particles for two adjacent spherical aggregate particle. (c) LDPM cell distretization for a typical dogbone specimen.[]{data-label="LDPMFigures"}](DogbonePRTC.pdf "fig:")
[0.3]{} ![(a) Spherical aggregate system for a typical dogbone specimen. (b) LDPM polyhedral particles for two adjacent spherical aggregate particle. (c) LDPM cell distretization for a typical dogbone specimen.[]{data-label="LDPMFigures"}](PolyCellsParticle.pdf "fig:")
[0.3]{} ![(a) Spherical aggregate system for a typical dogbone specimen. (b) LDPM polyhedral particles for two adjacent spherical aggregate particle. (c) LDPM cell distretization for a typical dogbone specimen.[]{data-label="LDPMFigures"}](Dogbone-Cell.pdf "fig:")
LDPM Kinematics {#LDPM-Kinematics}
---------------
The triangular facets forming the rigid polyhedral particles are assumed to be the potential material failure locations. Each facet is shared between two polyhedral particle and is characterized by a unit normal vector $\bf{n}$ and two tangential vectors $\bf{m}$ and $\bf{l}$. Accordingly, three strain components are defined on each triangular facet using Equations \[eps\] and \[curvature\], which for LDPM gives
$$\label{LDPMstr}
\epsilon_{N} = \frac{\mathbf{n}^T \llbracket {\mathbf{u}_{C}} \rrbracket}{r}; \hspace{0.25 in} \epsilon_{M} = \frac{\mathbf{m}^T \llbracket {\mathbf{u}_{C}} \rrbracket}{r}; \hspace{0.25 in} \epsilon_{L} = \frac{\mathbf{l}^T \llbracket {\mathbf{u}_{C}} \rrbracket}{r}$$
where $\llbracket {\mathbf{u}_{C}} \rrbracket$ is the displacement jump vector calculated at the facet centroid. One should consider that the LDPM constitutive equations explained in the next section are independent of facet curvatures.
LDPM constitutive equations {#LDPM-Constitutive}
---------------------------
This section reviews the specific constitutive equations governing the response of LDPM. First of all, it must be mentioned that LDPM assumes zero couple stresses at the meso-scale in both elastic and inelastic regime. This implies $m_\alpha=0$ for $\alpha=N,M,L$
In the elastic regime, the normal and shear stresses are proportional to the corresponding strains: $t_{N}= E_N \epsilon_{N};~ t_{M}= E_T \epsilon_{M};~ t_{L}= E_T \epsilon_{L}$, where $E_N=E_0$, $E_T=\alpha E_0$, $E_0=$ effective normal modulus, and $\alpha=$ shear-normal coupling parameter. Beyond the elastic regime, the vectorial constitutive relations are meant to reproduce three distinct sources of nonlinearity as described below.
### Fracture and cohesion due to tension and tension-shear {#LDPM-tens}
For tensile loading ($\epsilon_N>0$), fracturing and cohesive behavior due to tension and tension-shear are formulated through an effective strain, $\epsilon = \sqrt{\epsilon _{N}^{2}+\alpha (\epsilon _{M}^{2} + \epsilon _{L}^{2})}$, and stress, $t = \sqrt{{ t _{N}^2+ (t_{M}+t_{L})^2 / \alpha}}$, which define the normal and shear stresses as ; ; . The effective stress $t$ is incrementally elastic ($\dot{t}=E_0\dot{\epsilon}$) and must satisfy the inequality $0\leq t \leq \sigma _{bt} (\epsilon, \omega) $ where $\sigma_{bt} = \sigma_0(\omega) \exp \left[-H_0(\omega) \langle \epsilon-\epsilon_0(\omega) \rangle / \sigma_0(\omega)\right]$, $\langle x \rangle=\max \{x,0\}$, and $\tan(\omega) =\epsilon _N / \sqrt{\alpha} \epsilon _{T}$ = $t_N \sqrt{\alpha} / t_{T}$. The post peak softening modulus is defined as $H_{0}(\omega)=H_{t}(2\omega/\pi)^{n_{t}}$, where $n_t$ is the softening exponent, $H_{t}$ is the softening modulus in pure tension ($\omega=\pi/2$) expressed as $H_{t}=2E_0/\left(l_t/l_e-1\right)$; $l_t=2E_0G_t/\sigma_t^2$; $l_e$ is the length of the tetrahedron edge; and $G_t$ is the mesoscale fracture energy. LDPM provides a smooth transition between pure tension and pure shear ($\omega=0$) with parabolic variation for strength given by $\sigma_{0}(\omega )=\sigma _{t}r_{st}^2\Big(-\sin(\omega)+ \sqrt{\sin^2(\omega)+4 \alpha \cos^2(\omega) / r_{st}^2}\Big)/ [2 \alpha \cos^2(\omega)]$, where $r_{st} = \sigma_s/\sigma_t$ is the ratio of shear strength to tensile strength.
### Compaction and pore collapse from compression {#LDPM-comp}
For compressive loading ($\epsilon_N<0$), the normal stress evolves incrementally elastically and is subjected to the inequality $-\sigma_{bc}(\epsilon_D, \epsilon_V)\leq t_N \leq 0$ where $\sigma_{bc}$ is a strain-dependent boundary function of the volumetric strain, $\epsilon_V$, and the deviatoric strain, $\epsilon_D$. The function expressing $\sigma_{bc}$ models pore collapse for $-\epsilon_V \leq\epsilon_{c1} = \kappa_{c0} \epsilon_{c0}=\kappa_{c0} \sigma_{c0}/E_0$, and it is formulated as $\sigma_{bc} = \sigma_{c0} + \langle-\epsilon_V-\epsilon_{c0}\rangle H_c(r_{DV})$ where $H_c(r_{DV})=H_{c0}/(1 + \kappa_{c2} \left\langle r_{DV} - \kappa_{c1} \right\rangle)$, $r_{DV}=\epsilon_D/\epsilon_V$, $\sigma_{c0}$ is the mesoscale compressive yield stress; and $\kappa_{c0}$, $\kappa_{c1}$, $\kappa_{c2}$ and $H_{c0}$ are material parameters. Compaction and rehardening occur beyond pore collapse for $-\epsilon_V \geq \epsilon_{c1}$. In this case one has $\sigma_{bc} = \sigma_{c1}(r_{DV})$ $\exp \left[( -\epsilon_{V}-\epsilon_{c1} ) H_c(r_{DV})/\sigma_{c1}(r_{DV}) \right]$ and $\sigma_{c1}(r_{DV}) = \sigma_{c0} + (\epsilon_{c1}-\epsilon_{c0}) H_c(r_{DV})$.
### Friction due to compression-shear {#LDPM-shear}
The evolution of shear stresses simulate frictional behavior due to compression-shear. The incremental shear stresses are computed as $\dot{t}_M=E_T(\dot{\epsilon}_M-\dot{\epsilon}_M^p)$ and , where , , and $\lambda$ is the plastic multiplier with loading-unloading conditions $\varphi \dot{\lambda} \leq 0$ and $\dot{\lambda} \geq 0$. The plastic potential is defined as , where the nonlinear frictional law for the shear strength is assumed to be $\sigma_{bs} = \sigma_s + (\mu_0 - \mu_\infty)\sigma_{N0}[1 - \exp(t_N / \sigma_{N0})] - \mu_\infty t_N$; $\sigma_{N0}$ is the transitional normal stress; $\mu_0$ and $\mu_\infty$ are the initial and final internal friction coefficients. Detailed description of model behavior in the nonlinear range can be found in Ref. [@cusatis-ldpm-1].
Concrete Mix-Design and Model Parameters Used in the Numerical Simulations
--------------------------------------------------------------------------
Minimum and maximum spherical aggregate size are $d_0=$ 4 mm and $d_a=$ 8 mm, respectively; cement content c = 612 $\text{kg/m}^\text{3}$; water to cement ratio w/c = 0.4; aggregate to cement ratio a/c = 2.4; Fuller curve coefficient $n_f$ = 0.42.
The following LDPM parameters are used: $E_N = 60$ GPa, $\sigma_t = 3.45$ MPa, $\sigma_{c0} = 150$ MPa, $\alpha = 0.25$, $n_t=0.4$, $l_t=500$ mm, $r_{st}=2.6$, $H_{c0}/E_0 = 0.4$, $\mu_0 = 0.4$, $\mu_\infty = 0$, $k_{c1} = 1$, $k_{c2}=5$, $\sigma_{N0} = 600$ MPa, $\alpha=E_T/E_N=0.25$.
Asymptotic Expansion of Strains and Curvatures {#exp-strains-details}
==============================================
In order to obtain multiple scale definition of facet strain vector, one should first plug macroscopic Taylor series expansion of displacement and rotation of particle $J$ around particle $I$, Eqs. \[taylor-1-J\] and \[taylor-2-J\], into facet strain definition, Equation \[eps\]. In addition, equation ${\mathbf}{x} = \eta{\mathbf}{y}$ is used to change the length type variables to fine-scale quantities; $\eta y^{IJ}_j = x^{IJ}_j$, $\eta \bar{c}^{I}_k = c^{I}_k$ and $\eta \bar{c}^{J}_k = c^{J}_k$. Equation \[eps\] writes
$$\begin{aligned}
\label{eps-expansion-1}
\begin{aligned}
\epsilon_{\alpha}=\eta^{-1} \bar{r}^{-1} \bigg[& u_i^J+ \eta u^J_{i,j} y^{IJ}_j + \eta^2 \frac{1}{2}u^J_{i,jk} y^{IJ}_j y^{IJ}_k - u_i^I \\
& +\eta \varepsilon_{ijk} \bigg( \theta_j^{J}+\eta \theta_{j,m}^J y^{IJ}_m +\frac{1}{2} \eta^2 \theta^J_{j,mn} y^{IJ}_m y^{IJ}_n \bigg) \bar{c}_{k}^{J} -\eta \varepsilon_{ijk} \theta_j^I \bar{c}_{k}^{I} \bigg] e^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Spatial derivatives of displacement and rotation in equation above are partial derivative with respect to $x$. So, first and second order partial derivative of displacement and rotation asymptotic expansions, Eqs. \[disp-expansion\] and \[rot-expansion\], with respect to $x$ are as follows
$$u_{i,j} \approx u^0_{i,j} +\eta u^1_{i,j} \hspace{0.25in} u_{i,jk} \approx u^0_{i,jk} +\eta u^1_{i,jk}
\label{disp-asymp-der}$$
$$\theta_{i,j} \approx \eta^{-1} \omega^0_{i,j} + \varphi^0_{i,j} + \omega^1_{i,j} + \eta \varphi^1_{i,j} \hspace{0.25in} \theta_{i,jk} \approx \eta^{-1} \omega^0_{i,jk} + \varphi^0_{i,jk} + \omega^1_{i,jk} + \eta \varphi^1_{i,jk}
\label{rot-asymp-der}$$
Using asymptotic expansion of displacement and rotation of a particle, Eqs. \[disp-expansion\] and \[rot-expansion\], along with their macro-scale derivatives, Eqs. \[disp-asymp-der\], \[rot-asymp-der\], and replacing them into Equation \[eps-expansion-1\], one obtains
$$\begin{aligned}
\label{eps-expansion-2}
\begin{aligned}
\epsilon_{\alpha} = \eta^{-1} \bar{r}^{-1} \bigg[& u_i^{0J} + \eta u_i^{1J} + \eta u^{0J}_{i,j} y^{IJ}_j + \eta^2 u^{1J}_{i,j} y^{IJ}_j + \eta^2 \frac{1}{2}u^{0J}_{i,jk} y^{IJ}_j y^{IJ}_k + \eta^3 \frac{1}{2}u^{1J}_{i,jk} y^{IJ}_j y^{IJ}_k - u^{0I}_i - \eta u^{1I}_i \\
& + \eta \varepsilon_{ijk} \bigg( \eta^{-1} \omega_j^{0J} + \varphi_j^{0J} + \omega_j^{1J} + \eta \varphi_j^{1J} + \omega_{j,m}^{0J} y^{IJ}_m + \eta \varphi_{j,m}^{0J} y^{IJ}_m + \eta \omega_{j,m}^{1J} y^{IJ}_m + \eta^2 \varphi_{j,m}^{1J} y^{IJ}_m \\
& ~~~~~~~~~~~ + \eta \frac{1}{2} \omega^{0J}_{j,mn} y^{IJ}_m y^{IJ}_n + \eta^2 \frac{1}{2} \varphi^{0J}_{j,mn} y^{IJ}_m y^{IJ}_n + \eta^2 \frac{1}{2} \omega^{1J}_{j,mn} y^{IJ}_m y^{IJ}_n + \eta^3 \frac{1}{2} \varphi^{1J}_{j,mn} y^{IJ}_m y^{IJ}_n\bigg) \bar c_{k}^{J} \\
& - \eta \varepsilon_{ijk} \bigg( \eta^{-1} \omega_j^{0I} + \varphi_j^{0I} + \omega_j^{1I} + \eta \varphi_j^{1I} \bigg) \bar c_{k}^{I} \bigg) \bigg] e^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Regrouping terms of the same order in above equation, one would get multiple scale definition of facet strain $$\begin{aligned}
\label{eps-expansion-3}
\begin{aligned}
\epsilon_{\alpha} = \bar{r}^{-1} \bigg[ & \eta^{-1} \bigg( u_i^{0J} - u^{0I}_i + \varepsilon_{ijk} \omega_j^{0J} \bar c_{k}^{J} - \varepsilon_{ijk} \omega_j^{0I} \bar c_{k}^{I} \bigg) \\
& + \eta^0 \bigg( u_i^{1J} + u^{0J}_{i,j} y^{IJ}_j - u^{1I}_i + \varepsilon_{ijk} \bigg( \varphi_j^{0J} + \omega_j^{1J} + \omega_{j,m}^{0J} y^{IJ}_m \bigg) \bar c_{k}^{J} - \varepsilon_{ijk} \bigg( \varphi_j^{0I} + \omega_j^{1I} \bigg) \bar c_{k}^{I} \bigg) \\
& + \eta \bigg( u^{1J}_{i,j} y^{IJ}_j + \frac{1}{2}u^{0J}_{i,jk} y^{IJ}_j y^{IJ}_k + \varepsilon_{ijk} \bigg( \varphi_j^{1J} + \varphi_{j,m}^{0J} y^{IJ}_m + \omega_{j,m}^{1J} y^{IJ}_m + \frac{1}{2} \omega^{0J}_{j,mn} y^{IJ}_m y^{IJ}_n \bigg) \bar c_{k}^{J} \\
& ~~~~~~ - \varepsilon_{ijk} \varphi_j^{1I} \bar c_{k}^{I} \bigg)
\bigg] e^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
In equation above, terms of order two and higher are neglected. Multiple scale definition of facet strain is derived, which consists of three classes of terms of $\mathcal{O}(-1)$, $\mathcal{O}(0)$, and $\mathcal{O}(1)$. Multiple scale definition of facet curvature vector will be obtained subsequently. Taylor series definition of rotation of particle $J$ with respect to particle $I$ in macro coordinate system, Equation \[taylor-2-J\], should be inserted into definition of facet curvature, Equation \[curvature\]
$$\label{cur-exp-1}
\chi_{\alpha}=\eta^{-1} \bar{r}^{-1} \left[ \theta_i^J+\eta \theta_{i,j}^J y^{IJ}_j + \frac{1}{2} \eta^2 \theta_{i,jk}^J y^{IJ}_j y^{IJ}_k - \theta^I_i \right] {e}^{IJ}_{\alpha i}$$
Asymptotic expansion of rotation, Equation \[rot-expansion\], along with its macroscopic first and second order derivatives, Equation \[rot-asymp-der\], are inserted into Equation \[cur-exp-1\]
$$\begin{aligned}
\label{cur-exp-2}
\begin{aligned}
\chi_{\alpha}= \eta^{-1} \bar{r}^{-1} \bigg[ & \eta^{-1} \omega_i^{0J} + \varphi_i^{0J} + \omega_i^{1J} + \eta \varphi_i^{1J} + \omega_{i,m}^{0J} y^{IJ}_m + \eta \varphi_{i,m}^{0J} y^{IJ}_m + \eta \omega_{i,m}^{1J} y^{IJ}_m + \eta^2 \varphi_{i,m}^{1J} y^{IJ}_m \\
& \eta \frac{1}{2} \omega^{0J}_{i,mn} y^{IJ}_m y^{IJ}_n + \eta^2 \frac{1}{2} \varphi^{0J}_{i,mn} y^{IJ}_m y^{IJ}_n + \eta^2 \frac{1}{2} \omega^{1J}_{i,mn} y^{IJ}_m y^{IJ}_n + \eta^3 \frac{1}{2} \omega^{1J}_{i,mn} y^{IJ}_m y^{IJ}_n \\
& - \eta^{-1} \omega_i^{0J} - \varphi_i^{0J} - \omega_i^{1J} - \eta \varphi_i^{1J}
\bigg] {e}^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Collecting the terms of the same order and neglecting the ones of order more than zero, one can restate above equation as
$$\begin{aligned}
\label{cur-exp-3}
\begin{aligned}
\chi_{\alpha}=\bar{r}^{-1} \bigg[ & \eta^{-2} \bigg( \omega_i^{0J} - \omega_i^{0I} \bigg) +\\
& \eta^{-1} \bigg( \varphi_i^{0J}+ \omega_i^{1J} + \omega_{i,j}^{0J} y^{IJ}_j - \varphi_i^{0I} - \omega_i^{1I} \bigg) \\
& + \eta^0 \bigg(\varphi_i^{1J} + \omega_{i,j}^{1J} y^{IJ}_j + \varphi_{i,j}^{0J} y^{IJ}_j + \frac{1}{2} \omega_{i,jk}^{0J} y^{IJ}_j y^{IJ}_k - \varphi_i^{1I} \bigg)
\bigg] {e}^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Equation \[cur-exp-3\] is the multiple scale definition of facet curvature vector, which consists of terms of $\mathcal{O}(-2)$, $\mathcal{O}(-1)$, and $\mathcal{O}(0)$.
Asymptotic Expansion of Facet Strain and Curvature using definition of rigid body motion of RVE {#Revised-strain-curvature}
===============================================================================================
Multiple scale definition of facet strain, Equation \[eps-expansion-3\], and facet curvature, Equation \[cur-exp-3\], can be rewritten regarding the definition of ${\mathbf}{u}^0$, Equation \[U0\]. One can calculate first and second partial derivative of $\mathbf{u}^0$ with respect to $\mathbf{x}$ as follows
$$\label{UJ0-der}
u^{0J}_{i,j} = v^{0J}_{i,j} +\varepsilon_{imn} \omega ^{0J}_{m,j} y^J_n \hspace{0.5 in} u^{0J}_{i,jk} = v^{0J}_{i,jk} +\varepsilon_{imn} \omega ^{0J}_{m,jk} y^J_n$$
Using Eqs. \[U0\], \[UJ0-der\] along with the fact that ${\mathbf}{v}^0$, $\mb{\omega}^0$ and $\mb{\varphi}^0$ are constant over the RVE: $\mathbf{v}^{0I} = \mathbf{v}^{0J} = \mathbf{v}^{0}$, $\mb {\omega}^{0I} = \mb{\omega}^{0J} = \mb{\omega}^{0}$ and $\mb {\varphi}^{0} = \mb{\omega}^{0}$, one can revise Equation \[eps-expansion-3\]
$$\begin{aligned}
\label{eps-expansion-4}
\begin{aligned}
\epsilon_{\alpha}=\bar{r}^{-1} \bigg[ & \eta^{-1} \bigg( v_i^{0} - v^{0}_i + \varepsilon_{ijk} \omega_j^{0} ( \bar y_{k}^{J} - \bar y_{k}^{I} ) + \varepsilon_{ijk} \omega_j^{0} ( \bar c_{k}^{J} - \bar c_{k}^{I} ) \bigg) \\
&+ \eta^0 \bigg( u_i^{1J} - u^{1I}_i + \varepsilon_{ijk} \omega_j^{1J} \bar c_{k}^{J} - \varepsilon_{ijk} \omega_j^{1I} \bar c_{k}^{I} \\
& ~~~~~~~ + v^{0}_{i,j} y^{IJ}_j + \varepsilon_{ijk} \omega_j^{0} ( \bar c_{k}^{J} - \bar c_{k}^{I} ) + \varepsilon_{imn} \omega ^{0}_{m,j} y^{IJ}_j y^J_n + \varepsilon_{ijk} \omega_{j,m}^{0} y^{IJ}_m \bar c_{k}^{J} \bigg) \\
& + \eta \bigg( u^{1J}_{i,j} y^{IJ}_j + \frac{1}{2}v^{0}_{i,jk} y^{IJ}_j y^{IJ}_k + \frac{1}{2} \varepsilon_{imn} \omega^{0}_{m,jk} y^{IJ}_j y^{IJ}_k y^{J}_n + \frac{1}{2} \varepsilon_{ijk} \omega^{0J}_{j,mn} y^{IJ}_m y^{IJ}_n \bar c_{k}^{J} \\
& ~~~~~~ + \varepsilon_{ijk} \bigg( \varphi_j^{1J} + \omega_{j,m}^{0} y^{IJ}_m + \omega_{j,m}^{1J} y^{IJ}_m \bigg) \bar c_{k}^{J}
- \varepsilon_{ijk} \varphi_j^{1I} \bar c_{k}^{I} \bigg) \bigg] e^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Using $\mathbf{y}^{IJ} = \mathbf{y}^{J} - \mathbf{y}^{I}$ and $\mathbf{y}^{IJ} = \mathbf{\bar c}^{I} - \mathbf{\bar c}^{J}$ in above equation along with ${\mathbf}{y}^J+{\mathbf}{\bar c}^J={\mathbf}{y}^c$ , one would get
$$\begin{aligned}
\label{eps-expansion-5}
\begin{aligned}
\epsilon_{\alpha}=\bar{r}^{-1} \bigg[ & \eta^0 \bigg( u_i^{1J} - u^{1I}_i + \varepsilon_{ijk} \omega_j^{1J} \bar c_{k}^{J} - \varepsilon_{ijk} \omega_j^{1I} \bar c_{k}^{I} + v^{0}_{i,j} y^{IJ}_j - \varepsilon_{ijk} \omega_j^{0} y^{IJ}_k + \varepsilon_{ijk} \omega_{j,m}^{0} y^{IJ}_m y_{k}^{c} \bigg) \\
& + \eta \bigg( u^{1J}_{i,j} y^{IJ}_j + \varepsilon_{ijk} \varphi_j^{1J} \bar c_{k}^{J} + \varepsilon_{ijk} \omega_{j,m}^{1J} y^{IJ}_m \bar c_{k}^{J}
- \varepsilon_{ijk} \varphi_j^{1I} \bar c_{k}^{I} \\
& ~~~~~~ + \frac{1}{2}v^{0}_{i,jk} y^{IJ}_j y^{IJ}_k + \frac{1}{2} \varepsilon_{ijk} \omega^{0}_{j,mn} y^{IJ}_m y^{IJ}_n y_{k}^{c} + \varepsilon_{ijk} \omega_{j,m}^{0} y^{IJ}_m \bar c_{k}^{J} \bigg) \bigg] e^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Multiple scale definition of facet curvature can also be revised by using $\mb {\omega}^{0I} = \mb{\omega}^{0J} = \mb{\omega}^{0}$ and $\mb {\varphi}^{0} = \mb{\omega}^{0} $ along with $\mathbf{y}^{IJ} = \mathbf{y}^{J} - \mathbf{y}^{I}$ , one can rewrite Equation \[cur-exp-3\]
$$\begin{aligned}
\label{cur-exp-4}
\begin{aligned}
\chi_{\alpha}=\bar{r}^{-1} \bigg[ & \eta^{-1} \bigg( \omega_i^{1J} + \omega_{i,j}^{0J} y^{IJ}_j - \omega_i^{1I} \bigg) \\
& + \eta^0 \bigg(\varphi_i^{1J} + \omega_{i,j}^{1J} y^{IJ}_j - \varphi_i^{1I} + \omega_{i,j}^{0} y^{IJ}_j + \frac{1}{2} \omega_{i,jk}^{0J} y^{IJ}_j y^{IJ}_k \bigg)
\bigg] {e}^{IJ}_{\alpha i}
\end{aligned}\end{aligned}$$
Macroscopic Translational and Rotational Equilibruim Equations {#MacroEquil-Derivation}
==============================================================
In order to derive macroscopic RVE translational equation of motion, one should consider the terms of $\mathcal{O}(1)$ in Equation \[motion-1-sep\]
$$\label{macro-trans-derv-1}
\bar{M}_u^I\ddot {u}_i^{0I} = \sum_{\mathcal{F}_I}{\bar{A}\, t^1_{\alpha} {e}_{\alpha i}^{IJ}} + \bar{V}^I b_i^0$$
Scaling back Equation \[macro-trans-derv-1\] by multiplying both sides of the equation by $\eta^3$ and using the definition of $t^1_{\alpha}$ presented in Equation \[ZeroOne-Terms-Def\], one can get
$$\label{macro-trans-derv-2}
{M}_u^I \ddot {u}_i^{0I} = \eta \sum_{\mathcal{F}_I}{{A} \frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + {V}^I {b}^{0}_i$$
where $t_i^{IJ}=t^0_\beta e_{\beta i}^{IJ}$. Equation \[macro-trans-derv-2\] represents the $\mathcal{O}(1)$ translational equilibrium equation for each particle inside the RVE. One can derive the RVE macroscopic translational equilibrium equation by summing up Equation \[macro-trans-derv-2\] over all RVE particles and dividing by the RVE volume $V_0$
$$\label{macro-trans-derv-3}
\frac{1}{V_0}\sum_I {M}_u^I (\ddot {v}_i^{0I} + \varepsilon_{imn} \ddot {\omega} ^{0I}_{m} y^I_n ) = \frac{1}{V_0}\sum_I \sum_{\mathcal{F}_I}{\eta A \frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac{1}{V_0}\sum_I {V}^I {b}^{0}_i$$
In above equation ${u}_i^{0I}$ is replaced by its definition, Equation \[U0\]. Considering the fact that ${v}_i^{0I}$ and ${\omega}_m^{0I}$ are equal for all RVE particles and the body force $b_i^0$ is considered to be constant over the RVE, Equation \[macro-trans-derv-3\] can be written as
$$\label{macro-trans-derv-4}
\ddot {v}_i^{0} \bigg(\frac{1}{V_0}\sum_I {M}_u^I \bigg) + \varepsilon_{imn} \ddot {\omega} ^{0}_{m} \bigg(\frac{1}{V_0}\sum_I {M}_u^I y^I_n \bigg) = \frac{1}{V_0}\sum_I \sum_{\mathcal{F}_I}{\eta A \frac {\partial {t}^{IJ}_{i}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + {b}^{0}_i \bigg(\frac{1}{V_0}\sum_I {V}^I \bigg)$$
Second term on the left hand side of the Equation \[macro-trans-derv-4\] is equal to zero considering the assumption that the local system of reference is the mass center of the particle system within the RVE; $\sum_I {M}_u^I y^{I}_i=0$. Final form of the Equation \[macro-trans-derv-4\] is presented in Equation \[macro-1-1-averaged\] in Section \[macro-derivation\].
Macroscopic RVE rotational equation of motion can be derived by considering the terms of $\mathcal{O}(1)$ in Equation \[motion-2-sep\]. To have a consistent formulation for all particles and RVEs, one should consider the moment of all forces with respect to a fixed point in space, say the origin of a global coordinate system as shown in Figure \[TwoScaleAnalysis\]b, which implies that the moment of Equation \[macro-trans-derv-1\] should be taken into account. Therefore, one can write $\mathcal{O}(1)$ moment equilibrium equation of particle $I$ as
$$\label{macro-rot-derv-1}
\bar{M}_u^I \varepsilon_{ijk} Y^{I}_j \ddot {u}_k^{0I} + \bar {M_\theta^I}\ddot{\omega}_i^{0I} = \sum_{\mathcal{F}_I} \bar{A}\, (p^{1}_{\alpha}{e}_{\alpha i}^{IJ} + q^{1}_{\alpha}{e}_{\alpha i}^{IJ}) + \bar{V}^I \varepsilon_{ijk} Y^{I}_j {b}_k^{0}$$
where $Y^I_j$ is the position vector of particle $I$ in the fine-scale global coordinate system $\mathbf{Y}=\mathbf{X}/\eta$; $p^{1}_{\alpha} {\mathbf e}_\alpha^{IJ} = {\mathbf{Y}}^C \times t^{1}_{\alpha} {\mathbf e}_\alpha^{IJ}$ is the moment of the facet traction with respect to the origin of the fine-scale global coordinate system, in which ${\mathbf{Y}}^C = {\mathbf{X}}^C/\eta$ is the position vector of the contact point $C$ between particles $I$ and $J$ in the global coordinate system. Scaling back Equation \[macro-rot-derv-1\] by multiplying both sides of the equation by $\eta^4$ and using the definition of $p^1_{\alpha}$ and $q^1_{\alpha}$ presented in Equation \[ZeroOne-Terms-Def\], one can get
$$\label{macro-rot-derv-2}
{M}_u^I \varepsilon_{ijk} X^{I}_j \ddot {u}_k^{0I} + \eta^{-1} {M_\theta^I} \ddot{\omega}_i^{0I} = \eta \sum_{\mathcal{F}_I} A \left( {\frac {\partial {w}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} \right) + {V}^I \varepsilon_{ijk} X^{I}_j {b}_k^{0}$$
Equation \[macro-rot-derv-2\] represents the $\mathcal{O}(1)$ rotational equilibrium equation for each particle inside the RVE. RVE macroscopic rotational equilibrium equation can be obtained by summing up Equation \[macro-rot-derv-2\] over all RVE particles and dividing by the RVE volume $V_0$
$$\label{macro-rot-derv-3}
\begin{aligned}
\frac{1}{V_0}\sum_I {M}_u^I \varepsilon_{ijk} X^{I}_j (\ddot {v}_k^{0I} + & \varepsilon_{kmn} \eta^{-1} \ddot {\omega} ^{0I}_{m} x^I_n ) + \frac{1}{V_0}\sum_I \eta^{-1} {M_\theta^I} \ddot{\omega}_i^{0I} = \\
& \frac{\eta }{V_0}\sum_I \sum_{\mathcal{F}_I} A \left( {\frac {\partial {w}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} \right) + \frac{1}{V_0}\sum_I {V}^I \varepsilon_{ijk} X^{I}_j {b}_k^{0}
\end{aligned}$$
In above equation ${u}_i^{0I}$ is replaced by its definition, Equation \[U0\]. Considering equality of ${v}_i^{0I}$ and ${\omega}_m^{0I}$ for all RVE particles along with $X^I_j = X_j + x^I_j$, Equation \[macro-rot-derv-3\] can be written as
$$\label{macro-rot-derv-4}
\begin{aligned}
& \frac{1}{V_0}\sum_I {M}_u^I \varepsilon_{ijk} X_j \ddot {v}_k^{0} + \frac{1}{V_0}\sum_I \left( {M_\theta^I} \delta_{im}+ {M}_u^I \varepsilon_{ijk} \varepsilon_{kmn} x^I_j x^I_n\right) \eta^{-1} \ddot{\omega}_m^{0} \\
& + \frac{1}{V_0}\sum_I {M}_u^I \varepsilon_{ijk} x^I_j \ddot {v}_k^{0} + \frac{1}{V_0}\sum_I {M}_u^I \varepsilon_{ijk} \varepsilon_{kmn} X_j x^I_n \eta^{-1} \ddot {\omega} ^{0}_{m} = \\
& \frac{\eta }{V_0}\sum_I \sum_{\mathcal{F}_I} A \left( {\frac {\partial {w}_{i}^{IJ}}{\partial \epsilon^0_{\alpha}} \epsilon^1_{\alpha}} + \frac {\partial {m}_{i}^{IJ}}{\partial \psi^{0}_{\alpha}} \psi^1_{\alpha} \right) + \frac{1}{V_0}\sum_I {V}^I \varepsilon_{ijk} X_j {b}_k^{0} + \frac{1}{V_0}\sum_I {V}^I \varepsilon_{ijk} x^I_j {b}_k^{0}
\end{aligned}$$
Considering $\sum_I {M}_u^I x^{I}_i=0$ along with the equality of ${v}_i^{0}$, ${\omega}_m^{0}$, $X_j$ for all RVE particles, one can conclude that the third and the forth terms on the left hand side and the last term on the right hand side of the Equation \[macro-rot-derv-4\] is equal to zero. Final form of the Equation \[macro-rot-derv-4\] is presented in Equation \[macro-2-averaged-init\] in Section \[macro-derivation\].
[^1]: Research Assistant, Northwestern University, CEE Department, 2145 N Sheridan Rd Evanston, IL 60208, USA.
[^2]: Corresponding author. Email: [email protected]. Associate Professor, Northwestern University, CEE Department, 2145 N Sheridan Rd Evanston, IL 60208, USA.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We directly compare the mean-field and the many-body approach in a one-dimensional Bose system in a harmonic trap. Both contact and dipolar interactions are considered. We propose a multi-atom version of the phase imprinting method to generate dark solitons in the system. We begin with a general analysis of system dynamics and observe the emergence of a dark soliton and a shock wave. Center of mass and soliton motion become decoupled because the shock wave oscillates with the trap frequency and soliton does not. A detailed investigation of frequencies reveals significant differences between results obtained in the mean-field and the many-body pictures.'
author:
- 'Micha[ł]{} Kowalski'
- 'Rafa[ł]{} O[ł]{}dziejewski'
- Kazimierz Rzżewski
bibliography:
- 'artbib.bib'
title: 'Breakdown of the mean field for dark solitons of dipolar bosons in a one-dimensional harmonic trap'
---
Introduction
============
Solitons – solutions of non-linear integrable differential equations which propagate without dispersion – appear across many areas of science that range from physics to biology and medicine [@dauxois2006]. Among a number of known equations supporting solitonic solutions an important example constitutes of the non-linear Schrodinger equation called also the Gross-Pitaevski equation (GPE). Although mainly used in the context of weakly interacting ultracold bosons [@frantzeskakis2010dark], it also describes the properties of the electric field of light in the non-linear media [@kivshar1998dark].\
If atoms in a Bose-Einstein condensate repel each other by point-like interactions, a resulting solitonic solution of the GPE takes on form of a dark soliton – a density dip with a phase jump across its density minimum. An analytical expression describing dark solitons in a uniform gas characterized by the GPE were already found in the 70s [@zakharov71; @zakharov73]. Shortly after achieving the first condensation in 1995, dark solitons were successfully generated by the phase imprinting method in the experimental setups with ultracold [@burger1999dark] and [@denschlag2000generating] atoms trapped in a harmonic confinement.\
One of the most classic results about dark solitons in a BEC interacting only by short-range forces concerns its dynamics in a harmonic trap. In the so-called Thomas-Fermi regime for ultracold bosons, a characteristic frequency of soliton’s oscillations is expressed by the trapping frequency $\omega$ and equals $\omega/\sqrt{2}$ [@busch2000motion] that was also observed in the experiment [@weller2008]. However, this remarkably robust results for short-range interactions, does not hold for the dipole-dipole atomic interactions [@bland2017]. In this case, the oscillation frequency of solitonic structures in a repulsive dipolar BEC – predicted only in 2015 [@pawlowski2015dipolar; @bland2015] – highly depends on the strength of the atomic interactions and the interplay between local and non-local contributions to the total energy. With the recent progress on quantum gases consisting of atoms with considerable magnetic moments like [@lu2011; @maier2014] and Er [@aikawa2012], dipolar systems are now within experimental reach and call for deeper analysis.\
The non-linear mean-field (MF) theory of ultracold bosons provides an approximate description of interacting cold atoms. The underlying many-body (MB) model is linear and the state of the system is given by the many- body wave-function depending on positions of all particles. The discussion of correspondence between dark solitons present in the MF and many-body solutions of the full Hamiltonian has a long history. The best known example refers to the link between dark solitons moving on the circumference of a ring and type-II excitations from the seminal Lieb-Liniger model [@LiebLiniger1963; @Lieb1963] in the context of contact interacting particles, see [@Kulish1976; @ishikawa1980; @KanamotoCarr2008; @martin2010prl; @Fialko2012; @Sato2012; @syrwid; @Syrwid2016; @Katsimiga2017bent; @Brand2019; @Kaminishi2019] and references therein. However, little is known about many-body states possesing features of dark solitons in a one-dimensional harmonic trap both for contact and purely dipolar interactions. Here, we aim to partially bridge this gap by studying many-body dark solitons in weakly interacting trapped systems of only few atoms far beyond the Thomas-Fermi regime. In particular, we investigate the oscillations of the many-body solitons and compare them to the dark solitons described by the GPE. Our results not only establish a link between the MF approach and the full many-body theory, but also can be verified in modern experiments with a precise control over only a few atoms in optical lattices or single traps (see for instance [@greiner2002; @serwane2011; @meinert2015; @baier2016; @baier2018]).\
The work is organized as follows. In Sec. \[model\], we introduce our many-body model for atoms interacting repulsively by contact or purely dipolar forces in a quasi-1D harmonic trap. We also remind the corresponding GPE. Then, in Sec. \[solutions\], we analyse the many-body eigenstates of the system. Surprisingly, there is no good candidate for a many-body solitonic state among them, in a stark contrast to the ring geometry. Therefore, we introduce the many-body phase imprinting method of creating dark solitons in a harmonic trap. In Sec. \[results\], we finally present our results. We investigate dynamics of the many-body solitons. We calculate the oscillations frequencies for different coupling strengths for both types of interactions and compare our findings with the GPE. The most important result is a significant disagreement between frequencies obtained in the MB and the MF approaches.
The Model {#model}
=========
We investigate a system of N repulsive bosons trapped in a harmonic potential $$\begin{aligned}
U(x, y, z) = \frac{1}{2}m\omega^2 x^2 + \frac{1}{2}m\omega^2_\bot(y^2+z^2).\end{aligned}$$ We assume that the transverse confinement is tight, so the wave function stays in the lowest energy level in the Y- and Z-directions meaning our system is quasi-1D. The aim of our study is to compare contact and dipolar solitons in many-body and mean-field approaches. The system of N repulsive bosons is often approximated by the using the quasi-1D GPE
$$\begin{aligned}
\begin{split}
i\frac{\partial \psi_{\operatorname{GPE}}(\mathbf{r},t)}{\partial t} = \psi_{\operatorname{GPE}}(\mathbf{r},t)\Big(-\frac{1}{2}\frac{\partial^2 }{\partial x^2}+U(\mathbf{r})+\\+\left(N-1\right)\int d\mathbf{r'} \lvert \psi_{\operatorname{GPE}}(\mathbf{r},t)\rvert^2 V(\mathbf{r-r'})\Big),
\end{split}\\
\int d\mathbf{r}\lvert \psi_{\operatorname{GPE}}(\mathbf{r},t)\rvert^2 = N\end{aligned}$$
\[eq:gpe\]
where $U(\mathbf{r})$ is the trapping potential and $V(\mathbf{r})$ is an interaction potential. Throughout the paper we use oscillatory units with $\hbar \omega$, $\sqrt{\frac{\hbar}{m\omega}}$, $\sqrt{\hbar m \omega}$, $\frac{1}{\omega}$ as units of energy, length, momentum and time respectively. In the context of ultracold atoms, this equation is also called the MF description of weakly interacting bosons. This approach provided correct predictions of many properties of the Bose-Einstein condensate, including its shape, energy, normal modes of excitations and many other nonlinear phenomena. Moreover, the GPE supports dark solitons in a form of solutions with a density notch and a quickly changing phase, which have also been experimentally produced with the phase imprinting method. However, the MF model assures a simplified description of system of N repulsive bosons based on a naive assumption that every atom is in the same state. It is only an approximation of the more fundamental many-body approach. As the nonlinear GPE supports solitons, it is important to look for solitons in the linear many-body approach and compare them. It has been done in the Lieb-Liniger model [@LiebLiniger1963; @Lieb1963] but, to the best of our knowledge, this is the first paper where multi-atomic solitons are considered for the harmonically trapped system.
In order to describe a system of N bosons following the many-body approach, one needs to derive the wave function depending on positions of all particles. One of possible ways of deriving the many-body wave function is to diagonalize a Hamiltonian matrix. The Hamiltonian of the system under investigation can be written in a form: $$\begin{aligned}
\label{eq:ham0}
H=\sum_i^N T_i + \sum_i^N \frac{1}{2} x_i^2 + \sum_{i<j}^NV_{ij},\end{aligned}$$ with $T_i$ being the single-particle kinetic energy operator and $V_{ij}$ the two-body interaction operator. The Hamiltonian can be rewritten as a sum of the Hamiltonian of the noninteracting quantum harmonic oscillator and the interaction term. Therefore, in the second-quantization the Hamiltonian reads: $$\begin{aligned}
\label{eq:hamiltonian}
H= H_{\operatorname{osc}} + \frac{1}{2}\int\hat{\Psi}^\dagger(x)\hat{\Psi}^\dagger(y)V(x-y)\hat{\Psi}(x)\hat{\Psi}(y)dxdy,\end{aligned}$$ with $\hat{\Psi(x)}$ being a bosonic field operator and $H_{\operatorname{osc}} =\frac{1}{2} \int\hat{\Psi}^\dagger(x)(-\frac{\partial^2}{\partial x^2} +x^2)\hat{\Psi}(x)dx$. We study systems where V(r) is either a short range $V_{\operatorname{sr}}(r)$ or a long range dipolar $V_{\operatorname{dd}}(r)$ potential. The short range potential is $V_{\operatorname{sr}}(r)=g_{\operatorname{sr}}\delta(r)$ with parameter $g_{\operatorname{sr}} = \int V_{\operatorname{sr}}(r) dr$ defining strength of the interaction. We study repulsive systems with $g>0$. In order to obtain the explicit formula describing $V_{\operatorname{dd}}$, we follow the procedure described in paper [@pawlowski2015dipolar].\
We introduce an aspect ratio of the trap $\sigma = \frac{\omega_{\perp}}{\omega}$ and a dipolar coupling strength $A_{\operatorname{dd}}$ yielding $$\begin{aligned}
\label{v_int}
V_{\operatorname{dd}}(r) = \frac{A_{\operatorname{dd}}}{\sigma^2} \frac{1}{\sigma} V_{\operatorname{eff}}(\frac{r}{\sigma})\end{aligned}$$ with a term $$\begin{aligned}
V_{\operatorname{eff}}(u)=\frac{3}{4}[-2|u| + \sqrt{2\pi}(1+u^2)e^{u^2/2} \operatorname{Erfc}(\frac{|u|}{\sqrt{2}}).\end{aligned}$$ This effective quasi-1D potential comes from the integration of the full 3D dipolar interaction over both transverse variables. The area of our interest is a long range part of interaction, so we assume that the contact term of the effective dipolar potential is exactly cancelled possibly with the help of Feshbach resonances. As we want to investigate similarities and differences between systems interacting via contact and dipolar forces, we define dipolar strength parameter $g_{\operatorname{dd}}$ $$\begin{aligned}
g_{\operatorname{dd}} = \int V_{\operatorname{dd}}(r) dr = \frac{3 A_{\operatorname{dd}}}{\sigma^2}.\end{aligned}$$ From now on, we will keep $\sigma = 0.1$ and compare systems described by the same strength parameters $g_{\operatorname{dd}} = g_{\operatorname{sr}} = g$. It is important to mention that in the case of the MF approach only a gas parameter $(N-1)g$ defines the system. However, in the multi-atom approach both $N$ and $g$ are separately relevant.
From the numerical point of view, the most optimal basis to describe the system are quantum harmonic oscillator eigenstates. We use the second quantization formalism and define a basis of Fock states. We take into account all states with a given number of particles, which energies are smaller than a cut-off energy. Defining the cut-off in the energy space, rather than in the momentum space is more efficient method of obtaining numerical convergence [@sowinski].
In order to obtain eigenstates and energies of the system, we diagonalize the Hamiltonian matrix $$\begin{aligned}
H_{ij}={\langle i|}\hat{H}{|j\rangle},\end{aligned}$$ where ${|i\rangle}, {|j\rangle}$ are states belonging to the Fock space.
Solutions
=========
Having access to both eigenenergies and eigenstates we are ready to look for solitons in the system. Following recent papers investigating many-body solitons in the Lieb-Liniger model [@syrwid; @oldziejewski2018a], one could think that also in this case dark solitons could be identified among eigenstates of the Hamiltonian .\
Studying the repulsive case, we are interested in properties of dark solitons. In this situation density forms a single notch and in the area of the notch phase exhibits a jump of $\pi$. Keeping this in mind, one can ask a question if any single particle state fulfil these conditions.
We start our analysis with the ideal gas. In this case, the excited state of quantum harmonic oscillator (QHO),$\bigotimes^N {|0,1,0,...\rangle} = {|0,N,0,...\rangle}$, seems like a reasonable candidate because it has both the density dip and the phase jump, so one can try to find the eigenstate of the interacting system with the highest contribution of the aforementioned state among all the Fock states.
This turnes out to be non-trivial even for weak interactions. We expected the maximum ${|0,N,0,...\rangle}$ occupation tend to one as the interactions become weaker. Instead, it was approaching different values depending on a number of particles considered. This is a peculiar property of the harmonic trap caused by evenly spaced energies of $H_{\operatorname{osc}}$. Once the interactions become weaker, the energy of the eigenstate with the highest ${|0,N,0,...\rangle}$ contribution approaches $\frac{3N}{2}$ but there are multiple other states with the same energy leading to degeneracy. For example in the case of N=2, ${|0,2,0\rangle}$ has the same energy as ${|1,0,1\rangle}$ namely 3. Hence in contrast with the Lieb-Liniger model, these eigenstates remain the combination of several other states with the energy equal to $\frac{3N}{2}$ even for vanishing interactions.
Even if the excited eigenstate would form the dark soliton, still it would be hard to realize this state in the experiment. Therefore we decided to follow a different approach and try to replicate the experimental procedure of phase imprinting. This method creates a dark soliton in a BEC via a pulse of a far detuned laser applied on one half of the condensate and so creates a phase difference between the left and the right side. The length of this pulse is tuned to create a phase difference of $\pi$ and hence causes emergence of the dark soliton. There are not many papers discussing the phase-imprinting method in the many-body approach [@schmelcher2015many]. To the best of our knowledge it was only applied together with density engineering, which is not the case in the experimental realisation. As in real life, our implementation of phase imprinting modifies only the phase of the wave function and is equivalent to multiplying the ground state wave function by an arbitrary phase factor $$\begin{aligned}
\Psi(x_1,x_2,...,x_N)=\Phi(x_1,x_2,...,x_N)e^{i\phi(x_1,x_2,...x_N)},\end{aligned}$$ where $\Psi$ is the many-body wave function of a solitonic state, $\Phi$ is the ground state wave function and $\phi$ is an arbitrary phase factor. For the numerical convenience we choose $\phi(x_1,x_2,...x_N) = \sum_j^N \tan^{-1}(\alpha x_j)$, where $\alpha$ is the parameter changing sharpness of phase jump which can also be controlled in experiments. The optimal sharpness of the phase jump has to be tuned to fit healing length of the soliton. It means that the stronger the interaction the narrower the phase jump has to be. As for now, the solitonic wave function $\Psi$ is merely an initial condition. We derive the time evolution of the system by expressing $\Psi$ in the basis of eigenstates of the system as follows: $$\begin{aligned}
\Psi(x_1,x_2,...,x_N, t) = \sum _{i} \beta_{i} \psi_{i}( x_{1} ,...,x_{N}) \exp\left( -iE_{i} t\right)\\
\beta _{i} =\int dx_{1} ...dx_{N} \psi ^{*}_{i}( x_{1} ,...,x_{N}) \Psi ( x_{1} ,...,x_{N}),\end{aligned}$$ where $\psi_{i}( x_{1} ,...,x_{N})$ and $E_i$ are the eigenstates and the eigenvalues of the Hamiltonian . In order to visualise a soliton, we derive a one-particle density $$\begin{aligned}
\rho (x_1,t) =\int dx_{2} ...dx_{N} \Psi^* ( x_{1} ,...,x_{N} ,t)\Psi(x_1,x_2,...,x_N, t).\end{aligned}$$ As we aim to obtain the MF dark solitons from Eq. and compare them with MB solutions, we employ an analogous scheme. Firstly, we find a ground state $\psi_{\operatorname{GPE}}(x)$ for given parameters by using the well-known imaginary time evolution (ITE) technique. At this point, we can compare ground states obtained in MF and MB approaches. Both density profiles and ground-state energies (up to 2 % difference for the highest $g$) are in a very good agreement for both dipolar and contact interactions and for all coupling strenghts considered in this work. Then, we imprint the same phase as in the MB calculations, namely $\Psi_{\operatorname{GPE}}(x)=\psi_{\operatorname{GPE}}(x)e^{i\phi_{MF}(x)}$, with $\phi_{MF}=\tan^{-1}(\alpha x)$. Finally, we evolve Eq. in a standard real-time evolution with $\Psi_{\operatorname{GPE}}(x)$ as an initial condition.
Note that we can calculate the quantum depletion for any many-body state, in particular for a many-body ground state before and after phase imprinting, by diagonalazing a single particle density matrix constructed from the many-body wave function. It provides a tool for comparing mean-field and many-body results.
Results
=======
Having a model of the experimental method of phase imprinting and being able to calculate the evolution of dark solitons, we can focus on properties of contact and dipolar many-body solitons and compare them with the MF results. Firstly, we would like to focus our attention on general aspects of the evolution of many-body solitons in the harmonic trap. In order to study the evolution of the system we plot the one-particle density as a function of time and space in Fig. \[fig:evo\] for $N=6$ dipolar bosons and $g=0.3$. It reveals that the phase imprinting method causes not only the dark soliton to emerge but also a shock wave in the form of a density peak initially moving in the opposite direction to the soliton. Plots of density profiles in consecutive time steps shown in Fig. \[fig:ani\] reveal more details of soliton evolution. The local density minimum moves from the center of the trap towards the left side as long as the density of the notch is greater than zero. When the soliton becomes black, namely when the one-particle density in the dip reaches zero, its velocity also equals zero, both indicating a turning point. The soliton begins to move right and becomes shallow in the center of the trap. The relation between the depth of the dark soliton and its velocity is one of the fundamental properties of solitons and have been studied in a number of papers [@zakharov71; @zakharov73; @parker2010dark].
![(color online) Sequence of images showing a spatial density profile in consecutive time-steps for the situation from Fig. \[fig:evo\] ($N=6$ dipolar particles and $g=0.3$). Time $t$ is given in units of $\frac{1}{\omega}$. Red dot indicates the position of the soliton. The soliton becomes deeper until density reaches zero – the soliton bounces from the trap and begins moving to the other side of the trap. Once moving towards the center of the trap it becomes shallower.[]{data-label="fig:ani"}](map.eps){width="45.50000%"}
![(color online) Sequence of images showing a spatial density profile in consecutive time-steps for the situation from Fig. \[fig:evo\] ($N=6$ dipolar particles and $g=0.3$). Time $t$ is given in units of $\frac{1}{\omega}$. Red dot indicates the position of the soliton. The soliton becomes deeper until density reaches zero – the soliton bounces from the trap and begins moving to the other side of the trap. Once moving towards the center of the trap it becomes shallower.[]{data-label="fig:ani"}](grid.eps){width="45.50000%"}
As we pointed before the phase imprinting method creates a soliton but also a shock wave. This effect has been already observed in experiments implementing phase imprinting [@burger1999dark; @denschlag2000generating]. Both the shock wave and the soliton oscillate in the trap harmonically but the shock wave oscillates with the trap frequency. It is then worth to study the frequency of the soliton movement as it was one of the factors differentiating contact and dipolar solitons in the mean-field approach [@bland2017].
One of properties of contact solitons revealed by number of studies [@fedichev1999; @busch2000motion; @parker2010dark] is that the frequency of oscillation does not depend on the strength of interactions and equals $\omega_{\operatorname{TF}} = \frac{1}{\sqrt{2}}\omega$. However, this result is obtained in the Thomas-Fermi (TF) limit assuming that the background density varies slowly on the scale of the soliton. The kinetic energy of particles in the TF limit can be neglected compared to the potential energy. However, satisfying this condition in the case of small systems would demand very strong interactions causing atoms to deplete the ground state and thus making our and the mean-field result incomparable. Our studies focused on the case of small systems in the many-body approach far from the TF limit and with the quantum depletion of the ground state before phase imprinting not exceeding 5%.
In order to analyse the frequency of oscillation, we trace the position of a local minimum of the one-particle density (in the many-body approach) or condensate wave function (in the mean-field picture) in consecutive time steps. We trace the minimum until it crosses the center of the trap which gives us half of the period. Recent paper investigating dipolar solitons in the mean-field approach revealed significant differences between contact and dipolar solitons in the TF regime [@bland2017], one of them exhibited by the frequency of oscillations. In contrast to previously described mean-field contact solitons, the frequency of dipolar solitons depends on the interaction strength and, in general, the dipolar soliton frequency is smaller than the one obtained for contact solitons. It is then worth asking if many-body solitons exhibit similar behaviour already for weak interactions. To answer this question we directly compare many-body and mean-field solitons for contact and dipolar interactions. We plot frequencies obtained for system of N=6 particles for the increasing coupling strength $g$ in Fig. \[fig:comp\]. Firstly, we note that the frequencies of MF and MB solitons differ significantly. In the MF picture, contact and dipolar solitons are almost indistinguishable. In opposition, MB dipolar solitons oscillate much slower than contact ones which agrees with the recent MF analysis within the TF approximation in [@bland2017].
It is then important to ask why the MF model far away from the TF regime is inconsistent when applied to the studied system. We can indicate two factors that may play a significant part. On the technical level, it seems like rapidly changing excited QHO states contribute to the difference in the energy between short-range and dipolar interactions in the many-body picture. On the other hand, the MF wave-function does not vary at the scale given by the range of dipolar interaction. Hence, for systems far from the TF regime, the dipolar interacting scenario almost does not differ from the short-range interaction case. The other factor contributing to the difference between MB and MF solutions is the phase imprinting method. Before imprinting, the systems are comparable as the depletion of the MB ground state does not exceed 5%. After the procedure, it rises by 10-20% depending on the sharpness of the phase jump. It means that the excited fraction cannot be neglected anymore in the MB calculations while it is not present at all in the MF case. This is a very important observation as the depletion of the ground state is fundamentally bound with phase-imprinting method and need to be taken into consideration when studying small systems both theoretically and experimentally.
Having discussed the differences between MF and MB results, we focus on the properties of MB contact and dipolar solitons. The system proves to be interaction-sensitive as the frequency varies significantly with both the strength and the range of the interaction. In both cases, the frequency decreases with the increasing coupling strength $g$. The dipolar solitons always oscillate slower than their short-range counterparts. As we are far from the TF limit, the frequency of contact solitons does not converge to $\frac{1}{\sqrt{2}}\omega$.
We have investigated not only the frequency of solitons oscillation but also their lifetimes. While it is hard to define a sharp condition for the soliton to be indistinguishable from the background, we noted that contact solitons live significantly longer than dipolar counterparts, with the lifetime strongly dependent on the coupling strength $g$.
![(color online) Comparison of frequency as a function of a coupling strength $g$ for contact and dipolar interactions in the many-body (circles) and mean-field (squares) approaches for $N=6$ particles. Mean-field solitons are almost indistinguishable with dipolar ones being only slightly slower. On the other hand, many-body solitons differ significantly. Dipolar solitons always oscillate slower as for the MF dark solitons in the TF regime studied in [@bland2017].[]{data-label="fig:comp"}](final1.eps){width="45.50000%"}
Conclusions
===========
The goal of this paper is to compare dark solitons in the mean-field and many-body approaches for contact and dipolar interactions. We have began our many-body analysis with calculating eigenstates and energies of the system via the numerical diagonalization of the Hamiltonian matrix. We have found that one cannot identify the dark solitons with a specific eigenstate of the system, in stark contrast to the well-known situation of atoms in a ring trap. This follows directly from quantum degeneracy of multi-particle eigenstates of the non-interacting gas in the harmonic trap because the single particle eigenenergies are spaced evenly.
In order to study many-body solitons in the harmonic trap we have introduced the multi-atom version of the phase imprinting method. Just as in the classic experiment it causes not only the soliton but also the shock wave to appear. Those waves oscillate with different frequencies and thus movements of the soliton and the center of mass are decoupled. We investigate the frequency of oscillation of dark solitons to reveal similarities and differences between contact and dipolar solitons and compare our many-body analyses with mean-field results.
Although calculated for small and weakly interacting systems, our studies comparing many-body contact and dipolar solitons uncover similar features as previously discussed mean-field results within the Thomas-Fermi regime [@bland2017]. The frequency of oscillations for dipolar solitons strongly depends on the coupling strength and is lower compared to contact solitons for the corresponding interaction strength. For comparison, we also analyzed contact and dipolar solitons in our small system induced by the phase imprinting method at the MF level. The MF approach fails in the case of our system as the dipolar and contact solitons are almost identical and their properties differ significantly from the MB solitons. We can define two factors that cause significant differences between our MB and MF solitons. Firstly, the quickly oscillating excited QHO states play an important role in the case of dipolar interaction. Secondly, the phase imprinting method enlarges a depletion of the ground state and the excited fraction is no longer negligible.
We thank K. Pawłowski for his careful and critical reading of the manuscript and fruitful discussions. This work was supported by the (Polish) National Science Center Grants 2016/21/N/ST2/03432 (RO and MK) and 2015/19/B/ST2/02820 (KR). Center for Theoretical Physics is a member of KL FAMO.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We find a consistency between two different approaches of hard diffraction, namely the QCD dipole model and the Soft Colour Interaction approach. A theoretical interpretation in terms of S-Matrix and perturbative QCD properties in the small $x_{Bj}$ regime is proposed.'
author:
- 'H. Navelet and R. Peschanski[^1]'
title: Unifying approach to hard diffraction
---
ł
[**1.**]{} In the present paper, we focus on two different theoretical approaches to hard diffraction, which are shown to be compatible and complementary. The first one is based on an extension [@bi96] to hard diffractive processes of the “QCD dipole” approach in the small $x_{Bj}$ regime of perturbative QCD. In this picture, the hard photon is supposed to probe the parton structure of the Pomeron considered as an hadronic particle. In the original Ingelman-Schlein formulation , this hard probe was formulated in the parton model, later complemented by QCD evolution equations [@blu]. In the dipole model, this hard interaction is described via the Balitsky Fadin Kuraev Lipatov (BFKL) resummation [@bfkl]. A major uncertainty for this model is the relative normalization of diffractive over non-diffractive cross-sections which stays beyond the perturbative framework.
A second approach [@in96] is the Soft Colour Interaction (SCI) model, where hard diffraction is described as the superposition of two processes. At short time, the hard probe initiates a typical deep-inelastic interaction with colour quantum numbers exchange. Then, at large times/distances, a “soft” colour interaction is assumed to rearrange the colour quantum numbers and gives rise to singlet exhanges -and thus diffraction- with a probability of order ${\bf \frac
1{N_c^2}},$ where $N_c$ is the number of colours[^2]. While in this approach, the relative normalization is fixed, the exact nature of the interplay between soft and hard components is not known.
In the present paper, we propose a way to relate these two approaches which allows one to consider them as compatible and complementary. This provides a definite prediction for both the normalization and the analytic form of the amplitude. Our main results are the following:
[**i)**]{} The interplay between hard and soft components of hard diffraction is expressed via “effective” parameters of a Pomeron interaction determined from leading log perturbative QCD resummation. It is found to depend not only on $Q^2$ but also on the ratio $\eta =
\left(Y-y\right) /y,$ where Y (resp. y) are the total (resp. gap) rapidity interval.
We obtain $$F_{T,L}^{Diff}(Q^2,Y,y) = {\bf \frac 1{N_c^2}}\
\frac {{\cal N}^{tot}}{x_P}
\ \frac {e^{2y\D}}{4\pi \D ^{''} y}
\ \sqrt {\frac 2
{1\!+\!2\eta}}
\exp \left\{ (Y\!-\!y)\ {\bf \epsilon} _s \right\}\ \left(\frac
Q{Q_0}\right)^{2{\bf \g_s}}
\exp{\left( - \frac {2\log^2\!\left(\frac
Q{Q_0}\right)}{D_s
(Y\!-\!y)}\right)} \ ,
\l{biczy10}$$ with $$\D(x) \equiv \frac {\a
N_c}{\pi}\left\{2\psi(1)-\psi(x)-\psi(1-x)\right\}
\ \sim \D + \frac
{\ \D^{''}}2 \left(1/2\!-\!x \right)^2
\l{chi}$$ is the BFKL evolution kernel [@bfkl] (together with its gaussian approximation near the minimum at $x=1/2$) and $Q_0$ a non-perturbative scale associated with the proton. We have $${\bf \g_s}= \frac {\eta}{1+2\eta}\ ;\
{\bf \epsilon}_s =\D + \frac {\D^{''}}{8 (1+2\eta)}\ ;\ D_s = \frac
{1+2\eta}{\eta}\ \D^{''}\ .
\l{epsilon}$$
Note that one may write $
F_{T,L}^{Diff} \ \sim
\ \sigma^{tot}_{\gamma^*-P}\times e^{2y\Delta }/(4\pi \D ^{''} y) ,$ which is the known “triple Pomeron” formula [@ka79] where $\sigma^{tot}_{\gamma^*\!-\!P}$ defines the effective interaction cross-section of a virtual photon with a BFKL Pomeron $e^{y\Delta }/\sqrt {4\pi \D ^{''} y}$ derived from the QCD dipole formalism. By analogy with BFKL [@bfkl], $\g_s, \epsilon_s$ and $D_s$ can be defined, respectively, as the anomalous dimension, intercept and diffusion parameter of an “effective” BFKL $\gamma^*\!-\!P$ cross-section.
The normalization, which remains unknown in the QCD dipole model description, is determined as the product of the factor ${\bf \frac 1{N_c^2}}$ and the normalization factor ${\cal N}^{tot}$ of the non-diffractive structure function, according to the SCI prescription, see below.
[**ii)**]{} In SCI models, the following relation between the total structure function and the overall contribution of hard diffraction at fixed value reads of $x_{Bj}:$
$$F_{T,L}^{Diff/tot}\!\! \equiv \!\int_{x_{Bj}}^{x_{gap}} dx_P \ F_{T,L}^{Diff} =
{\bf \frac 1{N_c^2}}\
F_{T,L}^{tot}
\l{biczy4}$$
where $\log 1/{x_{gap}}$ is the minimal rapidity gap. If we insert the QCD dipole prediction for $F_{T,L}^{Diff}$ in formula (\[biczy4\]), we find $$F_{L,T}^{tot} = \frac {{\cal N}}{\bf {N_c^2}}\left(\frac
Q{Q_0}\right)^{2\g^{*}}
\frac {\exp \left(Y\D (\g^{*})\right)}{\sqrt {2\pi\D ^{''} \ Y}} \ ,
\l{tot1}$$ which is equivalent to a canonical BFKL expression for non-diffractive structure functions, apart the substitution of the BFKL effective anomalous dimension $\g_{BFKL}$ by $\g^{*},$ namely $$\g_{BFKL} = \ 1/2- 2\frac
{\log\left(\frac Q{Q_0}\right)}{\D ^{''} \ Y}
\ \rightarrow \ \g^{*} = cst. \sim 0.175\ ,
\l{value}$$ where the “universal” value $\g^{*}$ is solution (for $0<\g<1/2$) of the implicit equation $
2\D\left(\frac{1\!-\!\g}{2}\right)-\D(\g)=0\ $. Here, it is interesting to note that the shift (\[value\]) may be useful to avoid the objection to SCI models [@dok] based on “Low’s theorem” following which soft colour radiation cannot be emmitted from inside a partonic process. The differences we find with the original model means that, in our dipole formulation, the soft colour interaction indeed seems to modify the initial parton kinematics.
[**iii)**]{} Using S-Matrix properties of triple-Regge contributions, a relation is found between discontinuities of a $3 \to 3$ amplitude and the two approaches to hard diffraction we consider. Following old results of S-Matrix theory in the Regge domain [@mu], and as sketched in Fig.1, one may consider three types of discontinuities of a $3 \to 3$ amplitude representing hard diffraction. A single discontinuity over the diffractive $mass^2$ describing the hard Pomeron interaction, a double discontinuity taking into account the analytic discontinuity in the subenergy variable of one of the incident Pomeron exchanges (and its complex conjugate) for the SCI model and the full triple discontinuity including those of the two Pomeron exchanges, which is characteristic of the QCD dipole model description [@bi96]. An interesting new feature is thus the S-Matrix interpretation of the SCI approach as a specific double discontinuity of the $3 \to 3$ forward amplitude, which formulates the model in terms of simultaneous exchanges of a soft and a hard Pomeron.
[**2.**]{} Let us sketch the derivation of our results.
Our starting point is a triple-Regge formula for the diffractive structure function for longitudinal and transverse photon in the QCD dipole formalism: F\_[T,L]{}\^[Diff]{}(Q\^2,Y,y)\~ \_[c-i]{}\^[c+i]{} (1-\_1-\_2-) (Q[Q\_0]{})\^[2]{} {y ((\_1)+(\_2)) + (Y-y) ()} , ł[biczy]{} where $\D(\g)$ is the BFKL evolution kernel (\[chi\]) and ${\cal N}^{Diff}$ is a normalization containing both QCD perturbative and non-perturbative factors [@ba96]. Strictly speaking [@ba96] the $\delta$-function is the unique contribution in the differential diffractive structure function at momentum transfer $t=0.$ However, it can be shown that this is the dominant perturbative contribution even at non zero transfer due to specific properties [@na02] of the analytic QCD triple-Pomeron couplings [@na01] .
The first step of the computation of formula (\[biczy\]) is to use the saddle-point approximation [@ba96] at large $y$ to integrate over the difference $\g_1\!-\!\g_2.$ One easily gets F\_[T,L]{}\^[Diff]{}= [N]{}\^[Diff]{}1[x\_P ]{} \_[c-i]{}\^[c+i]{} {2y ( 2) + (Y-y) () + 2 Q[Q\_0]{}} . ł[biczy1]{}
Using the gaussian approximation (\[chi\]) for the BFKL kernels $\D(\g)$ and $\D\left(\frac {1\!-\!\g} 2\right)$ in the relevant interval $0 <\g<1/2,$ and again a saddle-point approximation at large rapidity gap $y$, one obtains, up to a normalization factor, formula (\[biczy10\]), with $\g_s, \epsilon_s$ and $D_s$ defined as in (\[epsilon\]). The normalization ${\cal N}^{Diff}$ is not yet specified at this stage.
The derivation of the normalization is coming from the comparison with the SCI approach. Inserting (\[biczy\]) in the integral (\[biczy4\]), one is led to perform a two-dimensional saddle-point approximation in the $y,\g$ complex plane. The saddle-point equations read: -y \^[’]{}( 2) + (Y-y) \^[’]{}() + 2 Q[Q\_0]{}=0 2()-()=0 , ł[cols]{} whose solution $(y^*,\g^*)$ is y\^\* &=& (Y+) (1+)\^[-1]{} (\^\*)&=&2() ł[solution]{} resulting in a value of $\g^*\simeq 0.175 ,$ which is “universal”, i.e. independent of the kinematics of the reaction.
After computation of the prefactors to the saddle-point approximation, one finds: F\_[T,L]{}\^[Diff/tot]{} =\^[Diff]{} 1 [\^[’]{}(2)+\^[’]{}(\^\*)]{} (Q[Q\_0]{})\^[2\^\*]{} . ł[biczy7]{} Note that it is the linearity in $y$ of the saddle-point equation (\[cols\]) which allows one to get such an elegant form for the integrals. Using (\[biczy7\]) and by comparison with the canonical BFKL formula we identify the “hard” component of the SCI model by the substitution $
\g_{BFKL} \rightarrow \g^{*} ,$ see (\[value\]). Then using the SCI ansatz (\[biczy4\]), we obtain the relation \^[Diff]{} [\^[’]{}(2)+\^[’]{}(\^\*)]{} ł[norma]{} which fixes the relative normalization of the diffractive vs. non diffractive structure functions. This ends the derivation of formulae (\[biczy10\]-\[value\]).
[**3.**]{} Let us finally come to the S-Matrix interpretation of our approach. For every fixed but arbitrary value of the parameters $\g,\g_1,\g_2,$ the triple-Regge formula (\[biczy\]) can be obtained from the canonical formulalism[^3] corresponding to the vertex of three Regge pole singularities[^4] in the complex plane of angular momentum [@ka79]. As such, one can make use of the important S-Matrix Mueller-Regge relation [@mu], valid in kinematical regions including the triple-Regge limit, between semi-inclusive amplitudes and specific discontinuity contributions of forward elastic $3 \to 3$ amplitudes. It naturally applies to hard diffraction initiated by a virtual photon, as sketched in Fig.1, namely \^\* + pp + X Disc\_1{\^\* |p p \^\* |p p } . \[3to3\] Quite interestingly, the existence of Regge phase factors allows one to relate other discontinuities of $A(3 \to 3)$ to $Disc_1 A.$ As sketched in Fig.1, one may also consider a double discontinuity $Disc_2 A(3 \to 3)$ taking into account also the analytic discontinuity in the subenergy of one of the incident Pomeron exchanges and a triple discontinuity $Disc_3 A(3 \to 3)$ including the discontinuity over the two Pomeron exchanges. The expression of the discontinuities, through generalized unitarity relations, is obtained through the imaginary part of the relevant Regge phase factors [@mu]. Moreover, one finds an equality relation $Disc_1 A \! = \!Disc_2 A\! = \!Disc_3 A $ which is due to the fact that the discontinuity taken over the mass variable (corresponding to diffractively produced states) is common to all three cases in Fig.1 and factorizes the same $p\bar p$ vertex in $A(3\! \to \!3)$ (cf. the classical derivation in the last paper of Ref.[@mu]).
Let us now take advantage of the hard probe in the process, allowing one to introduce in the game the (resummed) perturbative QCD expansion at high energy (small $x_{Bj}$). In a generic S-Matrix approach, the analytic discontinuities of scattering amplitudes are related to a summation over a complete set of asymptotic [*hadronic*]{} final states. If however, the underlying microscopic field theory is at work with small renormalized coupling constant due to the hard probe, it is possible in some cases to approximate the same discontinuity using a complete set of [ *partonic*]{} states. In particular, at high energy and within the approximation of leading logs (and also large $N_c$), QCD dipoles can be identified as providing such a basis .
The discontinuity $Disc_3$ appears naturally in the dipole formulation of hard diffraction [@bi96]. Indeed, hard diffraction calculations make use of the probability distribution for finding two dipoles in the wave-function of one initial dipole in the virtual photon through the exchange over a BFKL Pomeron exchange described by its discontinuity in the $Y-y$ rapidity range. Then, each of these dipoles interact with the target through two perturbative (BFKL) Pomeron exchanges described by their own discontinuities over the $y$ range. Thus the calculation of scattering amplitudes implies the full discontinuity over three Pomerons described by intermediate dipole interactions, which is nothing but $Disc_3.$
The appearance of $Disc_2$ is natural in the Regge formalism [@mu]. However a physical interpretation involving perturbative QCD contributions has not been previously noticed. In the framework of our study, it corresponds to the superposition of a hard perturbative cut Pomeron interaction with a soft one described by a non-cut Pomeron (with complex conjugate contribution, see Fig.1). This is similar to the superposition of hard and soft interactions which characterizes the SCI approach. We are thus led to propose $Disc_2$ as a way to get quantitative predictions, in particular the relative normalization of diffractive over non diffractive cross-sections. Indeed, it appears as a “hard” partonic interaction very similar to the one describing ordinary deep-inelastic processes, in parallel with a “soft” correction evolving during a long time, corresponding to the uncut Pomeron singularity in the middle graphs (including complex conjugate) of Fig.1.
The equality between the different discontinuities allows the connection between the different models, leading to the results given in section [**1**]{}.
**FIGURE**
epsf =8.truecm =8.cm
[**Figure 1**]{}
[*S-Matrix interpretation of the three approaches to hard diffraction.*]{} Upper graph: Description of $Disc_1 A(3 \to 3)$; Middle graph: Description of $Disc_2 A(3 \to 3)$ and its complex conjugate (candidates for the SCI approach); Lower graph: Description of $Disc_3 A(3 \to 3)$ (QCD dipole approach).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank A.Bialas and W.Buchmuller for valuable remarks.
References {#references .unnumbered}
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[^1]: CEA, Service de Physique Theorique, CE-Saclay, F-91191 Gif-sur-Yvette Cedex, France
[^2]: More recent versions of the model consider a probability factor of the same order but not necessarily connected with colour.
[^3]: These triple Regge formulae, relevant for [*inclusive cross-sections*]{}, have to be distinguished from the AGK cutting rules which are relevant for the two and multi Pomeron contributions to the $2 \to 2$ [ *elastic*]{} amplitudes in both the S-Matrix framework [@AGK] and perturbative QCD at leading log level [@AGK1].
[^4]: We neglect at this stage the complications due to Regge cuts or other more sophisticated singularities and deal with simple effective Regge poles.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the hierarchical structures of countries based on electricity consumption and economic growth by using the real amounts of their consumption over a certain time period. We use of electricity consumption data to detect the topological properties of 60 countries from 1971 to 2008. These countries are divided into three subgroups: low income group, middle income group and high income group countries. Firstly, a relationship between electricity consumption and economic growth is investigated by using the concept of hierarchical structure methods (minimal spanning tree (MST) and hierarchical tree (HT)). Secondly, we perform bootstrap techniques to investigate a value of the statistical reliability to the links of the MST. Finally, we use a clustering linkage procedure in order to observe the cluster structure more clearly. The results of the structural topologies of these trees are as follows: i) we identified different clusters of countries according to their geographical location and economic growth, ii) we found a strong relation between energy consumption and economic growth for all the income groups considered in this study and iii) the results are in good agreement with the causal relationship between electricity consumption and economic growth.'
author:
- |
Ersin Kantar$^{a, b}$, Alper Aslan$^{c}$, Bayram Deviren$^{d}$ and Mustafa Keskin$^{*, a}$\
$^{a}$Department of Physics, Erciyes University, 38039 Kayseri, Turkey\
$^{b}$Institute of Science, Erciyes University, 38039 Kayseri, Turkey\
$^{c}$Faculty of Economics and Business, Nevsehir University, 50300, Nevşehir,Turkey\
$^{d}$Department of Physics, Nevsehir University, 50300 Nevşehir, Turkey
title: Hierarchical structure of the countries based on electricity consumption and economic growth
---
C10, C52, Q40, Q43
\[sec:level1\]Introduction
==========================
Electricity consumption has become a topic of immense importance. The growing interest in developed and developing countries has largely been triggered by the growing demand for energy across the world fueled mainly by increasing economic activities, particularly in emerging countries. Estimating electricity consumption in advance is crucial in the planning, analysis and operation of power systems in order to ensure an uninterrupted, reliable, secure and economic supply of electricity. Moreover, modeling and predicting electricity consumption play a vital role in developed and developing countries for policy makers and related organizations.
The causal relationship between electricity consumption and economic growth has been investigated and the empirical literature has focused on four hypotheses when dealing with the causal relationship between electricity consumption and economic growth: conservation, growth, feedback, and neutrality. The first is the conservation hypothesis which is supported if an increase in economic growth causes an increase in electricity consumption. Under this scenario, an increase in economic growth would have a negative impact on electricity consumption. The second is the growth hypothesis which supposes that electricity consumption can directly impact on economic growth and indirectly as a complement to labor and capital in the production process. The growth hypothesis verified if there is a unidirectional causality from electricity consumption to economic growth. If this is the case, an increase in electricity consumption has a positive impact on economic growth; energy conservation oriented strategies that decrease electricity consumption may have a harmful impact of economic growth. The feedback hypothesis highlights the interdependent relationship between electricity consumption and economic growth. The existence of bidirectional causality between electricity consumption and economic growth provides support for the feedback hypothesis. Fourth, the neutrality hypothesis suggests that energy consumption provides a relatively trivial position in the determination of economic growth.
Payne [@Payne2010] compares the various hypotheses associated with the causal relationship between electricity consumption and economic growth using a survey of the empirical literature. The results illustrate that 31.15 % supported the neutrality hypothesis; 27.87 % of studies the conservation hypothesis; 22.95 % the growth hypothesis; and 18.03 % the feedback hypothesis.
There are several studies in the empirical literature on the causal relationship between electricity consumption and economic growth. The results in the literature, however, are ambiguous and are presented in Table 1.
[ | l | l | p[2.4in]{} |]{}**Author(s)** & **Period/Countries** & **Methodology**\
\
\
Ghosh [@Ghosh2002] & 1950-1997/India & Johansen-Juselius;Granger causality-VAR\
Narayan et al. [@Narayan2005] & 1966-1999/Australia & ARDL bounds testing;Granger causality\
Yoo and Kim [@Yoo2006] & 1971-2002/Indonesia & Engle-Granger;Johansen-Juselius;Hsiao’ s causality\
Ho and Sui [@Ho2007] & 1966-2002/Hong Kong & Johansen-Juselius;Granger causality\
Mozumder and Marathe [@Mozumder2007] & 1971-1999/Bangladesh & Johansen-Juselius;Granger causality\
Jamil and Ahmad [@Jamil2010] & 1960-2008/Pakistan & Johansen-Juselius;Granger causality\
Shahbaz and Feridun [@Shahbaz2012] & 1971-2008/Pakistan & Toda Yamamoto Wald-test causality tests\
\
Aqeel and Butt [@Aqeel2001] & 1955-1996/Pakistan & Engle-Granger;Hsiao’s causality\
Shiu and Lam [@Shiu2004] & 1971-2000/China & Johansen-Juselius;Granger causality\
Altinay and Karagol [@Altinay2005] & 1950-2000/Turkey & Dolado-Lutkepohl test for causality\
Lee and Chang [@Lee2005] & 1954-2003/Taiwan & Johansen-Juselius;Weak exogeneity test\
Yoo [@Yoo2005] & 1970-2002/Korea & Johansen-Juselius;Granger causality\
Narayan and Singh [@Narayan2007] & 1971-2002/Fiji Islands & ARDL bounds testing;Granger causality\
Yuan et al. [@Yuan2007] & 1978-2004/China & Johansen-Juselius;Granger causality\
Odhiambo [@Odhiambo2009a] & 1971-2006/ Tanzania& ARDL bounds testing;Granger causality-VECM\
Abosedra et al. [@Abosedra2009] & 1995-2005/ Lebanon & Granger causality\
Chandran et al. [@Chandran2010] & 1971-2003/Malaysia & ARDL bounds testing;Engle-Granger;Johansen-Juselius;Granger causality\
Narayan and Narayan [@Narayan2010] & 1980-2006/93 Countries & Granger causality\
Ahamad and Nazrul [@Ahamad2011] & 1971-2008/Bangladesh & Granger causality\
Bildirici and Kayikci [@Bildirici2012] & 1990-2009/11 Commonwealth Independent States & Fully Modified Ordinary Least Squares and Panel ARDL\
\
Yang [@Yang2000] & 1954-1997/Taiwan & Engle-Granger;Granger causality-VAR\
Jumbe [@Jumbe2004] & 1970-1999 Malawi & Engle-Granger;Granger causality-VECM\
Zachariadis and Pashourtidou [@Zachariadis2007] & 1960-2004/Cyprus & Johansen-Juselius;Granger causality-VECM\
Tang [@Tang2008] & 1972-2003/Malaysia & Granger causality\
Tang [@Tang2009] & 1970-2005/Malaysia & ARDL bounds testing;Granger causality\
Odhiambo [@Odhiambo2009b] & 1971-2006/South Africa & Johansen-Juselius;Granger\
Lean and Smyth [@Lean2010] & 1971-2006/Malaysia & A RDL bounds testing;Johansen-Juselius\
Ouedraogo [@Oue2010] & 1968-2003/Burkina Faso & ARDL bounds testing;Granger\
Shahbaz et al. [@Shahbaz2011] & 1971-2009/Portugal & Granger causality\
Kouakou [@Kouakou2011] & 1971-2008/Cote d’Ivoire & Granger causality\
Gurgul and Lach [@Gurgul2011] & Q1 2000-Q4 2009/Poland & Toda-Yamamoto\
Shahbaz and Lean [@ShahbazLean2012] & 1972-2009/Pakistan & Granger causality\
\
Wolde [@Wolde2006] & 1971-2001/17 African countries & ARDL Bounds testing;Toda-Yamamoto’s causality\
Chen et al. [@Chen2007] & 1971-2001/10 Asian countries & Johansen-Juselius;Granger causality\
Narayan and Prasad [@Narayan2008] & 1960-2002/30 OECD countries & Toda-Yamamoto’s causality test\
Payne [@Payne2009] & 1949-2006 US & Granger causality\
Ozturk and Acaravci [@Ozturk2010] & 1980-2006/4 European countries & ARDL Bounds test and Granger causality\
Ozturk and Acaravci [@Ozturk2011] & 1971-2006 MENA countries & ARDL Bounds test-VECM\
The topic of the causal relationship between energy consumption and economic growth has been well studied in the energy economics literature. Different studies have focused on different countries, time periods, proxy variables and different econometric methodologies have been used to determine the energy consumption and growth relationship. Moreover, a literature survey on the relationship between energy consumption and economic growth is given in detailed by Ozturk [@Ozturk].
Complex networks provide a very general framework, based on the concepts of statistical physics, for studying systems with large numbers of interacting assets. These networks have been able to successfully describe the topological properties and characteristics of many real-life systems such as multilocus sequence typing for analyses of clonality [@Chen2006], scientific collaboration in the European framework programs [@Garas2008], taxonomy of correlations of wind velocity [@Bivona2008], Brazilian term structure interest rates [@Tabak2009], the international hotel industry in Spain [@Brida2010a], and foreign trade [@Kantar2011]. Moreover, the most recent literature has studied networks generated by correlations of stock prices [@Mantegna1999; @Mantegna2000; @Bonanno2003; @Bonanno2000; @Onnela2003a; @Onnela2003b; @Jung2006; @Tumminello2007a; @Jung2008; @Feng2010; @Keskin2010a; @Keskin2010b; @Kantar2011a; @Kantar2011b]. In this paper, we focus on the electricity consumption and the main objective is to characterize the topology and taxonomy of the network of the countries. To the best of the authors’s knowledge, this is the first study on electricity consumption and economic growth by using the hierarchical structure methods.
The aim of the present paper is to examine relationships among countries, based on low income group, middle income group and high income group countries, by using the concept of the minimal spanning tree (MST) and hierarchical tree (HT) over the period between 1971-2008. From these trees, both geometrical (through the MST) and taxonomic (through the HT) information about the correlation between the elements of the set can be obtained. Note that the MST and then the HT are constructed using the Pearson correlation coefficient as a measure of the distance between the time series. Moreover, we use the bootstrap technique to associate a value of reliability to the links of the MST. We also use average linkage cluster analysis to obtain the HT. These methods give a useful guide to determining the underlying economic or regional causal connections for individual countries.
The remainder of the paper is structured as follows. The next section briefly introduces the set of empirical data we work with. Sec. III is targeted at presenting the method. Sec. IV presents the empirical results. Finally, Sec. IV provides some final considerations.
The data
========
We chose data on the electricity consumption of 60 low income group, middle income group and high income group countries. We used the data period from 1971 to 2008 and listed the countries and their corresponding symbols in Table 2. The annual amounts were downloaded from the World Bank database (http://data.worldbank.org/).
The method
==========
In this section, we describe the methodology used for the analysis of the data. Recent empirical and theoretical analysis have shown that useful economic information can be detected in a correlation matrix using a variety of methods [@Mantegna1999; @Mantegna2000; @Mizuno2006; @Ortega2006; @Brida2009a; @Feng2010; @Keskin2010a; @Naylor2007; @Bonanno2004; @Vandewalle2001; @Brida2010; @Eom2007; @Brida2007; @Brida2009b; @Garas2007; @Brida2010c; @Bonanno2000; @Coelho2007a; @Gilmore2008; @Sieczka2009; @Tabak2010; @Onnela2003a; @Onnela2003b; @Onnela2002; @Onnela2003c; @Micciche2003; @Coelho2007b]. In this paper, we use three different approaches, based on hierarchical methods (MST and HT), the bootstrap technique, and the ALCA technique. We will briefly describe the basic aspects of these three different methods in the subsections.
Minimal spanning tree (MST) and hierarchical tree (HT)
------------------------------------------------------
In order to construct the MST following the method suggested by Mantegna [@Mantegna1999], the correlation coefficient between a pair of countries based on electricity consumption should be calculated in the first step. The correlation coefficient between a pair of countries based on electricity consumption defines a degree of similarity between the synchronous time evolution of a pair of assets between the countries.
$$\label{GrindEQ__1_}
C_{ij} =\frac{\left\langle R_{i} R_{j} \right\rangle -\left\langle R_{i} \right\rangle \left\langle R_{j} \right\rangle }{\sqrt{\left(\left\langle R_{i}^{2} \right. \rangle -\left\langle R_{i} \right\rangle ^{2} \right)\left(\left\langle R_{j}^{2} \right. \rangle -\left\langle R_{j} \right\rangle ^{2} \right)} } ,$$
where ${ R}_{{ i}}$ is the vector of the time series of log-returns, ${ R}_{{ i}} {(t)\; =\; ln\; P}_{{ i}} { (t\; +\; }\tau { )\; -\; ln\; P}_{{ i}} { (t)}$ is the log return, and ${ P}_{{ i}}(t)$ is the electricity consumption amount of a country i (i=1,..., N) at time t. We take $\tau$ as one annual in the following analysis throughout this paper.
We create a country network with a significant relationship between countries using the MST. The MST, a theoretical concept in graph theory [@West1996], is the spanning tree of the shortest length using the Kruskal algorithm [@Kruskal1956; @Cormen1990; @Prim1957]. Hence, it is a graph without a cycle connecting all nodes with links. This method is also known as the single linkage method of cluster analysis in multivariate statistics [@Everitt1974]. The MST is generated from the graph by selecting the most important correlations between foreign trade prices. The MST reduces the information space from $\textit{N}(\textit{N - 1})\textit{/2}$ separate correlation coefficients to ($\textit{N - 1}$) linkages, known as tree “edges”, while retaining the salient features of the system [@Gilmore2008]. Therefore, the MST is a tree which has $\textit{N - 1}$ edges that minimize the sum of the edge distances in a connected weighted graph of the *N* rates.
Mantegna [@Mantegna1999], and Mantegna and Stanley [@Mantegna2000] showed that the correlation coefficients can be transformed into distance measures, which can in turn be used to describe hierarchical organization of the group of analyzed assets. Distance measure $$\label{GrindEQ__2_}
{\rm d}_{{\rm ij}} =\sqrt{2(1-C_{ij} )} ,$$
where ${\rm d}_{{\rm ij}} $ is a distance for a pair of the rate i and the rate *j*, and it fulfills the three axioms of Euclidean distance [@Mantegna1999].
Now, one can construct an MST for a pair of countries using the N $\times$ N matrix of ${\rm d}_{{\rm ij}} $. Hence, a country network with a significant relationship between countries using the MST is obtained. The MST, a theoretical concept in graph theory [@West1996], is the spanning tree of the shortest length using the Kruskal algorithm [@Kruskal1956; @Cormen1990; @Prim1957]. Hence, it is a graph without a cycle connecting all nodes with links. This method is also known as the single linkage method of cluster analysis in multivariate statistics [@Everitt1974].
We also introduce the ultrametric distance or the maximal ${\rm d}_{{\rm ij}}^{{\rm \wedge }} $${}_{ }$ between two successive countries encountered in order to construct an HT, when moving from the first country *i* to the last country *j* over the shortest part of the MST connecting the two countries. (For a fuller technical discussion see [@Mantegna1999; @Mantegna2000; @Bonanno2003; @Onnela2003a; @Tumminello2007a; @Feng2010; @Keskin2010a; @Kantar2011a].) The hierarchical tree ranks the linkages between countries via the subdominant ultrametric distance, beginning with the pair exhibiting the shortest distance measure. Successive countries are added to the center of this tree in order of increasing distances. Thus, the last country added to the hierarchical tree are those with the most distant linkages to the center country or countries.
The stability of links with the bootstrap technique
---------------------------------------------------
The major weakness of the described methodology lies in the fact that the calculated MST and HT might be unstable. Moreover, without further statistical analysis, we cannot be sure whether the links present in the MST are actually the important links in the network or are rather a statistical anomaly, i.e. whether the results are sensitive to the sampling. We use a bootstrap technique proposed by Tumminello et al. [@Tumminello2007b; @Tumminello2007a; @Tumminello2010] specifically for MST and HT analysis to deal with the problem. The bootstrap technique, which was invented by Efron [@Efron1979], and has been widely used in phylogenetic analysis since the paper by Felsenstein [@Felsenstein1985] as a phylogenetic hierarchical tree evaluation method [@Efron1996]. This technique was used to quantify the statistical reliability of the hierarchical structures of Turkey’s foreign trade [@Kantar2011] and major international and Turkish companies [@Kantar2011a].
In the technique, by using the original MST and HT, we construct a bootstrapped time series from the original while keeping the the length of the time series fixed (i.e. the observations may repeat in the bootstrapped sample). MST and HT are then constructed for the bootstrapped time series and links are recorded. It is then checked whether the connections in the original MST are also present in the new MST based on bootstrapped time series. We repeat such procedure 1000 times so that we can distinguish whether the connections in the original MST and HT are the strong ones or statistical anomalies [@Keskin2010a]. The bootstrap value gives information about the reliability of each link of a graph.
Cluster analysis
----------------
The correlation matrix of the time series of a multivariate complex system can be used to extract information about aspects of the hierarchical organization of such a system. Correlation based clustering has been used to infer the hierarchical structure of a portfolio of stocks from its correlation coefficient matrix [@Mantegna1999; @Bonanno2001; @Bonanno2003]. The correlation based clustering procedure also allows a correlation based network to associate with the correlation matrix. For example, it is natural to select the MST as the correlation based network associated with single linkage cluster analysis. A different correlation based on networks can be associated with the same hierarchical tree putting emphasis on different aspects of the sample correlation matrix. Useful examples of correlation based networks apart from the minimum spanning tree are the planar maximally filtered graph [@Tumminello2005] and the average linkage minimum spanning tree [@Tumminello2007a; @Kantar2011; @Kantar2011a].
We use average linkage cluster analysis (ALCA) in order to observe more clearly the different clusters of countries according to their geographical location and economic growth. Since the ALCA, which is a hierarchical clustering method, and an account of the method was presented in detailed by Tumminello et al. [@Tumminello2010] and also [@Tumminello2007a; @Kantar2011; @Kantar2011a], we only give the obtained results. The constructions of the MST and HT will be elaborated in Section IV.
Numerical Results and Discussions
=================================
In this section, we present the MST, including the bootstrap values, and HT of 60 countries based on electricity consumption from 1971 to 2008. These countries are divided into three subgroups: low income group, middle income group and high income group countries. We also investigate cluster structures by using a clustering linkage procedure.
We construct the MST by using Kruskal’s algorithm [@Kruskal1956; @Cormen1990; @West1996] for the electricity consumption based on a distance-metric matrix. The amounts of the links that persist from one node (country) to the other correspond to the relationship between the countries in electricity consumption. We carried out the bootstrap technique to associate a value of the statistical reliability to the links of the MST. If the values are close to one, the statistical reliability or the strength of the link is very high. Otherwise, the statistical reliability or the strength of the link is lower [@Tumminello2007a; @Keskin2010a]. We also obtained the cluster structure of the hierarchical trees much better by using average linkage cluster analysis.
Fig. 1 shows the MST applying the method of Mantegna [@Mantegna1999], Mantegna and Stanley [@Mantegna2000] for electricity consumption based on a distance-metric matrix for the period 1971-2008. In Fig. 1, we observe different clusters of countries according to their geographical proximity and economic growth. In this figure, we detected three different clusters: mainly European Union countries formed the first cluster of countries with a GDP of over \$30,000; the second cluster was formed mainly by some European and South American countries; and mainly African countries formed the third cluster with a GDP of under \$5,000. It can also be clearly seen that in the MST, the European Union countries form the central structure. It is observed that DEU is at the center of the European Union countries and it is the predominant country for this period. The first cluster consists of DEU, AUT, FRA, ITA, DNK, NLD, ESP, LUX, BEL, IRL, FIN, GBR, SWE, NOR, GRC, USA, JPN, CAN and CHE, which are an European Union countries except USA, JPN, CAN and CHE; hence it is a heterogonous cluster. In this cluster, there are strong relationships among BEL - NLD, SWE - NOR, AUT - CHL and USA - JPN. We can establish this fact from the bootstrap values of the links between these countries, which are equal to 1.00, 0.91, 0.91 and 0.91 in a scale from zero to one, respectively; hence these countries are very closely connected with each other. The second cluster is composed of some European and South American countries, namely, HUN, POL, ROM, BGR, CZE, BRA, ARG, URY, MEX, OMN and NZL. In this cluster, there are strong relationships among HUN - MEX, BRA - OMN and POL - ROM. We can establish this from the bootstrap values of the links among the countries, which are equal to 1.00, 0.87 and 0.78 in a scale from zero to one, respectively. The third cluster was formed by mainly of African countries, and was separated four sub-groups. The first sub-group contains SEN, KEN and ETP, and there is strong relationships between SEN and KEN. We can establish this from the bootstrap value of the link between the SEN and KEN, which is equal to 1.00 in a scale from zero to one. The second sub-group consists of BEN, BGD and PAK, and the bootstrap values of the links between BEN - BGD and BGD - PAK are equal to 0.83 and 0.74, respectively in this sub-group; hence these countries are very closely connected with each other. (MAR, ZMB and CMR) and (YEM and NPL) formed the third and fourth sub-groups, respectively. On the other hand, the bootstrap values of the links between GHA - ZWE, LUX - CHL, TUR - IND, TUR - VNM and KEN - ETH are very low, as seen in Fig. 1. This means that these links could only demonstrate a statistical fluctuation. It is worth mentioning that in comparison with other regions, such as Latin America, the Middle East, Europe, and North America, Africa has one of the lowest per capita consumption rates. Modern energy consumption in Africa is very low and heavily reliant on traditional biomass.
The HT of the subdominant ultrametric space associated with the MST is shown in Fig. 2. Two countries (lines) link when a horizontal line is drawn between two vertical lines. The height of the horizontal line indicates the ultrametric distance at which the two countries are joined. To begin with, in Fig. 2, we can observe three clusters. The first cluster is composed of countries with a GDP per capita of over \$30,000 and consists of three sub-groups, namely European Union countries (DEU, AUT, FRA, ITA, BEL, NLD and FIN), USA and JPN, and SWE and NOR. The distance between ITA and BEL is the smallest of the sample, indicating the strongest relationship between these two countries. The second cluster is mainly made up of countries from Europe and South America. In this cluster, the distance between ROU and CZE is the smallest of the sample, indicating the strongest relationship between these two countries. The third cluster is composed of mainly African countries; it also includes of the two sub-groups, namely YEM and NPL, and PAK and BGD.
In the HT, we used average linkage cluster analysis (ALCA) in order to observe the cluster structure more clearly. The HT seen in Figs. 3 is obtained from data based on electricity consumption for the period 1971-2008. When comparing the HT and ALCA, similar cluster structures were observed; however, the number of countries in the ALCA cluster was found to be more than in the HT. For example, seven countries in the cluster with a GDP per capita of over \$30,000 were seen in the HT, but seventeen countries were seen in ALCA, as can be verified by comparing Fig. 2 with Fig. 3. In addition the groups of African countries are more clearly. Thus, we see that the cluster structures are obtained more efficiently by using ALCA.
Overall results of the study show that even there is a strong relationship between energy consumption and economic growth for some individual countries, and also three different clusters are detected: mainly European Union countries formed the first cluster of countries with a GDP of over \$30,000; the second cluster was formed mainly by some European and South American countries; and mainly African countries formed the third cluster with a GDP of under \$5,000. In other words, there is an evidence indicating that energy consumption leads economic growth in some of the three income groups considered in this study. Therefore, a stronger energy conservation policy should be pursued in all countries. In addition, policymakers should take into consideration the degree of economic growth in each country when energy consumption policy is formulated.
SUMMARY AND CONCLUSION
======================
There is a growing literature that examines the relationship between energy consumption and economic growth. The bulk of this literature focuses on developing, developed and emerging countries. It is important for policymakers to understand the relationship between energy consumption and economic growth in order to design effective energy and environmental policies. A general conclusion from these studies is that there is no consensus either on the existence of the relationship or the direction of causality between energy consumption and economic growth in the literature.
In this paper attempts were made to re-examine the strong relationship between energy consumption and economic growth and vice versa in 60 countries by using the concept of the MST, including the bootstrap values, and the HT for the 1971-2008 period. We also divided these countries into three subgroups: low income group, middle income group and high income group countries. We obtained the clustered structures of the trees and identified different clusters of countries according to their geographical proximity and economic growth. From the topological structure of these trees, we found that the European Union countries are at the center of the network and the bootstrap values show that they are closely connected to each other. We also found that these countries play an important role in world electricity consumption. Moreover, African countries have low energy consumption compared to other regions such as Latin America, the Middle East, Europe, and North America. We performed the bootstrap technique to associate a value of statistical reliability to the links of MST to obtain information about the statistical reliability of each link of the trees. From the results of the bootstrap technique, we can see that, in general, the bootstrap values in the MST are highly consistent with each other. We also used average linkage cluster analysis to obtain the cluster structure of the hierarchical trees more clearly. The results are in good agrement with the causal relationship between the electricity consumption and economic growth along a survey of the empirical literature. The findings of this study have important policy implications and it shows that this issue still deserves further attention in future research. Finally, it is important for policymakers to understand the relationship between energy consumption and economic growth in order to design effective energy and environmental policies.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We have studied how the equation of state of thermal QCD with two light flavours is modified in strong magnetic field by calculating the thermodynamic observables of hot QCD matter up to one-loop, where the magnetic field affects mainly the quark contribution and the gluonic part is largely unaffected except for the softening of the screening mass due to the strong magnetic field. To begin with the effect of magnetic field on the thermodynamics, we have first calculated the pressure of a thermal QCD medium in strong magnetic field limit (SML), where the pressure at fixed temperature increases with the magnetic field faster than the increase with the temperature at constant magnetic field. This can be envisaged from the dominant scale of thermal medium in SML, which is the magnetic field, like the temperature in thermal medium in absence of strong magnetic field. Thus although the presence of strong magnetic field makes the pressure of hot QCD medium harder but the increase of pressure with respect to the temperature becomes less steeper. Corroborated to the above observations, the entropy density is found to decrease with the temperature in the ambience of strong magnetic field which resonates with the fact that the strong magnetic field restricts the dynamics of quarks in two dimensions, hence the phase space gets squeezed resulting the reduction of number of microstates. Moreover the energy density is seen to decrease and the speed of sound of thermal QCD medium is increased in the presence of strong magnetic field. These crucial findings in strong magnetic field could have phenomenological implications in heavy ion collisions because the expansion dynamics of the medium produced in noncentral ultrarelativistic heavy ion collisions is effectively controlled by both the energy density and the speed of sound.'
author:
- |
Shubhalaxmi Rath[^1] and Binoy Krishna Patra[^2]\
Department of Physics,\
Indian Institute of Technology Roorkee, Roorkee 247667, India
title: '**[One-loop QCD thermodynamics in a strong homogeneous and static magnetic field]{}**'
---
Introduction
============
The last three decades had witnessed hectic activities towards recreating the conditions similar to those existing shortly after the big bang, known as Quark-gluon Plasma (QGP), in the terrestrial laboratory by carrying out collisions of ultra-relativistic heavy ions RHIC, BNL and LHC at CERN, where only few events are truly head-on, indeed most occur under a finite impact parameter or centrality. As a consequence, the two highly charged ions impacting with a small offset may produce extremely large magnetic fields $ \sim m_{\pi}^2$ ($\simeq 10^{18}$ Gauss) at RHIC and $\sim 15 m_{\pi}^2$ at LHC [@Skokov:IJMP24'2009; @Bzdak:PLB710'2012]. Naive classical estimates of the lifetime of these magnetic fields indicate that they only exist for a small fraction of the lifetime of QGP [@Kolb:PRC67'2003; @QGP]. However depending on the transport coefficients of the medium, the magnetic field may be near to its maximum strength and also be stationary [@Tuchin:PRC82'2010] - [@Fukushima:PRD86'2012] in its lifetime. Moreover the magnetic field may be assumed uniform because even though the spatial distribution of the magnetic field is globally inhomogeneous but in the central region of the overlapping nuclei the magnetic field in the transverse plane varies very smoothly, which is found in the simulations of hadron-string-dynamics model [@Homogeneous] for ${\rm{Au}}$-${\rm{Au}}$ collisions at $\sqrt{s_{NN}}=$ 200 GeV with impact parameter, b=10 fm. Therefore, it is worthwhile to investigate the QCD physics in a strong homogeneous and static magnetic field, such as chiral magnetic effect related to the generation of electric current parallel to magnetic field due to difference in number of right and left-handed quarks [@KDH] - [@Kharzeev:PPNP75'2014], axial magnetic effect due to the energy flow by the axial magnetic field [@Braguta:PRD89'2014; @MAAKM], chiral vortical effect due to an effective magnetic field in rotating QGP [@DD; @Kharzeev:PPNP88'2016], magnetic catalysis and inverse magnetic catalysis at finite temperature arising due to the breaking and restoration of chiral symmetry [@VVI] - [@AFA], thermodynamic properties [@Andersen:JHEP1210'2012] - [@MVD], refractive indices and decay constant [@Fayazbakhsh:PRD86'2012; @Fayazbakhsh:PRD88'2013] of mesons in hot magnetized medium, conformal anomaly and production of soft photons [@Basar:PRL109'2012; @AJCL] at RHIC and LHC, dispersion relation in a magnetized thermal QED [@Sadooghi:PRD92'2015], synchrotron radiation [@Tuchin:PRC87'2013], dilepton production from both weakly [@Tuchin:PRC87'2013] - [@Sadooghi:AP376'2017] and strongly [@Mamo:JHEP1308'2013] coupled plasma.
A variety of studies of the effects of strong magnetic fields on QCD thermodynamics have been recently resurrected by the possibility to achieve the magnetic fields at RHIC and LHC. Since the magnetic field breaks the translational invariance in space, so the pressure becomes anisotropic arising due to the difference between the pressures that are transverse and longitudinal to the direction of background magnetic field, which is illustrated for an ensemble of spin one-half particles [@MVD]. Recent lattice QCD calculations [@GFGSA] delve into the effects of background magnetic fields on the equation of state (EoS) by calculating the thermodynamic observables, namely transverse and longitudinal pressure, magnetization, energy density, entropy density etc. and inferred that the transition temperature gets reduced by the magnetic field [@Agasian:PLB663'2008] - [@Ayala:JPCS720'2016]. For the hadronic matter too, the phase structure and the phase transitions in strong magnetic fields and zero quark chemical potentials have been reviewed in [@JWA] through the low-energy effective theories and models, where the thermodynamic quantities are also found to increase with the magnetic field [@AANA]. Thus the EoS is expected to be changed due to the magnetic field and this fascinates us to study and explore the modification of the EoS in presence of strong magnetic field. In the present work, we also aim to study the thermal QCD equation of state perturbatively up to one-loop in a background of strong and homogeneous magnetic field.
For thermal medium the free energy of non-abelian gauge theories has been analytically computed up to $\mathcal{O}\left(g^4\right)$ in [@PC; @PC1] ($g$, the coupling constant) and up to $\mathcal{O}\left(g^5\right)$ in [@CB]. The values of pressure obtained by the addition of successive higher order contributions oscillate haphazardly and seem to diverge, hence the reorganization of perturbative expansion of thermodynamic quantities becomes necessary. In this process, various renormalization scales and effective field theory methods have been emerged, such as evaluation of free energy by the finite temperature effective field theory methods [@EA], where the contributions coming from various momentum scales, [*viz.*]{} $T$, $gT$, and $g^2T$ are separated in weak-coupling regime. However, in presence of magnetic field, a thermal medium possesses an additional scale related to the magnetic field and depending on the strength of magnetic field compared to the temperature of thermal medium and the quark masses, QCD thermodynamics has been studied in two scenarios : In weak field limit ($T^2 \gg |q_fB|$, $T^2 \gg m^2_f$, where $m_f$ and $|q_f|$ are the mass and the absolute charge of the quark with flavor $f$, respectively), the temperature remains the dominant scale of the system, so the hard thermal loop (HTL) perturbation theory remains the best theoretical tool in calculating the free energy of hot quark gluon plasma [@1JEM] - [@ANM]. On the other hand, in strong magnetic field limit ($|q_fB|\gg T^2$, $|q_fB|\gg m^2_f$), we calculate the thermodynamic observables by replacing the upper limit of loop momentum by the magnetic field, which is the most prominent scale available now (precisely $\sqrt{eB}$), like the temperature in thermal medium in absence of strong magnetic field in HTL perturbation theory.
In presence of magnetic field, the quark momentum $\mathbf{p}$ is separated into components transverse and longitudinal to the direction of magnetic field (say, $z$-direction), hence the dispersion relation for quarks is modified quantum mechanically into $$\begin{aligned}
E_n(p_z)=\sqrt{p_z^2+m_f^2+2n\left|q_fB\right|}
\quad,\end{aligned}$$ where $n=0$,$1$,$2$,$\cdots$ are the quantum numbers specifying the Landau levels. In strong magnetic field, the quarks are rarely excite thermally to the higher Landau levels, only the lowest Landau levels (LLL) $(n=0)$ are populated ($E_0=\sqrt{p_z^2+m_f^2}$). Thus the dynamics of quarks are effectively restricted to $(1+1)$ dimensions. In addition, the quark propagator is also modified in the magnetic field, which was first derived in coordinate space by Schwinger [@JS] using the proper-time method and later by Tsai [@W] in the momentum space. With the modifications discussed above in strong magnetic field, our aim will be to calculate the one-loop contribution to the thermodynamic observables of a hot strongly magnetized QCD matter to analyze the behaviour of QGP phase in strong magnetic field.
Our work proceeds in the following way. In section 2, we have derived the effective quark propagator in a thermal QCD medium in strong magnetic field limit. For that purpose, we first revisit the vacuum quark propagator in the presence of a strong magnetic field and obtain both the quark and gluon propagators at finite temperature in real-time formalism (RTF), in sections 2.1 and 2.2, respectively. This helps us to compute the one-loop quark self-energy in section 2.3 at finite temperature in strong magnetic field limit. Similarly we have derived the effective gluon propagator in the similar environment in section 3, where we first calculate the one-loop gluon self-energy in terms of screening mass in section 3.1. The effect of magnetic field enters through the screening mass, so we have calculated the screening mass in strong magnetic field limit in section 3.2 for both massless and physical quark masses by the static limit of real part of gluon self-energy. Having thus obtained the effective propagators for quarks and gluons in sections 2 and 3, respectively, we have calculated the quark and gluon free energies and then, the thermodynamic observables for a strongly magnetized QCD matter have been calculated in section 4. Finally we conclude in section 5.
Thermalized effective quark propagator in a strongly-magnetized hot QCD medium
==============================================================================
The effective quark propagator in a strongly-magnetized hot QCD medium is obtained from the Schwinger-Dyson equation : $$\begin{aligned}
S^{-1}(P) &=& S^{-1}_0(P)-\Sigma(P)
\quad,\label{defeffquarkprop}\end{aligned}$$ where $S_0(P)$ and $\Sigma(P)$ are the free propagator and quark self-energy in a strongly-magnetized hot QCD medium, respectively. As mentioned earlier, the strong magnetic field affects the quark propagator [*via*]{} the projection operator and the dispersion relation, which will, in turn affect the quark self-energy. In addition, the QCD coupling will now run with both the magnetic field and temperature, however, in strong magnetic field limit ($eB \gg T^2$), it runs exclusively with the magnetic field and is almost independent of the temperature, because the most dominant scale available is the magnetic field, not the temperature of medium anymore. For this purpose we closely follow the results in [@Ferrer:PRD91'2015], where the coupling is split into terms dependent on the momentum parallel and perpendicular to the magnetic field, separately. In our case of magnetic field ($\mathbf{B}=B\hat{z}$), we will use the coupling which depends on the longitudinal component only, because the energies of Landau levels for quarks in SML depend only on the longitudinal component of momentum. In fact, the coupling dependent on the transverse momentum does not depend on magnetic field at all, thus the relevant coupling is given by [@Ferrer:PRD91'2015] $$\alpha_{s}^\|(eB)=\frac{g^2}{4\pi}=\frac{1}{{\alpha_s^0(\mu_0)}^{-1}+\frac{11N_c}{12\pi}
\ln\left(\frac{\Lambda_{QCD}^2+M^2_B}{\mu_0^2}\right)+\frac{1}{3\pi}\sum_f \frac{|q_f B|}{\tau}}
~,$$ where $$\alpha_s^0(\mu_0) = \frac{12\pi}
{11N_c\ln\left(\frac{\mu_0^2+M^2_B}{\Lambda_V^2}\right)}
~,$$ $M_B$ is taken $\sim~1$ GeV as an infrared mass and the string tension is taken as $\tau=0.18 ~{\rm{GeV}}^2$.
We now first revisit the vacuum quark propagator in a strong magnetic field and then thermalize both quark and gluon propagators in a hot QCD medium, which are the ingredients to compute the quark self-energy.
Vacuum propagators in strong magnetic field
-------------------------------------------
The magnetic field breaks the translational invariance of space, as a result the quark propagator becomes function of separate components of momentum transverse and longitudinal to the magnetic field direction. Schwinger’s proper-time method [@JS] computes the quark propagator in coordinate-space as $$\label{S(X,Y)}
S(x,y)=\phi(x,y)\int\frac{d^4K}{(2\pi)^4}e^{-iK(x-y)}S(K)\quad,$$ where the phase factor $\phi(x,y)$ is expressed as $$\phi(x,y)=e^{i|q_f|\int^x_y A^\mu(\zeta)d\zeta_\mu}.$$ The above phase factor is the gauge-dependent part and is responsible for breaking of translational invariance. In a single fermion propagator, for a symmetric gauge, i.e. $A^\mu(x)=\frac{B}{2}(0,-x_2,x_1,0)$ in a magnetic field directed along the $z$ axis ($\mathbf{B}=B\hat{z}$), it is possible to gauge away the phase factor by an appropriate gauge transformation and one can work with the momentum-space representation of the propagator [@AJMMR] as an integral over the proper-time ($s$) $$\begin{aligned}
\nonumber{S(K)} &=& i\int^{\infty}_0{ds}e^{-is{m_f}^2}\exp\left(isk_\parallel^2
-\frac{ik_\perp^2\tan(|q_fBs|)}{|q_fB|}\right) \\ && \times\left[\left(m_f+
\gamma^\parallel\cdot{k}_\parallel\right)
\left(1+\gamma^1\gamma^2\tan(|{q_fBs}|)\right)
-\gamma^\perp\cdot{k}_\perp\left(1+\tan^2(|q_fBs|)\right)\right].\end{aligned}$$ The quantities in above equation are defined as follows $$\begin{aligned}
&&k_\parallel\equiv(k_0,0,0,k_3),~~ k_\perp\equiv(0,k_1,k_2,0),\nonumber\\
&&\gamma^\parallel\equiv(\gamma^0,~~\gamma^3),~~
\gamma^\perp\equiv(\gamma^1,\gamma^2),\nonumber\\
&&g^{\mu\nu}=g^{\mu\nu}_\parallel+g^{\mu\nu}_\perp, \nonumber\\
&&g^{\mu\nu}_\parallel={\rm{diag}}(1,0,0,-1),~~
g^{\mu\nu}_\perp={\rm{diag}}(0,-1,-1,0), \nonumber\\
&&\gamma^\parallel\cdot{k}_\parallel=\gamma^0
k_0-\gamma^3k_3,~~ \gamma^\perp\cdot{k}_\perp=\gamma^1k_1+\gamma^2k_2,
\nonumber\\
&&k^2_\parallel\equiv{k}^2_0-k^2_3,~~k^2_\perp\equiv{k}^2_1+k^2_2.\nonumber\end{aligned}$$ After integration over the proper-time, $s$, $S(K)$ can be written in discrete notation $$\label{S(K)}
S(K)=ie^{-\frac{k^2_\perp}{|q_fB|}}\sum^\infty_{n=0}
(-1)^n\frac{D_n(|q_fB|,K)}{k^2_\parallel-m^2_f-2|q_fB|n}
\quad,$$ where $D_n(|q_fB|,K)$ can be expressed in terms of generalized Laguerre polynomials labelling the Landau levels [@W; @AKD; @NK] as $$\begin{aligned}
\label{D_n(|q_fB|,K)}
\nonumber{D_n(|q_fB|,K)} &=& \left(\gamma^\parallel\cdot{k}_\parallel
+m_f\right)\left[\left(1-i\gamma^1\gamma^2\right)L_n\left(\frac{2k^2_\perp}{|q_fB|}\right)-\left(1+i\gamma^1\gamma^2\right)L_{n-1}\left(\frac{2k^2_\perp}
{|q_fB|}\right)\right] \\ && +4\gamma^\perp\cdot{k}_\perp{L^{(1)}_{n-1}}
\left(\frac{2k^2_\perp}{|q_fB|}\right)
\quad.\end{aligned}$$ In presence of a strong magnetic field ( $k^2_\parallel,k^2_\perp\ll|q_fB|$), the transitions to the higher Landau levels ($n\geq1$) are suppressed and only LLL ($n=0$) is occupied. Putting $n=0$ in equation ($\ref{D_n(|q_fB|,K)}$) yields the quark propagator in strong magnetic field in momentum-space $$\label{Q.P. in S.M.F.A.}
S_{LLL}(K)=ie^{-\frac{k^2_\perp}{|q_fB|}}\frac{\left(\gamma^\parallel
\cdot{k}_\parallel+m_f\right)}{k^2_\parallel-m^2_f}\left(1
-\gamma^0\gamma^3\gamma^5\right).$$ However, the gluons remain unaffected by the presence of magnetic field, hence the form of the vacuum gluon propagator remains the same even in presence of magnetic field.
Thermalization of propagators in strong magnetic field
------------------------------------------------------
The vacuum quark and gluon propagators in presence of strong magnetic field discussed above get thermalized in a thermal QCD medium using the RTF, where the medium effects are conceived through the distribution functions. In this formalism, propagators are manifestly separated into vacuum and thermal parts and the degrees of freedom has also been doubled so the propagators acquire a $2\times{2}$ matrix structure. We also note that, in an equilibrium system, to evaluate the real part of the one-loop quark self energy, it is adequate to calculate the 11-components of the quark and gluon propagators.
### Quark propagator
Since quarks are only populated in the lowest Landau levels in strong magnetic field, so LLL quark propagator in equation (\[Q.P. in S.M.F.A.\]) is used in the thermalization process. Denoting $S_{LLL}(K) =S_0(K)$, the vacuum quark propagator gets matrix structure in thermal medium as $$\begin{aligned}
\label{Q.P.}
S(K)=U_F(k_0)\begin{pmatrix}S_0(K) & 0 \\ 0 & S^*_0(K)\end{pmatrix}U_F(k_0)
\quad,\end{aligned}$$ where $U_F(k_0)$ is the unitary matrix which brings the temperature dependence through the distribution function and it has the following form. $$\begin{aligned}
U_F(k_0)=\begin{pmatrix}\sqrt{1-n_F(k_0)} & -\sqrt{n_F(k_0)}
\\ \sqrt{n_F(k_0)} & \sqrt{1-n_F(k_0)}\end{pmatrix}
\quad,\end{aligned}$$ with the distribution function for quarks $$\begin{aligned}
n_F(k_0)=\frac{1}{e^{\beta|k_0|}+1}
\quad.\end{aligned}$$ Substituting the unitary matrix in equation (\[Q.P.\]) and simplifying, we get $$\begin{aligned}
S(K)=\begin{pmatrix}n^2_2S_0(K)-n^2_1S^*_0(K) & -n_1n_2(S_0(K)+S^*_0(K))
\\ n_1n_2(S_0(K)+S^*_0(K)) & n^2_2S^*_0(K)-n^2_1S_0(K)\end{pmatrix}
\quad,\end{aligned}$$ where $n_1=\sqrt{n_F(k_0)}$ and $n_2=\sqrt{1-n_F(k_0)}$.
Now from the above matrix, the 11 - component of the quark propagator in a strongly magnetized thermal medium is obtained as $$\begin{aligned}
\label{11 Q.P.}
\nonumber{S_{11}}(K) &=& ie^{-\frac{k^2_\perp}{|q_fB|}}\left(\gamma^0k_0-\gamma^3k_3+m_f\right)\left(1
-\gamma^0\gamma^3\gamma^5\right) \\ && \times\left[\frac{1}
{k_{\parallel}^2-m_f^2+i\epsilon}+2\pi{i}n_F(k_0)
\delta\left(k_{\parallel}^2-m_f^2\right)\right],\end{aligned}$$ which is found to be modified by the strong magnetic field.
### Gluon propagator
For gluon, the unitary matrix needed to thermalize the propagator is of the form $$\begin{aligned}
U_B(q_0)=\begin{pmatrix}\sqrt{1+n_B(q_0)} & \sqrt{n_B(q_0)}
\\ \sqrt{n_B(q_0)} & \sqrt{1+n_B(q_0)}\end{pmatrix}
\quad,\end{aligned}$$ with the distribution function for gluons $$\begin{aligned}
n_B(q_0)=\frac{1}{e^{\beta|q_0|}-1}
\quad.\end{aligned}$$ In matrix form, the gluon propagator is expressed as $$\begin{aligned}
D^{\mu\nu}(Q)=U_B(q_0)\begin{pmatrix}D^{\mu\nu}_0(Q) & 0 \\ 0 & D^{*\mu\nu}_0(Q)\end{pmatrix}U_B(q_0).\end{aligned}$$ Now proceeding like the quark case, the 11 - component of the gluon propagator can be read from the above matrix as $$\begin{aligned}
\label{11 G.P.}
D^{\mu\nu}_{11}(Q)=ig^{\mu\nu}\left[\frac{1}{Q^2+i\epsilon}
-2\pi{i}n_B(q_0)\delta\left(Q^2\right)\right]
.\end{aligned}$$
One-loop quark self-energy in strongly magnetized medium
--------------------------------------------------------
Using Feynman rules and 11-components of the quark and gluon propagators (equations (\[11 Q.P.\]) and (\[11 G.P.\])), the one-loop quark self energy (in figure 1) in presence of a strong magnetic field is given as $$\begin{aligned}
\Sigma(P) = -\frac{4}{3} g^2~i\int{\frac{d^4K}{(2\pi)^4}}\left[\gamma_\mu
{S_{11}(K)}\gamma^\mu{D_{11}(P-K)}\right]\quad,\end{aligned}$$ where the factor $4/3$ is associated with the fundamental representation of ${SU(3)}_c$ gauge group through the relation: $C_F=\frac{N^2_c-1}{2N_c}$ and $g$ is the running coupling constant.
{width="4.9cm"}
The momentum integration can be factorized into parallel and perpendicular components with respect to the direction of magnetic field $$\begin{aligned}
\Sigma(P) &=& \frac{4g^2i}{3(2\pi)^4}
\int{d^2k_{\perp}}{d^2k_{\parallel}e^{-\frac{k_{\perp}^2}{\mid{q_fB}\mid}}}\left[\gamma_\mu\left(\gamma^0k_0-\gamma^3k_3
+m_f\right)\left(1-\gamma^0\gamma^3\gamma^5\right)\gamma^\mu\right] \nonumber\\
&& \times\left[\frac{1}{k_{\parallel}^2-m_f^2+i\epsilon}+2\pi{i}n_F(k_0)\delta
\left(k_{\parallel}^2-m_f^2\right)\right] \nonumber\\ &&
\times\left[\frac{1}{(P-K)^2+i\epsilon}-2\pi{i}n_B(p_0-k_0)\delta
\left((P-K)^2\right) \right]~.
\label{qsefirst}\end{aligned}$$ In strong magnetic field limit, the external quark momentum $P$ can be assumed to be purely longitudinal [@VA], i.e. $p_\perp=0$, so the internal gluon momentum squared becomes $$\begin{aligned}
(P-K)^2=\left(p_\parallel-k_\parallel\right)^2
-k_\perp^2
~.\end{aligned}$$ In LLL approximation, $k_\parallel^2$, $k_\perp^2$ are assumed to be much smaller than $|q_fB|$, hence, the form factor $e^{-\frac{k_{\perp}^2}{|q_fB|}}$ can be set equal to $1$ and the upper limits of all the momenta integrals should be cut off at $|q_fB|$. By evaluating the product of gamma matrices in (\[qsefirst\]) as $$\begin{aligned}
\gamma_\mu\left(\gamma^0k_0-\gamma^3k_3+m_f\right)\left(1
-\gamma^0\gamma^3\gamma^5\right)\gamma^\mu=-2\left(\gamma^0k_0
-\gamma^3k_3-2m_f\right)~,\end{aligned}$$ the quark self energy can thus be separated into the vacuum and medium components as $$\begin{aligned}
\label{Total sigma}
\nonumber\Sigma(p_{\parallel}) &=& \frac{-8g^2i}{3(2\pi)^4}\int{d^2k_{\perp}{d^2k_\parallel}}\left(\gamma^0k_0-\gamma^3k_3-2m_f\right)
\\ && \nonumber\times\left[\frac{1}{k_{\parallel}^2-m_f^2+i\epsilon}+2\pi{i}n_F(k_0)\delta(k_{\parallel}^2-m_f^2)\right]
\\ && \nonumber\times\left[\frac{1}{(p_\parallel-k_\parallel)^2
-k_\perp^2+i\epsilon}-2\pi{i}n_B(p_0-k_0)
\delta((p_\parallel-k_\parallel)^2-k_\perp^2)\right] \\
&\equiv& \Sigma_V(p_{\parallel})+\Sigma_n(p_{\parallel})
+\Sigma_{n^2}(p_{\parallel}),\end{aligned}$$ where $\Sigma_V(p_{\parallel})$ represents the quark self-energy in vacuum, $\Sigma_n(p_{\parallel})$ is the quark self energy in a medium containing both quark and gluon distribution functions and $\Sigma_{n^2}(p_{\parallel})$ represents the quark self energy in a medium containing product of quark and gluon distribution functions.
### Vacuum part
The vacuum contribution to the one-loop quark self energy is given by $$\begin{aligned}
\nonumber\Sigma_V(p_{\parallel})=\frac{-8g^2i}{3(2\pi)^4}
\int{d^2k_{\perp}{d^2k_\parallel}}\left(\gamma^0k_0-\gamma^3k_3
-2m_f\right)\\\times\left[\frac{1}{k_{\parallel}^2-m_f^2
+i\epsilon}\right]\left[\frac{1}{(p_\parallel-k_\parallel)^2
-k_\perp^2+i\epsilon}\right]
.\end{aligned}$$ Separating the (real) principal value and imaginary part by the identity: $$\label{identity}
\frac{1}{x\pm{y}\pm{i\epsilon}}=\rm{P}\left(\frac{1}
{x\pm{y}}\right)\mp{i\pi{\delta(x\pm{y})}}
~,$$ we obtain the real part of vacuum contribution, $\Sigma_V(p_{\parallel})$ with the following form $$\begin{aligned}
\label{$Sigma_v$}
\Sigma_V(p_{\parallel})=\frac{-8g^2i}{3(2\pi)^4}
\int{d^2k_{\perp}d^2k_{\parallel}}\left(\gamma^\parallel
\cdot{k}_\parallel-2m_f\right)\left[\frac{1}{k_{\parallel}^2
-m_f^2}\right]\left[\frac{1}{(p_\parallel-k_\parallel)^2-k_\perp^2}\right].\end{aligned}$$ Using the Feynman parametrization and Wick rotation, the parallel momentum ($k_\parallel$) integration can be recast into the form $$\begin{aligned}
\label{K.P. integration}
I &=& i\int^1_0~dz~d^2k^\prime~\frac{z\gamma^\parallel\cdot{p_\parallel}-2m_f}{\left[{k^\prime}^2-z(1-z)p^2_\parallel+zk^2_\perp+(1-z)m^2_f\right]^2} \nonumber\\
&=& -\frac{i\pi}{|q_fB|}\int^1_0dz\left(z\gamma^\parallel
\cdot{p}_\parallel-2m_f\right)+i\pi\int^1_0dz\frac{z\gamma^\parallel
\cdot{p}_\parallel-2m_f}{-z(1-z)p^2_\parallel+zk^2_\perp+(1-z)m^2_f}
\nonumber\\
&\equiv& I_1+I_2
\quad,\end{aligned}$$ where $I_1$ is solved as $$\begin{aligned}
\nonumber{I_1} &=& -\frac{i\pi}{|q_fB|}\int^1_0dz\left(z\gamma^\parallel
\cdot{p}_\parallel-2m_f\right) \\
&=& -\frac{i\pi}{2|q_fB|}\left(\gamma^\parallel\cdot{p}_\parallel-4m_f\right).\end{aligned}$$ For solving $I_2$, we expand it in a Taylor series around the mass-shell condition : $\gamma^\parallel\cdot{p}_\parallel=m_f$ in a strong magnetic field, $$\begin{aligned}
I_2=A+B\left(\gamma^\parallel\cdot{p}_\parallel-m_f\right)
+C\left(\gamma^\parallel\cdot{p}_\parallel-m_f\right)^2+\cdots\end{aligned}$$ Dropping the higher-order terms, the integral, $I_2$ is given by $$\begin{aligned}
\label{I_2}
I_2=A+B\left(\gamma^\parallel\cdot{p}_\parallel-m_f\right),\end{aligned}$$ where $A$ and $B$ are given by the following expressions $$\begin{aligned}
A &=& \left.I_2\right\vert_{\gamma^\parallel
\cdot{p}_\parallel=m_f} \nonumber\\
&=& i\pi{m_f}\int^1_0dz\frac{z-2}{m^2_f(1-z)^2+zk^2_\perp} \\
B &=& \left.\frac{\partial{I_2}}{\partial\left(\gamma^\parallel
\cdot{p}_\parallel\right)}\right
\vert_{\gamma^\parallel\cdot{p}_\parallel=m_f} \nonumber\\
&=& {i\pi}\int^1_0dz\frac{z}{(1-z)^2m^2_f+zk^2_\perp}+i\pi\int^1_0dz
\frac{2m^2_fz(1-z)(z-2)} {\left((1-z)^2 m^2_f+zk^2_\perp\right)^2}
~,\end{aligned}$$ respectively. At least, for light flavours we may drop terms proportional to $m^2_f$ and higher orders in the solutions of above integrals to get $A$ and $B$ as $$\begin{aligned}
\label{A}
A &=&\frac{i\pi}{2m_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right), \\
B&=&-\frac{i\pi}{2m^2_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right).\end{aligned}$$ Substituting the values for A and B in equation (\[I\_2\]), it yields $$\begin{aligned}
\nonumber{I_2} &=& \frac{i\pi}{2m_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right)-i\pi\left(\gamma^\parallel\cdot{p}_\parallel-m_f\right)
\frac{1}{2m^2_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right) \\ &=& \frac{i\pi}{m_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right)-\left(\gamma^\parallel\cdot{p}_\parallel\right)\frac{i\pi}{2m^2_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right)
~.\end{aligned}$$ Now the integral involving the $k_\parallel$ integration (\[K.P. integration\]) yields $$\begin{aligned}
I &=& -\frac{i\pi}{2|q_fB|}\left(\gamma^\parallel\cdot{p}_\parallel
-4m_f\right)+\frac{i\pi}{m_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right)-\left(\gamma^\parallel\cdot{p}_\parallel\right)\frac{i\pi}{2m^2_f}\ln\left(\frac{k^2_\perp}{m^2_f}\right)
~.\end{aligned}$$ Finally inserting the integral, $I$ in $\left(\ref{$Sigma_v$}\right)$ and then performing the remaining $k_\perp$ integration, we get the real part of the vacuum contribution of one-loop quark self-energy, $$\begin{aligned}
\label{QSE in vacuum}
\Sigma_V(p_{\parallel})
&=& \nonumber\frac{\left(\gamma^\parallel
\cdot{p}_\parallel\right)g^2}{6\pi^2}\left[-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left\lbrace\ln\left(\frac{|q_fB|}{m^2_f}\right)
-1\right\rbrace\right] \\ && +\frac{g^2}{6\pi^2}\left[2m_f
+\frac{|q_fB|}{m_f}\left\lbrace\ln\left(\frac{|q_fB|}{m^2_f}\right)-1\right\rbrace\right]
.\end{aligned}$$
### Medium part
In a medium, both $\Sigma_{n}(p_{\parallel})$ and $\Sigma_{n^2}(p_{\parallel})$, which contain single quark and gluon distribution and product of quark and gluon distribution functions, contribute to the one-loop quark self-energy. Now, using the identity (\[identity\]), we obtain the real-part of one-loop quark self energy due to single distribution function, $$\begin{aligned}
\label{S.D.F.P}
\nonumber\Sigma_n(p_{\parallel}) &=& \frac{8g^2}{3(2\pi)^3}\int{d^2k_{\perp}dk_3dk_0}\left(\gamma^0k_0
-\gamma^3k_3-2m_f\right) \\ && \nonumber\times\left[\frac{\delta
\left(k_{\parallel}^2-m_f^2\right)n_F(k_0)}{(p_\parallel-k_\parallel)^2-k_\perp^2}+\frac{\delta\left(\left(p_\parallel-k_\parallel\right)^2
-k_\perp^2\right)\left[-n_B(p_0-k_0)\right]}{k_\parallel^2-m^2_f}\right]
\\ &\equiv& \Sigma_{n_F}(p_{\parallel})+\Sigma_{n_B}(p_{\parallel}),\end{aligned}$$ with the quark and gluon contributions are now separable and the quark part is $$\begin{aligned}
\Sigma_{n_F}(p_{\parallel}) &=& \frac{8g^2}{3(2\pi)^3}
\int{d^2k_{\perp}dk_3dk_0}\left(\gamma^0k_0-\gamma^3k_3
-2m_f\right)\frac{\delta\left(k_0^2-\omega^2_k\right)n_F(k_0)}
{(p_\parallel-k_\parallel)^2-k_\perp^2}
~,\end{aligned}$$ with $\omega^2_k=k^2_3+m^2_f$. After performing the $k_0$ integration using the property of Dirac delta function, we find $$\begin{aligned}
\label{Sigma pp.}
\Sigma_{n_F}(p_{\parallel}) &=& \frac{-8g^2}{3(2\pi)^3}
\int{dk_3}\frac{n_F\left(\omega_k\right)}{\omega_k}
\left(\gamma^3k_3+2m_f\right)\int{d^2k_\perp}\frac{1}
{\left(p_\parallel-m_f\right)^2-k^2_\perp}
~,\end{aligned}$$ which involves two independent integrations over $k_3$ and $k_\perp$ momenta. The integral involving $k_3$ integration is solved into [@LR] $$\begin{aligned}
\label{$k_3$ integration}
I_{k_3} &=& \int^{+\infty}_{-\infty}{dk_3}
\frac{n_F\left(\omega_k\right)}{\omega_k}\left(\gamma^3k_3+2m_f\right)
\nonumber\\
&=&4m_f\left[-\frac{1}{2}\ln\left(\frac{m_f}{\pi{T}}\right)-\frac{1}{2}\gamma_E+\mathcal{O}\left(\frac{m^2_f}{T^2}\right)\right]
,\end{aligned}$$ where $\gamma_E$ is the Euler-Mascheroni constant. For a thermal medium considered here, $m^2_f$ for light flavours is much less than $T^2$, so the term, $\mathcal{O}\left({m^2_f}/{T^2}\right)$ can be dropped and $I_{k_3}$ turns out to be $$\begin{aligned}
I_{k_3}=-2m_f\left[\ln\left(\frac{m_f}{\pi{T}}\right)+\gamma_E\right].\end{aligned}$$ Now the $k_\perp$ integration in equation $\left(\ref{Sigma pp.}\right)$ is performed after taking the upper limit of the integration by $|q_fB|$ compatible to LLL approximation $$\begin{aligned}
I_{k_\perp} &=& -\pi\left[i\pi+\ln\left(\frac{|q_fB|}
{\left(p_\parallel-m_f\right)^2}\right)\right].\end{aligned}$$ Substituting the values of $I_{k_3}$ and $I_{k_\perp}$ integrations in equation (\[Sigma pp.\]) and keeping the real-part only, we obtain $$\begin{aligned}
\label{QSE in medium}
\Sigma_{n_F}(p_{\parallel}) &=& -\frac{2g^2m_f}{3\pi^2}
\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)
\left[\ln\left(\frac{m_f}{\pi{T}}\right)+\gamma_E\right].\end{aligned}$$ Similarly the part involving gluon distribution in equation $\left(\ref{S.D.F.P}\right)$ is $$\begin{aligned}
\nonumber\Sigma_{n_B}(p_{\parallel}) &=& -\frac{8g^2}{3(2\pi)^3}
\int{d^2k_{\perp}dk_3dk_0}\left(\gamma^0k_0-\gamma^3k_3-2m_f\right)
\\ && \times\frac{\delta\left(\left(p_\parallel-k_\parallel\right)^2
-k_\perp^2\right)}{k^2_0-\omega^2_k} n_B(p_0-k_0).\end{aligned}$$ Simplifying the argument of Dirac delta function in the above integration for small $k_\perp$, the $k_0$ and $k_\perp$ integrations have been facilitated to yield the contribution of gluon distribution as $$\begin{aligned}
\nonumber\Sigma_{n_B}(p_{\parallel}) &=& -\frac{4\pi|q_fB|g^2}{3(2\pi)^3}
\int^{+\infty}_{-\infty}{dk_3}~\frac{n_B(p_3-k_3)}{\left(p_3-k_3\right)}
\left[\frac{\gamma^0p_0-2m_f}{\left(p_0+p_3-k_3\right)^2-\omega^2_k}\right. \\ && \left.\nonumber+\frac{\gamma^0\left(p_3-k_3\right)}{\left(p_0+p_3-k_3\right)^2
-\omega^2_k}-\frac{\gamma^3k_3}{\left(p_0+p_3-k_3\right)^2-\omega^2_k}+\frac{\gamma^0p_0-2m_f}{\left(p_0-p_3+k_3\right)^2
-\omega^2_k}\right. \\ && \left.-\frac{\gamma^0\left(p_3-k_3\right)}{\left(p_0-p_3+k_3\right)^2-\omega^2_k}-\frac{\gamma^3k_3}{\left(p_0-p_3+k_3\right)^2
-\omega^2_k}\right].\end{aligned}$$ Finally the above $k_3$ integration can be integrated out to give $$\begin{aligned}
\Sigma_{n_B}(p_{\parallel})=-\frac{4\pi|q_fB|g^2}{3(2\pi)^3}\left(I^1+I^2+I^3+I^4+I^5+I^6\right)
,\end{aligned}$$ where $I^1$, $I^2$, $I^3$, $I^4$, $I^5$ and $I^6$ are found as follows $$\begin{aligned}
&&I^1=\frac{i\pi\left(\gamma^0{p_0}-2m_f\right)}{2\left(p_0+p_3\right)}
\left[\frac{\beta(a-p_3)-2}{2\beta(p_3-a)^2}+\frac{n_B(a-p_3)}{a-p_3}\right]~, \\
&&I^2=\frac{-i\pi\gamma^0}{2\left(p_0+p_3\right)}
\left[\frac{1}{\beta(p_3-a)}+n_B(a-p_3)\right]~, \\
&&I^3=\frac{-i\pi\gamma^3}{2\left(p_0+p_3\right)}
\left[\frac{-2a-\beta{p_3}(p_3-a)}{2\beta(p_3-a)^2}+\frac{an_B(a-p_3)}{a-p_3}\right]~, \\
&&I^4=\frac{-i\pi\left(\gamma^0{p_0}-2m_f\right)}{2\left(p_0-p_3\right)}
\left[\frac{\beta(b-p_3)-2}{2\beta(p_3-b)^2}+\frac{n_B(b-p_3)}{b-p_3}\right]~, \\
&&I^5=\frac{-i\pi\gamma^0}{2\left(p_0-p_3\right)}
\left[\frac{1}{\beta(p_3-b)}+n_B(b-p_3)\right]~, \\
&&I^6=\frac{i\pi\gamma^3}{2\left(p_0-p_3\right)}
\left[\frac{-2b-\beta{p_3}(p_3-b)}{2\beta(p_3-b)^2}+\frac{bn_B(b-p_3)}{b-p_3}
\right]~,\end{aligned}$$ where $a$ and $b$ are given by $$\begin{aligned}
&&a=\frac{\left(p_0+p_3\right)^2-m^2_f}{2\left(p_0+p_3\right)}~, \\
&&b=\frac{\left(p_0-p_3\right)^2-m^2_f}{2\left(p_3-p_0\right)}
~.\end{aligned}$$ Therefore, $\Sigma_{n_B}(p_{\parallel})$ cannot contribute to the real part of one-loop quark self-energy.
Finally, the medium contribution to the quark self energy involving product of quark and gluon distribution functions (from equation $(\ref{Total sigma})$) is $$\begin{aligned}
\nonumber\Sigma_{n^2}(p_{\parallel}) &=& \frac{-8g^2i}{3(2\pi)^4}
\int{d^2k_{\perp}{d^2k_\parallel}}4{\pi^2}n_F(k_0)n_B(p_0-k_0)\left(\gamma^0k_0-\gamma^3k_3-2m_f\right)
\\ && \times\left[\delta(k_{\parallel}^2-m_f^2)
\delta((p_\parallel-k_\parallel)^2-k_\perp^2)\right],\end{aligned}$$ which however does not contribute to the real part of quark self energy.
Thus the vacuum (\[QSE in vacuum\]) and the medium contributions (\[QSE in medium\]) are added together to give the real part of one-loop quark self energy of a thermal QCD medium in strong magnetic field $$\begin{aligned}
\label{Q.S.E.}
\nonumber\Sigma(p_{\parallel}) &=& \frac{\left(\gamma^\parallel
\cdot{p}_\parallel\right)g^2}{6\pi^2}\left[-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left\lbrace\ln\left(\frac{|q_fB|}{m^2_f}\right)
-1\right\rbrace\right] \\ && \nonumber+\frac{g^2}{6\pi^2}
\left[2m_f+\frac{|q_fB|}{m_f}\left\lbrace\ln\left(\frac{|q_fB|}
{m^2_f}\right)-1\right\rbrace\right] \\ && -\frac{2g^2m_f}{3\pi^2}
\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)
\left[\ln\left(\frac{m_f}{\pi{T}}\right)+\gamma_E\right],\end{aligned}$$ which enables us to compute the effective quark propagator from Dyson-Schwinger equation (\[defeffquarkprop\]).
Thermalized effective gluon propagator in a strongly-magnetized hot QCD medium
==============================================================================
This section is attributed to the evaluation of effective gluon propagator in a thermal medium in presence of a strong magnetic field. In general, the effective gluon propagator can be obtained from the Schwinger-Dyson equation $$\begin{aligned}
D^{-1}_{\mu\nu}(P)=D^{-1}_{0\mu\nu}(P)+\Pi_{\mu\nu}(P)
~.\end{aligned}$$ At finite temperature, the effective gluon propagator gets decomposed into longitudinal and transverse components in thermal medium. Although gluons are not affected directly by the presence of magnetic field but the dependence of magnetic field enters directly through the Debye mass and indirectly through the running strong coupling. To evaluate the components of gluon propagator, first we revisit how to decompose the gluon self-energy in a thermal medium in the coming subsection.
One-loop gluon self-energy in a hot QCD medium
----------------------------------------------
In vacuum, the gluon self-energy tensor is the linear combination of available four-momentum of the particle ($P_\mu$) and the metric tensor $(g_{\mu \nu})$. Being a Lorentz invariant quantity, self-energy depends on $P^2$ and further restriction by the Ward identity : $P^\mu\Pi_{\mu\nu}(P)=0$ imposes the structure of the tensor as $$\begin{aligned}
\Pi_{\mu\nu}(P)&=&\left(g_{\mu \nu} - \frac{P_\mu{P_\nu}}{P^2}\right)
\Pi(P^2)\nonumber\\
&\equiv &P_{\mu \nu} \Pi(P^2),\end{aligned}$$ where $P_{\mu\nu}$ is the transverse projection operator. However, at finite temperature, the Lorentz invariance is broken due to the direction of heat bath, which is introduced in terms of a four-velocity, $u_\mu$ in the rest frame of heat bath. Now with the available four vectors, $P_\mu$, $u_\mu$ and the tensor, $g_{\mu\nu}$, two orthogonal tensors, which are most adopted to the physical degrees of freedom and project on the subspace transverse and parallel to the three momentum, $\mathbf{p}$, respectively, are constructed $$\begin{aligned}
P^T_{\mu\nu}&=&g_{\mu \nu}-\frac{P_\mu P_\nu}{P^2} -\frac{P^L_{\mu \nu}}
{-N^2}~, \\
P^L_{\mu\nu}&=& -N_\mu N_\nu, {\rm{with}}~~ N_\mu=
\frac{P_\mu(P.u)-u_\mu P^2}{{(P.u)}^2-P^2}
~,\end{aligned}$$ as the tensorial basis to decompose the gluon self-energy tensor in thermal medium $$\begin{aligned}
\label{G.S.E.}
\Pi_{\mu\nu}(p_0,\mathbf{p})=P^T_{\mu\nu}\Pi_T(p_0,\mathbf{p})+P^L_{\mu\nu}
\Pi_L(p_0,\mathbf{p})\quad.\end{aligned}$$ The above functions $\Pi_T$ and $\Pi_L$ are known as transverse and longitudinal self-energies, respectively, which depends on both energy and three momentum in the rest frame of medium due to lack of Lorentz invariance as $$\begin{aligned}
p_0&=&u^\mu\cdot{P_\mu}~, \\
|\mathbf{p}|&=&\sqrt{\left(u^\mu\cdot{P_\mu}\right)^2-P^2}
\quad.\end{aligned}$$ By using the properties of the above projection operators, $P^L_{\mu \nu}$, and $P^T_{\mu \nu}$, the transverse and longitudinal self energies can be obtained as $$\begin{aligned}
\label{S.T.G.S.E.}
\Pi_L(P)&=&-\Pi_{00}(P)~, \\
\Pi_T(P)&=&\frac{1}{2}\left(\Pi_\mu^\mu(P) - \frac{P^2}{p^2} \Pi_L(P)\right)
~,\end{aligned}$$ which are obtained in HTL perturbation theory [@JEEM]. The HTL gluon self-energy tensor in a thermal medium determined by the angular average over the spatial directions of light-like vectors is $$\begin{aligned}
\nonumber\Pi_{\mu\nu}(P) &=& m^2_D\left[\int\frac{d\Omega}{4\pi}
K_\mu{K_\nu}\frac{P_\mu\cdot{u^\mu}}{P_\mu\cdot{K^\mu}}-u_\mu{u_\nu}\right]
\\ &=& m^2_D\left[\int\frac{d\Omega}{4\pi}
K_\mu{K_\nu}\frac{p_0}{p_0+\mathbf{p}\cdot{\mathbf{\hat{k}}}}-u_\mu{u_\nu}\right]
,\end{aligned}$$ where $K_\mu=(1,\mathbf{\hat{k}})$ is a light like four-vector and $m_D$ is the Debye mass. Therefore, the transverse and longitudinal components, $\Pi_T(P)$ and $\Pi_L(P)$ become $$\begin{aligned}
\label{T.G.S.E.}
\Pi_T(P) &=& \frac{m^2_D}{2}\frac{p^2_0}{p^2}
+\frac{m^2_D}{4}\frac{p_0}{p}\left(1-\frac{p^2_0}{p^2}\right)
\ln\left(\frac{p_0+p}{p_0-p}\right)\\
\Pi_L(P) &=& m^2_D-\frac{m^2_D}{2}\frac{p_0}{p}
\ln\left(\frac{p_0+p}{p_0-p}\right),
\label{L.G.S.E.}\end{aligned}$$ respectively. Thus the Schwinger-Dyson equation gives the dressed thermal gluon propagator as $$\begin{aligned}
\label{G.P.}
D_{\mu\nu}=P^T_{\mu\nu}~\Delta_T+P^L_{\mu \nu} \frac{P^2}{p^2} \Delta_L
~,\end{aligned}$$ where the transverse and longitudinal components of gluon propagator are given by $$\begin{aligned}
\Delta_T&=&\frac{-1}{P^2+\Pi_T(P)}\label{T.G.P.}~, \\
\Delta_L&=&\frac{1}{p^2+\Pi_L(P)}\label{L.G.P.}
~.\end{aligned}$$ Physically, $\Delta_T$ describes the propagation of two transverse vacuum modes in thermal medium whereas $\Delta_L$ does not exist in the vacuum and thus represents collective modes of medium.
Now when the medium becomes strongly magnetized, the dependence of strong magnetic field in $\Pi_{T}(P)$ and $\Pi_L(P)$ originates from the magnetic field dependence of the Debye mass. Therefore we are going to derive the Debye mass in strong magnetic field by the static limit of the longitudinal component of gluon self energy in the next subsection.
Screening mass in strong magnetic field
---------------------------------------
The Debye screening manifests in the collective oscillation of the medium via the dispersion relation and is obtained by the static limit of the longitudinal part (“00” component) of gluon self-energy, i.e. $$\begin{aligned}
\label{Definition}
\Pi_L (p_0=0,\mathbf{p}\rightarrow 0)=m^2_D
~.\end{aligned}$$ Out of four contributing diagrams (tadpole, gluon loop, ghost loop and quark loop) of the gluon self energy, only quark-loop (in figure 2) gets influenced by the magnetic field.
{width="4.9cm"}
In real-time formalism, using Schwinger’s proper-time propagator in strong magnetic field (\[11 Q.P.\]) for internal quark line, the 11-component of the gluon self-energy for the quark-loop is written as $$\begin{aligned}
\nonumber\Pi^{\mu\nu}(P) &=& -\frac{ig^2}{2}
\int\frac{d^4K}{(2\pi)^4}tr\left[\gamma^\mu{S_{11}(K)}
\gamma^\nu{S_{11}(K-P)}\right] \\ &=& \nonumber\frac{ig^2}
{2(2\pi)^4}\sum_f\int{d^2k_\perp}{d^2k_\parallel}{tr}\left[\gamma^\mu
\left(\gamma^0k_0-\gamma^3k_3+m_f\right)\gamma^\nu
\left(\gamma^0q_0-\gamma^3q_3+m_f\right)\right]
\\ && \nonumber\times\left[\frac{1}{k^2_\parallel-m^2_f+i\epsilon}+2\pi{i}n_F\left(k_0\right)\delta\left(k^2_\parallel-m^2_f\right)\right]e^{-\frac{k^2_\perp}{|q_fB|}}
\\ && \times\left[\frac{1}{q^2_\parallel-m^2_f+i\epsilon}+2\pi{i}n_F\left(q_0\right)\delta\left(q^2_\parallel-m^2_f\right)\right]e^{-\frac{q^2_\perp}{|q_fB|}}
\quad,\end{aligned}$$ where the factor $1/2$ enters due to the trace over colour indices and we use $Q=\left(q_0,\mathbf{q}\right)$ in place of $K-P=\left(k_0-p_0,\mathbf{k}-\mathbf{p}\right)$. The momentum integration is factorized into parallel and perpendicular components with respect to the direction of magnetic field, where the $k_\perp$ integration, $\Pi_{k_\perp}(p_\perp)$ is separated out to give $$\begin{aligned}
\label{P.C.G.S.E.}
\nonumber{\Pi_{k_\perp}(p_\perp)} &=& \int{dk_1}{dk_2}e^{-\frac{k^2_\perp}
{|q_fB|}}e^{-\frac{q^2_\perp}{|q_fB|}} \\ &=& \frac{\pi|q_fB|}{2}
e^{-\frac{p^2_\perp}{2|q_fB|}}.\end{aligned}$$ Thus the gluon self energy becomes completely separable into the components of momentum which are parallel and perpendicular to the magnetic field as $$\begin{aligned}
\Pi^{\mu\nu}(P) &=& \frac{\pi|q_fB|}{2}e^{-\frac{p^2_\perp}
{2|q_fB|}}\Pi^{\mu\nu}(p_\parallel).\end{aligned}$$ Denoting the trace over gamma matrices by $L^{\mu \nu}$ as $$\begin{aligned}
L^{\mu\nu}=8\left[k^\mu_\parallel\cdot{q^\nu_\parallel}
+k^\nu_\parallel\cdot{q^\mu_\parallel}-g^{\mu\nu}_\parallel\left(k^\mu_\parallel\cdot{q}_{\parallel\mu}
-m^2_f\right)\right]
~,\end{aligned}$$ the self energy depending only on the parallel component of momentum is decomposed into vacuum and thermal contributions as $$\begin{aligned}
\label{G.S.E.S.M.F.A.}
\nonumber\Pi^{\mu\nu}(p_\parallel) &=& \frac{ig^2}
{2(2\pi)^4}\sum_f\int{dk_0dk_3}L^{\mu\nu}\left[\frac{1}
{k^2_\parallel-m^2_f+i\epsilon}+2\pi{i}n_F\left(k_0\right)
\delta\left(k^2_\parallel-m^2_f\right)\right]
\\ && \nonumber\times\left[\frac{1}{q^2_\parallel-m^2_f
+i\epsilon}+2\pi{i}n_F\left(q_0\right)\delta
\left(q^2_\parallel-m^2_f\right)\right]
\\ &\equiv & \Pi^{\mu\nu}_V(p_\parallel)+\Pi^{\mu\nu}_n(p_\parallel)
+\Pi^{\mu\nu}_{n^2}(p_\parallel)
\quad,\end{aligned}$$ where $\Pi^{\mu \nu}_V(p_{\parallel})$, $\Pi^{\mu \nu}_n(p_{\parallel})$ and $\Pi^{\mu \nu}_{n^2} (p_{\parallel})$ are the vacuum and medium contributions to the gluon self energy containing single and double distribution functions, respectively, and are given by $$\begin{aligned}
\Pi^{\mu \nu}_V(p_{\parallel}) &=& \frac{ig^2}{2(2\pi)^4}
\int dk_0 dk_3 L^{\mu\nu}\left[\frac{1}{(k_{\parallel}^2
-m_f^2+i\epsilon)}\frac{1}
{(q_{\parallel}^2-m_f^2+i\epsilon)}\right],
\label{G.S.E.V.} \\
\Pi^{\mu \nu}_n(p_{\parallel}) &=& -\frac{g^2}
{2(2\pi)^3}\int dk_0 dk_3 L^{\mu\nu}\left[\frac{n_{F}(k_0)
\delta(k_{\parallel}^2-m_f^2)}{(q_{\parallel}^2
-m_f^2+i\epsilon)}
+\frac{n_{F}(q_0)\delta(q_{\parallel}^2-m_f^2)}
{(k_{\parallel}^2-m_f^2+i\epsilon)}\right],
\label{G.S.E.S.D.} \\
\Pi^{\mu \nu}_{n^2} (p_{\parallel}) &=& -\frac{ig^2}{2(2\pi)^2}
\int dk_0 dk_3 L^{\mu\nu}\left[n_{F}(k_0)n_{F}(q_0)
\delta(k_{\parallel}^2-m_f^2)
\delta(q_{\parallel}^2-m_f^2)\right]~.
\label{G.S.E.D.D.}\end{aligned}$$
We will now first evaluate the vacuum part (\[G.S.E.V.\]) for which the real part (using the identity (\[identity\])) is given by $$\Pi^{\mu\nu}_V(p_\parallel)=\left(g_{\parallel}^{\mu\nu}
-\frac{p_{\parallel}^{\mu}p_{\parallel}^{\nu}}{p_{\parallel}^2}
\right)\Pi(p_\parallel^2),$$ where $\Pi (p_\parallel^2)$ is given by $$\begin{aligned}
\Pi (p_\parallel^2)=\frac{g^2}{2\pi^3}\sum_{f} \left[\frac{2m_{f}^2}
{p_{\parallel}^2}
\left(1-\frac{4m_{f}^2}{p_{\parallel}^2}\right)^{-1/2}
\ln \left\lbrace \frac{{\Big(1-\frac{4m_{f}^2}
{p_{\parallel}^2}\Big)}^{1/2}+1}
{{\Big(1-\frac{4m_{f}^2}{p_{\parallel}^2}
\Big)}^{1/2}-1} \right\rbrace +1\right].\end{aligned}$$ The real part of 00 - component of vacuum contribution to the one-loop gluon self-energy tensor becomes $$\Pi^{00}_V (p_0, p_3)=-\frac{p_{3}^2}{p_{\parallel}^2}
~\Pi (p_\parallel^2)
~.$$
Thus multiplying the transverse component (\[P.C.G.S.E.\]) of gluon self-energy, the vacuum part of one-loop gluon self energy for massless flavours in the static limit ($p_0=0$, $p_1, p_2, p_3 \rightarrow 0$) is simplified into $$\Pi^{00}_V =\frac{g^2}{4\pi^2}\sum_{f}|q_{f}B| ,
\label{Massless case}$$ whereas for physical quark masses, it vanishes in the static limit, $$\Pi^{00}_V = 0
~.\label{Massive case}$$
Similarly the real part of $00$ - component of the thermal contribution to the gluon self-energy, $\Pi^{\mu\nu}_n(p_\parallel)$ with a single distribution function for $p_0=0$ is given by $$\begin{aligned}
\label{D.M.S.(p_3)}
\nonumber\Pi^{00}_n(p_0=0,p_3) &=& -\frac{g^2}
{2(2\pi)^3}\sum_f\int{dk_3}\left[\frac{L^{00}(k_0=\omega_k)n_F\left(k_0=\omega_k\right)}{2\omega_k\{\omega_k^2-\omega^2_q\}}\right. \\ && \left.\nonumber+\frac{L^{00}(k_0=-\omega_k)n_F\left(k_0=-\omega_k\right)}{2\omega_k\{-\omega_k^2-\omega^2_q\}}
+\frac{L^{00}(k_0=\omega_q)n_F\left(k_0=\omega_q\right)}{2\omega_k\{\omega_q^2-\omega^2_k\}}
\right. \\ && \left.+\frac{L^{00}(k_0=-\omega_q)n_F\left(k_0=-\omega_q\right)}{2\omega_k\{(-\omega_q)^2-\omega^2_k\}}\right]
,\end{aligned}$$ where the different factors in above equation are given by $$\begin{aligned}
&&L^{00}=8\left(k^0q^0+k^3q^3+m^2_f\right), \nonumber\\
&&\omega^2_k=k^2_3+m^2_f,~\omega^2_q=q^2_3+m^2_f, \nonumber\\
&&L^{00}(k_0=\omega_k)=8\left(2\omega^2_k-k_3p_3\right), \nonumber\\
&&L^{00}(k_0=-\omega_k)=8\left(2\omega^2_k-k_3p_3\right), \nonumber\\
&&L^{00}(k_0=\omega_q)=8\left(\omega^2_k+\omega^2_q-k_3p_3\right), \nonumber\\
&&L^{00}(k_0=-\omega_q)=8\left(\omega^2_k+\omega^2_q-k_3p_3\right), \nonumber\\
&&n_F\left(k_0=\omega_k\right)=n_F\left(k_0=-\omega_k\right)=\frac{1}{e^{\beta|\omega_k|}+1}, \nonumber\\
&&\nonumber{n_F\left(k_0=\omega_q\right)}=n_F\left(k_0=-\omega_q\right)=\frac{1}{e^{\beta|\omega_q|}+1}
.\end{aligned}$$ For massless quarks, the medium contribution, $\Pi^{00}_n(p_0=0,p_3)$ reduces to $$\label{Massless}
\Pi^{00}_n(p_0=0,p_3)=\frac{8g^2}{2(2\pi)^3}
\left[-1-\frac{T}{p_3}\ln(2)
+\frac{T}{p_3}\ln\left(1+e^{\frac{p_3}{T}}\right)\right]
,$$ whereas for the physical quark masses, it becomes $$\begin{aligned}
\label{Massive}
\Pi^{00}_n(p_0=0,p_3) &=& -\frac{g^2}{2(2\pi)^3}
\int dk_3 \left[ \frac{8k_{3}n_F(\omega_k)}{\omega_{k}p_{3}}\right. \nonumber \\
&& \left. -\frac{8(k_{3}-p_{3})n_F(\omega_q)}{\omega_{q}p_{3}}+\frac{16m_{f}^2n_F(\omega_k)}
{\omega_{k}p_{3}(2k_{3}-p_{3})}-\frac{16m_{f}^2n_F(\omega_q)}
{\omega_{q}p_{3}(2k_{3}-p_{3})} \right]
.\end{aligned}$$
The medium contribution to the self-energy with the square of quark distribution function (\[G.S.E.D.D.\]) is purely imaginary so it does not have the real-part, i.e. $$\begin{aligned}
\Pi^{\mu \nu}_{n^2}(p_{\parallel})=0
~.\end{aligned}$$
Again multiplying the transverse component (\[P.C.G.S.E.\]) to parallel component (\[Massless\]) gives the real part of $00$ - component of medium contribution to one-loop gluon self-energy tensor for massless quarks, which, in the static limit ($p_0=0$, $p_3 \rightarrow 0$), is simplified into $$\label{Massless (M.C)}
\Pi^{00}_n=-\frac{g^2}{4\pi^2}\sum_{f}
|q_{f}B|+\frac{g^2}{8\pi^2}\sum_{f}|q_{f}B|
~,$$ whereas the real part of $00$ - component of medium contribution to one-loop gluon self-energy tensor for physical quark masses in the static limit is reduced to $$\label{Massive (M.C.)}
\Pi^{00}_n=\frac{g^2}{4\pi^{2}T}\sum_{f}|q_{f}B|\int_{0}^{\infty}dk_{3}\frac{e^{\beta\omega_{k}}}{(1+e^{\beta\omega_{k}})^2}
~.$$
Finally the vacuum (\[Massless case\]) and medium contributions (\[Massless (M.C)\]) are added together to give the real part of the $00$ - component of one-loop gluon self-energy in static limit for massless quarks $$\Pi^{00}= \frac{g^2}{8\pi^2}\sum_{f}
|q_{f}B|
~,\label{a}$$ whereas the vacuum (\[Massive case\]) and medium contributions (\[Massive (M.C.)\]) yield the real part of the $00$ - component of one-loop gluon self-energy tensor in static limit for physical quark masses $$\Pi^{00}=\frac{g^2}{4\pi^{2}T}
\sum_{f}|q_{f}B|\int_{0}^{\infty}
dk_{3}\frac{e^{\beta\omega_{k}}}{(1+e^{\beta\omega_{k}})^2}
~.\label{b}$$ Therefore the definition (\[Definition\]) gives the Debye mass for the massless quarks $$\label{$m_D^2$}
m_{D}^2=\frac{g^2}{8\pi^2}\sum_{f}|q_{f}B|
~,$$ which was recently obtained by one of us [@MBB] and by others by different approach [@Fukushima:PRD93'2016; @Bandyopadhyay:PRD94'2016]. It is found that the Debye mass of thermal QCD medium in the presence of strong magnetic field depends solely on the magnetic field and is independent of temperature, therefore the collective behaviour of the medium gets strongly affected by the presence of strong magnetic field. However, for physical quark masses, the Debye mass is obtained as $$\begin{aligned}
\label{D.M.S.}
m^2_D=\frac{g^2}{4\pi^2T}\sum_f|q_fB|\int^\infty_0dk_3
\frac{e^{\beta\sqrt{k^2_3+m^2_f}}}{\left(1+e^{\beta\sqrt{k^2_3
+m^2_f}}\right)^2}
~,\end{aligned}$$ which depends now on both magnetic field and temperature. However it becomes independent of temperature beyond a certain temperature for a particular strong magnetic field [@MBB].
Thermodynamic observables of QCD matter in strong magnetic field
================================================================
Free energy and pressure
------------------------
The one loop free energy for $N_f$ quarks with $N_c$ colours in hot QCD medium in a static and homogeneous strong magnetic field is given by the sum of free energies due to quarks ($\mathcal{F}_q$) and gluons ($\mathcal{F}_g$), which are obtained by the functional determinant of effective quark and gluon propagators, respectively. Finally the pressure for the quark matter is obtained by the negative of free energy in the thermodynamic limit. We are now in a position to calculate the free energies and hence the pressures due to quarks and gluons using their respective one-loop propagators.
### Quark contribution
The free energy due to $N_f$ quarks with $N_c$ colours is obtained by the effective quark propagator, $S(P)$ from equation (\[defeffquarkprop\]) $$\begin{aligned}
\label{Free energy due to quarks}
\mathcal{F}_q &=& N_cN_f\int\frac{d^4P}{(2\pi)^4}\ln\left[\det\left(S(P)
\right)\right] \nonumber\\
&=&-N_cN_f\int\frac{d^4P}{(2\pi)^4}\ln\left[\det\left(\gamma^\parallel
\cdot{p}_\parallel-m_f-\Sigma(p_\parallel)\right)\right]
.\end{aligned}$$ Due to the external magnetic field in $z$ direction, the momentum integration in the quark free energy is also factorized into the momentum parallel and perpendicular to the magnetic field, which is facilitated by the finding that the integrand (i.e. the effective propagator) depends only on the longitudinal momentum component. For the sake of simplicity, we first express the quark self-energy into dependent and independent terms on (parallel) momentum from equation (\[Q.S.E.\]) $$\begin{aligned}
\Sigma(p_\parallel)&=&\left(\gamma^\parallel
\cdot{p}_\parallel\right)C+D+E, ~{\rm{with}} \\
C&=&\frac{g^2}{6\pi^2}\left[-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left\lbrace\ln\left(\frac{|q_fB|}
{m^2_f}\right)-1\right\rbrace\right]\label{C}, \\
D&=&\frac{g^2}{6\pi^2}\left[2m_f+\frac{|q_fB|}{m_f}
\left\lbrace\ln\left(\frac{|q_fB|}{m^2_f}\right)-1\right\rbrace\right]\label{D}, \\
E&=&-\frac{2g^2m_f}{3\pi^2}\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f
\right)^2}\right)\left[\ln\left(\frac{m_f}{\pi{T}}\right)+\gamma_E\right]\label{E},\end{aligned}$$ and then evaluate the determinant, $$\begin{aligned}
\det\left[\gamma^\parallel\cdot{p}_\parallel-m_f-\Sigma(p_\parallel)\right]=\left[p^2_\parallel\left(1
-C\right)^2-\left(m_f+D+E\right)^2\right]^2.\end{aligned}$$ Thus after plugging the determinant into the integration (\[Free energy due to quarks\]), the thermodynamic free energy of QCD matter due to quark contribution is expressed as $$\begin{aligned}
\label{F.E.Q.C.}
\nonumber\mathcal{F}_q &=& -2N_cN_f
\int\frac{d^2p_\perp}{(2\pi)^2}\int\frac{d^2p_\parallel}
{(2\pi)^2}\ln\left[p^2_\parallel\left(1-C\right)^2
-\left(m_f+D+E\right)^2\right] \\ &=& \nonumber-2N_cN_f
\int\frac{d^2p_\perp}{(2\pi)^2}\left[\int\frac{d^2p_\parallel}{(2\pi)^2}
\ln\left(p^2_\parallel\right)+\int\frac{d^2p_\parallel}{(2\pi)^2}\ln\left[\left(1
-C\right)^2-\frac{1}{p^2_\parallel}\left(m_f+D+E\right)^2\right]\right]
\\ &\equiv& -\frac{N_cN_f|q_fB|}{2\pi}\left(I_{1p_\parallel}+I_{2p_\parallel}
\right)
.\end{aligned}$$ At finite temperature, the integrals of type $I_{1p_\parallel}$ have been frequently solved in $4$-dimension employing Matsubara frequency sum method, where the continuous energy integrals are replaced by discrete frequency sums. Thus we solve this analytically as $$\begin{aligned}
\nonumber{I_{1p_\parallel}} &=& \int\frac{dp_0}{2\pi}\int\frac{dp_3}
{2\pi}\ln\left(p^2_0-p^2_3\right) \\ &=& \frac{\pi{T^2}}{6}
~.\end{aligned}$$ The integral, $I_{2p_\parallel}$ cannot be evaluated analytically, so we compute it numerically $$\begin{aligned}
\nonumber{I_{2p_\parallel}} &=& \int\frac{d^2p_\parallel}
{(2\pi)^2}\ln\left[\left(1-C\right)^2-\frac{1}{p^2_\parallel}
\left(m_f+D+E\right)^2\right] \\
&=& \frac{1}{4\pi}\int^{|q_fB|}_0{dp^2_\parallel}\ln\left[\left(1-C\right)^2-\frac{1}{p^2_\parallel}\left\lbrace m_f+D+W\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)\right\rbrace^2\right]
,\end{aligned}$$ by reexpressing the (parallel) momentum-dependent term, $E$ as $$\begin{aligned}
E &=& W\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)~,~
{\rm{with}}\\
W &=& -\frac{2g^2m_f}{3\pi^2}\left[\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right]\label{W}
.\end{aligned}$$ Now, substituting the integrals $I_{1p_\parallel}$ and $I_{2p_\parallel}$ in equation (\[F.E.Q.C.\]), the free energy due to quark contribution of hot QCD matter in strong magnetic field is obtained as $$\begin{aligned}
\nonumber\mathcal{F}_q &=& -\frac{N_cN_f|q_fB|}{4}
\left[\frac{T^2}{3}+\frac{1}{2\pi^2}\int^{|q_fB|}_0{dp^2_\parallel}
\ln\left[\left(1-C\right)^2\right.\right. \\
&& \left.\left.-\frac{1}{p^2_\parallel}\left\lbrace m_f+D+W\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)\right\rbrace^2\right]\right].\end{aligned}$$ Hence the negative of the above free energy in the thermodynamic limit gives the quark contribution to the thermodynamic pressure $$\begin{aligned}
\nonumber{P_q} &=& \frac{N_cN_f|q_fB|}{4}\left[\frac{T^2}{3}
+\frac{1}{2\pi^2}\int^{|q_fB|}_0{dp^2_\parallel}\ln\left[\left(1
-C\right)^2\right.\right. \\ && \left.\left.-\frac{1}{p^2_\parallel}
\left\lbrace m_f+D+W\ln\left(\frac{|q_fB|}{\left(p_\parallel
-m_f\right)^2}\right)\right\rbrace^2\right]\right].\end{aligned}$$ After substituting the values of $C$, $D$ and $W$ (equations (\[C\]), (\[D\]) and (\[W\])) in above equation, we get the pressure due to quark contribution $$\begin{aligned}
\nonumber{P_q} &=& \frac{N_cN_f|q_fB|}{4}\left[\frac{T^2}{3}
+\int^{|q_fB|}_0\frac{dp^2_\parallel}{2\pi^2}\ln\left[\left(1
-\frac{g^2}{6\pi^2}\left\lbrace-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left(\ln(\frac{|q_fB|}
{m^2_f})-1\right)\right\rbrace\right)^2\right.\right.
\\ &-& \left.\left.\nonumber\frac{1}{p^2_\parallel}
\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right.
\\ &-& \left.\left.\left.\frac{2g^2m_f}{3\pi^2}
\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}{\left(p_\parallel
-m_f\right)^2}\right)\right)^2\right]\right].\end{aligned}$$
### Gluon contribution
The free energy due to gluons in adjoint representation of $SU(N_c)$ gauge theory is given by both transverse and longitudinal modes $$\begin{aligned}
\label{F.E.G.}
\nonumber\mathcal{F}_g &=& (N_c^2-1)\left[2\mathcal{F}^T_g
+\mathcal{F}^L_g\right] \\ &=& \nonumber(N_c^2-1)\left[
\int\frac{d^4P}{(2\pi)^4}\ln\left[-\Delta_T(P)\right]^{-1}
+\frac{1}{2}\int\frac{d^4P}{(2\pi)^4}\ln\left[\Delta_L(P)\right]^{-1}\right]
\\ &=& -\left(N_c^2-1\right)\left[\int\frac{d^4P}{(2\pi)^4}\ln\left[-\Delta_T(P)\right]+\frac{1}{2}\int\frac{d^4P}{(2\pi)^4}\ln\left[\Delta_L(P)\right]\right],\end{aligned}$$ where $\Delta_T(P)$ and $\Delta_L(P)$ are the transverse and longitudinal parts of the hard thermal loop gluon propagator, respectively, obtained earlier in equations (\[T.G.P.\],\[L.G.P.\]).
Then substituting the values of $\Delta_T(P)$ and $\Delta_L(P)$ in equation (\[F.E.G.\]), we obtain the gluon contribution to the thermodynamic free energy of QCD matter in strong magnetic field $$\begin{aligned}
\mathcal{F}_g &=& -\left(N_c^2-1\right)\left[
\int\frac{d^4P}{(2\pi)^4}\ln\left(\frac{1}{P^2+\Pi_T(P)}\right)\right.
\nonumber\\ &+& \left.\frac{1}{2}\int\frac{d^4P}{(2\pi)^4}\ln\left(\frac{1}{p^2
+\Pi_L(P)}\right)\right],\end{aligned}$$ where $\Pi_L$ and $\Pi_T$ depend on the magnetic field through the screening (Debye) mass (\[D.M.S.\]).
For gluon, the loop momenta may be hard (i.e. order of $T$) or soft (i.e. order of $gT$). In imaginary-time, the gluon energy $p_0$ is an integer multiple of $2\pi{T}$, so the soft region requires $p_0=0$. But for quark, the energy, $p_0=(2n+1)\pi{T}$ can never be zero even for $n=0$, so, the quark loop is always hard. Since the gluons are not affected by the presence of magnetic field, the highest scale for them in a medium is still the temperature. For the hard loop momenta, the self energy components act like perturbative corrections and thereby making the effective propagators to expand around them. In case of soft loop momenta, in the zero frequency (static) mode, $\Pi_T$ becomes zero and only $\Pi_L$ exists. Since this is as large as the inverse of free propagator, the effective propagator can not be expanded. The total free energy due to gluon contribution is therefore obtained by adding the free energies determined from both hard and soft scales. Up to $\mathcal{O}\left(g^4\right)$ in strong running coupling, the gluon free energy is calculated in [@JEEM] $$\begin{aligned}
\nonumber\mathcal{F}_g &=& -\left(N_c^2-1\right)\left[\frac{\pi^2T^4}{45}
-\frac{T^2m^2_D}{24}+\frac{Tm^3_D}{12\pi}\right. \\ && \left.+\left\lbrace\frac{1}{\epsilon}
+2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace
\frac{m^4_D}{128\pi^2}\right]
,\end{aligned}$$ where $\Lambda$ is the renormalization scale. The divergent term ($1/\epsilon$) is isolated by the dimensional regularization, which is taken care of by the vacuum counter term as $$\begin{aligned}
\Delta\mathcal{E}_0=\frac{N^2_c-1}{128\pi^2\epsilon}m^4_D
\quad.\end{aligned}$$ After adding the counter term, the renormalized gluon free energy for a strongly magnetized thermal QCD matter takes the following form $$\begin{aligned}
\label{$F_g$}
\nonumber\mathcal{F}_g &=& -\left(N_c^2-1\right)\left[\frac{\pi^2T^4}{45}
-\frac{T^2m^2_D}{24}+\frac{Tm^3_D}{12\pi}\right. \\ && \left.+\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace
\frac{m^4_D}{128\pi^2}\right].\end{aligned}$$ Hence the pressure is obtained from the negative of the above free energy as $$\begin{aligned}
\label{$P_g$}
\nonumber{P_g} &=& \left(N_c^2-1\right)\left[\frac{\pi^2T^4}{45}
-\frac{T^2m^2_D}{24}+\frac{Tm^3_D}{12\pi}\right. \\ && \left.
+\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7
+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace\frac{m^4_D}{128\pi^2}\right].\end{aligned}$$ After plugging the Debye mass in strong magnetic field limit (\[$m_D^2$\]) for light flavours, the magnetic field dependence in the gluon contribution will be explicitly seen as $$\begin{aligned}
\label{$P_g$}
\nonumber{P_g} &=& \left(N_c^2-1\right)\left[\frac{\pi^2T^4}{45}
-g^2\frac{T^2{eB}}{192\pi^2}+g^3\frac{T(eB)^{\frac{3}{2}}}{192\sqrt{2}\pi^4}\right. \\ && \left.
+g^4\frac{(eB)^2}{8192\pi^6}\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7
+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace\right].\end{aligned}$$
### Total pressure
The total one-loop pressure of hot QCD matter in a strong magnetic field is obtained by adding both quark and gluonic contributions and has the following expression. $$\begin{aligned}
\label{Total pressure}
\nonumber{P(T,eB)} &=& \frac{N_cN_f|q_fB|}{4}\left[\frac{T^2}{3}
+\int^{|q_fB|}_0\frac{dp^2_\parallel}{2\pi^2}\ln\left[\left(1
-\frac{g^2}{6\pi^2}\left\lbrace-\frac{1}{2}
\right.\right.\right.\right. \\ && \left.\left.\left.\left.\nonumber-\frac{|q_fB|}{2m^2_f}\left(\ln(\frac{|q_fB|}
{m^2_f})-1\right)\right\rbrace\right)^2\right.\right.
\\ && \left.\left.\nonumber-\frac{1}{p^2_\parallel}
\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right.
\\ && \left.\left.\left.-\frac{2g^2m_f}{3\pi^2}
\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}{\left(p_\parallel
-m_f\right)^2}\right)\right)^2\right]\right] \nonumber\\
&& \nonumber+\left(N_c^2-1\right)\left[\frac{\pi^2T^4}{45}
-\frac{T^2m^2_D}{24}+\frac{Tm^3_D}{12\pi}
\right. \\ && \left.+\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7
+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace\frac{m^4_D}{128\pi^2}\right]
,\end{aligned}$$ where the renormalization scale $\Lambda$ is set at $2\pi{T}$. The ideal component can thus be read as $$\begin{aligned}
\label{I.P.S.M.F.A.}
P_{ideal}(T,eB)=N_cN_f\frac{|q_fB|T^2}{12}
+\left(N_c^2-1\right)\frac{\pi^2T^4}{45}
~.\end{aligned}$$
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a b
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\[fig1\]
Before calculating the thermodynamic observables in the strong magnetic field limit, we should be careful in choosing the range of temperatures and magnetic fields compatible with the limit. [*For example*]{}, to observe the variation of pressure with magnetic field at a temperature $T=300$ MeV, the starting value of magnetic field has to be greater than $\sim 4.60~m_\pi^2$, however, we have taken the starting magnetic field, $eB=10~m_\pi^2$, which is almost twice the above marginal value. Similarly for calculating the pressure as a function of temperature up to $T=400$ MeV, we have fixed the magnetic fields at $eB=15~m_\pi^2$, $25~m_\pi^2$, and $50~m_\pi^2$.
To see how the pressure of hot QCD matter with two light flavours is affected by the ambient strong magnetic fields quantitatively, we have computed the pressure as a function of magnetic field at fixed temperatures, $T=200$ MeV and $300$ MeV of the medium in figure 3a, whereas the figure 3b denotes the variation of pressure with the temperature at different strong magnetic fields, $ eB= 15$ $m^2_\pi$, $25$ $m^2_\pi$, and $50$ $m^2_\pi$. It is found that the pressure increases rapidly with the magnetic field (figure 3a) compared to its much slower variation with the temperature (figure 3b). The above contrast in the behaviour of pressure with magnetic field compared to the temperature reflects the fact that the dominant scale of thermal medium in strong magnetic field limit is the magnetic field, not the temperature anymore happened to be in thermal medium in absence of magnetic field. To be more precise, although the variation of pressure with temperature is not steeper in the presence of strong magnetic field but the pressure for thermal QCD in strong magnetic field is larger than the pressure of thermal medium in absence of strong magnetic field (denoted by solid line in figure 3b), hence the strong magnetic field makes the equation of state for a thermal medium harder. These observations will facilitate in understanding the effects of strong magnetic field on the entropy density (figure 5). Recently, the thermodynamic pressure and other observables arising due to the presence of magnetic fields have also been studied extensively in lattice QCD with $2+1$ flavour [@GFGSA], where they have noticed the similar increasing trend of the longitudinal pressure with the increase of magnetic field strengths.
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a b
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\[fig1\]
To see how the pressure of interacting quarks and gluons in a hot QCD medium in presence of strong magnetic field approaches to the noninteracting (ideal) limit asymptotically both with the magnetic field and the temperature, we have computed the pressure in units of ideal pressure (\[I.P.S.M.F.A.\]) as function of magnetic fields (in figure 4a) and temperatures as well (in figure 4b). From the figure 4a, it is interesting to know that the deviation of pressure from its ideal value increases with the strong magnetic field, i.e. the thermal QCD medium never achieve its ideal limit asymptotically in the presence of strong magnetic field. On the other hand, it is found from figure 4b that, the pressure of the thermal medium at a fixed magnetic field approaches its ideal limit as expected, but in strong magnetic field limit one cannot arbitrarily increase the temperature due to its constraint ($eB \gg T^2$).
Entropy density
---------------
To see how the available microstates to a given macrostate of a thermal QCD medium are affected due to the presence of strong magnetic field, we calculate the entropy density of hot QCD matter in a strong magnetic field by partially differentiating the pressure (\[Total pressure\]) with respect to the temperature $$\begin{aligned}
S&=&\frac{\partial{P}}{\partial{T}} \nonumber\\
& \equiv & S_q + S_g
~,\end{aligned}$$ where the entropy density due to quark contribution is calculated as $$\begin{aligned}
\nonumber{S_q} &=& \frac{N_cN_f|q_fB|}{6}
\left[T-\frac{m_fg^2}{T\pi^4}\int^{|q_fB|}_0dp^2_\parallel
~\frac{1}{p^2_\parallel}\ln\left(\frac{|q_fB|}
{\left(p_\parallel-m_f\right)^2}\right)\right.
\\ && \left.\nonumber\times\left[\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right. \\ && \left.\left.\left.\nonumber-\frac{2g^2m_f}{3\pi^2}\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}
{\left(p_\parallel-m_f\right)^2}\right)\right)\right.\right. \\ && \left.\left. \nonumber\Big{/}\left(\left(1-\frac{g^2}{6\pi^2}\left\lbrace-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left(\ln(\frac{|q_fB|}
{m^2_f})-1\right)\right\rbrace\right)^2\right.\right.\right. \\ && \left.\left.\left.\nonumber-\frac{1}{p^2_\parallel}\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right.\right. \\ && \left.\left.\left.\left.-\frac{2g^2m_f}{3\pi^2}\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)\right)^2\right)\right]\right].\end{aligned}$$ Similarly, the entropy density due to gluonic contribution is calculated as $$\begin{aligned}
{S_g} &=& \left(N^2_c-1\right)\left[\frac{4\pi^2T^3}{45}
-\frac{Tm^2_D}{12}+\frac{m^3_D}{12\pi}\right], \end{aligned}$$ which will be seen to depend on magnetic field after substituting the Debye mass for two light flavours from (\[$m_D^2$\]) as $$\begin{aligned}
{S_g} &=& \left(N^2_c-1\right)\left[\frac{4\pi^2T^3}{45}
-g^2\frac{TeB}{96\pi^2}+g^3\frac{(eB)^{\frac{3}{2}}}{192\sqrt{2}\pi^4}\right].\end{aligned}$$
In calculating the above expression, we set the partial derivative of the square of screening mass with respect to the temperature to zero, i.e. $\frac{\partial\left(m^2_D\right)}{\partial{T}}\simeq0$, because the screening mass in strong magnetic field limit solely depends on the magnetic field for massless flavours, it may however depend on both magnetic field and temperature for realistic physical quark masses but the temperature dependence is still negligible.
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a b
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\[fig1\]
In figure 5a, we have shown how the entropy density of a hot QCD medium at fixed temperatures, $T=200$ MeV and $300$ MeV has been affected by the stronger magnetic fields, where we found that it increases almost linearly with the magnetic field. Similarly figure 5b depicts the variation of entropy density of a hot QCD medium with the increasing temperatures in strong magnetic field backgrounds, $eB=15 m_\pi^2$, $25 m_\pi^2$, and $50 m_\pi^2$. It is also seen in figure 5b that both the entropy density and its rate of increase with temperature in presence of strong magnetic field get waned compared to the medium in absence of strong magnetic field (denoted by the solid line, $B$=0). This crucial observation can be understood qualitatively: As we know that the strong magnetic field restricts the dynamics of quarks in momentum space from 4-dimension to 2-dimension, hence the phase space gets shrunk to 2-dimension only. Since the entropy is a measure of number of possible microstates in the phase space, so the entropy of thermal medium gets decreased in the presence of strong magnetic field. Thus both figures of the entropy density corroborate the observations in the variation of pressure with magnetic field and temperature (figures 3a and 3b), respectively.
Energy density
--------------
We are now in a position to calculate the energy density in baryonless ($\mu_q=0$) hot QCD medium in a strong magnetic field. This can be obtained from the following thermodynamic relation, $$\begin{aligned}
\label{Total energy density}
\varepsilon &=& -P+TS\nonumber\\
& \equiv& \varepsilon_q+ \varepsilon_g
~,\end{aligned}$$ where the energy density due to quark contribution is calculated as $$\begin{aligned}
\nonumber\varepsilon_q &=& -P_q+TS_q \\ &=& \nonumber\frac{N_cN_f|q_fB|}{6}
\left[\frac{T^2}{2}-\frac{3}{4\pi^2}\int^{|q_fB|}_0{dp^2_\parallel}
~\ln\left[\left(1-\frac{g^2}{6\pi^2}\left\lbrace-\frac{1}{2}
\right.\right.\right.\right. \\ && \left.\left.\left.\left.\nonumber-\frac{|q_fB|}{2m^2_f}\left(\ln(\frac{|q_fB|}
{m^2_f})-1\right)\right\rbrace\right)^2\right.\right. \\ && \left.\left.\nonumber-\frac{1}{p^2_\parallel}\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right. \\ && \left.\left.\left.\nonumber-\frac{2g^2m_f}{3\pi^2}\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)\right)^2\right]\right. \\ && \left.\nonumber-\frac{m_fg^2}
{\pi^4}\int^{|q_fB|}_0dp^2_\parallel~\frac{1}{p^2_\parallel}
\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}\right)\right. \\ && \left.\nonumber\times\left[\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right. \\ && \left.\left.\left.\nonumber-\frac{2g^2m_f}{3\pi^2}\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}
{\left(p_\parallel-m_f\right)^2}\right)\right)\right.\right. \\ && \left.\left. \nonumber\Big{/}\left(\left(1-\frac{g^2}{6\pi^2}\left\lbrace-\frac{1}{2}
-\frac{|q_fB|}{2m^2_f}\left(\ln(\frac{|q_fB|}
{m^2_f})-1\right)\right\rbrace\right)^2\right.\right.\right. \\ && \left.\left.\left.\nonumber-\frac{1}{p^2_\parallel}\left(m_f+\frac{g^2}{6\pi^2}\left\lbrace2m_f+\frac{|q_fB|}{m_f}
\left(\ln(\frac{|q_fB|}{m^2_f})-1\right)\right\rbrace\right.\right.\right.\right. \\ && \left.\left.\left.\left.-\frac{2g^2m_f}{3\pi^2}\left\lbrace\ln\left(\frac{m_f}{\pi{T}}\right)
+\gamma_E\right\rbrace\ln\left(\frac{|q_fB|}{\left(p_\parallel-m_f\right)^2}
\right)\right)^2\right)\right]\right].\end{aligned}$$ Similarly, the energy density due to gluonic contribution has been calculated as $$\begin{aligned}
\nonumber\varepsilon_g &=& -P_g+TS_g \\ &=& \left(N^2_c-1\right)
\left[\frac{\pi^2T^4}{15}-\frac{T^2m^2_D}{24}-\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace
\frac{m^4_D}{128\pi^2}\right].\end{aligned}$$ The magnetic field dependence can be seen after replacing the Debye mass for massless flavours from (\[$m_D^2$\]) $$\begin{aligned}
\nonumber\varepsilon_g &=& \left(N^2_c-1\right)
\left[\frac{\pi^2T^4}{15}-g^2\frac{T^2{eB}}{192\pi^2}\right. \\ && \left.-g^4\frac{(eB)^2}{8192\pi^6}\left\lbrace2\ln\left(\frac{\Lambda}{4\pi{T}}\right)-7+2\gamma_E+\frac{2\pi^2}{3}\right\rbrace\right].\end{aligned}$$
{width="5.5cm" height="5cm"}
To see how the energy density of a hot QCD medium has been affected by the presence of external strong magnetic field, we have computed the energy density as a function of temperature at different (strong) magnetic fields in figure 6, where the energy density increases with the temperature as expected but it increases with the temperature much faster for the medium in absence of magnetic field (B=0, denoted by solid line) which resonates with the observation of entropy density with temperature in figure 5b. In brief, the strong magnetic field reduces the energy density of thermal QCD medium.
Speed of sound
--------------
The speed of sound in a medium depends on the nature of equation of state, whether it is soft or hard and is related to the thermodynamic pressure and energy density through the following equation $$\begin{aligned}
C^2_s=\frac{\partial{P}}{\partial\varepsilon}=\frac{{\partial{P}}/{\partial{T}}}
{{\partial\varepsilon}/{\partial{T}}}
~,\end{aligned}$$ where the partial derivatives of pressure and energy density with respect to the temperature are obtained from equations (\[Total pressure\]) and (\[Total energy density\]), respectively. Since the existence of strong magnetic field modifies the thermodynamic observables, $c^2_s$ is also expected to deviate from its value in the presence of strong magnetic field.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
{width="5.5cm" height="5cm"} {width="5.5cm" height="5cm"}
a b
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
To see the effects of the magnetic fields on the equation of state, we have computed the speed of sound of a hot QCD medium as a function of external magnetic fields in figure 7a. It is found that the speed of sound at a fixed temperature increases with the strength of magnetic fields, which can be understood by the fact that, as the strength of magnetic field increases, the energy density decreases and the pressures increases, hence the speed of sound gets increased. From the original perspective of how the speed of sound of a thermal QCD is now modified in the presence of magnetic field, we have computed $c_s^2$ as a function of temperature at different strengths of magnetic fields in figure 7b. We found that $c_s^2$ decreases with the temperature as expected and reaches asymptotically to the ideal value 1/3 for the case when there is no magnetic field.
Interestingly when we plot the speed of sound in terms of ideal limit in strong magnetic field in figure 8, we have found that the speed of sound shows a dip in specific temperature-magnetic field combination. The above crucial observations in strong magnetic field could have phenomenological implications in heavy ion collisions, because the speed of sound modulates the hydrodynamic expansion of the medium (QGP) produced in noncentral ultrarelativistic heavy ion collisions.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
{width="5.5cm" height="5cm"} {width="5.5cm" height="5cm"}
a b
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Conclusions
===========
In this work, we have explored how the thermodynamic observables of a hot QCD medium in one-loop have been affected in an ambience of very strong magnetic field, which may be produced in noncentral events of ultrarelativistic heavy ion collisions. All thermodynamic observables have been contributed both by quarks and gluons through their respective one-loop self energies, where the quark contribution has been affected strongly by the strong magnetic field whereas, the gluonic part is largely unaffected except for the softening of the screening mass in strong magnetic field. As a result, even the pressure for the noninteracting quarks in thermal medium gets enhanced in strong magnetic field and overall an increase in total pressure of thermal medium is observed compared to the thermal medium in the absence of strong magnetic field. As a consequence, the entropy density gets decreased due to the presence of strong magnetic field, so the energy density too decreases with respect to pure thermal medium. Finally we obtain the equation of state by calculating the speed of sound of thermal QCD medium, which is found to increase due to the presence of strong magnetic field and shows a dip in a specific range of temperatures and strong magnetic fields. The above observations could have interesting implications on the expansion dynamics of the medium produced at RHIC and LHC in presence of strong magnetic field.
Acknowledgements
================
We convey our sincere thanks to Bhaswar Chatterjee and Mujeeb Hasan for their constant help during this work.
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[^1]: [email protected]
[^2]: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The super-Earth GJ1214b transits a nearby M dwarf that exhibits $1\%$ intrinsic variability in the near-infrared. Here, we analyze new observations to refine the physical properties of both the star and planet. We present three years of out-of-transit photometric monitoring of the stellar host GJ1214 from the MEarth Observatory and find the rotation period to be long, mostly likely an integer multiple of 53 days, suggesting low levels of magnetic activity and an old age for the system. We show such variability will not pose significant problems to ongoing studies of the planet’s atmosphere with transmission spectroscopy. We analyze 2 high-precision transit light curves from ESO’s Very Large Telescope along with 7 others from the MEarth and FLWO 1.2 meter telescopes, finding physical parameters for the planet that are consistent with previous work. The VLT light curves show tentative evidence for spot occultations during transit. Using two years of MEarth light curves, we place limits on additional transiting planets around GJ1214 with periods out to the habitable zone of the system. We also improve upon the previous photographic $V$-band estimate for the star, finding $V=14.71\pm 0.03$.'
author:
- 'Zachory K. Berta, David Charbonneau, Jacob Bean, Jonathan Irwin, Christopher J. Burke and Jean-Michel Désert'
- Philip Nutzman
- 'Emilio E. Falco'
title: 'The GJ1214 Super-Earth System: Stellar Variability, New Transits, and a Search for Additional Planets'
---
Introduction
============
The transiting exoplanet GJ1214b offers an unparalleled opportunity to explore the physical properties of super-Earth planets. With a mass ($M_p$ = 6.6 [$\rm M_{\earth}$]{}) and radius ($R_p$ = 2.7 [${\rm R_{\earth}} $]{}) between those of Earth and Neptune, and a likely equilibrium temperature ($T_{eq} = 500K$) cooler than for most transiting planets, GJ1214b represents an intriguing new kind of world with no Solar System analog [@charbonneau.2009.stnls]. Given intrinsic degeneracies in the mass-radius diagram in this regime [@seager.2007.mrse; @adams.2008.optamrsewma; @rogers.2010.fqdeic], the bulk composition of the planet cannot be uniquely determined from current measurements of the mass and radius alone. For example, @rogers.2010.tpol1 can explain the observed mass and radius to within $1\sigma$ with any of three generic physical models: (i) a mini-Neptune that accreted and maintained a low-mass H/He layer from the primordial nebula, (ii) a superfluid water-world with a sublimating ${\rm H_2O}$ envelope, or (iii) a rocky planet with an H-dominated atmosphere formed by recent outgassing. [ Detailed calculations of GJ1214b’s thermal evolution by @2010arXiv1010.0277N favor a metal-enriched H/He/[H$_2$O]{} envelope, finding that a water-only atmosphere would require an implausibly large water-to-rock ratio in the planet’s interior.]{}
Fortunately, because GJ1214b transits a very nearby (13 pc), bright ($K=8.8$), low-mass M dwarf (0.16 [$\rm M_{\sun}$]{}), it is amenable to follow-up observations that could distinguish among these hypotheses. In particular, the large ($D=1.4\%$) transit depth enables transit studies of the planet’s atmosphere. @miller-ricci.2009.assdbhha show that measuring the amplitude of the planet’s transmission spectrum (i.e., the wavelength-dependence of the transit depth $\Delta D(\lambda)$ caused by absorption at the limb of the planet) constrains the mean molecular weight of its atmosphere and, in turn, the hydrogen content of its outer envelope. Cases (i) or (iii) of @rogers.2010.tpol1 would produce $\Delta D(\lambda)\approx0.1\%$ variations in the transit depth across wavelengths accessible from the ground as well as [*Hubble*]{} and [*Spitzer Space Telescopes*]{}, while case (ii)’s dense atmosphere would result in variations below the sensitivity of current instruments [@miller-ricci.2010.nats1].
Providing a potential complication, however, the host star GJ1214 shows roughly sinusoidal photometric modulations that are presumably due to an asymmetric distribution of spots on a rotating star. Such spots can bias planetary parameters as measured from transit light curves whether or not they are occulted by the planet [e.g. @pont.2007.hsttppt1mrs; @desert.2011.tse1isom], partially decoupling the observed transit depth $D$ from the actual planet-to-star radius ratio $R_p/R_{\star}$. Of particular importance for transmission spectroscopy studies, the change in transit depth induced by spots can vary with both time and wavelength, potentially mimicking the signal of a planetary atmosphere.
Stellar spots have been observed in several transiting exoplanet systems around active stars [see @strassmeier.2009.s]. [*Hubble Space Telescope*]{} ([*HST*]{}) photometry [@rabus.2009.csstpts] and later ground-based follow-up [@dittmann.2009.tdsdctepfgedtpsat] of TrES-1b has shown evidence for spot occultations in transit light curves. The high photometric precision and continuous coverage provided by the [ CoRoT]{} satellite enabled detailed modeling of spotty transit and out-of-transit light curves for the hot Jupiters [ CoRoT-2b]{} [@wolter.2009.tmscpsswpt; @czesla.2009.saaseep; @huber.2010.pemcedrsm] and [ CoRoT-4b]{} [@aigrain.2009.npcdpp; @lanza.2009.parpsc]. For the former, joint fits to the transit and out-of-transit flux showed that initial estimates of the planet’s $R_p/R_{\star}$ were 3% (9$\sigma$) too low [@czesla.2009.saaseep]. The interpretation of the transiting super-Earth [ CoRoT-7b]{} is obfuscated by the fact that both the transit depth and the reflex motion are well below the amplitude of activity-induced modulations [@leger.2009.tefcsmvcfswmr; @queloz.2009.cpsos]. Reanalyses of the [ CoRoT-7]{} radial velocities find changing values for the mass of CoRoT-7b [@hatzes.2010.iirvvc; @ferraz-mello.2010.pmdcsoasccs; @lanza.2010.parrvvpsc] and call into question the significance of the mass measurements for both [ CoRoT-7b]{} and the claimed outer planet [ CoRoT-7c]{} [@pont.2010.rrepac].
Like GJ1214b, the well-studied hot Jupiter HD189733b [@bouchy.2005.emstjivjtbs1] is an ideal system for characterization studies, but requires corrections for stellar activity. The host HD189733 is an active K2 dwarf [@moutou.2007.sotpsd1] with 2% peak-to-peak variability in the optical [@croll.2007.ls1sstmsp; @miller-ricci.2008.msptes1ptmtaas]. @henry.2008.rpps1 undertook a long-term photometric monitoring campaign from which they measured the 12 day stellar rotation period of HD189733. Extrapolation from their out-of-eclipse photometric spot characterization was useful for interpreting transmission spectroscopy results of individual transits from [*Hubble*]{} [@pont.2007.hsttppt1mrs; @swain.2008.pmaep] and measurements of the thermal phase curve from [*Spitzer*]{} [@knutson.2007.dcep1; @knutson.2009.mcdcp1]. Understanding the time-variable surface of the star was even more crucial for broadband transmission spectroscopy studies that rely on comparing transit depths at different epochs [e.g., @desert.2009.scmate1; @desert.2010.tsehisom; @sing.2009.tse1iswfhwn]; interpretation of these data rely heavily on the photometric monitoring of @henry.2008.rpps1.
To aid ongoing and future studies of GJ1214b, we present new data (§\[Observations\]) to characterize the star GJ1214’s variability and estimate its rotation period (§\[Rotation\]). We compare the measured variability to a simultaneous analysis of 2 high-precision transit light curves from ESO’s Very Large Telescope with 7 other new or previously published transits (§\[Transits\]). Additionally, we place upper limits on the radii of other possible transiting planets in the system (§\[Search\]) and present a refined estimate of the star’s $V$ flux, which bears directly upon its metallicity as estimated using $M_K$ and $V-K$ relations. Finally, we discuss the implications of the measured variability for the properties of the star and for transmission spectroscopy studies of GJ1214b’s atmosphere (§\[Discussion\]).
We also note the following correction. In @charbonneau.2009.stnls, we quoted a systemic radial velocity for GJ1214 that had a typo in the sign; the actual velocity is $\gamma=+21.1\pm1.0$ km s$^{-1}$ (i.e. a redshift).
Observations and Data Reduction {#Observations}
===============================
MEarth Photometry
-----------------
We monitored the brightness of the GJ1214 system at a variety of cadences with the MEarth Observatory at Mt. Hopkins, AZ throughout the 2008, 2009, and 2010 spring observing seasons. As described in @nutzman.2008.dcgtshpod, the MEarth Observatory was designed to detect transiting exoplanets around nearby M dwarfs, and consists of eight identical 40-cm telescopes on German Equatorial mounts in a single enclosure at the Fred Lawrence Whipple Observatory (FLWO). Each telescope is equipped with a thinned, back-illuminated 2048x2048 CCD with a pixel scale of 0.757”/pixel for a 26’ field of view. For the bulk of the data presented in this work telescopes were equipped with a fixed, custom, 715 nm long-pass filter; the response is similar to a combination of the Sloan $i + z$ bandpasses and will be hereafter referred to as the “MEarth” bandpass. The MEarth Observatory is almost fully automated and operates on every clear night, observing target stars selected from a list of 2000 nearby late M dwarfs [@nutzman.2008.dcgtshpod]. In typical operating mode, each telescope observes its own list of 20-30 stars per night when they are above airmass 2 with the cadence and exposure times necessary to detect a transiting planet as small as 2 [${\rm R_{\earth}} $]{} in each star’s habitable zone.
Light curves are extracted automatically from MEarth images by a modified version of the Monitor pipeline [@irwin.2007.mpdplcp], using nightly flat field (dawn and dusk), dark, and bias exposures for calibration. [ A differential photometry correction for each frame was calculated from a robust, weighted fit to 78 automatically selected field comparison stars within 2.3 instrumental magnitudes of GJ1214. The mean MEarth - $K$ color of these stars is 0.98; none were as red as GJ1214 (MEarth - $K$ = 2.19).]{} Predicted uncertainties for each measurement are calculated from a standard CCD noise model.
GJ1214 was observed with three main cadences. “Low” cadence (20-30 minutes between exposures) was that associated with the normal survey mode, and was employed in the 2008 and 2009 seasons. “Medium” cadence (5-10 minutes) was implemented after the discovery of GJ1214b and was intended to boost sensitivity both to other transiting planets and to characterizing the out-of-transit variability of the star. “High” cadence (40 seconds) was employed at predicted times of transit to determine the system parameters. High cadence transits were observed simultaneously with 7 or 8 MEarth telescopes for greater precision, as the systematic noise sources and scintillation patterns among pairs of MEarth telescopes appear to be largely uncorrelated, so the S/N improvement scales with the square root of the number of telescopes. In Table \[tab-LC\], we present one new MEarth transit light curve, along with the four MEarth and two KeplerCam light curves that were analyzed but not made electronically available in @charbonneau.2009.stnls.
While we include data from 2008 for the rotation analysis, we caution that these observations took place during MEarth’s early commissioning, before the observing strategy and software were finalized. Changes to the telescope throughout the season may have corrupted the season-long stability. Importantly, a field acquisition loop designed to mitigate flat-fielding errors by bringing each star back to the same pixel was not implemented until the late spring of 2008. During the 2008 season, a Bessell-prescription I filter [@bessell.1990.up] was used instead of the custom MEarth bandpass.
[lrrlcc]{} 2454964.8926699 & 0.99622 & 0.00317 & 1.1197 & MEARTH & 0\
2454964.8933879 & 0.99777 & 0.00321 & 1.1198 & MEARTH & 0\
2454964.8941049 & 1.00578 & 0.00325 & 1.1199 & MEARTH & 0\
\
2454980.7148766 & 0.99733 & 0.00185 & 1.6190 & FLWO & 9\
2454980.7153966 & 1.00044 & 0.00185 & 1.6140 & FLWO & 9\
2454980.7158946 & 0.99959 & 0.00185 & 1.6080 & FLWO & 9\
\
2455315.7660750 & 0.99786 & 0.00014 & 1.2125 & VLT & 221\
2455315.7668492 & 0.99762 & 0.00014 & 1.2105 & VLT & 221\
2455315.7702432 & 0.99763 & 0.00012 & 1.2020 & VLT & 221\
V-band Photometry from KeplerCam.
---------------------------------
An independent out-of-transit light curve was obtained through Harris $V$ and $I$ filters[^1] with KeplerCam on the 1.2 meter reflector at FLWO atop Mt. Hopkins, AZ. The observations discussed here were gathered in service mode from 26 March 2010 until [ 17 June 2010]{}, after which date the mirror was taken off and put back on the telescope, introducing an uncorrectable systematic offset to the field light curves so later data had to be discarded.
Given the large night-to-night positional shifts of the field, we made an effort to quantify and ameliorate flat-fielding errors by sampling multiple regions of the detector, with each observation consisting of a set of three exposures offset by 3 arcminute dithers. Individual exposures had theoretical noise limits ranging from 0.3% to 2%, but the scatter among dither points suggested that calibration errors from flat-fielding introduced a 1% noise floor to the light curve. Dark sky flats were generated and corrected over time for changes at high spatial frequencies (i.e. dust donuts) by nightly dome flat exposures.
We measured calibrated $V$ and $I$ magnitudes (Table \[tab-vi\]) to improve on previously published photographic estimates [@lepine.2005.cnswapmlt0lc]. Standard fields [@landolt.1992.upssmr1ace] were observed on the nights of 26, 27, and 28 March. Conditions were clear, although seeing as poor as 10" FWHM was witnessed. We estimate the calibration uncertainties for the nights from the scatter in multiple standard exposures.
Light Curves from VLT-FORS2
---------------------------
Spectra of GJ1214 and 6 comparison stars were gathered during three transits of GJ1214b using ESO Director’s Discretionary Time on the VLT (Prog. ID \#284.C-5042 and 285.C-5019). As described by @bean.2010.temp, the primary purpose for obtaining these data was to measure the transmission spectrum of GJ1214b’s atmosphere by generating multiwavelength transit light curves and determining the wavelength-dependence of the transit depth. In this work, we generate and analyze high precision “white” light curves by summing together all the photons collected in each spectrum.
Observations were performed in queue mode with the multiobject, low dispersion spectrograph FORS2 [@appenzeller.1998.scffoi] on VLT/UT1. The spectrograph was configured with the 600z+23 grism with a central wavelength of 900 nm and the red-sensitive (MIT) CCD in the standard 2x2 read mode. Exposure times were 20-40 seconds, and the readout time was 37 seconds. A custom slit mask was used; each slit was a rectangle 12” in the dispersion direction and 15-30” in the spatial direction, small enough to isolate GJ1214 and the comparison stars but large enough that changing slit losses due to variable seeing were negligible. Wavelength calibration exposures with a He, Ne, Ar emission lamp were taken through a 1” slit the day after each set of observations. Given the position of the comparison stars on the chip, the wavelength range 780 to 1000 nm was used in this analysis. The CCD response governs the red edge of this range, and the spectral response is similar to that of the MEarth bandpass.
After bias subtraction and flat fielding, we used the comparison stars to correct for the time varying zeropoint of the system. For each exposure, we extracted 1D spectra from the images using the optimal extraction of @horne.1986.oeas, and divided the total flux (summed over wavelengths) of GJ1214 by the total flux in all the comparison stars. Theoretical error bars calculated from photon statistics alone were assigned to each point. Each exposure yielded $1-3\times10^8$ photons from GJ1214 and twice as many from comparison stars.
The corrected GJ1214 light curves exhibit systematic trends which we correct for by fitting a second-order polynomial function of time. In §\[Transits\] we propagate the uncertainty from the systematics corrections through to the transit parameters. We searched for correlations between the relative flux and airmass, seeing, and positional shifts in the dispersion and cross-dispersion directions. The relationships were more complicated than low-order polynomials, so we did not attempt to remove them using common decorrelation techniques [e.g. @burke.2010.notjx].
We also tested whether the observed drifts in flux could be caused by a changing color-dependence of the atmospheric extinction along the line of sight. To do so, we applied differential photometry corrections to individual spectral channels [*before*]{} combining them, to allow each wavelength its own extinction. This procedure did not remove the systematic trends.
We suggest the following as a more probable explanation for the systematics. @moehler.2010.cfrfscsuavd found that the linear atmospheric dispersion corrector (LADC) on the telescope has surface features that affect is sensitivity across the field of view. Because the LADC is positioned before the field rotator in the optical path and rotates relative to the sky, individual stars can drift across these features and encounter throughput variations that are not seen by the other comparison stars. No rotationally-dependent flat-fields were applied to these data, although @moehler.2010.cfrfscsuavd provide a route to a possible correction.
The first two “white” light curves, normalized to their median out-of-transit flux level, are published in Table \[tab-LC\] and shown in Fig. \[fig-vlt\].
A third transit observation was attempted on 2010 Jul 22. The brightest comparison star could not be used because it saturated mid-transit. The exposure times were cut in half immediately after egress, and a notable offset is visible in the transit light curve, perhaps due to an uncorrected non-linearity in the detector. Although the light curves of the first two transits were robust to the choice of comparison stars, the third changed significantly depending on which set of comparison stars was used. We show this transit in Fig. \[fig-vlt\], but exclude it from all following analyses.
[ccc]{}
$V $ & $14.71 \pm 0.03$ & this work\
${I}$ & $11.52 \pm 0.03$ & this work\
${J}$ & $9.750 \pm 0.024$ & 2MASS\
${H}$ & $9.094 \pm 0.024$ & 2MASS\
${K}$ & $8.782 \pm 0.024$ & 2MASS\
Rotation Period of GJ1214 {#Rotation}
=========================
With a growing understanding of the systematic effects present in MEarth data, we revisit the issue of GJ1214’s intrinsic variability. In the discovery paper for GJ1214b [@charbonneau.2009.stnls], we stated that the dominant periodicity seen in the out-of-transit light curve of the star GJ1214 had an 83 day period, implying that this was the rotation period of the star. Here, we [ revisit the question of GJ1214’s rotation period with another season of observations.]{}
Semi-stable spot complexes on the surface of a star imprint photometric modulations that can be approximated as a sinusoid with a fundamental period that matches the stellar rotation period. [ We search each year’s light curve with a weighted, least-squares periodogram that has been modified to simultaneously fit for stellar variability along with scaled templates of known systematic effects. These systematics are discussed in the next several subsections. To account for the likely evolution of spots with time, we investigate the 2008, 2009, 2010 data sets separately and do not require a coherent sine curve to persist over multiple years’ data. ]{}
Avoiding Persistence in MEarth Light Curves
-------------------------------------------
The MEarth detectors are subject to image persistence; pixels that are illuminated in one exposure can show enhanced dark current in subsequent exposures, which decays exponentially with a half hour time scale. Because the dark current in a given pixel depends on how recently that pixel was illuminated, differential photometry light curves can show baseline shifts between observations taken at different cadences, as well as ‘ramps’ at the start of a high-cadence sequence of exposures.
Correcting for these changing baseline shifts would require a simultaneous modeling of the complete photon detection history of every pixel and is impractical. During MEarth’s normal survey mode, we purposely center subsequent targets on different pixels to avoid persistent charge stacking up. As the effect is most noticeable for data with cadence shorter than 5 minutes, we circumvent the problem by throwing out from the rotation period analysis all but the first point of any segment with such cadence.
![Periodograms showing the $\Delta \chi^2$ improvement achieved by fitting a sinusoid of a given period + common mode + meridian flip over a null model consisting only of common mode + meridian flip for MEarth ([*top 3 panels*]{}) and FLWO V-band photometric monitoring ([*bottom*]{}). Possible periods as short as 1 day ([*left*]{}) and a zoom in to longer rotational periods ([*right*]{}) are shown. Periods for which less than one full cycle is observed per season are shaded.[]{data-label="fig-periodograms"}](f1.eps){width="3.5in"}
![[ Out-of-transit light curves]{} of GJ1214 from the MEarth Observatory ([*top three panels*]{}) and in V from the FLWO 1.2m ([*bottom panel*]{}). Individual exposures ([*gray points*]{}) and nightly binned values with errors that include the systematic ‘jitter’ ([*black circles*]{}, see text for details) are shown for each. A sine curve at the proposed 53 day period (derived from 2010 MEarth data) is shown ([*solid lines*]{}) at the best-fit phase and semiamplitude for each light curve. The MEarth points have been corrected for the best-fit common mode and meridian flip decorrelation.[]{data-label="fig-rotation"}](f2.eps){width="3.5in"}
Adding a Systematic ‘Jitter’
----------------------------
Even having removed the highest cadence data, the sampling in the MEarth GJ1214 light curve can vary from N=1 to N=35 points per night, with a typical theoretical noise limit per point of 3 millimagnitudes. In a strict least-squares sense, if the noise in our data were accurately described by an uncorrelated Gaussian process (e.g. photon noise), then every exposure should be allowed to contribute on its own to the period search, meaning the uncertainty associated with each night would go down as $1/\sqrt{N}$. Although MEarth telescopes have achieved such white noise down to millimagnitude levels within individual nights [@charbonneau.2009.stnls], photometric variations between nights are most likely dominated by subtle changes in the telescope that are not corrected by our calibration efforts. The $1/\sqrt{N}$ weighting scheme would unfairly bias a period search to fit only a few well-sampled nights.
To account for this in each light curve, we remove all in-transit exposures and bin the data to a nightly time scale. For each night with $N$ data points, we calculate an inverse variance weighted mean flux and time, using the theoretical errors calculated for each exposure in the weighting. To each of these nightly bins we assign an error given by $$\sigma_{nightly} = \sqrt{\sigma_{bin}^2 + \sigma_{jitter}^2}$$ where $\sigma_{bin}$ is the intrinsic standard error on the mean of the nightly bin ($RMS/\sqrt{N-1}$) and $\sigma_{jitter}$ is a constant noise floor term to capture the night-to-night calibration uncertainty.
The signal lost in the binning process should be minimal. Preliminary searches of unbinned data and visual inspection of high cadence nights revealed no significant periodic signal at periods shorter than 1 day. [ Under the (untested) assumption that the stellar spin is roughly aligned with the orbital angular momentum, the upper limit on the projected rotation velocity of $v\sin i < 2$ km s$^{-1}$ would correspond to a rotation period $P_{rot} > 5$ days.]{}
[ We use observations on successive nights to estimate $\sigma_{jitter}$ for the MEarth and the V-band light curves for each observing season. This assumes, on the basis of the apparent lack of short term variability, that the flux difference between pairs of nights is dominated by systematics. We find $\sigma_{jitter}$ = 0.0052, 0.0058, 0.0038, 0.0067 magnitudes for the 2008, 2009, 2010 MEarth and 2010 V light curves. ]{}As these values are comparable to the predicted noise for most exposures, the quadrature addition of $\sigma_{jitter}$ means we weight most nights roughly equally.
Correcting for Meridian Flips in MEarth Light Curves
----------------------------------------------------
MEarth’s German Equatorial mounts rotate the detectors $180^\circ$ relative to the sky when switching from negative to positive hour angles. Thus, the target and comparison stars sample two different regions of the detector. Given imperfect flat field corrections, an offset is apparent in many MEarth light curves between exposures taken on either side of the meridian.
To account for this effect, we allow different sides of the meridian to have different zeropoints. We construct a meridian flip template $m(t_i)$, which for an unbinned light curve would consist of binary values corresponding to the side of the meridian at each time stamp $t_i$. By extension, for each night of the binned data $t_i$, we define $$m(t_i) = n_+/(n_+ + n_-)$$ where $n_+$ and $n_-$ are the number of data points with positive or negative hour angles in a given nightly bin. We allow a scaled version of this template to be fit simultaneously with the period search.
Correcting for Water Vapor in MEarth Light Curves
-------------------------------------------------
Because the wide MEarth bandpass overlaps significant water absorption features in the telluric spectrum, the color-dependence of the throughput of our observing system is sensitive to the precipitable water vapor (PWV) in the column overhead. The fraction of stellar photons lost to water vapor absorption from a typical MEarth target M dwarf is much larger than the fraction lost from the (typically solar-type) comparison stars. When PWV along the line of sight to a star varies, a crucial assumption of simple differential photometry - that stars are experiencing the same losses - is violated.
Although the variations in any particular light curve might come either from the PWV induced noise or from intrinsic stellar variability, we can harness the ensemble of M dwarfs observed by MEarth at any particular time to characterize and correct for this effect. To do so, we construct a “common mode” template by robustly (median) binning all the differential photometry light curves of all M dwarfs observed on all eight MEarth telescopes into half hour bins. This averages out uncorrelated stellar variability and serves as an estimate of the atmospheric variation that is common to all red stars observed at a given time. We only use data during times when we have $>50$ and $>30$ targets contributing to a bin. The strongest periodicities in the common mode templates are 25.1 days for spring 2009 and 14.5 days for spring 2010 [@irwin.2010.amefsrpfmfmts]. If left uncorrected, such periodicities could appear as spurious intrinsic stellar variability.
To correct the GJ1214 light curve for this effect, we interpolate the “common mode” to the unbinned time stamps. We then perform the nightly binning on it to construct a common mode template $c(t_i)$, which we use for simultaneous decorrelation (see next section). This binning is justified because the typical common mode variation within a night is typically at the level of 1-2 millimagnitudes, much smaller than the night-to-night or week-to-week changes we hope to correct.
Periodograms
------------
We generate a periodogram by calculating the $\chi^2$ of a weighted linear fit of the light curve $\Delta F(t_i)$ (in magnitudes) to a model $$\Delta F_{sine}(t_i) = A\sin\left(\frac{2\pi(t_i - t_0)}{P_{rot}}\right) + B\ m(t_i) + C \ c(t_i) + D
\label{eq-sine}$$ where A is the semiamplitude of the sinusoidal variability, $t_0$ is an epoch, $P_{rot}$ is the stellar rotation period, B and C are scale factors for the systematics, and D is a constant offset. We compare this $\chi^2$ to that of the null hypothesis that $\Delta F$ is explained by the systematics alone $$\Delta F_{null}(t_i) = B\ m(t_i) + C\ c(t_i) + D.
\label{eq-null}$$ Mathematically, this procedure would be identical to traditional least-squares periodograms [@lomb.1976.lfausd; @scargle.1982.satsasasausd] if we fixed $B=C=0$. In @irwin.2010.amefsrpfmfmts, we use a similar method to estimate photometric rotation periods for a sample of 41 MEarth M dwarf targets and test its sensitivity with simulations.
In Fig. \[fig-periodograms\] we plot the $\chi^2$ improvement ($\Delta \chi^2 = \chi^2_{\rm null} - \chi^2_{sine}$) between these two hypotheses for each of the three MEarth seasons and the short 2010 V-band campaign.
Periods for which less than one cycle would be visible have been masked.
The most prominent peak among all the periodograms in Fig. \[fig-periodograms\] is that at $53$ days from the 2010 MEarth data, which corresponds to a semiamplitude of $A=3.5\pm0.7$ millimagnitudes, where the uncertainty has been estimated from the covariance matrix of the linear fit. We estimate the false alarm probability (FAP) for this period by running the complete period search on $10^4$ time series that consist of the best-fit scaled versions of $m(t_i)$ and $c(t_i)$ and randomly generated Gaussian noise set by $\sigma_{nightly}$, recording the $\Delta \chi^2$ of the best peak from each iteration. We find that a FAP of $10^{-4}$ corresponds to $\Delta \chi^2=28$, much less than the achieved $\Delta \chi^2 =41$.
There is a nearby, but statistically insignificant, peak at 51 days in the 2009 MEarth data. Both 2008 and 2009 MEarth light curves are dominated by long-period trends that are unresolvable in each year. Of the resolved peaks in the 2008, the strongest is at 81 days (FAP $< 10^{-4}$). In spite of the formal significance of this last peak, we caution that mid-season changes to the then still uncommissioned observatory might also account for the variations seen. One conclusion is robust; our 3 years of MEarth light curves show no evidence for any rotational modulation with a period shorter than 25 days.
Given that 2010 had the most uniform sampling and cadence, we tentatively suggest $P_{rot} = 53$ days as our current best estimate of GJ1214’s likely rotation period. Fig. \[fig-rotation\] shows the 3 MEarth and 1 V binned light curves with a sinusoid whose period has been fixed to our estimated $P_{rot}=53$ days but whose amplitude and phase have been fitted to the data. The fit is acceptable for 2009 MEarth, but clearly fails for the 2008 MEarth data.
We stress the caveat that the true rotation period could instead be a longer multiple of our quoted 53 day period (e.g. $P_{rot} \approx 100$ days) if the star exhibits multiple, well-spaced active regions. This kind of harmonic confusion appeared and was addressed in studies of Proxima Cen [@benedict.1998.ppcbsuhstfgsspv; @kiraga.2007.ards]. Preliminary data for GJ1214 collected in 2011 while this paper was under review do not seem to show evidence for a 53 day period, preferring instead a much longer one. Due to this factor-of-n uncertainty in the true rotation period, we do not quote a formal error bar on our 53 day period estimate.
![For a sub-sample of nights measured in both MEarth and V bandpasses, the nightly bins plotted against each other with the assigned $\sigma_{bin}$ error bars. The best-fit slope ([*black line*]{}) and 1$\sigma$ interval ([*shaded region*]{}) are shown.[]{data-label="fig-mearth_v"}](f3.eps){width="3.5in"}
Chromatic Spot Variation
------------------------
By itself, the V-band light curve prefers a period of 41 days (Fig. \[fig-periodograms\]; FAP = 5$\times 10^-4$). When forced to fit a 53 day period (Fig. \[fig-rotation\]), these data show a phase offset of only 2 days relative to the simultaneous MEarth data. The semiamplitude of this fit is $7 \pm 3$ millimagnitudes, twice that seen in the MEarth bandpass.
There are 28 nights when observations were obtained in both MEarth and V band. In Fig. \[fig-mearth\_v\] we plot the nightly bins against each other; the apparent correlation suggests that the two instruments are observing the same stellar variability across two bands and not telescope systematics. We fit a line to the relation, accounting for errors in both $\Delta$MEarth and $\Delta$V [@press.2002.nrsc] and find a slope of 2.4$\pm$0.8. While the significance is marginal, we take this as further evidence that the amplitude of the variability in V is greater than that in the MEarth band.
If starspots have a temperature ($T_{\bullet}$) that is only modestly lower than the stellar effective temperature ($T_{\rm eff}$), the color-dependence will arise from the spectrum of the spot rotating in and out of view. The factor of 2 we see would be consistent with $T_{\bullet}/T_{\rm eff}\approx90-95\%$ as is commonly assumed in M dwarf eclipsing binaries [e.g. @morales.2009.aplebd; @irwin.2009.3bvmebsdmo]. Totally dark spots ($T_{\bullet} = 0$K) would produce less of a chromatic variation, but would still be sensitive to the spectral signature of the stellar limb-darkening [e.g. @poe.1985.saatinbswlc].
[cclrr]{}
0 & 2009 May 13 & MEarth & 2950 & 60\
9 & 2009 May 29 & FLWO & 1960 & 45\
9 & 2009 May 29 & MEarth & 1580 & (binned) 45\
11 & 2009 Jun 01 & FLWO & 2060 & 45\
11 & 2009 Jun 01 & MEarth & 1240 & (binned) 45\
21 & 2009 Jun 17 & MEarth & 1620 & (binned) 45\
221 & 2010 Apr 29 & VLT & 380 & 72\
228 & 2010 May 10 & MEarth & 1770 & (binned) 45\
233 & 2010 May 18 & VLT & 350 & 72\
Fitting the Transit Light Curves {#Transits}
================================
{width="7in"}
{width="7in"}
{width="6in"}
We perform a simultaneous fit to the 4 transit light curves from MEarth and 2 from KeplerCam published by @charbonneau.2009.stnls, 1 more transit collected by MEarth in spring 2010, and 2 high-precision transits from the VLT. This totals 9 light curves of 7 independent transits, as shown in Figures \[fig-vlt\] and \[fig-other\]. We employ a model corresponding to a circular planet transiting a smooth, limb-darkened star [@mandel.2002.alcpts] that has the following parameters: the planet-to-star radius ratio $R_{p}/R_{\star}$, the stellar radius $R_{\star}$, the total transit duration $t_{\rm 14}$, two quadratic stellar limb darkening parameters $u_1$ and $u_2$ for each of the 3 telescope systems used, and 7 mid-transit times $T_{c}$. The reparameterization of the scaled semimajor axis $a/R_{\star}$ and inclination $i$ in terms of $R_{\star}$ and $t_{\rm 14}$ substantially reduces the degeneracies in the problem, leading to an efficient exploration of the parameter space [@burke.2007.xtjmcpmb; @carter.2008.aatlouc].
We fix the orbital period to $P=1.5804043$ days [@sada.2010.rtseg]. [ Given the upper limit on eccentricity from radial velocities and the short circularization time for GJ1214b [$10^6$ years for $Q'_p = 100$ and $Q'_* = 10^6$, following @2008MNRAS.384..663R], we assume an eccentricity $e=0$ throughout.]{} Where necessary to derive physical parameters from the geometric parameters in the light curve fit, we adopt the the stellar mass $M_{\star} = 0.157$ [$\rm M_{\sun}$]{} [@charbonneau.2009.stnls]; we describe how we propagate the $0.019$ [$\rm M_{\sun}$]{} uncertainty on this value to the other errors in the next section.
To account for the systematic trends present in the VLT light curves, we introduce a correction to the baseline stellar flux parameterized as a parabola in time ($a + b t + c t^2$) to each transit. Most of the MEarth and FLWO light curves showed no significant systematic trends, so only a single out-of-transit baseline flux level was fit to each night. The MEarth light curve on the night of 2010 May 10 showed a strong correlation with airmass, so we also included a linear trend with airmass for this one night.
$\chi^2$ minimization {#fit}
---------------------
We determine the best-fit values of the 30 model parameters by using an implementation of the Levenberg-Marquadt (LM) routine called MPFIT [@markwardt.2009.nlfwm] to minimize the value of $$\begin{aligned}
\chi^2 &=& \displaystyle \sum\limits_{i=1}^{N}\frac{(F_i - F_{\rm model})^2}{\sigma_{\rm i}^2}
\label{chi}\end{aligned}$$ where $F_i$ are the $N=1495$ flux measurements (Table \[tab-LC\]), $\sigma_i$ are their uncertainties, $F_{\rm model}$ is the model described above.
After this initial fit, the uncertainty estimates for each light curve were increased until the reduced $\chi^2$ of the out-of-transit residuals was unity and the fit was repeated.
While the LM fit provides a linearized estimate of the covariance matrix and errors of the parameters, this estimate is too precise because the method a) does not fully sample along non-linear correlations between highly-degenerate parameters and b) does not account for correlations between data points. In what follows, we calculate more conservative and realistic errors through a bootstrap method that addresses both these issues.
Error Estimates by Residual Permutation {#prayer}
---------------------------------------
Although the noise in the MEarth and FLWO light curves is well described by a white Gaussian process, the much lower photon noise in the VLT light curves reveals underlying low level serial correlations among data points, or ‘red’ noise. The autocorrelation function of the VLT residuals is above 0.25 out to 4 and 2 data point lags respectively for the two transits, and binning the residuals by N points reduces the scatter more slowly than $1/\sqrt{N}$. To quantify the excess uncertainty in the fitted parameters due to this red noise, we perform a ‘residual permutation’ bootstrap simulation [@moutou.2004.armteo; @gillon.2007.asirmn4; @desert.2009.scmate1] that fits resampled data while preserving the correlations between data points. This analysis is carried out simultaneously for the VLT, FLWO and MEarth light curves.
After subtracting the best-fit model from the ensemble of light curves, we perform $2\times10^4$ iterations of the following procedure. Preserving the time stamps for all the exposures, we cyclicly permute the residuals for each light curve by a random integer (shifting along the series and wrapping back from the last exposure to the first), inject the best-fit model back into the set of shifted residuals, perform the LM fit on the simulated light curve ensemble, and record the results. To initialize the parameters for the LM fit, we select guesses drawn from a multivariate Gaussian distribution whose covariance matrix is scaled up by $2^2$ (i.e. $2 \sigma$) from the LM’s estimate. We excised those bootstrap samples that found a best fit with unphysical limb-darkening parameters [$u_1 < 0, u_1 + u_2 > 1, u_1 + 2u_2 < 0$, @burke.2007.xtjmcpmb].
[lcc]{}
$R_p/R_*$ &$ 0.1162 \pm 0.0007 $& $0.1171\pm0.0010$\
$a/R_{*}$ &$ 14.66 \pm 0.41$ &$14.93\pm0.24$\
$i$ (deg.) &$88.62^{+0.35}_{-0.28}$& $88.80^{+0.25}_{-0.20}$\
$b$ & $0.354^{+0.061}_{-0.082}$& $0.313^{+0.046}_{-0.061}$\
$\rho_* (g/cm^3)$ & $23.9\pm2.1$ & $25.2\pm1.2$\
$R_* (R_\sun)$ &$ 0.2110 \pm 0.0097$ & $0.2064^{+0.0086}_{-0.0096}$\
$R_p (R_\earth)$ & $2.68 \pm 0.13 $& $2.64\pm0.13$\
$t_{12} = t_{34}$ (minutes) & & $6.00^{+0.24}_{-0.25}$\
$t_{23}$ (minutes) & & $40.10^{+0.46}_{-0.43}$\
$t_{14}$ (minutes) & & $52.11^{+0.25}_{-0.22}$\
$P$ (days) & $1.5804043 \pm 0.0000005$ & $ 1.58040490 \pm 0.00000033$\
$T_0$ (${\rm BJD_{TDB}}$) & $2454966.52506 \pm 0.00006$ & $2454966.525042 \pm 0.000065$
\
.
[lrrr]{}
MEarth & 715-1000 &$0.53\pm0.13$ & $-0.08\pm0.21$\
VLT-FORS2 & 780-1000 & $0.34\pm0.31$ & $0.28\pm0.46$\
KeplerCam $z$ & 850-1000& $0.26\pm0.11$ & $0.26\pm0.19$\
In Table \[tab-params\], we quote the best-fit value for each parameter and uncertainty bars that exclude the lower and upper 15.9% of the bootstrap samples (i.e. the central 68.3% confidence interval), where parameters that were not directly fit have been calculated analytically from those that were [@seager.2003.uspspfeptlc]. Having fit quadratic limb-darkening parameters for each instrument, we present similar confidence intervals of the coefficients $u_1$ and $u_2$ in Table \[tab-ld\].
Fig. \[fig-matrix\] summarizes the correlations among the parameters $R_p/R_{\star}$, $a/R_{\star}$, $i$, and the linear combination of the limb-darkening parameters $u_1 + u_2$ for the three telescope systems. As the difference between central and limb specific intensities, $u_1+u_2$ correlates strongly with the transit depth and $R_p/R_{\star}$. We also show in Fig \[fig-matrix\] the bootstrap histograms for these parameters.
Although we use $R_{\star}$ as a fit parameter, the quantity that is actually constrained by the light curves is $\rho_{\star}$. To calculate the true uncertainty on $R_{\star}$ (and $R_p$), we calculate $\rho_{\star}$ for all of our bootstrap samples using the fixed $M_{\star}=0.157$[$\rm M_{\sun}$]{}, assign values of $M_{\star}$ drawn from the appropriate Gaussian distribution, and recalculate $R_{\star}$ from $\rho_{\star}$ and $M_{\star}$.
We find results consistent with @charbonneau.2009.stnls. Despite the high precision of the VLT light curves, the uncertainties for most parameters are comparable to the earlier work, and that on $R_p/R_{\star}$ is slightly larger. This is due in part to the correlated noise analysis we perform that @charbonneau.2009.stnls did not. More significantly, @charbonneau.2009.stnls fixed the quadratic limb-darkening parameters to theoretical values while we fit for them directly. The extra degrees of freedom allowed by our relaxation of astrophysical assumptions are known to increase the uncertainty on $R_p/R_{\star}$ [@burke.2007.xtjmcpmb; @southworth.2008.hstepila]. [ For comparison, @charbonneau.2009.stnls used coefficients appropriate for a 3000K, $\log g = 5$, PHOENIX atmosphere, specifically Cousins $I$ [tabulated in @bessell.1990.up] coefficients [$u_1 = 0.303, u_2=0.561$; @claret.1998.vmsnll] as an approximation for the MEarth bandpass and Sloan $z$ coefficients [$u_1 = 0.114, u_2=0.693$; @claret.2004.nlsamisfc2tssg] for the KeplerCam FLWO data. While our individual fitted values differ from these, the integral over the stellar disk ($1-u_1/3-u_2/6$) is very well reproduced.]{}
[rlc]{} 0 & $2454966.525207$ & $0.000351$\
9 & $2454980.748682$ & $0.000104$\
11 & $2454983.909507$ & $0.000090$\
21 & $2454999.713448$ & $0.000155$\
221 & $2455315.794564$ & $0.000066$\
228 & $2455326.857404$ & $0.000110$\
233 & $2455334.759334$ & $0.000066$\
Transit Timing Results {#transit-ttv-results}
----------------------
Mid-transit times (equivalent to times of inferior conjunction given the assumed circular orbit) are printed in Table \[tab-times\]. Our uncertainty estimate on the VLT transit times is $50\%$ larger than it would be if we had ignored correlations in the data, but still only 6 seconds. This uncertainty is within a factor of two of the uncertainty on the highest-precision transit times yet measured from either the ground or space: the 3-second measurements of HD189733b with [*Spitzer*]{} [@agol.2010.c1fftems]. For stars of comparable brightness to GJ1214, large aperture ground based telescopes offer a powerful tool for precision transit times.
A revised linear ephemeris is shown in Table \[tab-params\] derived with a weighted least squares method from our transit times and those of @sada.2010.rtseg. Residuals are shown in Fig. \[fig-times\]. The reduced $\chi^2$ of the linear fit $\chi^2_{\nu}=0.62$ gives no indication of transit timing variations over the time scales probed, and a Bayesian model comparison test [see @burke.2010.notjx] does not show significant evidence for a model with linear and hypothetical sinusoidal components over a purely linear ephemeris. We note that a hypothetical 1.0 [$\rm M_{\earth}$]{} planet in a 2:1 mean-motion resonance with GJ1214b would introduce $\sim100$ second transit timing variations that would be easily detected in these data [following @bean.2009.attc].
![Deviations in the times of transit from the new best linear ephemeris, including transits from the VLT ([*filled black circles*]{}), MEarth ([*open black circles*]{}), and work by @sada.2010.rtseg ([*open grey circles*]{}). []{data-label="fig-times"}](f7.eps){width="3.5in"}
Occulted Spots
--------------
Occulted spots will appear in the transit light curve as a bump lasting roughly as long it takes the planet to move a distance $2(R_p + R_{\bullet})$ across the spot, assuming a circular spot with radius $R_{\bullet}$. For small spots ($R_{\bullet} << R_p$), this is roughly the transit ingress/egress time, or 6 minutes in the case of GJ1214b. The amplitude of a spot crossing event is determined by the fractional deviation in surface brightness occulted by the planet from the star’s mean and could in principle be comparable to the transit depth itself for totally dark spots larger than the planet.
We see evidence for spot occultations in the first two VLT transits shown in Fig. \[fig-vlt\], in which the residuals from a smooth model show a 0.1% brightening at the start of the first transit and near the middle of the second transit. These features last 5-10 minutes, as expected, and persist with comparable amplitudes in light curves generated separately from the blue and red halves of the FORS spectra. Their presence is robust to choice of comparison stars.
For the level of precision afforded by current data, we treat these possible spot occultations as excess red noise with correlations on the scale of the 6 minute spot crossing time. As such, the residual permutation method (§\[prayer\]) accounts for the uncertainty introduced by these features, regardless of their physical interpretation. Spots that are not occulted will not be accounted for in these errors estimates, but we discuss their influence in $\S\ref{implications}$.
Unocculted Spots {#spots-results}
----------------
Unocculted spots will also have an effect on the planetary parameters. By diminishing the overall flux from the star while leaving the surface brightness along the transit chord unchanged, increasing the coverage of unocculted spots on the star will make transits deeper. This is opposite the effect of occulted spots, which tend to fill in transits and make them shallower. Because we must apply a nightly normalization to each light curve to avoid systematics and compare across telescopes, this depth change carries through to the implied $R_p/R_{\star}$.
To test whether unocculted spots are biasing individual measurements of $R_p/R_{\star}$, we repeat the fit and uncertainty estimation described in §\[fit\] and §\[prayer\] but allow the 8 transit epochs to have different values of $R_p/R_{\star}$. As the visible spot coverage changes when the star rotates, we might detect changes in the inferred values of $R_p/R_{\star}$. For this experiment we are interested in relative changes among the $R_p/R_{\star}$ values, so we fix the other geometric parameters $R_{\star}$ and $t_{\rm 14}$ and the limb-darkening parameters to their best fit values to effectively collapse along those dominant degeneracies.
The inferred $R_p/R_{\star}$ for each transit is shown in Fig. \[fig-k\], with uncertainties estimated from the residual permutation method. Because the other parameters were fixed in this fit, the ensemble of points are free to move up and down slightly on this plot; the uncertainties shown are more relevant to comparisons between epochs. In §\[implications\], we will discuss our calculation of the predicted variations induced by unocculted spots consistent with the observed variability.
![Estimates of the apparent planet-to-star radius ratio at each epoch ([*black circles*]{} for MEarth/FLWO ([*open symbols*]{}) and VLT transits ([*filled symbols*]{})). Via the residual permutation estimate, the error bars include the uncertainty due to possible presence of occulted spots. The predicted variation in the apparent $R_{p}/R_{\star}$ due to the presence of unocculted spots (see §\[implications\]) is shown ([*grey*]{}), calculated directly from the nightly-binned MEarth photometry ([*points*]{}) or the best-fit sine curves to those data ([*lines*]{}).[]{data-label="fig-k"}](f8.eps){width="3.5in"}
Limits on Additional Transiting Planets {#Search}
=======================================
After clipping out known transits of GJ1214b, we investigate the light curve of GJ1214 for evidence of other transiting planets in the system. The light curve contains 3218 points spanning spring 2009 and 2010.
To remove structured variability from the light curve before searching for transits, we employ an iterative filtering process that combines a 2-day smoothed median filter [@aigrain.2004.ppp] with a linear decorrelation against both external parameters (primarily the common mode, meridian flip, seeing, and pixel position) and light curves of other field stars [@kovacs.2005.tfawvs; @tamuz.2005.cselplc; @ofir.2010.sadclcwsusep]. Such filtering decreases the light curve RMS from 9.6 to 4.7 millimagnitudes per point.
Using a variant of the box-fitting least squares (BLS) algorithm [@kovacs.2002.baspt], we search the filtered light curve for periodic rectangular pulses over a grid of period and transit phase. At each test period, we fix the transit duration to that of a mid-latitude transit of a circularly orbiting planet for GJ1214’s estimated stellar mass (0.157 [$\rm M_{\sun}$]{}) and radius (0.207 [$\rm R_{\sun}$]{}). [ With the typical MEarth precision and cadence, our sensitivity depends only weakly on transit duration; violations of these assumptions will not substantially penalize our detection efficiency.]{}
At every grid point, we determine the best fit transit depth $D$ and the improvement $\Delta \chi^2$ over the $D=0$ null hypothesis, using a weighted least-squares method that includes red noise estimated from the light curve itself using the $\sigma_r$ formalism of @pont.2006.enptd. Following @burke.2006.stepssilfswpoc1 we characterize candidates in terms of $\Delta \chi^2$ and $f=\max(\Delta\chi^2_{\rm each~transit}/\Delta\chi^2_{\rm total})$, which is the largest fraction that any one transit event contributes to the signal.
{width="7in"}
The best candidate found in the clipped GJ1214 light curve exhibited a $\Delta \chi^2$ of 31.3 and is shown in Fig. \[fig-inject\]. To estimate the significance of this value, we employ the bootstrap method of @jenkins.2002.stecpdtp. [ Strictly speaking, the presence of remaining correlated noise in our filtered light curve means this method gives an overestimate of the false alarm probability, but the complicated correlation structure of the light curve make a more accurate significance estimate difficult to calculate. In practice, we have found the Jenkins method to provide an appropriate limit, even for light curves with substantial red noise.]{} We generate $10^3$ (so we can estimate the $\chi^2$ associated with a $1\%$ false alarm probability from 10 samples) transit-less light curves with Gaussian white noise and time sampling identical to that of the real light curve and performing the BLS search on these fake light curves. We find $20\%$ of them show values of $\chi^2 > 31.3$, suggesting our best candidate shown in Fig. \[fig-inject\] is not significant.
These data place limits on the presence of other transiting planets in the GJ1214 system. Like @burke.2006.stepssilfswpoc1, @croll.2007.ls1sstmsp and @ballard.2010.sapneoesg, we simulate our sensitivity by injecting 8000 randomly phased, limb-darkened transits of 1.0, 1.5, 2.0, and 4.0 [${\rm R_{\earth}} $]{} planets with random periods. We then attempt to recover them with our transit search using objective detection criteria. To account for possible suppression from the filtering, we inject the transits into the raw light curves and reapply the filter at each iteration. We adopt two criteria for detection: $\Delta \chi^2>50$ (corresponding to a formal probability of false alarm of $10^{-5}$ by the bootstrap analysis) and $f<1.0$ (to ensure at least two events contribute).
To demonstrate this visually, we show in Fig. \[fig-inject\] examples of injected transits for each relevant radius, randomly selected from among the simulated planets with injected periods of $10 \pm 1$ days whose BLS results satisfied the detection criteria. Planets with periods near $10$ days are of particular interest; given GJ1214’s low luminosity, they would be in the star’s habitable zone. Fig. \[fig-sensitivity\] plots the fraction of injected planets that crossed the detection threshold as a function of period for each input planetary radius. The shape of the 4.0 [${\rm R_{\earth}} $]{} curve in Fig. \[fig-sensitivity\] is driven largely by the $f<1.0$ criterion requiring multiple events are observed, whereas the 1.5 [${\rm R_{\earth}} $]{} curve is dominated by the need for sufficient in-transit S/N to get to $\Delta\chi^2 > 50$. With these criteria, our sensitivity falls below 50% beyond periods of 15, 8, and 2 days for 4.0 [${\rm R_{\earth}} $]{}, 2.0 [${\rm R_{\earth}} $]{}, and 1.5 [${\rm R_{\earth}} $]{} transiting planets. We had no sensitivity to 1.0 [${\rm R_{\earth}} $]{} planets.
Fig. \[fig-sensitivity\] also shows the recovery fraction for 4.0 [${\rm R_{\earth}} $]{} planets without requirement that more than one event be observed. Neptune-sized transits of GJ1214 are so dramatic that they could be confidently identified by a single event (see Fig. \[fig-inject\]). Our simulations show that we are 90% sensitive to transiting Neptunes around GJ1214 out to 10 days, and 80% sensitive out 20 days. For smaller planets requiring multiple events, however, a significant volume of parameter space remains unconstrained.
![The recovery fraction as a function of period for the labeled planetary radii ([*solid, colored lines*]{}) assuming the detection criteria $\Delta \chi^2>50$ and $f<1.0$. For the 4.0 [${\rm R_{\earth}} $]{} case, we also show the recovery fraction after lifting the $f<1.0$ constraint ([*black dashed line*]{}); this is an estimate of the sensitivity to deep transits where only one event is necessary for a robust detection.[]{data-label="fig-sensitivity"}](f10.eps){width="3.5in"}
Discussion {#Discussion}
==========
GJ1214 as a Spotted Star
------------------------
The slow rotation period we find implies a projected rotational velocity of 0.2 km s$^{-1}$ that is well below the $v \sin i \simeq 1$ km s$^{-1}$ detection limit of high-resolution rotation studies [@browning.2010.rmasm; @delfosse.1998.rcafd; @reiners.2007.nmlprcvsr; @west.2009.flrild] but not inconsistent with long photometric periods detected for other field M dwarfs. @benedict.1998.ppcbsuhstfgsspv estimated a rotation period of 83 days for Proxima Cen, and recent photometric work with ASAS [@kiraga.2007.ards], HATnet [@hartman.2009.pvsfdswh], and MEarth [@irwin.2010.amefsrpfmfmts] has confirmed the presence of many field M dwarfs with $P_{rot}>10$ days.
Our $P_{rot}$ further implies that GJ1214 should exhibit signs of only weak magnetic activity [e.g. @reiners.2007.fdmsmfvms; @reiners.2007.nmlprcvsr]. Indeed, across three seasons of photometric monitoring, we see no evidence for flares in the MEarth bandpass, although their amplitude would be expected to be small in the near-IR. Activity induced chromospheric emission is not detected in either H$\alpha$ or the [Na I]{} D doublet in the HARPS spectra used to measure the radial velocities presented in @charbonneau.2009.stnls. The relation between magnetic activity and kinematic age [@west.2008.carcssdsdrlsss] suggests that GJ1214 is $>3$ Gyr old.
We calculate GJ1214’s $(U, V, W)$ space velocities [@johnson.1987.cgsvtuwaumg] in a left-handed system where $U > 0$ in the direction of the Galactic anti-center to be (-47,-4,-40) km s$^{-1}$. These motions are consistent with membership in the Galactic old disk [@leggett.1992.icls], lending further credence to an old age for GJ1214.
Implications for Transmission Spectroscopy {#implications}
------------------------------------------
When inferring the transmission spectrum of a planet, one hopes to attribute changes in the transit depth across different wavelengths to atmospheric absorption by the limb of the planet. If the transmission spectrum is sensitive to 5 scale heights ($H$) of the planetary atmosphere, the amplitude of the transit depth variations are $\Delta D_{\rm planet}(\lambda) = 10HR_p/R_{\star}^2$ or $0.001$ if GJ1214 has an hydrogen-rich atmosphere [@miller-ricci.2010.nats1]. However, the presence of unocculted spots on the stellar surface can introduce transit depth variations $\Delta D_{\rm spots} (\lambda, t)$ that are a function of both wavelength and time. To aid ongoing and future work to study GJ1214b’s atmosphere, we use a simple model to estimate the amplitude of the spot-induced contamination $\Delta D_{\rm spots} (\lambda, t)$.
We assume a fraction $s(t)$ of the star’s Earth-facing hemisphere is covered with spots; $s(t)$ will change as the star rotates and the spots evolve. The observed out-of-transit spectrum $F_{\rm o.o.t.}(\lambda, t)$ is a weighted average of the spectrum of the unspotted photosphere $F_{\circ}(\lambda)$ and that of the presumably cooler spotted surface $F_{\bullet}(\lambda)$: $$F_{\rm o.o.t.}(\lambda, t) = \left[1-s(t)\right]F_{\circ}(\lambda) + s(t)F_{\bullet}(\lambda).$$ To simplify calculations, we neglect limb-darkening and treat each of the two components as having uniform surface brightness; tests with a limb-darkened spot model [@dorren.1987.fsmcss] indicate only pathological cases could change the following results by more than $10\%$.
When a planet with no atmosphere blocks light across a spot-free transit chord, it changes the relative weight of the two sources, causing the observed in-transit spectrum $F_{\rm i.t.}(\lambda, t)$ to shift away from that of the unspotted photosphere toward that of the spots: $$\begin{aligned}
\frac{F_{\rm i.t.}(\lambda, t)}{F_{\rm o.o.t.}(\lambda, t)} &=& 1 - D(\lambda, t) \\
&=& 1- \left(\frac{R_p}{R_{\star}}\right)^2 - \Delta D_{\rm spots}(\lambda, t) \nonumber \\
&=& \frac{\left[1-s(t)-\left(\frac{R_p}{R_{\star}}\right)^2\right]F_{\circ}(\lambda) + s(t)F_{\bullet}(\lambda)}{ \left[1-s(t)\right]F_{\circ}(\lambda) + s(t)F_{\bullet}(\lambda)}. \nonumber\end{aligned}$$ Making the assumption that the total fraction of flux lost to the presence of spots is small ($s(t)[1-F_{\bullet}(\lambda)/F_{\circ}(\lambda)] << 1$), we solve to find $$\Delta D_{spots}(\lambda, t) \approx s(t)\left[1-\frac{F_{\bullet}(\lambda)}{F_{\circ}(\lambda)}\right]\times\left(\frac{R_p}{R_{\star}}\right)^2.
\label{eq-dspots}$$ If the quantity $s(t)[1-F_{\bullet}(\lambda)/F_{\circ}(\lambda)]$ were not small, we probably would have observed larger amplitude and more frequent spot occultation events in the transit light curves in §\[Transits\]. We note the significant possibility that $s(t)$ never reaches 0; that is, there may persist a population of symmetrically distributed spots that never rotates out of view. Such an unchanging population could cause us to overestimate the true value of $R_p/R_{\star}$ by up to several percent.
To match our observations of GJ1214’s variability ($\Delta F_{\rm o.o.t.}(\lambda, t)/\overline{F_{\rm o.o.t.}}$) to this model, we write $s(t) = \overline{s} + \Delta s(t)$ where $\overline{s}$ is the mean Earth-facing spot covering fraction and $\Delta s(t)$ can be positive or negative. With the same assumption as above, we find $$\frac{\Delta F_{\rm o.o.t.}(\lambda, t)}{\overline{F_{\rm o.o.t.}}} \approx - \Delta s(t) \left( 1 - \frac{F_{\bullet}(\lambda)}{F_{\circ}(\lambda)} \right).
\label{eq-var}$$ In §\[Rotation\] we measured $\Delta F_{\rm o.o.t.}(\lambda, t)/\overline{F_{\rm o.o.t.}}$ to have a peak-to-peak amplitude of $1\%$ in the MEarth bandpass ($715 < \lambda < 1000$ nm). Eq. \[eq-dspots\] and \[eq-var\] are equivalent to assuming a value of $\alpha = -1$ in @desert.2011.tse1isom’s $\alpha f_{\lambda}$ formalism.
@desert.2011.tse1isom show that measurements that rely on comparing photometric transit depths across multiple transits could potentially mistake time-variability of $\Delta D_{\rm spots}(t)$ for a feature in the transmission spectrum. From Eq. \[eq-dspots\] and \[eq-var\], we estimate the peak-to-peak time-variability of $\Delta D_{\rm spots}(t)$ to have an amplitude of $0.0001$ in MEarth wavelengths over the rotation period of the star. This spot-induced variability is comparable to, but smaller than, the estimated uncertainty from the transits analyzed in this work and corresponds to an apparent change in planetary radius of $\Delta R_p = 70~{\rm km}$ or 1/2 the scale height of an H$_2$-dominated atmosphere on GJ1214b [@miller-ricci.2010.nats1]. In Fig. \[fig-k\] we show the expected variation in the apparent planetary radius from unocculted spots and our individual $R_p/R_{\star}$ measurements.
Using blackbody spectra for $F_{\bullet}(\lambda)$ and $F_{\circ}(\lambda)$, we can extrapolate with Eq. \[eq-dspots\] from the MEarth observations ($\lambda \approx 0.85 ~{\rm \mu m}$) to wavelengths accessible to [*Warm Spitzer*]{}. If we assume the spots are 300K cooler than the $T_{\rm eff}=3000$K stellar photosphere, we find $\Delta D(t)$ variability amplitudes of 0.00004 and 0.00003 in [*Spitzer*]{}’s 3.6 and 4.5 ${\rm \mu m}$ bandpasses. This allows robust comparison of transit depths between these wavelengths (Désert et al., in prep.). In the conservative limit that $T_{\bullet}=0$K, the variability amplitude would be achromatic.
Even when comparison across different epochs is unnecessary, as is the case for spectroscopic observations of individual transits [e.g. @charbonneau.2002.depa; @pont.2008.dahep0mts1wh], unocculted spots can still introduce spurious wavelength features into the transmission spectrum. To place an upper limit on the amplitude of chromatic changes in $\Delta D_{\rm spots}(\lambda)$ within a given transit, we imagine an extreme scenario where $F_{\bullet}(\lambda)$ is identical to $F_{\circ}(\lambda)$ but with a very deep absorption line that does not appear in the unspotted spectrum, so the first factor in Eq. \[eq-dspots\] can vary with $\lambda$ between 0 and $s$. Making the fairly conservative assumption that the population of spots that rotates in and out of view is comparable to the symmetric population ($s < 4\max[\Delta s]$) we find that $\Delta D_{\rm spots}(\lambda) < 0.0003$ for GJ1214 at wavelengths near $1~{\rm \mu m}$. In practice, most features will show significantly lower amplitudes, although precise calculations of them and extrapolation to other wavelengths will require knowledge of the spot temperature and reliable model atmospheres in the 2500-3000K temperature regime.
Given the sensitivity of current generation instruments, the known population of spots on GJ1214 do not pose a significant problem for ongoing studies of GJ1214b’s atmosphere. Future transmission spectroscopy studies of GJ1214b, perhaps with the [*James Webb Space Telescope*]{}, comparing multiepoch, multiwavelength transit depths and aiming to reach a precision of $\sigma_{D(\lambda)} = 0.0001$ [see @deming.2009.dctseuatsfjwst] will have to monitor and correct for the stellar variability.
Limiting Uncertainties of GJ1214b
---------------------------------
As shown in Fig. \[fig-k\], stellar spots currently play a very small role in limiting our understanding of the bulk mass and radius of GJ1214b. Here we address what other factors provide the limiting uncertainties in the planet’s physical parameters.
The 0.98 [$\rm M_{\earth}$]{} ($15\%$) uncertainty on the planetary mass $M_p$ is the quadrature sum of 0.85 [$\rm M_{\earth}$]{} propagated from the measured radial velocity semiamplitude and 0.5 [$\rm M_{\earth}$]{} from the stellar mass uncertainty. Further spectroscopic monitoring may reduce the former, but to fully reap the benefits of the radial velocities, the 12% error on $M_{\star}$ should be improved. The current mass estimate is derived from an 2MASS photometry (2% uncertainty), an empirical $M_K$-mass relation [$\sim10\%$ scatter; @delfosse.2000.amvmsiimr] and the published system parallax [$77.2 \pm 5.4$ mas; @van-altena.1995.gctsp]. Both improving $M_K$-mass relation for low-mass dwarfs and confirming GJ1214’s parallax will be necessary to reduce GJ1214b’s mass uncertainty.
The 0.12 [${\rm R_{\earth}} $]{} ($5\%$) uncertainty on the planetary radius $R_p$ is already dominated by the $5\%$ uncertainty in the stellar radius $R_{\star}$, rather than the light curve parameters. This is currently constrained by the combination of the stellar density $\rho_{\star}$ that is measured directly from transit light curves [@seager.2003.uspspfeptlc] and the estimated mass. Even though $R_{\star} \propto (M_{\star}/\rho_{\star})^{1/3}$, and so is relatively insensitive to uncertainty in $M_{\star}$, if we fix $M_{\star} = 0.157 \pm 0.019$[$\rm M_{\sun}$]{} we find the errors on $R_{\star}$ and $R_p$ shrink by a factor of two. Improving the estimate of GJ1214’s mass is also the best way to improve our measurement of the radius of the planet.
We reiterate here that our measured out-of-transit photometric modulation probes only the spatially asymmetric component of the stellar spot distribution rotating around the star. If the star hosts a subtantial, unchanging, spatially symmetric population of unocculted spots, it will bias estimates of the true planetary radius [@czesla.2009.saaseep] too high. If the symmetric and asymmetric components of the spots are comparable, such a bias will be at the percent level of in $R_p$, smaller than the current uncertainties.
Metallicity of GJ1214
---------------------
Several authors have recently developed empirical photometric calibrations to estimate M dwarf metallicities from absolute ${\rm M_K}$ magnitude and $V - K$ color [@bonfils.2005.mdipcimrbms; @johnson.2009.mrdwp; @schlaufman.2010.ppcdm]. As exoplanet surveys like MEarth lavish more attention on M dwarfs as exoplanet hosts, such studies hope to address whether the giant planet vs. stellar metallicity correlation seen for AFGK stellar hosts [@fischer.2005.pc; @johnson.2010.gposmp] extends to smaller planets and smaller stars. @sousa.2008.spshpspsffe suggest the correlation does not persist down to Neptune-mass planets, but more data are needed.
Our improved estimate of $V = 14.71 \pm 0.03$ differs significantly from the $V=15.1$ central value published in @charbonneau.2009.stnls but was the value used in the most recent analysis of this photometric metallicity calibration by @schlaufman.2010.ppcdm, who found \[Fe/H\] = +0.28 for GJ1214. This analysis agrees quite well with work by @rojas-ayala.2010.mmphmwks that estimates empirically calibrated metallicities from alkali metal lines in moderate resolution $K$-band spectra and finds \[Fe/H\] = $+0.39 \pm 0.15$ for GJ1214. This lends incremental evidence towards the persistence of the mass-metallicity correlation down to super-Earths around M dwarf hosts.
Conclusions
===========
We have measured long-term photometric variability on GJ1214 to have a 1% peak-to-peak amplitude [ in the MEarth bandpass (715-1000 nm)]{} and a long rotation period, most likely an integer multiple of 53 days. Fitting very high precision light curves from the VLT, we find likely instances of GJ1214b crossing small spots during transit. Treating these occultation events as correlated noise, we find parameters for the planetary system that are consistent with previous work.
We estimate the amplitude of time-variable changes in the apparent radius of the planet due to the observed stellar variability as $\Delta D_{\rm spots} (t) = 0.0001$ and place an upper limit of $\Delta D_{\rm spots}(\lambda) < 0.0003$ on possible spot-induced spectral features in the planet’s transmission spectrum. Stellar spots do not limit current studies [e.g. @bean.2010.temp; @desert.2011.oemrasg], but could be important for future studies of GJ1214b with [*JWST*]{}.
Using two years of MEarth data, we have placed limits on the presence of other transiting planets around GJ1214. With 90% confidence, we rule out the presence of Neptune-radius transiting planets in orbits shorter than 10 days but cannot place strong constraints on planets smaller than 2.0 [${\rm R_{\earth}} $]{} at such long periods. In a system where a 1.0[$\rm M_{\earth}$]{} planet in a 2:1 mean motion resonance would create 100 second perturbations to GJ1214b’s transit times, we find no evidence for transit timing variations larger than 15 seconds. Further searches of the GJ1214 system for potentially habitable planets smaller and cooler than GJ1214b continue to be warranted.
We are extremely grateful to the KeplerCam observers Margaret Mclean, Gil Esquerdo, Mark Everett, Pete Challis, and Joel Hartman who collected V-band monitoring observations, as well as Gaspar Bakos, Dave Latham, and Matt Holman for donated time and Lars Buchhave and Xavier Bonfils for correcting the sign in the published $\gamma$ velocity. [ We thank the referee for thoughtful and helpful comments that improved the paper significantly.]{} J.B. acknowledges funding from NASA through the Sagan Fellowship Program. The MEarth team gratefully acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering (awarded to D.C.). This material is based upon work supported by the National Science Foundation under grant number AST-0807690. The MEarth team is greatly indebted to the staff at the Fred Lawrence Whipple Observatory for their efforts in construction and maintenance of the facility, and would like to explicitly thank Wayne Peters, Ted Groner, Karen Erdman-Myres, Grace Alegria, Rodger Harris, Bob Hutchins, Dave Martina, Dennis Jankovsky and Tom Welsh for their support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce a new type of quadrature, known as approximate Gaussian quadrature (AGQ) rules using $\epsilon$-quasiorthogonality, for the approximation of integrals of the form $\int f(x) \, {{\mathrm d}}\alpha(x)$. The measure $\alpha(\cdot )$ can be arbitrary as long as it possesses finite moments $\mu_n$ for sufficiently large $n$. The weights and nodes associated with the quadrature can be computed in low complexity and their count is inferior to that required by classical quadratures at fixed accuracy on some families of integrands. Furthermore, we show how AGQ can be used to discretize the Fourier transform with few points in order to obtain short exponential representations of functions.'
author:
- |
Pierre-David Létourneau,$^a$ Eric Darve,$^b$\
$^a$ Institute of Computational and Mathematical Engineering (ICME), Stanford University, CA\
$^b$ Mechanical Engineering Department, Stanford University, CA
bibliography:
- 'biblio.bib'
title: 'Gaussian Quadrature Rule using $\epsilon$-Quasiorthogonality'
---
Introduction {#Intro}
============
In this paper, we present a new kind of quadrature rule for approximating integrals by sums of the form, $$\label{discrete}
\int f(x) \, {{\mathrm d}}\alpha(x) \approx \sum_{i=1}^n w_i f(x_i)$$ having the following characteristics:
1. The measure $\alpha( \cdot )$ can be *arbitrary* (positive, signed, complex, ...) as long as it satisfies some weak condition.
2. The nodes and weights associated with the quadrature rule can be obtained in low computational complexity through a simple numerical algorithm.
3. The quadrature is at least as accurate as the Gaussian quadrature rule, and in many cases is significantly more accurate.
4. Low-order rules are able to integrate high-order polynomials with high accuracy.
The scheme presented in the current work uses a strategy similar to classical Gaussian quadrature rules (of which a few examples can be found in Table \[classicalGaussQuad\]). The Gaussian quadrature rule is designed to integrate exactly polynomials of degree at most $2n-1$ using $n$ quadrature points and weights: $$\int x^k \, {{\mathrm d}}\alpha(x)$$ for various weight functions $\frac{{{\mathrm d}}\alpha}{{{\mathrm d}}x}$ (see Table \[classicalGaussQuad\]).
Name Interval Measure (${{\mathrm d}}\alpha / {{\mathrm d}}x$ )
---------------------------- --------------------- -------------------------------------------------------------
Gauss-Legendre $[-1,1]$ $1$
Gauss-Laguerre $[0, \infty)$ $e^{-x}$
Gauss-Hermite $(-\infty, \infty)$ $e^{-x^2} $
Gauss-Jacobi $(-1,1)$ $(1-x)^{\alpha}(1+x)^{\beta} \; , \;\; \alpha, \beta > -1 $
Chebyshev-Gauss (1st kind) $(-1,1)$ $1/\sqrt{1-x^2}$
Chebyshev-Gauss (2nd kind) $[-1,1]$ $\sqrt{1-x^2}$
: Examples of classical Gaussian quadratures
\[classicalGaussQuad\]
The paper is structured as follows. In Section \[GQ\], a brief overview of classical Gaussian quadratures will be presented. In Section \[sec:agq\], the concept of quasiorthogonal polynomial and approximate Gaussian quadrature will be introduced together with an error analysis. This will be followed in Section \[NS\] by numerical results. In the same section, we will discuss representations of functions by short sums of exponentials.
Gaussian quadrature {#GQ}
===================
Gaussian quadratures are schemes used to approximate definite integrals of the form, $$\int_a^b f(x) \, {{\mathrm d}}\alpha(x)$$ by a finite weighted sum of the form, $$\sum_{n=0}^N w_n \, f(x_n)$$ where $a<b \in \mathbb{R}$. The coefficients $\{ w_n \}$ are generally referred to as the *weights* of the quadrature, whereas the points $\{ x_n \}$ are referred to as the *nodes*. An $(N+1)$-node Gaussian quadrature can integrate polynomials up to degree $2N+1$ *exactly* and is generally well-suited for the integration of functions that are well-approximated by polynomials.
In what follows, we will briefly describe how the nodes and weights of classical Gaussian quadratures can be obtained based on the classical theory of orthogonal polynomials. For this purpose, we shall denote the real and complex numbers by $\mathbb{R}$ and $\mathbb{C}$ respectively. $\alpha( \cdot )$ will represent an arbitrary measure (possibly complex) on $(\mathbb{R}, \mathcal{B})$ or $(\mathbb{C}, \mathcal{B})$ unless otherwise stated. Vectors are represented by lower case letter e.g., $v$. The $i^{th}$ component of a vector $v$ will be written as $v_i$, and we shall use super-indices of the form $v^{(j)}$ when multiple vectors are under consideration.
We begin by introducing four key objects: the orthogonal polynomials, the Lagrange interpolants, the moments of a measure $\alpha(\cdot)$ and the Hankel matrix associated with such a measure.
[**(Orthogonal polynomial)**]{} A sequence $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ of polynomials of degree $k$ is said to be a sequence of orthogonal polynomials with respect to a positive measure $\alpha(\cdot)$ if, $$\int p^{(k)}(x) p^{(l)} (x) \, {{\mathrm d}}\alpha(x) =
\left\{
\begin{array}{ll}
0 & \mbox{if } k \not = l \\
c_k & \mbox{if } k = l
\end{array}
\right.$$ If in addition $c_k = 1 \; \forall k \in \mathbb{N}$, then the sequence is called *orthonormal*.
We shall hereafter assume that all such polynomials are monic, i.e., that they can be written as, $$p^{(k)} (x) = x^k + \sum_{n=0}^{k-1} p^{(k)}_n x^n$$ where $\{ p^{(k)}_n \}_{n=0}^{k-1}$ are some (potentially complex) coefficients. We then introduce Lagrange interpolants,
[**(Lagrange interpolant)**]{} Given a set of $(d+1)$ data points $\{ (x_n, y_n) \}_{n=0}^d$, the Lagrange interpolant is the unique polynomial $L (x)$ of degree $d$ such that, $$L (x_n) = y_n , \; n = 0 ... d$$ It can be written explicitly as, $$L (x) = \sum_{n=0}^d y_n\, \ell_n (x)$$ where, $$\ell_n (x) = \prod_{\substack{{m=0}\\ {m \not = n}}}^d \frac{x-x_m}{x_n-x_m}$$ and $\ell_n (x)$ is referred to as the $n^{th}$ Lagrange basis polynomial.
Finally we introduce the moments as well as the Hankel matrix associated with a measure $\alpha(\cdot)$,
[**(Moment)**]{} Given an arbitrary measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$, its $n^{th}$ moment $\mu_n$ is defined by the following Lebesgue integral, $$\mu_n = \int x^n \, {{\mathrm d}}\alpha(x)$$ whenever it exists.
[**(Hankel matrix)**]{} An $(N+1) \times (M+1)$ matrix $H$ is called the $(N+1) \times (M+1)$ Hankel matrix associated with the measure $\alpha( \cdot )$ if its entries take the form, $$\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{M} \\
\mu_1 & \mu_2 & \cdots & \mu_{M+1}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{N} & \mu_{N+1} & \cdots & \mu_{N+M} \end{pmatrix}
\label{eq:hankel}$$ i.e., $H_{ij} = \mu_{i+j}$, where $(\mu_0, \mu_1, \ldots, \mu_{N+M} )$ are the first $(N+M)$ moments of $\alpha(\cdot)$ whenever they exist.
With these quantities we can now present the main results associated with classical Gaussian quadratures,
[**(Gaussian quadrature)**]{} Consider a positive measure $\alpha(\cdot)$ on $([a,b], \mathcal{B})$ (with $a,b \in \mathbb{R}$ potentially infinity) and a sequence of orthonormal polynomials $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ with respect to $\alpha( \cdot)$. Then, the quadrature rule with nodes $\{ x_n\}_{n=0}^k$ consisting in the zeros of $p^{(k+1)}(x)$ and weights $\{ w_n \}_{n=0}^k$ given by, $$w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x)$$ integrates polynomials of degree $\leq 2k+1$ exactly.
This is a classical result which can be found in [@Meurant] for instance. Explicit expression for the error incurred in the case of smooth integrand also exist.
To close this section, we introduce a further result characterizing the coefficients of the orthogonal polynomials $\{ p^{(k)}(x) \} $. As we shall see in the next section, this characterization lies at the heart of our scheme,
Consider a positive measure $\alpha(\cdot)$ on $([a,b], \mathcal{B})$ (with $a,b \in \mathbb{R}$ potentially infinity) and a sequence of orthogonal polynomials $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ with respect to $\alpha( \cdot)$. Then, the coefficients $\{ p^{(k+1)}_n \}_{n=0}^k$ of the $(k+1)^{th}$ orthogonal polynomial $p^{(k+1)}(x)$ satisfy the following Hankel system, $$Hp =
\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{k+1} \\
\mu_1 & \mu_2 & \cdots & \mu_{k+2}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{k} & \mu_{k+1} & \cdots & \mu_{2k+1}
\end{pmatrix}
\begin{pmatrix}
p_0^{(k+1)} \\
p_1^{(k+1)} \\
\cdots \\
p_{k}^{(k+1)}
\end{pmatrix} = 0$$ where $\{ \mu_n \}$ are the moments of the measure $\alpha ( \cdot ) $, whenever they exist. \[AGQ:characterization\]
First write, $$p^{(k+1)} (x) = \sum_{n=0}^{k+1} p^{(k+1)}_n x^n$$ Let $0 \leq j \leq k $. Then, from orthogonality we have, $$0 = \int p^{(k+1)} (x) \, x^j \, {{\mathrm d}}\alpha(x) = \sum_{n=0}^{k+1} p^{(k+1)}_n \int x^{n+j}\, {{\mathrm d}}\alpha(x) = \sum_{n=0}^{k+1} p^{(k+1)}_n \mu_{n+j}$$ Putting all these equations in matrix form provides the desired result.
The Hankel matrices associated with positive measures commonly encountered with classical Gaussian quadratures have been the subject of extensive study in the past (known as the moment problem). In some cases, they can be proved to be invertible although extremely ill-conditioned (see $\cite{Shohat}$ for details). On the other hand, less is known per regards to more general measures. In any case, in the event where the resulting Hankel matrix would be invertible, it can be expected to be ill-conditioned. Indeed, as an example it can be shown that for a large class of positive measures, the smallest eigenvalue of the $N \times N$ associated Hankel matrix scales like ${\mathcal{O}}\left (\frac{ \sqrt{N} }{\sigma^{2N}} \right ) $, where $\sigma$ depends only on the interval considered and is equal to $(1+\sqrt{2})$ for the interval $[-1,1]$ (see [@Widom:1966]).
The question we treat in the next section is whether such Hankel matrices arising from arbitrary measures can be used to derive Gaussian-like quadratures, and what this inherent ill-conditioning entails.
Approximate Gaussian quadrature (AGQ) {#sec:agq}
=====================================
In this section, we describe the concept of approximate Gaussian quadrature. For this purpose, we will need the concept of $\epsilon$-quasiorthogonal polynomial, which we introduce for the first time below. Before doing so however, we first point to the following key observation.
Let $H$ be a $N \times M$ with rank $0 < d < M $. Then, there exists $D \le d+1$ and a vector $a \not = 0$ such that $$Ha = 0, \quad \text{with $a_i = 0$ for all $i > D$}$$ \[AGQ:low\_rank\]
The rank of $H$ is $d$. Therefore if we consider the first $d+1$ columns for $H$ they are linearly dependent. Denote $D$ the smallest integer such that the first $D$ columns of $H$ are linearly dependent. We have $D \le d+1$ and, by definition, there is $a \neq 0$ such that $Ha = 0$ with $a_i = 0$, $i>D$.
We also have the following corollary,
\[AGQ:quasiortho\_cor\] Assume that the $N \times (N+1)$ Hankel matrix $H$ associated with the measure $\alpha(\cdot)$ exists. If $H$ has rank $d<N$ then there exists a nontrivial polynomial $p(x)$ with degree $(D-1)$ where $D \leq d+1$ such that, $$\int p(x) x^j \, {{\mathrm d}}\alpha(x) = 0$$ for all $j = 0,$ …, $N$.
Let $K$ be such that $$D = \inf \{ 0\leq n \leq N : \mathrm{rank}( H(:, 1:n) )= n \}$$ where $H(:, 1:n)$ is the matrix containing the first $n$ columns of $H$. By theorem \[AGQ:low\_rank\], there exists a vector $a \not = 0$ such that $Ha = 0$ and $a_i = 0$ for $i>D$.\
Let $p(x)$ be the polynomial with coefficients given by $a$, i.e. $$p(x) = \sum_{n=0}^D a_n x^n$$ Then, $$\begin{aligned}
\int p(x) x^j \, {{\mathrm d}}\alpha(x) &= \int \sum_{n=0}^{D} a_n x^{n+j} \, {{\mathrm d}}\alpha(x)\\
&= \sum_{n=0}^D a_n \mu_{n+j} \\
& = (H a)_j = 0\end{aligned}$$ since $a$ belongs to the null-space of $H$.
The consequences of this corollary are far-reaching and constitute the crux of the scheme presented here. Indeed, although we do not generally expect the Hankel matrix $H$ associated with some measure $\alpha( \cdot)$ to be *exactly* low-rank as in the case of Theorem \[AGQ:low\_rank\] (e.g., $H$ has full rank in the case of classical Gaussian quadratures) we can expect that in some cases $H$ will be *approximately* low rank. In other words, given $0 < \epsilon \ll 1 $ we expect, $$D \approx \max \{ 1 \leq i \leq N : \sigma_i > \epsilon \, \sigma_1 \}$$ where $\{ \sigma_i \}$ are the singular values of $H$, to be much smaller than $N$, i.e., $D \ll N$. We show for instance in Figure \[svd\] the first $50$ singular values of the Hankel matrix ($N=250$) associated with the Lebesgue measure in $[-1,1]$. The $y$-axis scales as a logarithm in base $10$, and it is seen that the singular values decay faster than exponentially.
In light of the above discussion, we might expect in these circumstances the existence of a polynomial $p(x)$ of degree $D \approx \max \{ 1 \leq i \leq N : \sigma_i > \epsilon \sigma_1 \}$ such that $$\left | \int p(x) x^j \, {{\mathrm d}}\alpha(x) \right | \lesssim \epsilon$$ for all $0 \leq j \leq N$, and this leads us to the introduction of the concept of $\epsilon$-quasiorthogonal polynomial which we now define,
A polynomial $p(x)$ is called $\epsilon$-quasiorthogonal of order $N$ with respect to the measure $\alpha(\cdot)$ and the basis $\{ L_n (x) \}$ if, $$\left | \int p(x) \, L_n(x) \, {{\mathrm d}}\alpha(x) \right | \leq \epsilon$$ for all $n=0,$ …, $N$.
Importantly, this definition imposes no restriction per regards to the measure $\alpha(\cdot)$, in opposition with orthogonal polynomials which demand the measure to be positive ([@Szego]). In this sense, the relation described is not one of orthogonality for it is not possible to define a nondegenerate inner-product unless $\alpha( \cdot )$ is positive. This is why we chose the name *quasi*-orthogonal. We also note that given $\epsilon \geq \sigma_N(H)$ such polynomial always exists for it suffices to pick $a$ aligned with the right singular vector associated with the smallest singular value $\sigma_N(H)$.
From a computational standpoint, there exists an efficient scheme to find such polynomials given a measure $\alpha( \cdot)$ and some $\epsilon>0$. This is the subject of Section \[AGQ:comp\]. For the remaining of this section, we will focus on demonstrating how such polynomials can be used to obtain efficient quadratures. As will be shown, the construction of the scheme shares a lot with that of classical Gaussian quadrature. This is what constitutes the origin of the denomination.
We will need the following technical lemma which proof is provided in appendix,
\[AGQ:tech\_lemma\] Let $\alpha(\cdot)$ be an arbitrary measure on $(\mathbb{R}, \mathcal{B})$, and $$p(x) =x^{d+1} + \sum_{n=0}^{d} p_n x^n$$ be a monic $\epsilon$-quasiorthogonal polynomial of degree $d+1$ and order $N$ associated with $\alpha(\cdot)$. Further, let, $$f(x) = \sum_{n=0}^{N+d} f_n \, L_n(x)$$ be some polynomial of degree $N+d$ and $ \tilde{q}(x) = \sum_{n=0}^{d} \tilde{q}_n x^n$ be the Lagrange interpolant of $q(x)$ associated with the zeros of $p(x)$. Finally, let $r(x) = \sum_{n=0}^{N} r_n x^n$ be the unique polynomial such that $ q(x) - \tilde{q}(x)= p(x) r(x) $. Then, $$\begin{aligned}
\sum_{n=0}^N |r_n| \leq \lVert\Gamma^{-1} \bar{q} \rVert_{1} \end{aligned}$$ where $\bar{q} =\left( q_{d+1}, q_{d+2}, \ldots, q_{N+d}\right )^T $ and $\Gamma$ is the $N \times N$ Toeplitz matrix such that $[ \Gamma ]_{i,j} = p_{j-i}$ if $0\leq j-i \leq d$ and $0$ otherwise. \[AGQ:errorlemma\]
We are now ready to prove our main theorem.
[**(Approximate Gaussian quadrature)**]{} \[AGQ:AGQ\_thm\] Consider an arbitrary measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$. Let $p(x)$ be a monic $\epsilon$-quasiorthogonal polynomial of degree $d+1$ and order $N$ with respect to $\alpha( \cdot )$, where $0<\epsilon<1$. Then, the quadrature rule with nodes $\{ x_n \}_{n=0}^{d}$ consisting in the zeros of $p(x)$ and weights $\{ w_n \}_{n=0}^{d}$ given by, $$\label{AGQ:AGQ_thm:weight}
w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x)$$ where $\ell_n (x)$ is the $n^{th}$Lagrange basis polynomial associated with the nodes, integrates polynomials $q(x)$ of degree $\leq N+d$ with an error bounded by, $$\left | \int q(x) \, {{\mathrm d}}\alpha(x) - \sum_{n=0}^d w_n \, q(x_n) \right | \leq \lVert\Gamma^{-1} \bar{q} \rVert_{1} \, \epsilon$$ where $\{ q_n \}_{n=0}^{N+d}$ are the coefficients of $q(x)$, $\bar{q} =\left( q_{d+1} , q_{d+2} , \ldots, q_{N+d}\right )^T $ and $\Gamma$ is the $N \times N$ Toeplitz matrix such that $[ \Gamma ]_{i,j} = p_{j-i}$ if $0\leq j-i \leq d$ and $0$ otherwise.
Let $q(x)$ be a polynomial of degree $ N + d$ and consider the Lagrange interpolant at the nodes $\{ x_n \}$, $$\tilde{q} (x) = \sum_{n=0}^d q(x_n) \ell_n (x)$$ Then consider, $$\begin{aligned}
I = \int \left [ q(x) - \tilde{q}(x) \right ] \, {{\mathrm d}}\alpha(x)\end{aligned}$$ The quantity $[q(x) - \tilde{q}(x)]$ is a polynomial of degree at most $(N+d)$ and has zeros located at each of the nodes $\{ x_n \}_{n=0}^d$. Therefore, by the factorization theorem for polynomials we can write, $$q(x) - \tilde{q}(x) = \prod_{n=0}^d (x - x_n) \, r(x)$$ where $r(x)$ is a polynomial of degree at most $N$. We further note that $\prod_{n=0}^d (x - x_n)$ is a monic polynomial of degree $d+1$ with zeros at $\{ x_n \}_{n=0}^d$ just as $p(x)$. Since monic polynomials are uniquely characterized by their roots we have, $$\prod_{n=0}^d (x - x_n) = p (x)$$ Therefore, $$\begin{aligned}
|I| &= \left | \int p(x) r(x) \, {{\mathrm d}}\alpha(x) \right | \leq \sum_{n=0}^N | r_n | \,\left | \int p(x) \, x^n \, {{\mathrm d}}\alpha(x) \right | \leq \sum_{n=0}^N | r_n | \, \epsilon \\\end{aligned}$$ where we used the $\epsilon$-quasiorthogonality of $p(x)$. Finally, thanks to Lemma \[AGQ:tech\_lemma\] we get, $$\left | \int q(x) \, {{\mathrm d}}\alpha(x) - \sum_{n=0}^d w_n \, q(x_n) \right | \leq \lVert \Gamma^{-1} \bar{q} \rVert_{1} \epsilon$$
Interestingly, the above analysis reveals that an AGQ of order $d$ is in fact *exact* for polynomials of degree $\leq d$.
Some advantages of AGQ is that there is no need for the measure $\alpha( \cdot )$ to have any specific properties beyond the existence of moments of high-enough order. Furthermore, the problem of the existence and uniqueness of the solution to the Hankel system is of no importance; in fact, the larger the null-space of $H$ the better it is.
Both characteristics are in sharp contrast with common wisdom regarding classical Gaussian quadratures. First, the positivity of the measure is key in proving the existence of a sequence of orthogonal polynomials necessary to build a classical quadrature (see [@Meurant], Theorem 2.7). Secondly, the notion of orthogonality is at the heart of modern numerical schemes used to obtain nodes and weights for it gives rise to a three-term recurrence relation that is thoroughly exploited computationally (see [@Golub:1969; @Meurant]).
Computational considerations {#AGQ:comp}
----------------------------
The first computational issue we describe here is that of finding an adequate $\epsilon$-quasiorthogonal polynomials of order $N$ given a measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$, some $N \in \mathbb{N}$ and some value $ 0< \epsilon$. For this purpose, we note that a sufficient condition for a monic polynomial $p(x)$ of degree $(d+1)$ to fall within this category is to satisfy the following inequality, $$\lVert H(N,d) \, \bar{p} + h(d) \rVert_{\infty} \leq \epsilon$$ where $\bar{p} = [p_0, \, p_1 \, , \ldots, p_{d}]^T$, $p(x) = x^{d+1} + \sum_{n=0}^d p_n x^n$, $ h(d) = [\mu_{d+1}, \, \mu_{d+2} \, , \ldots, \mu_{d+N+1}]^T$ and $H(N,d)$ is the $(N+1)\times (d+1)$ Hankel matrix associated with the measure, i.e. $$H(N,d) =
\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{d} \\
\mu_1 & \mu_2 & \cdots & \mu_{d+1}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{N} & \mu_{N+1} & \cdots & \mu_{d+N}
\end{pmatrix}$$ The proof is analogous to that of Corollary \[AGQ:quasiortho\_cor\] and uses the definition of quasiorthogonal polynomials.
This inequality provides a constructive way for finding an $\epsilon$-quasiorthogonal polynomial of small degree. This is described in Algorithm \[poly\_alg\]; note that we replace the $\lVert \cdot \rVert_{\infty}$ norm by the more computationally-friendly $\lVert \cdot \rVert_2$ norm which is equivalent.
Let $(d+1) = \mathrm{rank}(H,\delta)$\
Solve $ \min_p \lVert H(N,d) \, p + h(d) \rVert_2 $\
[**Note:** ]{} The quadrature obtained from $p(x)$ integrates polynomials of degree $N+d$ with error prescribed by Theorem \[AGQ:AGQ\_thm\]. This error term involves the norm of the inverse of a matrix $\Gamma$ which is upper-triangular, Toeplitz with diagonal entries all equal to $1$ and remaining entries depending on the coefficients of the polynomial $p(x)$. In order to guarantee that an AGQ integrates polynomials of degree $\leq N+d$ with accuracy $\delta$ say, it is sufficient to set $\epsilon \leq \frac{\delta}{C}$ and constrain $p(x)$ to be such that $\lVert \Gamma^{-1} \rVert_{\infty} \leq C$ for some $C>0$. Upon obtaining some characterization of the set $\mathcal{S}_C := \{ p(x) : \lVert \Gamma^{-1} \rVert_{\infty} \leq C \}$, one could potentially carry out the steps described in Algorithm \[poly\_alg\] while restraining the solution to $\mathcal{S}_C$. One would thus guarantee the accuracy of the AGQ *a priori*. Unfortunately, such characterization is not readily available so one is left with the *a posteriori* estimates of Theorem \[AGQ:AGQ\_thm\]. On the other hand, numerical experiments point to the fact that the product $\lVert \Gamma^{-1} \rVert_{\infty} \, \epsilon$ does indeed decay in a fast manner as a function of the degree of $p(x)$, for $\bar{p}$ the solution of the least-squares problem having the smallest norm in Algorithm \[poly\_alg\]. In short, although AGQ in its current state performs well, some improvements are still possible. This constitutes a topic for future research.
Once such polynomial has been obtained, its roots constitute the nodes of the approximate Gaussian quadrature as per Theorem \[AGQ:AGQ\_thm\]. The cost of solving a thin $(N+1) \times (d+1)$ least-squares problem is ${\mathcal{O}}( [N+1) + (d+1)/3] (d+1)^2 ) $ (see [@Golub]). Since in general we expect $d \ll N$ the cost is *linear* in $N$. Also, each step of the while loop constitutes a rank-1 update of the system, so $p$ can be recomputed cheaply.
Another great computational aspect of the scheme is the availability of a simple analytical formula for the computation of the weights. Indeed, from Theorem \[AGQ:AGQ\_thm\] we have, $$w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x) = \int \sum_{k=0}^d [\ell_n]_k x^k \, {{\mathrm d}}\alpha(x) = \sum_{k=0}^d [\ell_n]_k \, \mu_k$$ where $[\ell_n]_k $ is the $k^{th}$ coefficient of the $n^{th}$ Lagrange basis polynomial $ \ell_n (x)$, which can be obtained cheaply from the zeros of $ \ell_n (x)$, i.e., the nodes of the quadrature. We also noticed that it is generally possible to neglect nodes associated with small weights when such are present. This further reduces the cost of the method.
As a final comment, the accuracy of the scheme is highly dependent on the accuracy of the nodes. For this reason, we recommend performing the computations in extended arithmetic. In this paper, we used $Maple^\copyright$ in order to compute the nodes and weights of each approximate quadrature with high precision.
Numerical simulations {#NS}
=====================
In this section, we demonstrate the efficiency and the versatility of the scheme through a few numerical examples. In section \[NS:classical\], we compare fixed-order approximate Gaussian quadratures (AGQ) with two types of classical Gaussian quadratures (Gauss-Legendre and Gauss-Chebyshev) on monomials $x^n$ of increasing degree and show how it quickly becomes advantageous to use an approximate quadrature in those cases. Then in Section \[NS:sing\], we give examples related to functions with an integrable singularity at the origin.
In section \[NS:trig\], we show how the scheme can be applied to monomials on the complex circle, i.e., functions of the form $e^{{\imath}n x}$ where $0 \leq n$. The resulting quadratures are then used in Section \[NS:Beylkin\] to obtain approximations of functions through short exponential sums which is related to the method of Beylkin & Monzón [@Beylkin:2005; @Beylkin:2010].
Comparison with classical quadratures {#NS:classical}
-------------------------------------
In this section, we compare results between the approximate Gaussian quadrature scheme, the Gauss-Legendre $\left ( {{\mathrm d}}\alpha(x) = {{\mathrm d}}x \right ) $ and Gauss-Chebyshev $\left ( {{\mathrm d}}\alpha(x) = \frac{1}{\sqrt{1-x^2} }{{\mathrm d}}x \right ) $ quadrature.
### Integration of monomials
For this benchmark, we fix the order ($N$ in Section \[AGQ:comp\]) and study the error in approximating integrals of the form, $$\int_{-1}^1 x^n \, {{\mathrm d}}\alpha (x)$$ through quadratures involving different number of nodes ($d$ in Section \[AGQ:comp\]) where $n$ varies between $0$ to $700$.
Numerical results are shown in Figure \[GLcompare\] and \[GCcompare\]. They were obtained using $N=350$. The results need to be interpreted carefully. The choice of $N$ represents in effect the polynomial order that would be required to approximate a given function $f(x)$ to some accuracy $\epsilon$. A numerical quadrature will then be able to approximate the integral of $f(x)$ if it can integrate all monomials of degree less than $N$ with accuracy $\epsilon$. In Fig. \[GLcompare\] for example, we see that the Gauss-Legendre quadrature is exact to machine precision up to $n=39$. However the error increases rapidly to reach $10^{-3}$ near $n=350$. In contrast, although AGQ is not exact for $n \le 39$, the error up to $n \le 350$ remains lower than $10^{-4}$ with only 20 nodes. As we increase the number of nodes (middle and bottom plots) the gain below $n= 350$ is even more significant.
The behavior of AGQ in the top plot around $n \approx 40$ where Gauss-Legendre seems to outperform AGQ is not significant. Indeed if a polynomial of order $n \approx 40$ is sufficient to approximate $f$, we would reduce $N$. This would result in an AGQ quadrature much more accurate in the range $n \in [0,40]$.
On Figure \[bound:Legendre\] and \[bound:Chebyshev\], we also compare the theoretical bound obtained in Theorem \[AGQ:AGQ\_thm\] with the actual absolute error obtained through a 30-node AGQ for both the Lebesgue and Chebyshev measures respectively. In both cases, it is seen that the bound provides a reasonable estimate for the behavior of the error.
Finally, an interesting thing to be noted is that in both cases the nodes associated with the approximate Gaussian quadratures were *real* and the weights were *real and positive*; it is a known fact that this should be the case for classical Gaussian quadratures. However, this is by no means obvious for the case of approximate Gaussian quadratures, and we currently have no theory demonstrating that it is always the case for real positive measures.
### General integrands
An important difference between AGQ and Gaussian quadratures is that AGQ takes $N$ as a parameter. $N$ represents in effect the order of a polynomial that can approximate $f(x)$ to the desired accuracy. This is function-dependent and therefore may need to be adjusted in AGQ depending on the integrand, if one wishes to have a near optimal quadrature.
Generally speaking, AGQ should be able to outperform a classical Gaussian quadrature in all cases since a Gaussian quadrature is a special case of AGQ, by basically choosing $d=N-1$ where $d$ is the degree of the polynomial. Indeed, this is what we observed in our numerical tests. Whenever the classical Gaussian quadrature or CGQ performs well, no gain is obtained with AGQ. We note that, in this case, the usual numerical techniques to evaluate Gaussian quadrature nodes should be more effective than the numerical procedure we are advocating for AGQ (due to ill-conditioning for too stringent a tolerance as mentioned in the introduction).
Conversely, when the convergence of CGQ is slow, AGQ provides a significant improvement. This corresponds to situation where expanding $f$ using polynomials requires terms of high degree and then the approximation of AGQ for high order monomials makes a difference. This is illustrated in the examples below.
We used the following integrands to investigate the accuracy of AGQ: $$\begin{aligned}
\log \left ( 1- \frac{x}{1.05} \right ) &= - \sum_{n=0}^{\infty} \frac{1}{n (1.05)^n} \,x^n ,\;\;\; |x| \leq 1 \\
\frac{1}{ 1- \frac{x}{1.05} } &= - \sum_{n=0}^{\infty} \frac{1}{(1.05)^n} \,x^n ,\;\;\; |x| \leq 1 \\
e^{-10\,x} &= - \sum_{n=0}^{\infty} \frac{(-10)^n}{n!} \,x^n ,\;\;\; 0\leq x \leq 1\end{aligned}$$ The first two integrand have slowly-decaying coefficients and can be approximated in the interval $[-1,1]$ through a sum containing ${\mathcal{O}}( \log_{1.05}(1/\epsilon) ) $ terms for an accuracy of $\epsilon$. At $\epsilon$-machine ($\epsilon = 10^{-15}$) this implies approximately $700$ terms. The third integrand has very fast decay, and in this case only $50$ terms are sufficient.
For each case, we varied the number of nodes in the quadrature. Then for AGQ, we selected the integer $N$ that gave us the most accurate result. In practice, an algorithm would be required to estimate $N$ numerically but we will not address this question here. Results are show in Table \[general:log\]–\[general:exp\].
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- -----------------------
10 75 $2.16 \cdot 10^{-8}$ $1.39 \cdot 10^{-4}$
15 100 $1.08 \cdot 10^{-8}$ $3.94 \cdot 10^{-6}$
20 150 $2.05 \cdot 10^{-11}$ $1.26 \cdot 10^{-7}$
25 200 $3.99 \cdot 10^{-14}$ $4.31 \cdot 10^{-9}$
30 250 $1.61 \cdot 10^{-15}$ $1.54 \cdot 10^{-10}$
: Absolute error incurred by an AGQ and a Gauss-Legendre quadrature for the integration of $f(x) = \log \left ( 1- \frac{x}{1.05} \right ) $ over the interval $[-1,1]$ for various number of nodes.[]{data-label="general:log"}
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- ----------------------
10 75 $5.81 \cdot 10^{-5}$ $8.15 \cdot 10^{-3}$
15 100 $2.20 \cdot 10^{-6}$ $3.60 \cdot 10^{-4}$
20 150 $4.26 \cdot 10^{-9}$ $1.56 \cdot 10^{-5}$
25 200 $1.58 \cdot 10^{-11}$ $6.76 \cdot 10^{-7}$
30 250 $4.01 \cdot 10^{-13}$ $2.92 \cdot 10^{-8}$
35 300 $1.77 \cdot 10^{-15}$ $1.25 \cdot 10^{-9}$
: Absolute error incurred by an AGQ and a Gauss-Legendre quadrature for the integration of $f(x) = \frac{1}{ 1- \frac{x}{1.05} }$ over the interval $[-1,1]$ for various number of nodes.[]{data-label="general:geo"}
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- -----------------------
5 15 $1.09 \cdot 10^{-6}$ $8.82 \cdot 10^{-5}$
7 7 $1.29 \cdot 10^{-7}$ $1.29 \cdot 10^{-7}$
10 10 $1.02 \cdot 10^{-12}$ $1.02 \cdot 10^{-12}$
12 12 $4.44 \cdot 10^{-16}$ $4.44 \cdot 10^{-16}$
: Absolute error incurred for $e^{-10\,x} $ over the interval $[0,1]$. In that case, Gauss-Legendre converges very fast and AGQ simply provides a quadrature with the same accuracy. The two methods become essentially identical.[]{data-label="general:exp"}
We observe the superior accuracy of AGQ. The first two cases are challenging for CGQ and AGQ does significantly better. For the last case, CGQ converges extremely fast and then AGQ simply finds that the optimal choice is CGQ and provides an estimate with the same accuracy.
In summary, $N$ shoud be adjusted depending on the type of integrand. If the integrand is such that expansions in a polynomial basis possess slowy-decaying coefficients, AGQ will provide significantly greater accuracy. If on the contrary, a polynomial expansion converges very rapidly, both AGQ and CGQ will provide essentially identical (and fast) convergence.
We also stress that AGQ can be constructed for a wide range of measures whereas CGQ is restricted to positive measures (weight function) only.
Singular functions {#NS:sing}
------------------
We show how AGQ can be used to integrate functions with integrable singularities. For this purpose, we consider integrand of the form $x^n \log(x)$ for $x \in (0,1]$ and $0 \leq n \leq 700$. In this case, the integral of interest takes the form, $$\int_{0}^1 x^n \, \log(x) \, {{\mathrm d}}x$$ This quantity can either be seen as the integration of $x^n \, \log(x)$ with respect to Lebesgue measure or as the integration of the monomial $x^n$ with respect to the measure ${{\mathrm d}}\alpha(x) = \log(x) \, {{\mathrm d}}x$. Considering the latter, we build an AGQ of order $N=350$ with different number of nodes and display the absolute error as a function of the degree $n$ and the number of quadrature points. This is shown in Figure \[NS:log\]. Note that the bound is not plotted beyond $N=350$ for it is no more valid past this point.
We note that in this case we cannot perform a comparison with a classical Gaussian quadrature for no such quadrature exists as is the case with most measures but a few.
Quadrature for polynomials on the complex circle {#NS:trig}
------------------------------------------------
In this section, we are interested in integrands that take the form of trigonometric monomials, i.e., functions of the form, $$f(x) = e^{{\imath}n x}$$ where $0 \leq n$. As their name conveys, such functions are just homogeneous polynomials $z^n$ in the complex plane which have been restricted to the boundary of the unit circle, i.e., $z = e^{ix}$. Thanks to this close relationship with polynomials on the real axis, one can also develop approximate Gaussian quadratures for such functions as well. In fact it suffices to replace the moments $\mu_n$ by the trigonometric moments, $$\tau_n = \int (e^{{\imath}x})^n \, {{\mathrm d}}\alpha(x)
= \int z^n \, {{\mathrm d}}\alpha(z)$$ in all that has been presented above and similar results follow.
As an example, we built an AGQ of order $N=350$ for trigonometric polynomials with respect to the Lebesgue measure over the interval $[-1,1]$. The absolute error between our approximation and the exact value of the integral, $$\int_{-1}^1 e^{{\imath}n x} \, {{\mathrm d}}x = \frac{e^{{\imath}n} - e^{-{\imath}n}}{{\imath}n}$$ are presented in Figure \[NS:trigplot\]. There, it is seen that as little as $30$ quadrature points are necessary to integrate a complex exponential with frequency $n=500$ with $\approx 10^{-6}$ accuracy. We also plotted the theoretical bound of Theorem \[AGQ:AGQ\_thm\]. Again, it appears to be a good estimate.
It is interesting to look at the location of the nodes for such quadratures. An example is displayed in Figure \[NS:trignodes\]. The nodes are shown in the complex plane and appear to lie along a curve which rapidly moves upward from $-1$, slowly moves across, and rapidly moves back to $1$ on the real axis. This does not appear to be a coincidence given the fact that functions of the form $e^{{\imath}n x}$ decay exponentially and do not oscillate along the positive imaginary axis. Thus, the underlying curve could be some sort of path of *least oscillation* in an average sense over $0 \leq n \leq N$. At this point, this is a mere qualitative observation, but might be worth investigating in the future.
![Location in the complex plane of the nodes of a 20-node AGQ for trigonometric polynomials. The nodes appear to lie on a smooth curve with positive imaginary part.[]{data-label="NS:trignodes"}](trignodes.png){width="10cm"}
Approximation of functions through short exponential sums {#NS:Beylkin}
---------------------------------------------------------
In this section, we are interested in the approximation of functions by a short sum of exponentials. That is, given a function $f(x)$ defined over an interval $[a,b]$, we seek some approximation in the form, $$f(x) \approx \sum_{m=0}^d \alpha_m e^{ \beta_m x}$$ for $x \in [a,b]$, and where $d$ should be as small as possible. Such expansions can be viewed as more efficient representations of functions compared to Fourier transforms as they typically require fewer terms. They can form the starting point for various fast algorithms such as the fast multipole method, hierarchical matrices ($\mathcal H$-matrices), etc. Such techniques are particularly desirable when it comes to the solution of integral equations with translation-invariant kernels (see e.g., [@Letourneau:2012; @Beylkin:2005]). Very powerful techniques based on dynamical systems and recursion ideas were recently introduced by Beylkin & Monzón [@Beylkin:2005; @Beylkin:2010] in order to approach this problem. As was mentioned earlier, the latter inspired the current work.
We will show how AGQ can be used to derive similar approximations through the discretization of the Fourier transform. The final formulation shares some characteristics with the problem of Beylkin & Monzón that can be stated as follows: given the accuracy $\epsilon > 0$, for a smooth function $f(x)$ find the minimal number of complex weights $w_n$ and nodes $e^{t_m}$ such that, $$\left | f(x) - \sum_m w_m e^{t_m x} \right | < \epsilon$$ for $x \in I$, $I$ being some interval in $\mathbb{R}$.
Their scheme is based on an important result regarding Hankel matrices. Consider a Hankel matrix $H$ associated with a sequence $h_k$ where $h_k = f(x_k)$ are uniform samples of $f$. Assume that the null space of $H$ is non-trivial and consider the polynomial whose coefficients are given by a vector in the null space of $H$. The zeros of this polynomial, $\lambda_i$, satisfy the following property (see e.g., [@Boley:1998]), $$h_k = \sum_{i=1}^r \lambda_i^k \,d_i$$ for some $\{ d_i \}$, where $r$ is at most the number of columns of $H$. With our choice for $h_k$, one obtains, $$f(x_k) = \sum_{i=1}^r d_i \, e^{ \log( \lambda_i ) k }$$ which naturally extends to an interpolation formula for $f(\cdot)$.
In [@Beylkin:2005; @Beylkin:2010], the authors search for an approximate formula since in general the matrix $H$ is full rank and therefore no efficient representation, that would yield exactly $f(x_k)$, is possible. To achieve this, Beylkin et al. [@Beylkin:2005; @Beylkin:2010] show how $\lambda_i$ can be obtained as the roots of a polynomial whose coefficients are given as the entries of a con-eigenvector $u$, i.e., a vector such that, $$H u = \sigma \overline{u}$$ $\sigma$ being real and nonnegative. The error is then on the order of $\sigma$. They also show that the weights satisfy a well-conditioned Vandermonde system.
As will be seen, both our method and theirs involve a Hankel matrix with entries given by the uniform samples of the function to be approximated over the interval considered. However, the current approach avoids the solution of a con-eigenvalue problem altogether and allows for the direct computation of the weights rather than their computation through the solution of a Vandermonde system. Furthermore, since the quasi-orthogonal polynomial obtained through our scheme has small degree, the number of zeros that must be computed is also much smaller. This results in significant computational savings compared to the former method.
The resulting error estimates for both methods are different. Indeed, in the case of [@Beylkin:2005] one expects the error to be bounded *uniformly* by an expression on the order of the modulus of the small con-eigenvalue $\sigma$ (Theorem 2, [@Beylkin:2005]), and such value can be determined *a priori*. In our case however, the error in *not* uniform (as can be seen from the numerical examples). Furthermore, our current error estimate is *a posteriori*.
To begin with, consider a function $f(x) \in \mathcal{L}^2 ( \mathbb{R})$ uniformly sampled at $x_n = a + \frac{n (b-a)}{N}$, $n = 0... (N-1)$ for some $N \in \mathbb{N}$ and $a,b \in \mathbb{R}$, and use the Fourier transform to write, $$f(x_n) = \int_{-\infty}^{\infty} e^{2 \pi {\imath}x_n \xi} \hat{f} (\xi) \, {{\mathrm d}}\lambda( \xi)=\frac{N}{(b-a)}\, \int_{-\infty}^{\infty} e^{2 \pi {\imath}n \zeta} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right ) \, {{\mathrm d}}\lambda( \zeta)$$ where $\hat{f} (\xi)$ denotes the Fourier transform of $f(x)$, and $\lambda( \cdot )$ is the Lebesgue measure. We note that $$\frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right )$$ can be seen as a Radon-Nykodym derivative of a certain measure $\alpha( \cdot)$ absolutely continuous with respect to Lebesgue measure (see [@Cohn]), i.e., $$\frac{{{\mathrm d}}\alpha }{ {{\mathrm d}}\lambda} (\zeta) = \frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right )$$ With this measure we have, $$f(x_n) = \int_{-\infty}^{\infty} e^{2 \pi {\imath}n \zeta} \, {{\mathrm d}}\alpha(\zeta) , \;\; n = 0... N$$ which is perfectly well-suited for discretization through an approximate Gaussian quadrature as described in the previous section. To find such quadrature, we first need the trigonometric moments of the measure. These moments turn out to have a very simple form. Indeed, a quick look at their definition shows that, $$\tau_n = \int_{\mathbb{T}} e^{{\imath}n \zeta} \, {{\mathrm d}}\alpha(\zeta) = \int_{\mathbb{T}} e^{{\imath}n \zeta} \, \left [ \frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right ) \right ] {{\mathrm d}}\lambda(\zeta) = f \left( a + n \frac{(b-a)}{N} \right )$$ At this point, we note that the Hankel matrix arising from such moments is exactly the same as the one described in [@Beylkin:2005] as previously mentioned.
Finally, the nodes $\{ w_n \}$ can be obtained through Eq.. In the end, we obtain $$f(x_n) \approx \sum_{m=0}^d w_m \, e^{ {\imath}n \zeta_m } , \;\;\; n=1,...,N$$ with error bounded by the expression provided in Theorem \[AGQ:AGQ\_thm\]. To obtain an approximation to $f(x)$ in all of $[a,b]$, we simply allow $\frac{n}{N}$ to vary continuously so that $$\frac{n}{N} = \frac{x-a}{b-a}$$ for $x \in [a,b]$ and write, $$\begin{aligned}
f(x) &\approx \sum_{m=0}^d \alpha_m \, e^{ {\imath}\beta_m x } \\
\alpha_m &= w_m \, e^{-{\imath}\frac{a}{b-a} N \xi_m } \\
\beta_m &= \frac{1}{b-a} N \xi_m\end{aligned}$$
When $x$ corresponds to a sample, i.e., $x = x_n$ for some $n$, this reduces to the previous expression. However, when $x$ lies between two samples this last formula should be seen as an interpolation. We do not currently have the complete theory describing the interpolation error. However, it was observed numerically that such error is generally of the same order as that associated with the closest sample whenever the function $f(x)$ is sufficiently oversampled. Numerical examples are provided below.
At this point, we describe an algorithm for the construction of such an approximation. The description can be found in pseudo-code in Algorithm \[approx\_alg\].
Pick $N \in \mathbb{N}$ sufficiently large (beyond the Nyquist rate)\
Compute $ \tau_n = f \left( a + n \frac{(b-a)}{N} \right )$\
Build the Hankel matrix $H_{i,j} = \tau_{i+j}$ for $i,j = 0 .. N$\
Proceed as described in Algorithm \[poly\_alg\] to find $p(x)$\
Compute $\{ x_n \}$, the nodes/zeros of $p(x)$\
Compute weights $w_n$ following Eq.\
Build approximation: $ \sum_{n} w_n \, e^{ {\imath}\frac{(x-a)}{(b-a)} N \, \frac{\log(x_n)}{{\imath}} }$
We now provide a few examples for the representation of some oscillatory functions: the Bessel functions of the first kind $J_{\nu} (100 \pi \, x )$ over the interval $[0,1]$ and for orders $ \nu \in \{ 0, 25 \}$. Such functions are relevant in problems involving the scattering of waves in two dimensions for instance. In both cases, the order of the AGQ is $N=400$ (note that the spectrum of both functions is bounded by about $400 \approx 100 \pi$) and a $40$-terms approximation is obtained using the scheme just introduced. The results are presented in Figure \[bessel0\] and \[bessel25\] respectively. Agreement within $10^{-10}$ and $10^{-7}$ absolute error is observed in each cases respectively. It should also be noted that the number of terms lies much below what should be expected with a standard Fourier series given the nature of the oscillations.
As a final example, we chose to represent the Dirichlet kernel, $$D_N (x) = \sum_{k=-N}^N e^{{\imath}k x} = \frac{\sin \left ( \pi( N + 1/2) \, x \right ) }{\sin \left ( \pi/2 \, x \right )}$$ over the interval $[-1,1]$. When applied through convolution, the Dirichlet kernel acts as a low-frequency filter. In this sense, a short exponential sum approximation can be used to speed up the filtering process.
We picked $N = 200$. To obtain the approximation, we proceeded as described in [@Beylkin:2005] and went on to first approximate, $$G_{200} (x) = \sum_{k \geq 0 } \frac{\sin(200 \pi (x+k))}{200 \pi (x+k)}$$ through a 40-term exponential sum and then built the Dirichlet kernel through the identity, $$D_{200} (x) = G_{200} (x) + G_{200} (1-x)$$ resulting in a 80-term approximation. It is shown in Figure \[dirichlet\]. The error is non-uniform as expected from Theorem \[AGQ:AGQ\_thm\] but still remains below $10^{-7}$ for all values in the interval.
Conclusion
==========
We have introduced a new type of quadrature closely related to Gaussian quadratures but which use the concept of $\epsilon$-quasiorthogonality to reduce the number of quadrature nodes and weights. Such quadratures have desirable computational properties and can be applied to a family much broader than that targeted by classical Gaussian quadratures. We have provided the theory for the existence of such quadratures and have provided error estimates together with practical ways of constructing them. We have also carried out various numerical examples displaying the versatility and performance of the method. Finally, we have described how AGQ can be used to approximate functions through short exponential sums and provided further numerical examples in these cases.
Acknowledgements
================
The authors would like to thank Professor Ying Wu from King Abdullah University of Science and Technology (KAUST) for supporting this research through her grant as well as the National Sciences and Engineering Research Council of Canada (NSERC) for their financial support.
[**Proof of Lemma \[AGQ:errorlemma\].**]{} First, thanks to the factorization theorem for polynomials (see e.g., [@Hungerford]) $$\label{error:1}
q(x) - \tilde{q} (x) = \sum_{i=0}^{d} (q_i - \tilde{q}_i ) x^i + \sum_{i=d+1}^{n+d} q_i x^i = \left( \sum_{i=0}^d p_i x^i \right ) \left ( \sum_{i=0}^n r_i x^i \right ) = p(x) r(x)$$ and from the Cauchy product, we have, $$\label{error:2}
\left( \sum_{i=0}^d p_i x^i \right ) \left ( \sum_{i=0}^N r_i x^i \right ) = \sum_{i=0}^{N+d} \left( \sum_{k=0}^i r_k p_{i-k} \right ) x^i$$ where it is understood that coefficients corresponding to indices outside the original range of definition of $p(x)$ and $r(x)$ are $0$. By matching coefficients of like powers in Eq. and and putting the linear system thus obtained in matrix form, one gets $$\Gamma r = \Gamma
\begin{pmatrix}
r_0 \\
r_1 \\
r_2 \\
\vdots \\
r_N
\end{pmatrix}
=
\begin{pmatrix}
q_0 - \tilde{q}_0 \\
\vdots \\
q_d - \tilde{q}_d \\
q_{d+1}\\
\vdots \\
q_{N+d}
\end{pmatrix}
= \kappa$$ where, $$\Gamma =
\begin{pmatrix}
p_0 & 0 & 0 & \cdots & 0 &0 \\
p_1 &p_0 & 0 & \cdots & 0 & 0\\
p_2 & p_1 & p_0 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & p_{d}\\
0& 0 & 0 & \cdots & 0 & 1
\end{pmatrix}$$ $\Gamma$ is an $(N+d+1) \times (N+1)$ Toeplitz matrix characterized by the coefficients of the known quasi-orthogonal polynomial $p(x)$. We know form the existence and uniqueness theorem for the factorization of polynomials that there exists a unique solution to the above system. We further write, (assuming $N>d$) $$\Gamma =
\begin{pmatrix}
\Gamma_1 \\
\Gamma_2
\end{pmatrix}$$ where $\Gamma_1$ is a $d \times N$ matrix containing the first $d$ rows of $\Gamma$ and $\Gamma_2$ is a $N \times N$ matrix containing the last $N$ rows of $\Gamma$. It is to be noted that $\Gamma_2$ is an upper triangular matrix with diagonal entries all equal to $1$. Therefore, all eigenvalues of $\Gamma_2$ are equal to $1$. In particular, $\Gamma_2$ is invertible and we can write, $$\begin{pmatrix}
r_0 \\
r_1 \\
r_2 \\
\vdots \\
r_N
\end{pmatrix}
= \Gamma_2^{-1}
\begin{pmatrix}
q_{d+1}\\
\vdots \\
q_{N+d}
\end{pmatrix}$$ where $\Gamma_2^{-1} $ is also an upper triangular Toeplitz matrix with diagonal entries all equal to $1$. Therefore, $$\sum_{n=0}^N |r_n| = \lVert r \rVert_1 = \lVert \Gamma_2^{-1} \bar{q} \rVert_1$$
| {
"pile_set_name": "ArXiv"
} |
---
address:
- 'Osaka University, Osaka, Japan'
- 'The University of Tokyo, Tokyo, Japan'
- 'Nara Institute of Science and Technology, Nara, Japan'
- RIKEN Center for Advanced Intelligence Project
author:
- Koki Kishimoto
- Katsuhiko Hayashi
- Genki Akai
- Masashi Shimbo
bibliography:
- 'ref.bib'
title: Binarized Canonical Polyadic Decomposition for Knowledge Graph Completion
---
Knowledge graph completion ,Tensor factorization ,Model compression
Introduction {#intro}
============
Related Work {#sec:related}
============
Notation and Preliminaries {#sec:notation}
==========================
Tensor Decomposition for Knowledge Graphs {#sec:tensor}
=========================================
Proposed Method {#sec:proposed}
===============
Experiments {#sec:exp}
===========
Conclusion {#sec:conclusion}
==========
Proof of Theorem \[thm:expressiveness\] {#sec:proof}
=======================================
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
One of the most challenging and long-standing problems in computational biology is the prediction of three-dimensional protein structure from amino acid sequence. A promising approach to infer spatial proximity between residues is the study of evolutionary covariance from multiple sequence alignments, especially in light of recent algorithmic improvements and the fast growing size of sequence databases. In this paper, we present a simple, fast and accurate algorithm for the prediction of residue-residue contacts based on regularized least squares. The basic assumption is that spatially proximal residues in a protein coevolve to maintain the physicochemical complementarity of the amino acids involved in the contact. Our regularized inversion of the sample covariance matrix allows the computation of partial correlations between pairs of residues, thereby removing the effect of spurious transitive correlations. The method also accounts for low number of observations by means of a regularization parameter that depends on the effective number of sequences in the alignment. When tested on a set of protein families from Pfam, we found the RLS algorithm to have performance comparable to state-of-the-art methods for contact prediction, while at the same time being faster and conceptually simpler.
The source code and data sets are available at <http://cms.dm.uba.ar/Members/slaplagn/software>
author:
- Massimo Andreatta
- Santiago Laplagne
- Shuai Cheng Li
- Stephen Smale
bibliography:
- 'mybib\_plain.bib'
date: 'March 26, 2014'
title: 'Prediction of residue-residue contacts from protein families using similarity kernels and least squares regularization'
---
Introduction {#section:introduction}
============
A major problem in computational biology is the prediction of the 3D structure of a protein from its amino acid sequence. Anfinsen’s dogma suggests that, in principle, the amino acid sequence contains enough information to determine the full three-dimensional structure [@anfinsen1973principles]. However, a few decades on, the mechanisms of protein folding are still not satisfactorily explained [@dill2012protein]. In particular, the space of possible spatial configurations given a certain amino acid 1D sequence is immense (the “Levinthal paradox”), yet an unfolded polypeptide chain is driven to its native 3D structure in a finite time, typically milliseconds to seconds, upon shifting to folding conditions [@rose2006backbone].
Such enormous search space poses important challenges to the development of *ab initio* methods for structure prediction. Therefore, it is essential to exploit different kinds of information that can help reduce the degrees of freedom in the configurational search space. A powerful way of inferring distance constraints is the prediction of residue-residue contacts from multiple sequence alignments (MSA). The underlying assumption is that contacting residues coevolve to maintain the physicochemical complementarity of the amino acids involved in the contact. That is, if a mutation occurs in one of the contacting residues, the other one is also likely to mutate, lest the fold of the protein may be disrupted. Methods based on residue coevolution aim at inferring spatial proximity between residues (contacts) from such signals of correlated mutations (Figure \[fig:contacts\]).
![Illustration of a residue-residue contact. The contact imposes a constraint on the evolution of residues $i$ and $j$. Vice versa, coevolution of $i$ and $j$ can be used to infer their physical proximity.[]{data-label="fig:contacts"}](Contact_diagram-eps-converted-to.pdf){width="45.00000%"}
Thanks to the recent exponential growth in sequence data collected in databases such as Pfam [@PFAM], algorithms for the prediction of contacting residues from MSA have enjoyed increasing attention. Different kinds of approaches have been recently applied for contact prediction, from mutual information (MI) between pairs of positions [@buslje2009correction; @dunn2008mutual; @wang2013predicting], to Bayesian network models [@burger2010disentangling], direct-coupling analysis [@balakrishnan2011learning; @morcos2011direct; @marks2011protein] and sparse inverse covariance matrix estimation [@jones2012psicov]. See also [@marks2012protein] and [@de2013emerging] for recent reviews. In particular, the more sophisticated and successful methods attempt to disentangle direct and indirect correlations, that is the artifactual correlations emerging from transitive effects of covariance analysis [@lapedes1999correlated; @weigt2009identification]. Morcos et al. [@morcos2011direct] and Marks et al. [@marks2011protein] tackle this problem using a maximum-entropy approach, whereas Jones et al. [@jones2012psicov] estimate partial correlations by inverting the covariance matrix. A very recent pseudo-likelihood method based on 21-state Potts models [@ekeberg2013improved] was shown to outperform other approaches for direct-coupling analysis. Kamisetty et al. [@kamisetty2013assessing] systematically analyzed the conditions under which predicted contacts are likely to be useful for structure prediction, and found several hundred families that meet their criteria.
Here, we propose a new approach for computing direct correlations that employs regularized least squares (RLS) regression to invert a sample covariance matrix $S$. We compute the regularized inverse by the formula $$\label{eq:theta}
\Theta = (S^2 + \eta \operatorname{Id})^{-1} S,$$ with fixed $\eta > 0$. It proves to be a very simple, direct and fast approach, and requires no assumption on probabilities distributions or sparsity in the correlations.
The RLS algorithm described in this paper was applied to three different sets of protein families, and we compared its performance to state-of-the-art methods for contact prediction. The RLS method achieves precision rates superior to PSICOV [@jones2012psicov] and comparable to plmDCA [@ekeberg2013improved] but it is considerably faster than either.
Approach
========
The covariance matrix {#section:covariance}
---------------------
Let ${\mathscr{A}}$ be the set of $20$ amino acids plus the gap symbol $-$ and $\PP = \{p^m = (p_1^m, \dots, p_L^m)\}_{m=1,\dots,M}$ a given Pfam family of $M$ aligned protein sequences, possibly with gaps, where $L$ denotes the length of the protein domains. On this set of proteins, the covariance between any pair of columns $(i,j)$ for the amino acids pair $(a,b)$ is given by $$S^0_{ij}(a,b) = f_{ij}(a,b) - f_i(a) f_j(b)$$ where the corrected frequencies are calculated as $$\label{eq:pseudo}
f_i (a) = \frac{1}{\lambda + M_{\text{eff}}} \Big( \frac{\lambda}{21} + \sum_p w(p) \delta(a, p_i) \Big)$$ $$f_{ij} (a,b) = \frac{1}{\lambda + M_{\text{eff}}} \Big( \frac{\lambda}{21^2} + \sum_p w(p) \delta(a, p_i) \delta(b, p_j) \Big)$$ The delta kernel takes value $\delta(a,b)=1$ if $a = b$ and $\delta(a,b)=0$ otherwise. $w(p)$ is the weight of protein $p$ and ${M_{\text{eff}}}= \sum_p w(p)$ (see section \[section:measure\] for details on sequence weighting).
The parameter $\lambda$ is the so-called pseudocount, a regularization parameter that accounts for non-observed pairs. We note that the same, or similar, constructions for the corrected amino acid frequencies have been proposed previously by other authors [@ekeberg2013improved; @jones2012psicov; @morcos2011direct].
### Modified covariance matrix
We set $S^0_{ii}(a,b) = 0$ for $a \neq b$, and call $S$ this new matrix. This modification also appears in the code of PSICOV [@jones2012psicov] although it is not stated in their paper. By setting those values to $0$, the resulting matrix contains in general negative eigenvalues (see Figures and ) and hence is not anymore semi-definite positive, but it is still symmetric. We do not fully understand this step, but it is noteworthy that Equation \[eq:theta\] still makes sense for any $\eta > 0$.
In general, working with $S$ instead of $S^0$ gives better results in our experiments. See Table S1 for the effect of this step on predictive performance.
Regularized inverse – the key algorithm
---------------------------------------
As we mentioned in the Introduction, the covariance between our random variables does not distinguish between direct and indirect correlations. To overcome this problem, a technique used by statisticians is to compute the so-called partial correlations, which can be obtained from the inverse of the covariance matrix using its associated correlation matrix.
Since the covariance matrix is usually singular or ill conditioned, regularization techniques must be used to compute a regularized inverse $\Theta$. We achieve this by solving the following optimization problem $$\label{eq:optimization}
\Theta = \operatorname*{argmin}_{X \in {\mathbb{R}}^{20L \times 20L}} \| SX - \operatorname{Id}\|_2^2 + \eta \|X\|_2^2,$$ where $\| \cdot \|_2$ denotes the Frobenius norm, and $\eta$ is a regularization parameter to be determined. Observe that the first term is minimized by the inverse of $S$ when it exists.
The problem has a unique solution for any $\eta > 0$ as we see in the next proposition.
\[prop:regularization\] For a symmetric matrix $S \in {\mathbb{R}}^{n \times n}$ and a regularization parameter $\eta > 0$, the optimization problem has a unique solution, which is also symmetric and given by equation \[eq:theta\]. When $S$ is semidefinite positive, then the solution also is.
Since the norms involved are coordinate norms, the problem can be decoupled into independent problems for each column of $X$: $$\Theta^{(i)} = \operatorname*{argmin}_{x \in {\mathbb{R}}^{n \times 1}} \| S^tx - e^{(i)} \|_2^2 + \eta \|x\|_2^2,$$ where $\Theta^{(i)}$ is the $i$-th column of $\Theta$ and $e^{(i)}$ is the $i$-th column of the identity matrix.
This is a well studied problem known as regularized least squares (also called Tikhonov regularization or Ridge regression in different areas, see [@tikhonov1943stability] and [@hoerl1962application]). The unique solution is $\Theta^{(i)} = (S^tS + \eta \operatorname{Id})^{-1} S^t e^{(i)}$.
Hence, the solution to our matrix problem is $\Theta = (S^tS + \eta \operatorname{Id})^{-1} S^t$. Since we are assuming $S$ symmetric, we get $$\Theta = (S^2 + \eta \operatorname{Id})^{-1} S.$$
The matrix $S$ is diagonalizable with all of its eigenvalues real. The eigenvalues of $S$ are transformed by the same formula defining $\Theta$. If $\lambda_k$, $1 \le k \le 20L$, are the eigenvalues of $S$ then the eigenvalues of $\Theta$ will be $$\gamma_k = f(\lambda_k) = \frac{\lambda_k}{\lambda_k^2 + \eta}$$ This function is well defined for all $\lambda \in {\mathbb{R}}$ when $\eta$ is positive, which proves that the matrix $S^2 + \eta \operatorname{Id}$ is invertible. The resulting matrix $\Theta$ is symmetric by standard matrix theory. Finally, $f$ preserves the sign of the eigenvalue and hence $\Theta$ will be a semidefinite positive matrix whenever $S$ is.
Note that $\Theta$ can be computed by solving the linear system $(S^2 + \eta \operatorname{Id}) \Theta = S$, which is faster and more accurate than inverting the matrix $S^2 + \eta \operatorname{Id}$.
For a better understanding of our regularization formula, we study the function $f$ in more detail. The derivative of $f$ is $f'(\lambda) = \frac{-\lambda^2 + \eta}{(\lambda^2+\eta)^2}$. Hence $f$ is increasing for $|\lambda| < \sqrt{\eta}$ and decreasing for $|\lambda| > \sqrt{\eta}$, with maximum value at $\lambda = \sqrt{\eta}$ and minimum value at $\lambda = -\sqrt{\eta}$. We show in Figure the plot of this function for $\eta = \eta'/{M_{\text{eff}}}= 1000/3912$ (see Section \[subsec:regularization\] for the choice of $\eta$).
As mentioned in the proof of Proposition \[prop:regularization\], the function is smooth at $0$, so using this regularization formula we deal in a simple way with the conditioning problem of inverting the covariance matrix.
Aggregation
-----------
The matrix $\Theta$ obtained is a $20L \times 20L$ matrix. Its entries are estimates of the partial correlation between pairs of columns $(i,j)$ for [*all*]{} pairs of amino acids $(a,b)$. Since our goal is to detect relations between pairs of columns in the alignment, we compute a coupling score aggregating the values of $\Theta$ using the $l_1$-norm on the $20 \times 20$ sub-matrices, as in [@jones2012psicov]. That is, $$P(i, j) = \sum_{1 \le a, b \le 20} | \Theta_{ij}(a, b) |.$$
The $l_2$-norm for aggregation showed poorer performance than the $l_1$-norm above. Finally, following[@dunn2008mutual] and [@jones2012psicov] we define a corrected score $P_{\text{APC}}(i, j) = P(i, j) - \frac{P(\cdot, j)P(i, \cdot)}{P(\cdot, \cdot)}$, where $\cdot$ stands for the average over all positions.
The prediction of contacts between pairs of residues can now be obtained by ranking the $P_{\text{APC}}(i, j)$, where higher scores identify more likely residue-residue contacts.
Method details
==============
In this section we give more details on the actual implementation of the algorithm described above.
Sequence weighting {#section:measure}
------------------
Families from the Pfam database contain some degree of redundancy. A common strategy to overcome this problem is sequence weighting, which weighs down groups of similar sequences and assigns higher weights to isolated sequences.
We first define a similarity measure between proteins, following [@smale2013introduction]. We start from the BLOSUM90 frequency substitution matrix $B_{90}(a,b)$ defined in [@henikoff1992amino] and call $\hat{B}_{90}(a,b)$ for a pair of amino acids $(a,b)$ the normalized matrix $$\hat{B}_{90}(a,b) = \frac{B_{90}(a,b)}{\sqrt{B_{90}(a,a) B_{90}(b,b) }}$$
We then proceed to construct a similarity kernel between pairs of proteins $$K^3(p, q) = \sum_{k=1}^{10} \left(\sum_{i=1}^{L-k+1} K^2\left((p_i\dots p_{i+k-1}), (q_i\dots q_{i+k-1})\right)\right)$$ where $$K^2\left((p_i\dots p_{i+k-1}), (q_i\dots q_{i+k-1})\right) = \prod_{j=1}^k \hat{B}_{90}(p_{i+j-1}, q_{i+j-1}),$$ for $p, q \in \PP$, $1 \le k \le L$ and $1 \le i \le L-k+1$;
The normalized version of $K^3$ is obtained using $$\hat K^3(p, q) = \frac{K^3(p, q)}{\sqrt{K^3(p, p)K^3(q,q)}}.$$
Note that, since Pfam families consist of pre-aligned sequences, our $K^3$ kernel definition differs slightly from [@smale2013introduction] as it only compares aligned amino acid $k$-mers. Also, we limit the $k$-mers considered in the construction of $K^3$ to lengths smaller or equal to $10$. This implies a substantial improvement in computation time, with no significant loss in predictive power.
We fix a threshold $\theta$ (in this paper, $\theta = 0.7$) and for any protein $p \in \PP$ we define the equilibrium measure $$\pi(p) = \sum_{\substack{q \in \PP \\ \hat K^3(p, q) > (1-\theta)}} \hat K^3(p, q).$$
The weight of a protein $p$ is then defined as the reciprocal of the equilibrium measure $w(p) = (\pi(p))^{-1}$ , and the effective number of sequences in the alignment is $M_{\text{eff}} = \sum_p w(p)$. The kernel $ \hat K^3(p, q)$ is a measure of similarity between pairs of proteins, and for a given protein $p$ the quantity $\pi(p)$ effectively counts the number of sequences with similarity larger than a threshold $1 -\theta$, thereby weighing down sequences that are over-represented in the data set.
#### Hamming distance weighting
As a term of comparison, we also applied a more traditional sequence weighting scheme based on the hamming distance between pairs of sequences. For each sequence $p$, we count the number of other sequences in the alignment that share more than $\theta$% sequence identity with $p$ $$m^p = | \{ b \in \{ 1, \dots, M \} : \text{\%id}(p,b) > \theta \} |$$ and then assign a weight $w(p) = 1 / m^p$ to sequence $p$.
This approach was used previously by several authors such as [@morcos2011direct], [@jones2012psicov] and [@ekeberg2013improved], but we note that these authors use different values for the theshold $\theta$ (respectively 0.8, 0.62 and 0.9). In this paper we choose $\theta=0.62$. See See Tables S2-S3 for the optimization of the two weighting schemes with respect to the regularization parameter $\eta'$, and Table S4 for a comparison of their performance.
Regularization parameter {#subsec:regularization}
------------------------
The matrix inversion in Equation \[eq:theta\] contains a regularization parameter $\eta$. We observed that families containing few sequences, where the number of sequences $M$ is comparable in size to the number of random variables ($20L$) require a larger regularization parameter compared to bigger families ($M \gg 20L$). We use then a regularization parameter of the form $\eta = \eta' / {M_{\text{eff}}}$, where ${M_{\text{eff}}}$ is the effective number of sequences defined above.
We tried different values of $\eta'$ over the 15 families from [@marks2011protein], and observed that in general roughly the same $\eta'$ appears to be optimal across families with different ${M_{\text{eff}}}$. Thus the normalization $\eta = \eta' / {M_{\text{eff}}}$ appears appropriate.
In Figure we show how the actual eigenvalues of the modified covariance matrix corresponding to PFAM family PF00028 are transformed when computing the regularized inverse.
Pseudocounts {#subsec:pseudocounts}
------------
The pseudocounts parameter $\lambda$ in Equation \[eq:pseudo\] accounts for non-observed pairs of amino acids. Following [@morcos2011direct], we set $\lambda=44$. However, we observed that the performance gain is small compared to setting $\lambda=0$ (see Table S5). In fact, pseudocounts have a similar regularizing effect to the parameter $\eta'$ described in the previous section, and probably for this reason the contribution of $\lambda$ is minimal.
Results and Conclusion
======================
The method and estimation of parameters described above were first applied to the 15 families studied in Marks et al. [@marks2011protein]. Performance was estimated in terms of the fraction of correct predicted contacts among the $L/5$, $L/3$, $L/2$ and $L$ pairs with highest $P_{\text{APC}}$ score, where $L$ is the length of the alignment. We considered as a true contact a pair of amino acids with beta-carbons (C$\beta$) with distance $< 8$ [Å]{} and at least 5 residues apart along the length of the protein. We find that on these 15 families the optimal value for the regularization parameter is around $\eta'=1000$ (see Table S2).
Table \[table:results15\] compares the performance of the RLS algorithm with PSICOV version 1.11 [@jones2012psicov] and the plmDCA method [@ekeberg2013improved]. We observe that on this set our method outperforms both PSICOV and DCA on the majority of families. Additionally, in Table S6 in shown the positive predictive value of the methods with respect to short range ($5 \leq i - j \leq 11$), medium range ($12 \leq i - j \leq 23$) and long range ($>$ 23) interactions for the prediction of the top $L/5$ contacts.
Next, we applied the three methods using the same parameters on an additional set of families from [@ekeberg2013improved]. This set partially overlapped with the families from [@marks2011protein] studied in Table \[table:results15\], and after removing the duplicates we are left with a set of 22 families. Table \[table:results22\] shows the positive predictive value of the RLS algorithm compared to PSICOV and plmDCA. On this set plmDCA obtains the highest average performance on three out of four ranking categories. However, note that plmDCA was optimized on this set of families, so there may be a bias in favor of this method.
Finally, we constructed an independent set of 10 families, selected randomly from Pfam with the only condition of containing at least 1,000 unique sequences. This set had not been used in the optimization of the algorithms, therefore constitutes a fair ground for comparison. The results (see Table \[table:results10\]) show that RLS outperforms the other two methods in the $L/5$ and $L/3$ subsets, whereas DCA obtains highest average performance for the prediction of the top $L/2$ and $L$ contacts.
A comparison of the running times of the two best algorithms (RLS and DCA) shows that RLS is at least one order of magnitude faster than DCA (Table \[table:timings\]). Note that we used the latest fast version of plmDCA [@ekeberg2014fast], termed “asymmetric plmDCA”, which improves considerably on previous pseudolikelihood methods in terms of speed. Our fast regularized inversion of the covariance matrix allows contact prediction on hundreds of amino acids-long domains in a matter of seconds, practically removing the limitations on the length of proteins that can be analyzed. In fact, the slowest step in the predictions is sequence weighting (not accounted in Table \[table:timings\] for either method), in particular the $K^3$-based weighting can be slow for very large families, but we showed that a simpler and faster weighting strategy does not affect too dramatically the performance (Table S4).
In general, we observed that the performance depends on the effective number of sequences ${M_{\text{eff}}}$ in the alignment. For instance, families PF00390 or PF00793 are composed of several thousand sequences, but they contain much redundancy, which brings down ${M_{\text{eff}}}$ to a few hundred units. Roughly, it appears that at least 1000 non-redundant sequences (${M_{\text{eff}}}> 1000$) are necessary to achieve a reasonable precision for contact prediction. This is in agreement with previous estimates [@marks2011protein; @kamisetty2013assessing] which place this number at about $5L$, where $L$ is the length of the alignment.
In conclusion, we demonstrated how our simple regularization scheme for covariance matrix inversion allows the fast and accurate prediction of residue-residue contacts. Currently, a major restriction to this kind of approach is the fairly high number of non-redundant sequences required to infer coevolution from a multiple sequence alignment, limiting the application to a relatively small subset of Pfam. However, as the number of protein sequences deposited in public databases increases, we expect a larger number of protein families to become accessible to our analysis, as well as improved performance on those that are already accessible.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work received funding from City University of Hong Kong grants RGC \#9380050 and \#9041544. Work by S.L. was partially supported by Ministerio de Ciencia, Tecnología e Innovación Productiva, Argentina.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Mariana Arantes[^1^]{} Flavio Figueiredo[^2^]{} Jussara M. Almeida[^1^]{}\
\
bibliography:
- 'bibs.bib'
subtitle: 'A Measurement Study on User Behavior, Popularity, and Content Properties'
title: 'Understanding Video-Ad Consumption on YouTube'
---
<ccs2012> <concept> <concept\_id>10002951.10003260.10003272.10003276</concept\_id> <concept\_desc>Information systems Social advertising</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003277.10003281</concept\_id> <concept\_desc>Information systems Traffic analysis</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Data Collection and Cleaning {#sec:data}
============================
User Skipping Behavior {#sec:user}
=======================
Video-Ad Popularity {#sec:pop}
===================
Video-Content to Video-Ad Pairs {#sec:match}
===============================
Discussion and Future Work {#sec:conclusions}
==========================
Acknowledgments {#acknowledgments .unnumbered}
===============
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The effects of regular S=1 dilution of S=1/2 isotropic antiferromagnetic chain are investigated by the quantum Monte Carlo loop/cluster algorithm. Our numerical results show that there are two kinds of ground-state phases which alternate with the variation of $S^1=1$ concentration. When the effective spin of a unit cell is half-integer, the ground state is ferrimagnetic with gapless energy spectrum and the magnetism becomes weaker with decreasing of the $S^1$ concentration $\rho = 1/M$. While it is integer, a non-magnetic ground state with gaped spectrum emerges and the gap gradually becomes narrowed as fitted by a relation of $\Delta \approx 1.25\sqrt{\rho}$.'
author:
- 'Fengping Jin, Zhaoxin Xu[^1], Heping Ying and Bo Zheng'
title: 'On effects of regular S=1 dilution of S=1/2 antiferromagnetic Heisenberg chains by a quantum Monte Carlo simulation'
---
Introduction
============
The effects of substitutions of magnetic impurities on the antiferromagnetic spin chain have attracted great interests in the past decade. It has been shown theoretically that the ground state properties vary with different dilution cases. For the random substitutions, the most interested case is of the S=1/2 impurities in Haldane chain [@Sorensen]. For example, inelastic neutron scattering experiment on the compound Y$_2$BaNiO$_5$ substituting Ca$^{2+}$ for Ni$^{2+}$ [@Ditusa] show a substantial increase of the spectral function below the Haldane gap to indicate the creation of states below the energy of the spin gap. This effects are also studied by numerical works by S. Wessel [@Wessel]. For regular substitutions, these systems are the mixed-spin chains which have been extensively studied by many authors in the past a few years. Analytical methods of non-linear sigma model, mean field theory and spin-wave method [@Yamamoto2; @Fukui; @Takano; @Wu] as well as numerical works by density matrix renormalization group [@Pati] and quantum Monte Carlo [@Tonegawa; @Yamamoto] have been applied extensively for such systems. So far, it is well known that the topology of spin arrangements in the mixed chains plays an essential role on the ground state properties and thermodynamics in the mixed-spin systems.
Experimently, many Quasi-1D mixed-spin materials have been synthesized in the past two decades, such as ACu(pba)(H$_2$ O)$_3$ $\cdot$ n(H$_2$ O) and ACu(pbaOH)(H$_2$ O)$_3$ $\cdot$ nH$_2$O (where pba=1,3-propylenebis(oxamato),pbaOH=2-hydroxo-1,3-propylenebis and A= Ni,Fe,Co,Mn,Zn). These materials contains two different transition metal ions per unit cell, and their properties were studied as ferrimagnetic chains [@Kahn1; @Kahn2; @Verdaguer; @Gleizes; @Pei; @Hagiwara2]. The experiment results imply that the magnetic properties of the mixed-spin compounds can all be described by a Heisenberg model with nearest-neighbor antiferromagnetic coupling as $$H=\sum_{i=1}^{N} J_i S_i \cdot S_{i+1},
\label{Hamiltonian}$$ where $S_i$ denotes a spin-$S$ moment at site $i$, $N$ is system size and $J_i>0$. T. Fukui and N. Kawakami [@Fukui] have studied spin chain composed by a periodic array of impurities $S^1$ embedded in the host $S^2 \neq
S ^1$ spin chain with the period $M$, i.e., $$\underbrace{S^1 \otimes S^2 \otimes S^2 \otimes \ldots \otimes
S^2}_{\mbox{M}} \otimes S^1 \otimes S^2 \otimes \ldots \otimes
S^2.
\label{spin_arrange}$$
The dilutions of the model denoted by the impurity concentration $\rho$ has two limits: ([*i*]{}) $\rho=0$, the undoped pure antiferromagnetic $S^2$ chain, it has a non-magnetic ground state; ([*ii*]{}) $\rho=0.5$, the alternating spin chain of $S^1$ and $S^2$. According to Marshall theorem and Lieb-Schultz-Mattis (LSM) theorem [@Lieb], ground state of the doped cases are specified by the spin quantum number $S=0(|S_1-S_2|N/M)$ for $M=odd$ or $even$, it is either a spin singlet or ferrimagnetic. If the effective spin in a unit composed of $M$ spins $S_{eff}$ is half-integer, so the system has a gapless energy spectrum. But when $S_{eff}$ is integer, LSM theorem fails to predict the energy spectrum to be gaped or gapless. By applying non-linear $\sigma$ model, it is found that the system has an energy gap when the $S_{eff}$ is integer[@Fukui; @Takano]. But details of ground state properties and thermodynamics can not be given by non-linear $\sigma$ model analyses.
The authors of present paper have recently studied the model (1) with the case of $S^1=1/2$ and $S^2=1$ by applying quantum Monte Carlo simulations [@Wang], where the numerical results reveal different non-trivial magnetic properties happened between two kinds of diluting cases, i.e. for $odd$ $S^2=1$ spins in a unit, system has magnetic ground state and it shows ferrimagnetic features; while for $even$ $S^2=1$ spins in a unit, systems behave non-magnetic ground states with antiferromagnetic-like features. For both the $odd-even$ cases, the ground states are [*gapless*]{} steadily. And the system gradually transits from the ferrimagnetic ground state of the alternating $S^1$-$S^2$ chain to the disordered ground state of pure $S=1$ chain in two different tendencies. In this Letter, we study an opposite case with $S^1=1$ and $S^2=1/2$. Previous analytical work predicted that if $odd$ $S^2=1/2$ spins in a unit, the effective spin $S_{eff}$ is half-integer, the ground state is ferrimagnetic with a gapless energy spectrum; while if $even$ $S^2=1/2$ in a unit, $S_{eff}$ is integer, the ground state is non-magnetic and the system has an energy gap. Our numerical study will focus on how the ground state properties depend on the concentration and the finite temperature magnetic properties evolute with decreasing of the $S^1=1$ concentration $\rho$.
Calculation and Results
=======================
We use the efficient continuous imaginary time version of loop cluster algorithm to perform the quantum Monte Carlo simulation [@Beard], which has been successfully applied for the other mixed-spin chains [@Xu; @Wang]. We confine our calculation to isotropic antiferromagnetic coupling cases, i.e. $J_i = J >0$ in equation (\[Hamiltonian\]), and the positions of spin $S_1=\frac{1}{2}$ and $S_2=1$ are arranged as represented in equation (\[spin\_arrange\]) with $M$ taking the values from 2 to 11. We carry out $10^5$ Monte Carlo steps for measuring physical quantities after $10^3$ Monte Carlo steps for the thermalization. In order to clearly explore the ground state properties, the simulations are performed at the very low temperature $\beta = 1/T= 200$ for system sizes $L>200$ in condition of even number of unit. The physical quantities we measure are the ground state energy $E_G$, the uniform magnetic susceptibility $\chi_u$ and staggered susceptibility $\chi_s$ by using the improved estimators in the loop cluster algorithm, e.g. , $$<\chi>=\frac{\beta}{4V} \Bigl \langle \sum_{cluster \\ c} w_t(c)^2 \Bigr \rangle_{MC},$$ $$<\chi_s>=\frac{1}{4V\beta} \Bigl \langle \sum_{cluster \\ c} |C|^2 \Bigr \rangle_{MC},$$ where $w_t(c)$ is winding number of cluster $c$, and $|C|$ is the cluster size. The magnetization and staggered magnetization are estimated by $$<M^2>= \Bigl \langle 3(\sum_i S_i^z)^2 \Bigr \rangle_{MC}$$ and $$<M_s^2>=\Bigl \langle 3(\sum_i(-1)^i S_i^z)^2 \Bigr \rangle_{MC}.$$ The energy gap $\Delta$ is also estimated in the way given by Todo, [@Todo] $$\bigtriangleup = \lim_{L \rightarrow \infty} \frac{1}{\xi_{\tau,0}(L)},$$ where $\xi_{\tau,0}$ is the correlation length in the imaginary time direction.
The results for magnetizations and uniform susceptibility are plotted in Fig. \[mz\] and Fig. \[sus\_u\]. We find that the magnetic properties are apparently different for two cases of $M=odd$ and $M=even$. When $M=even$, the magnetization is finite and approaches zero linearly with decreasing of $\rho$. While $M=odd$, the magnetization remains almost at zero value. On the other hand, it can be observed from our results that the uniform susceptibilities $\chi_u$ is finite for $M=even$, but it vanishes when $M=odd$.
Thus there is magnetic long-range order (LRO) in the ground state when $M=even$, but the order is absent when $M=even$.
We further estimate the staggered magnetization and its susceptibility as a function of concentration shown in Fig. \[smz\] and Fig. \[sus\_s\] respectively.
The two observables are both finite for $M=odd$ and $M=even$ cases, but the data for the cases of $M=even$ have much stronger values than the cases of $M=odd$.
In order to confirm the results observed above, we begin to investigate the finite temperature uniform magnetic susceptibility. As displayed in Fig. \[sus\_T\],
one can easily find that $\chi_u$ diverges when the temperature $T=1/\beta$ goes to zero in the cases $M=even$. This is the typical behavior of a system with magnetic LRO. In the cases $M=odd$, all the $\chi_u$ approach zero when $T \rightarrow 0$, a remarkable evidence to reveal the existence of the energy gap.
Up to now, our results verify numerically that there are magnetic LRO and antiferromagnetic LRO in the ground states when $M=even$. They clearly show that the ground states are ferrimagnetic in such cases. While for $M=odd$, there should exist of spin liquid phases denoted by the vanish of the magnetizations. Consequently we believe our numerical results consist correctly with the previous analytical predictions. More important, one can easily see that the magnetism decreases with decreasing of impurity concentration in the case of $M=even$. But there is not notable change of the magnetic properties when the $S^1=1$ concentration decreases as $M=odd$.
Next, we consider the feature of the energy gap $\Delta$ on different regular dilutions. Not surperised for us, the energy gap is closed when $M$ is $even$ and it opens again while $M$ is $odd$ as shown in Fig. \[gap\]. These results is consist with the prediction by non-linear $\sigma$ model and LSM theorem [@Fukui; @Takano].
It is interesting that the energy gap $\Delta$ tends to be narrow as decreasing of $S^1=1$ concentration when $M=odd$. We confirm such behavior by fitting $\Delta$ to the curve of $1.25\sqrt{\rho}$ as one can see in Fig. \[gap\].
Moreover, we show the finite-size effect of $\Delta$ results for several cases with $L$ increasing in Fig. \[size\]. In our estimations, although the gaps are not exact closed for $M=even$ due to the finite-size simulations, we find the data of the gaps decrease fast than $L^{-1}$, so it is obvious that the gaps will trend to zero as $L \rightarrow \infty$. For the cases $M=odd$, where the gap opens all the time, there is almost no finite-size effect.
In order to identify the ground state phases, we calculate the valence-bond-solid (VBS) [@Affleck] order parameter $$z\equiv \langle \exp[i\frac{2\pi}{N}\sum^N_{j=1}jS^z_j]\rangle,$$
According to the LSM theorem, $z$ vanishes in the gapless phase as system size $N\rightarrow \infty$. On the other hand, one expects that $z$ varies in between $\pm 1$ but $z\neq 0$ in a given gaped phase. In exact VBS states, $z=\pm 1$ [@Nakamura]. Our calculations are plotted in Fig. \[VBS\]. It is clear that $z
\approx -1$ for all cases of $M=odd$ to present the system located in a VBS phase; while $z \approx 0$, it reveals the gapless energy spectrum for all $M=even$ cases.
Especially, all these ground state phases can be understood under the scenario of VBS picture. In VBS picture, each impurity $S^1=1$ can be regarded as two spin-1/2 in a triplet state, these two spin-1/2 can form singlet with their nearest neighbor S=1/2 spin due to the antiferromagnetic coupling. When $M=odd$, each unit have [*even*]{} number of S=1/2 host spins, so they can fall into singlets with their nearest neighbors including the two spin-1/2 of $S=1$ to induce the VBS order as seen in Fig. \[illus\_1\] (a). As a result, the system now shows a gaped energy spectrum.
=3.5cm
But for $M=even$, [*odd*]{} number of spin-1/2 exist in a unit and there will be an active spin which is not used to form singlet as shown in Fig. \[illus\_1\](b), thus there is no VBS order and the system emerges no spin gap. Our results of the VBS order parameter $z$ clearly verify this picture, $z \approx -1$ when $M=odd$ and $z \approx 0$ when $M=even$ as shown in Fig. \[VBS\]. At last, we note that the VBS phase is stable with the variation of $S^1=1$ concentration $\rho$ when $M=odd$.
Discussion and Conclusion
=========================
Our Monte Carlo study verifies that two brunches of different magnetic behaviors emerge in cases of regular S=1 diluted S=1/2 host chains. According Marshall theorem, the cases with $M=even$ have the ferrimagnetic ground states which can be specified by quantum number $S_{total}=|S_1-S_2|N/M$, so the magnetization per site is finite and it decreases linearly as a function of $\rho$ to ${\cal M}=0$, the case of the pure S=1/2 antiferromagnetic Heisenberg chain. This feature can be easily observed from our results in Fig. \[mz\] and Fig. \[smz\]. When $M$ is $odd$, the ground state is singlet with $S_{total}=0$, thus the magnetization per site keeps zero and this is a non-magnetic state. As observed in our simulations there is no notable variations of the ground state magnetic properties in the cases of $M=odd$.
To compare our ground state results of the model in this Letter for $S^1=1$ and $S^2=1/2$ ([*system I*]{}) with the one we studied previously [@Wang] when $S^1=1/2$ and $S^2=1$ ([*system II*]{}), we collect the main points of the numerical calculations in Table I.
------------- --------- -------------- -------------- --------------
$M=odd$ $M=even$ $M=odd$ $M=even$
$S_{eff}$ integer half-integer half-integer half-integer
$<$M$>$ zero finite zero finite
$\chi_u$ zero large small large
$<$M$_s$$>$ finite finite finite finite
$\chi_s$ small large large large
$\Delta$ gaped gapless gapless gapless
$z$ -1.0 0.0 0.0 0.0
------------- --------- -------------- -------------- --------------
: Comparison of ground state properties of two model, where $S_{eff}$ is effective spin in a unit, $<$M$>$ is magnetization, $<$M$_s$$>$ is staggered magnetization, $\chi_u$ is uniform susceptibility, $\chi_s$ is staggered susceptibility, $\Delta$ is energy gap and $z$ is VBS order parameter.
One can easily see that both systems behave with two kinds of different ground state phases, magnetic or non-magnetic, respectively. If $M=even$, the ground states are ferrimagnetic for both [*system I*]{} and [*system II*]{}, and their magnetizations and staggered magnetizations are all finite and decrease linearly with decreasing of impurity concentration. However, for the cases of $M=odd$, there appears VBS order in [*system I*]{} which is gaped, but the order is absent in [*system II*]{} where the spin arrangements can not induce such order, so the gap is constantly closed. This feature reveals that this topological order plays an important role to the behavior of the energy gap in the mixed-spin system. We believe that the fitted relation of $\Delta \approx 1.25
\sqrt{\rho}$, to denote the energy gap as function of $S^1=1$ concentration, provides a good stuff to study how the topological order affects the energy gap in the mixed-spin systems.
In conclusion, we have studied the ground state and finite temperature magnetic properties of the regular $S^1=1$ diluted in $S^2=1/2$ antiferromagnetic chain. Our calculations show that there exist different phases in the ground state as a function of $S^1$ concentration. When there is one $S^1$ impurity and $odd$ number of host $S^2$ spins in a unit cell, the ground states are ferrimagnetic and the system has a gapless energy spectrum. The ferrimagnetism becomes weaker as the impurity concentration reduced. While for one $S^1$ and $even$ number of $S^2$ in one unit cell, the ground state is a VBS phase where there is a gaped energy spectrum and the energy gap gradually approaches to zero with decreasing the concentration $\rho$. An interesting observation is that the behavior of the energy gap can be numerically well fitted by $\Delta \approx 1.25 \sqrt{\rho}$. Further analytical work, for exmaple using the mean-field theory [@Wu], is required to explain why such dependence of the energy gap exist in VBS phases.
Acknowledgment
==============
The authors would like to thank Prof. Jianhui Dai for stimulating discussions and comments. This work was supported in part by the NNSF and SRFDP of China, and by the NSF of Zhejiang province.
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[^1]: E-mail: [email protected].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The spatial resolution along the pad-row direction was measured with a GEM-based TPC prototype for the future linear collider experiment in order to understand its performance for tracks with finite projected angles with respect to the pad-row normal. The degradation of the resolution due to the angular pad effect was confirmed to be consistent with the prediction of a simple calculation taking into account the cluster-size distribution and the avalanche fluctuation.'
title: |
Cosmic ray tests of a GEM-based TPC prototype\
operated in Ar-CF$_4$-isobutane gas mixtures: II
---
TPC,ILC,GEM,CF$_4$,Spatial resolution,Angular pad effect
29.40.Cs ,29.40.Gx
Introduction
============
In the previous paper [@ref1] we demonstrated the feasibility of a GEM-based Time Projection Chamber (TPC) operated in an Ar-CF$_4$-isobutane gas mixture as a central tracker (LCTPC) for the future linear collider experiments (ILC [@refLC1] and CLIC [@refLC2]). The spatial resolution along the pad-row direction was presented for tracks nearly perpendicular to the pad row in Ref. [@ref1] because the resolution in the $r$-$\phi$ plane better than $\sim$ 100 $\mu$m per pad row for stiff and radial tracks is of prime importance for the physics goals of the experiments. A TPC equipped with GEM readout is certainly an ideal main tracker, which is free from the $E \times B$ and the angular wire effects inherent in conventional TPCs with MWPC readout.
It should be noted, however, that the azimuthal resolution degrades with increasing projected track angle ($\phi$) measured from the pad-row normal because of the angular pad effect as far as conventional pads are employed for readout. As will be seen the angular pad effect adds an almost constant offset to the resolution with its amount depending on the pad height as well as the track angle. Therefore the requirement for the spatial resolution above would not be met for inclined tracks. The degraded resolution for slanted and/or low-momentum tracks provided by the central tracker could affect the physics capability of the whole detector system. An example is the need for good energy resolution for soft jets; thus it is important to understand whether such things can be affected by the design of the TPC.
In this paper the resolutions measured with cosmic rays for inclined tracks in a prototype TPC are presented and compared to the expectation in order to provide a basis for the optimization of the pad height of the LCTPC.
The expected deterioration of the resolution for inclined tracks, compared to that for right angle tracks, is estimated in Section 2. The comparison of the measured resolution with the expectation is presented in Section 3 after a brief description of the experiment. Section 4 is devoted to a discussion and Section 5 concludes the paper.
Expectation
===========
For right angle tracks ($\phi = 0^\circ$) the resolution along the pad-row direction ($\sigma_{\rm X}$) is approximately given by
$$\sigma_{\rm X}^2 = \sigma_{\rm X00}^2 + \frac{D^2}{n_{\rm eff}} \cdot z$$
where $\sigma_{\rm X00}$ is the intrinsic resolution[^1], $D$ is the diffusion constant, $n_{\rm eff}$ is the effective number of electrons per pad row, and $z$ is the drift distance [@ref2][^2]. It is worth noting that the value of $n_{\rm eff}$ is almost independent of the drift distance [@ref3][^3]. Even in the case of finite track angle the explicit drift-distance dependence (the second term) of the resolution is scarcely affected by practically small $\phi$ [@ref4]. The first term is, on the other hand, sensitive to the track angle. It may be expressed as
$$\sigma_{\rm X0}^2 = \sigma_{\rm X00}^2 + \frac{h^2 \cdot \tan^2 \phi}
{12 \cdot N_{\rm eff}}$$
where $h$ is the pad height[^4] and $N_{\rm eff}$ is the effective number of [*clusters*]{} per pad row (see footnote 3). The second term in Eq. (2) represents the contribution of the angular pad effect to the resolution, which is parametrized by $N_{\rm eff}$.
In fact, $N_{\rm eff}$ is a function of $\phi$, $\theta$, $z$ and $h$:
$$N_{\rm eff} = N_{\rm eff}(\phi, \theta, z, h)$$
where $\theta$ is the angle between the track and the readout pad plane[^5]. Let us consider first the $h$ dependence of $N_{\rm eff}$, i.e. $N_{\rm eff}(0, 0, z, h)$. The average number of clusters per pad row ($\left< N \right>$) is proportional to $h$. Furthermore the $z$ dependence of the effective number of clusters due to de-clustering is expected to be small [@ref4][^6]. Accordingly
$$N_{\rm eff}(0, 0, z, h) \sim N_{\rm eff}(0, 0, 0, h)
= N_{\rm eff}(\left< N \right>) \;.$$
For a fixed number of clusters $N$, $N_{\rm eff}$ is given by
$$N_{\rm eff}(N) = \left< \sum_{i=1}^N Q_i^2 \; /
\left( \sum_{i=1}^N Q_i \right) ^2 \right> ^{-1}$$
where $Q_i$ is the total charge of the cluster $i$ given by
$$Q_i = \sum_{j=1}^{n_i} q_j$$
with $q_j$ being the amplified signal of the $j$-th electron in the cluster $i$ of size $n_i$ (see Appendix). $N_{\rm eff}$ was estimated by numerical calculations, taking into account the cluster-size distribution for argon [@ref5], and is shown in Fig. 1, with (filled circles) and without (open circles) the typical avalanche fluctuation (a Polya distribution with $\theta =
0.5$[^7]) for each electron. The figure tells us that the effective number of clusters is considerably smaller than $N$ because of the large cluster-size fluctuation. Furthermore it is not a linear function of $N$; see Appendix for a qualitative estimation of $N_{\rm eff}$. In real case, $N$ is not a constant and obeys Poisson statistics with the average $\left< N \right>$. The curves in Fig. 1 show the effective number of clusters as a function of $\left< N \right>$.
![\[fig1\] Effective number of clusters ($N_{\rm eff}$) as a function of the total or average number of clusters: plots for fixed $N$, and curves for Poissonly distributed $N$. The filled (open) circles and the full (dotted) curve are calculated with (without) the avalanche fluctuation.](\figdir/fig1.eps)
From the curve with the avalanche fluctuation in Fig. 1 the effective number of clusters for a given track angle can be estimated since
$$\left< N \right> = \frac{d \cdot h}{\cos \phi \cdot \cos \theta}$$
with $d$ being the cluster density ($\sim$ 2.43/mm for minimum ionizing particles in argon [@ref6]), and
$$N_{\rm eff}(\phi, \theta, 0, h)
= N_{\rm eff}(\left< N \right>)
= N_{\rm eff}\left( \frac{d \cdot h}{\cos \phi \cdot \cos \theta} \right) \;.$$
Let us define $S_{\rm X00}$ as the square root of the second term in Eq. (2) at $z$ = 0:
$$S_{\rm X00} \equiv \frac{h \cdot \tan \phi}
{\sqrt{12 \cdot N_{\rm eff}(\left< N \right>)}}\;.$$
Fig. 2 shows $S_{\rm X00}$ as a function of the pad height ($h$) for $\phi$ = 5$^\circ$, 10$^\circ$, 15$^\circ$ and 30$^\circ$, calculated with $\theta$ fixed to 0$^\circ$. It should be noted that the resolutions shown in the figure are the best possible values expected to be obtained without diffusion (at $z$ = 0) and without contribution of the intrinsic term ($\sigma_{\rm X00}$).
![\[fig2\] Expected contribution of the angular pad effect ($S_{\rm X00}$) as a function of the pad height ($h$) for different track angles. Poisson statistics is assumed for the number of clusters and a Polya distribution ($\theta = 0.5$) is assumed for the avalanche fluctuation. The tracks are assumed to be minimum ionizing and parallel to the readout plane. The points plotted at $h$ = 6.3 mm are the measurements (see Section 3.2).](\figdir/fig2.eps)
Experiment
==========
Setup and analysis
------------------
We used a small GEM-based TPC prototype (MP-TPC) operated in a gas mixture of Ar (95%)-CF$_4$ (3%)-isobutane (2%) at atmospheric pressure. The MP-TPC is a small time projection chamber with a maximum drift length of 257 mm. Its gas amplification device is a triple GEM, 100 mm $\times$ 100 mm in size. The amplified electrons are collected by a readout plane placed right behind the GEM stack, having 16 pad rows at a pitch ($h$) of 6.3 mm, each consisting of 1.17 mm $\times$ 6 mm rectangular pads arranged at a pitch of 1.27 mm. The neighboring pad rows are staggered by half a pad pitch. The pad signals are then fed to readout electronics, a combination of preamplifiers, shaper amplifiers and digitizers. See Ref. [@ref1] for details of the experimental setup and the analysis procedure for the cosmic ray tests of the MP-TPC.
We re-analyzed the data taken for the previous paper on the normal incident tracks [@ref1] with different cuts on the track angles. Among the data sets the data collected with a drift field of 250 V/cm and $B$ = 0 T were selected because of its highest statistics and the negligible influence of the finite pad-pitch term in the absence of axial magnetic field [@ref1]. The offset to the resolution due to finite track angle is to be added quadratically as well in the presence of a magnetic field, depending on the local track angle, at drift distances where the finite pad-pitch term is negligible.
The track angle distributions are shown in Fig. 3. As mentioned in Introduction our primary concern in the cosmic ray tests with the MP-TPC was the resolution for right angle tracks. Therefore the acceptance to inclined tracks was limited by trigger-counter arrangement in order to reduce the trigger rate to the relatively slow readout electronics. The maximum available track angle is thus $|\phi| {\hbox{ \raise3pt\hbox to 0pt{$<$}\raise-3pt\hbox{$\sim$} }}10^\circ$ as seen in Fig. 3 (a).
![\[fig3\] Track angle distributions: (a) for $\phi$, and (b) for $\theta$.](\figdir/fig3.eps)
Results
-------
The spatial resolutions along the pad-row direction are shown in Fig. 4 for $|\phi|$ = 0$^\circ$, 5$^\circ$ and 10$^\circ$, for tracks nearly parallel ($|\theta| \leqq 10^\circ$) to the readout plane. The azimuthal angle cuts are the nominal values $\pm$ 2$^\circ$. The resolutions squared as function of the drift distance ($z$) were fitted by a function $\sigma_{\rm X}^2 = \sigma_{\rm X0}^2 + D^2 / n_{\rm eff} \cdot z$ for free parameters $\sigma_{\rm X0}$ and $n_{\rm eff}$, with the value of $D$ fixed to 315 $\mu$m/$\sqrt{\rm cm}$ given by Magboltz [@ref7]. $S_{\rm X00}$ and $N_{\rm eff}$ were then obtained using Eqs. (2) and (9) for each $\phi$, assuming $\sigma_{\rm X00}$ to be $\sigma_{\rm X0}$ measured for $\phi$ = 0$^\circ$. The resultant $\sigma_{\rm X0}$, $n_{\rm eff}$, $S_{\rm X00}$ and $N_{\rm eff}$ are summarized in Table 1 along with the values of $S_{\rm X00}$ and $N_{\rm eff}$ calculated for $h$ = 6.3 mm. The measured values of $S_{\rm X00}$ are plotted also in Fig. 2.
The measured values of $S_{\rm X00}$ and $N_{\rm eff}$ are consistent with those given by the calculation. In addition, the values of $n_{\rm eff}$ for inclined tracks are close to that for normal incident tracks as expected and are consistent with an estimation in Ref. [@ref2].
![\[fig4\] Resolution squared ($\sigma_{\rm X}^2$) as a function of the drift distance ($z$): (a) for $|\phi|$ = 0$^\circ$, (b) for $|\phi|$ = 5$^\circ$ and (c) for $|\phi|$ = 10$^\circ$. See text for the straight lines fitted through the data points.](\figdir/fig4.eps)
Discussion
==========
The figure of merit for the azimuthal spatial resolution of a cylindrical TPC is the resolution per projected track length in the $r$-$\phi$ plane along the radial direction. From Eqs. (1), (2) and (9), the resolution per pad row is expressed as
$$\sigma_{\rm X}^2 \sim \sigma_{\rm X00}^2 + S_{\rm X00}^2
+ \frac{D^2}{n_{\rm eff}} \cdot z$$
at drift distances where the finite pad-pitch term is negligible, and each of the three terms is a function of the pad height $h$.
We consider here the effect of splitting a pad row with $h$ = $H$ into a couple of identical pad rows with a height of $H$/2. Let us assume for simplicity that the combined track coordinate is given by the average of the two measurements provided by the neighboring pad rows with $h$ = $H$/2. Then the resolution per projected track length $H$ becomes
$$\sigma_{\rm X}^2 = \frac{{\sigma_{\rm X}^\ast}^2}{2}$$
where $\sigma_{\rm X}^\ast$ is the resolution obtained with a single pad row with the halved height. The diffusion contribution (the third term in Eq.(10)) is almost unaffected since $n_{\rm eff}$ is approximately proportional to the pad height [@ref3].
On the other hand, the angular pad effect ($S_{\rm X00}$) is reduced appreciably. We temporarily assume $N_{\rm eff}(\left< N \right>)$ to be proportional to the pad height[^8]. Then
$$S_{\rm X00}^2 \propto h$$
from Eq. (9), and the combined contribution of the angular pad effect ($S_{\rm X00}$) per projected track length $H$ is halved because of Eq. (11). Actually $S_{\rm X00}$ is reduced by more than a factor of 2 because Eq. (12) gives an overestimate for a smaller $h$ (see Fig. 2).
In addition, $\sigma_{\rm X00}^2$ at long drift distances can be shown mathematically to be
$$\sigma_{\rm X00}^2 = \frac{B_0^2}{n_{\rm eff}}$$
with a constant $B_0$ independent of the pad height if the contribution of the electronic noise is negligible (see Appendix B of Ref. [@ref1]). Similarly to the diffusion contribution, the intrinsic term ($\sigma_{\rm X00}$) in the combined resolution is expected to be close to the counterpart in the resolution for a single pad row with $h$ = $H$.
Consequently the net effect of halving the pad height on the resolution per projected track length is essentially the alleviation of the angular pad effect ($S_{\rm X00}$) by more than a factor of 2. For example, Eq. (9) gives $S_{\rm X00}$ $\sim$ 140 $\mu$m (450 $\mu$m) for $\phi$ = 10$^\circ$ (30$^\circ$) with $h$ = 6.3 mm, while the corresponding value for a couple of pad rows with $h$ = 3.15 mm is about 60 $\mu$m (200 $\mu$m). The spatial resolutions, and therefore the momentum resolutions improve significantly for slanted and/or low momentum tracks with the shorter pads.
The number of voxels in the sensitive volume of a TPC is doubled if the pad height is halved (with the electronics channel density doubled). This would enhance the pattern recognition capability and the d$E$/dx resolution of the TPC as well.
Conclusion
==========
The azimuthal spatial resolutions for inclined tracks were measured with a GEM-equipped prototype TPC as well as for right angle tracks. The angular pad effect contributes as a virtually constant offset to the spatial resolution to be added quadratically, depending on the track angle and the pad height. The offsets are found to be consistent with the predictions given by a simple model calculation taking into account the cluster-size distribution and the avalanche fluctuation.
The results are expected to be useful in optimizing the pad height of the LCTPC from the physics point of view.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the group at the KEK cryogenics science center for the preparation and the operation of the superconducting magnet. We are also grateful to many colleagues of the LCTPC collaboration for their continuous encouragement and support, and for fruitful discussions. This work was supported by the Creative Scientific Research Grant No. 18GS0202 and No. 23000002 of the Japan Society of Promotion of Science.
Behavior of $N_{\rm eff}(N)$
============================
The effective number of clusters ($N_{\rm eff}$) parametrizes the degradation of the resolution due to the angular pad effect (the second term in Eq. (2)). We consider here the behavior of $N_{\rm eff}$ at $z = 0$ as a function of the fixed total number of clusters per pad row ($N$), qualitatively for $N$ = 1, 2, 4 and $\infty$. The track coordinate ($X$) along the pad-row direction is assumed to be determined from the charge centroid of the clusters detected by the pad row having an infinitesimal pad pitch. The clusters are fully intact at $z = 0$ and are assumed to be point-like.
In order to estimate $N_{\rm eff}(N)$ it is necessary to evaluate the variance ($\equiv {S_N}^2$) of the charge centroid of $N$ clusters, each with charge $Q_i$ and coordinate $x_i$, which are randomly scattered over the lateral range on the pad row covered by an inclined track ($h \cdot \tan \phi$).
1. $N = 1$\
The resolution ($\equiv S_1$) does not depend on the cluster charge ($Q$).
$$\begin{aligned}
{S_1}^2 &=& \frac{h^2 \cdot \tan^2\phi}{12} \;, \;\; {\rm and} \\
N_{\rm eff}(1) &=& 1 \;\;\; {\rm by \;\; definition}.\end{aligned}$$
2. $N = 2$\
Let the coordinates and charges of the clusters be $(x_1, Q_1)$ and $(x_2, Q_2)$. Their weighted-mean coordinate is given by
$$X = \frac{x_1 Q_1 + x_2 Q_2}{Q_1 + Q_2} \;.$$
Its variance (${S_2}^2$) is given by
$$\begin{aligned}
{S_2}^2 &\equiv& \left< (X - \left< X \right>)^2 \right> \nonumber\\
&=& \left< \left( \frac{(x_1 - \left< X \right>)\cdot Q_1
+ (x_2 - \left< X \right>)\cdot Q_2}{Q_1 + Q_2} \right)^2 \right> \nonumber\\
&=& \left< \frac{(x_1 - \left< X \right>)^2\cdot {Q_1}^2
+ (x_2 - \left< X \right>)^2\cdot {Q_2}^2}{(Q_1 + Q_2)^2} \right> \nonumber\\
&=& \left< (x - \left< x \right>)^2 \right> \cdot
\left< \frac{{Q_1}^2 + {Q_2}^2}{(Q_1 + Q_2)^2} \right> \nonumber\\
&=& {S_1}^2 \cdot \left< \frac{(Q_1 + Q_2)^2 - 2Q_1Q_2}{(Q_1 + Q_2)^2} \right> \nonumber\\
&=& {S_1}^2 \cdot \left( 1 - 2 \cdot
\left< \frac{Q_1Q_2}{(Q_1 + Q_2)^2} \right> \right) \nonumber\\
&\geqq& {S_1}^2 \; / 2 \;.\end{aligned}$$
The third and fourth lines in the equation above are justified since the variables $x$ and $Q$ are not correlated, whereas the last line follows from
$$\begin{aligned}
\label{eqX} Q_1Q_2 \; /(Q_1 + Q_2)^2 &\leqq& 1/4 \\
\because \; (Q_1 - Q_2)^2 &=& (Q_1 + Q_2)^2 - 4Q_1Q_2 \geqq 0 \;. \nonumber \end{aligned}$$
The equality in Eq. (\[eqX\]) holds only when $Q_1 = Q_2$. Therefore, in a general case addressed here
$$N_{\rm eff}(2) \equiv {S_1}^2 \;/{S_2}^2 < 2 \;.$$
3. $N = 4$\
$$\begin{aligned}
X &=& \frac{x_1Q_1 + x_2Q_2 + x_3Q_3 +x_4Q_4}{Q_1+Q_2+Q_3+Q_4} \nonumber\\
&=& \frac{x_1^\prime Q_1^\prime + x_2^\prime Q_2^\prime}
{Q_1^\prime + Q_2^\prime} \end{aligned}$$
where
$$\begin{aligned}
x_1^\prime Q_1^\prime &\equiv& x_1Q_1 + x_2Q_2 \nonumber\\
x_2^\prime Q_2^\prime &\equiv& x_3Q_3 + x_4Q_4 \nonumber\end{aligned}$$
with $Q_1^\prime \equiv Q_1 + Q_2$ and $Q_2^\prime \equiv Q_3 + Q_4$.
$$\begin{aligned}
{S_4}^2 &\equiv& \left < (X - \left< X \right>)^2 \right> \nonumber\\
&=& \left< (x^\prime - \left< x \right>)^2 \right> \cdot
\left< \frac{{Q_1^\prime}^2 + {Q_2^\prime}^2}
{(Q_1^\prime + Q_2^\prime)^2} \right> \nonumber\\
&=& \left< (x^\prime - \left< x \right>)^2 \right> \cdot
\left( 1 - 2 \cdot \left< \frac{Q_1^\prime Q_2^\prime}
{(Q_1^\prime + Q_2^\prime)^2} \right> \right) \nonumber\\
&>& {S_2}^2 / 2 \;,\;\;{\rm with} \;\;
{S_2}^2 \equiv \left< (x^\prime - \left< x \right>)^2 \right> \;. \end{aligned}$$
Therefore
$$\begin{aligned}
\frac{{S_2}^2}{{S_4}^2} &<& 2 \;,\;\;{\rm and} \\
N_{\rm eff}(4) &<& 2N_{\rm eff}(2) \;.\end{aligned}$$
Consequently
$$N_{\rm eff}(1) = 1,\; N_{\rm eff}(2) < 2,\;
N_{\rm eff}(4) < 2 N_{\rm eff}(2), \;\; {\rm and \; so \; on.}$$
Thus $N_{\rm eff}(N) / N$ is expected to be a decreasing function of $N$.
4. $N = \infty$
$$\begin{aligned}
X &=& \frac{\sum_{i=1}^N x_i Q_i}{\sum_{i=1}^N Q_i} \\
{S_{\rm N}}^2 &\equiv& \left< (X - \left< X \right>)^2 \right> \nonumber\\
&=& \left< (x - \left< x \right>)^2 \right> \cdot
\left< \frac{\sum_{i=1}^N {Q_i}^2}{\left( \sum_{i=1}^N Q_i\right)^2} \right> \nonumber\\
&\sim& {S_1}^2 \cdot \frac{\left< \sum_{i=1}^N {Q_i}^2 \right>}
{N^2 \left< Q \right>^2} \nonumber\\
&\sim& {S_1}^2 \cdot \frac{1}{N} \cdot
\frac{\left< Q^2 \right>}{\left< Q \right>^2} \nonumber\\
&\sim& {S_1}^2 \cdot \frac{1}{N} \cdot \frac{\left< Q \right >^2
+ {\sigma_Q}^2}{\left< Q \right>^2} \nonumber\\
&\sim& {S_1}^2 \cdot \frac{1}{N} \cdot (1 + F^\prime) \end{aligned}$$
where the relative variance $F^\prime \equiv {\sigma_Q}^2\; / \left< Q \right>^2$ with $\sigma_Q$ being the standard deviation of the cluster charge, including the fluctuations in cluster size, and in avalanche gain for each electron in the cluster. Actually
$$F^\prime = F + \frac{1}{\left<n\right>} \cdot f$$
with $F$ ($f$) being the relative variance of the cluster-size (avalanche-size) fluctuation and $\left<n\right>$ the average cluster size. Therefore
$$\lim_{N \rightarrow \infty} \frac{N_{\rm eff}(N)}{N} = \frac{1}{1+F^\prime}
\sim \frac{1}{1+F}$$
because $F$ ($\sim$ 2000 for argon [@ref3]) is much greater than $f$ ($\sim$ 1).
[99]{} M. Kobayashi, [et al.]{}, Nuclear Instrumentation and Methods in Physics Research\
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[^1]: The values of $\sigma_{\rm X00}$ are measured to be about 100 $\mu$m without axial magnetic field ($B$ = 0 T) and $\sim$ 50 $\mu$m for $B$ = 1 T. The observed $B$-dependence of $\sigma_{\rm X00}$ is most likely due to the intrinsic track width. See Appendix C of Ref. [@ref1] for the possible contributors to the intrinsic term.
[^2]: The finite pad-pitch term [@ref1] is neglected here.
[^3]: In Refs. [@ref1; @ref2; @ref3], $n_{\rm eff}$ is denoted as $N_{\rm eff}$, which is reserved for the effective number of [*clusters*]{} per pad row (see below) in the present paper.
[^4]: More precisely $h$ should be understood as the pad-row pitch, which is usually slightly larger than the pad height when the readout plane is covered over with pads. The pad-row pitch and the pad height ($h$) are not distinguished in the present paper.
[^5]: $\theta$ is defined to be 0$^\circ$ when the track is parallel to the readout plane.
[^6]: The value of $\sigma_{\rm X0}$ given by Eq. (2) is therefore practically independent of $z$.
[^7]: The parameter $\theta$ for Polya distributions (see, for example, Ref. [@ref2]) should not be confused with the track angle $\theta$ defined above. We use the same symbol since they can be easily distinguished by their units.
[^8]: This is a bolder assumption than $n_{\rm eff} \propto h$ above (see Fig. 1).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The magnetic structure of BaFe$_2$As$_2$ was completely determined from polycrystalline neutron diffraction measurements soon after the ThCr$_2$Si$_2$-type FeAs-based superconductors were discovered. Both the moment direction and the in-plane antiferromagnetic wavevector are along the longer $a$-axis of the orthorhombic unit cell. There is only one combined magnetostructural transition at $\sim$140 K. However, a later single-crystal neutron diffraction work reported contradicting results. Here we show neutron diffraction results from a clean single crystal sample, grown by a self-flux method, that support the original polycrystalline work.'
address:
- '$^1$ Department of Physics, University of Virginia, Charlottesville, VA 22904, USA'
- '$^2$ NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA'
- '$^3$ Dept. of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA'
- '$^4$ Los Alamos National Laboratory, Los Alamos, NM 87545, USA'
- '$^5$ Hefei National Laboratory for Physical Science at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China'
author:
- 'M. Kofu$^1$, Y. Qiu$^{2,3}$, Wei Bao$^4$, S.-H. Lee$^1$, S. Chang$^2$, T. Wu$^5$, G. Wu$^5$ and X. H. Chen$^5$'
title: 'Neutron scattering investigation of the magnetic order in single crystalline BaFe$_2$As$_2$'
---
Introduction
============
Over the last two years, several superconductors have been discovered in fluorine-doped lanthanum oxypnictides LaFePO [@Kamihara2006], LaNiPO [@Watanabe2007] and LaFeAsO [@Kamihara2008]. The common ZrCuSiAs-type (1111) structure is composed of alternating Fe(Ni)As(P) and LaO layers. Band structure calculations indicate that electronic states at the Fermi level are contributed predominately by the transition metal Fe or Ni in these semimetals [@bs07; @A030429; @A031279; @A060750; @A071010; @A103274], thus the importance of the layer formed by the edge-sharing Fe(Ni) pnictide tetrahedra. The same edge-sharing tetrahedral layer is also the central structural component in the previously discovered $Ln$Ni$_2$B$_2$C superconductors [@NiBC_str]. However, a distinguishing feature of the LaFeAsO system is that its superconducting transition temperature $T_C$ increases from 26 K to above 40 K [@A033603; @A033790], and finally reaches about 55 K at optimal F doping [@A042053; @A042105] when La is replaced by magnetic Sm or Ce. Other magnetic lanthanide substitutions have also been shown to result in increased $T_C$ [@A034234; @A034283; @A034384; @A040835; @A042582; @A043727; @A044290; @A060926]. Therefore, an unconventional superconducting mechanism is suspected. Indeed while the FeP and NiP materials with their low $T_C\approx 3$-4 K may be accounted for by conventional electron-phonon interactions, the $T_C$ of the FeAs materials is too high for the phonon mechanism according to theoretical calculations [@A032703]. The first-principle phonon spectrum used in the calculations has since found support from neutron scattering, optical and resonant x-ray scattering measurements [@A051062; @A051321; @A073370; @A073172; @A073968].
While the $Ln$O layer ($Ln$=La, Sm, Ce, Nd, Pr, Gd, Tb or Dy [@Kamihara2008; @A033603; @A033790; @A042053; @A042105; @A034234; @A034283; @A034384; @A040835; @A042582; @A043727; @A044290; @A060926]) provides an excellent opportunity to investigate the interaction between superconductivity and rare-earth magnetism in the $Ln$FeAsO systems, its existence is not necessary for superconductivity. Superconductivity has been discovered in related materials with the ThCr$_2$Si$_2$-type (122) structure, where the $Ln$O layer is replaced by elemental Ba [@A054630], Sr [@A061209; @A061301] or Ca [@A064279] layer, and in Li$_{1-x}$FeAs [@A064688] and Fe$_{1+x}$(Se,Te) [@A072369; @A074775] with the PbO-type (11) structure, which does not contain the intervening layer, but Li [@A072228] or excess Fe [@A092058] occupies an interstitial site. Therefore, the multi-orbital theoretical model based on similar semimetallic electronic states from the common Fe structural layer is likely to capture the essential physics for understanding high $T_C$ superconductivity in these Fe-based materials [@A032740; @A033325; @A033982; @A034346; @A041113; @A044678; @A061933; @A063285; @A101476]. In addition, magnetic fluctuations have been proposed as the bosonic glue for Cooper pair formation. Currently, the extended $s$-wave superconductivity mediated by magnetic fluctuations is favored. Experiments supporting a nodeless superconducting gap have emerged [@A054616; @A070398; @A070419].
Stoichiometric $Ln$FeAsO and $A$Fe$_2$As$_2$ ($A$ = Ba, Sr or Ca) are not superconductors. LaFeAsO experiences a structural transition from tetragonal to orthorhombic symmetry at 150 K, which shows up as a strong anomaly in resistivity, and an antiferromagnetic transition at 137 K [@A040795; @A043569]. For BaFe$_2$As$_2$, the structural and magnetic transitions occur at the same temperature [@A062776]. The magnetic propagation vector is ($\pi,0,\pi$) in terms of the primitive tetragonal magnetic unit cell for both LaFeAsO [@A040795; @A063878] and BaFe$_2$As$_2$ [@A062776], although their crystal structures are different. When La is replaced by magnetic Nd or Pr, the magnetic wavevector changes to ($\pi,0,0$) in the combined Fe and rare-earth magnetic order at low temperature [@A062195; @A074441; @A074872]. For Ce substitutions, a different antiferromagnetic ordering of Ce ions was reported without the refinements provided [@A062528], but the Fe part of the magnetic order is still the same as we reported for the Nd compound and is characterized by ($\pi,0,0$) [@A062195].
The in-plane ($\pi,0$) magnetic wavevector is consistent with the nesting of electron and hole Fermi surfaces, which has been anticipated from band structure theory [@A033236; @A033325; @A033286]. It breaks the tetragonal symmetry of the high temperature structure and is consistent with the orthorhombic distortion at low temperature. The antiparellel moment alignment is determined by neutron diffraction to be along the longer of the in-plane axes, and the parallel alignment along the shorter axis of the orthorhombic unit cell in both NdFeAsO and BaFe$_2$As$_2$ [@A062195; @A062776]. This magnetostriction pattern is opposite to the usual case of single orbital magnetism and is explained by careful calculations taking into account the multi-orbital origin of the antiferromagnetic order [@A042252]. The moment direction has also been determined to be along the longer of the in-plane axes [@A062195; @A062776], and the same magnetostrictive expansion and contraction have been found later in SrFe$_2$As$_2$ [@A070632; @A071077], CaFe$_2$As$_2$ [@A071525] and PrFeAsO [@A074441; @A074872] in poly- and single-crystal studies. However, in a single crystal neutron diffraction study of BaFe$_2$As$_2$ [@A071743], results different from our polycrystalline work [@A062776] have been reported. To clarify the issue, we show single crystal results in section \[sec3\], which are consistent with our previous polycrystalline study.
Experimental details
====================
The single crystal sample of BaFe$_2$As$_2$ was grown using a self-flux method [@A062452]. A distinct feature of single crystals grown this way is that the resistivity shows a sharp drop at the phase transition at $\sim$140 K, similar to results from polycrystalline samples. The single crystal used by Su in their single crystal neutron diffraction work was grown in Sn flux [@A071743]. Not only is the transition temperature much reduced, the material becomes an insulator at low temperature, in contrast to the expected metallic behavior. We conducted single crystal neutron diffraction measurements with the cold neutron triple-axis spectrometer SPINS at NIST Center for Neutron Research. The sample was mounted in a He-filled Al can in a closed cycle refrigerator so that ($h0l$) was in the scattering plane. Neutrons of 5 meV were selected using pyrolytic graphite (002) as both monochromator and analyzer. A cold Be filter was placed in the neutron path to reduce contamination from higher order neutrons. The lattice parameters are $a=5.615$, $b=5.571$ and $c=12.97\AA$ at 12.5 K in the othorhombic structure.
Single crystal diffraction experiments {#sec3}
======================================
The mosaic of our BaFe$_2$As$_2$ single crystal sample is shown in (b). The composition uniformity is indicated by the nice peak in the $\theta-2\theta$ scan in (a). At 12.5 K, the orthorhombic distortion of the crystal structure is indicated by the well resolved (400) and (040) Bragg peaks due to twinning in (c). That $a>b$ is reflected in the shorter reciprocal length of the $|(400)|$ in comparison to $|(040)|$. The fact that only (101), not the twinning (011), peak exists in (d) is consistent with our previous determination of the (101), not (011), as the magnetic propagation vector with the definition of $a>b$ [@A062776]. It is opposite to what was reported by Su for their Sn-flux grown single crystal sample [@A071743]. The $l$ scan in (e), close to the rocking direction, further supports the commensurate assignment of the magnetic propagation vector.
![ (a-b) The $\theta-2\theta$ and rocking scans of the (004) structural Bragg peak. (c) The twinning (400) and (040) structural Bragg peaks measured using $\lambda/2$ neutrons near the (200) position. (d-e) Two perpendicular scans through the (101) magnetic Bragg peaks. (f) The (103) magnetic Bragg peak. The blue symbols represent measurements at 12.5 K, and the red at 150 K. The error bars in all the figures represent the standard deviation in the measurement.[]{data-label="fig1"}](fig1.ps "fig:") -.2 cm
Consistent with our previous data [@A062776], the (103) magnetic Bragg peak in (f) is stronger than that of (101), reflecting the moment orientation factor in the magnetic neutron diffraction cross-section when the moment points along the (100) direction. At 150 K above the simultaneous magnetostructural transition, magnetic Bragg peaks disappear completely as shown in (d) for (101).
While there is no dispute regarding the first order nature of the structural transition in BaFe$_2$As$_2$ [@A062776], the continuous appearance of the magnetic order parameter raises a scenario where the magnetic transition is of second order. If so, the lack of hysteresis in the presumed second order magnetic transition would indicate two phase transitions during a cooling/heating cycle, since obvious hysteresis in the structural transition has been observed, suggesting a similar situation to the double transition case of LaFeAsO [@A043569]. However, a continuous appearance of the magnetic order parameter does not preclude a magnetic transition with first-order hysteresis. This has been shown previously to occur in Ca$_3$Ru$_2$O$_7$ in a wide phase space of the temperature-magnetic field plane [@bao08a], where there exists a lattice contraction associating with a Mott transition. It has also been demonstrated for CaFe$_2$As$_2$ [@A071525], verifying that there is indeed only one simultaneous magnetic and structural transition in the 122 materials. In , the squared magnetic order parameter is shown during the cooling and warming cycle for the BaFe$_2$As$_2$ single crystal sample. The small difference between measurements using the two ramping rates indicates the rate is slow enough. A supercooling of about 20 K was observed, which is twice that observed previously for the polycrystalline sample [@A062776]. This larger hysteresis in the single crystal sample is expected due to larger structural strain to be dissipated in the single crystal at the first order transition.
![Temperature dependence of the magnetic (101) Bragg peak as a measure of the squared magnetic order parameter. The blue symbols were measured during cooling and the red heating. The temperature ramping rate was 2.3 K/min for , and 4.5 K/min for . []{data-label="fig2"}](fig2.ps "fig:") -.2 cm
Discussions
===========
Using a single crystal sample of BaFe$_2$As$_2$ grown by a self-flux method, the original magnetic structure determined using a polycrystal sample [@A062776] is confirmed. The single crystal sample used by Su has very different physical properties, likely due to inclusion of the Sn flux into the crystal. It is not clear whether the Sn inclusion is also responsible for the very different magnetic structure reported by them. Their sample also showed decoupled magnetic and structural transitions, and the orthorhombic distortion survived at high temperature in the heating cycle. This is different from our poly and now also single crystal results. Notice that except for the results reported by Su , magnetic and structural properties of the Ba, Sr and Ca 122 materials are very similar.
The prediction of the ($\pi,0$) in-plane magnetic wavevector from the nesting of quasi-two-dimensional Fermi surface before experiments [@A033236; @A033325; @A033286] certainly has boosted the credential of the spin-density-wave (SDW) mechanism for antiferromagnetism discovered in both the 1111 and 122 materials. The same SDW prediction for Fe$_{1+x}$Te [@A074312], however, differs from our observed magnetic structure characterized by a completely different magnetic propagation vector ($\delta\pi,\delta\pi,\pi$) with $\delta$ tunable from 0.346 to 0.5, namely from incommensurate to commensurate magnetic structure, by excess Fe composition [@A092058]. In addition, the customary energy gap from spin-density-wave order is absent in angle resolved photoemission spectroscopy (ARPES) measurements [@A062627]. A possible cause due to a topological constraint of degenerate orbitals has been advanced theoretically [@A053535]. On the other hand, there has been another theoretical approach from the strong correlation side which explains magnetic order in the Fe based materials from a localized magnetic moment picture [@A042252; @A042480]. Both localized and itinerant theories now exist for Fe$_{1+x}$Te [@A094732; @A103274; @A111294], and incommensurate magnetic order is also possible from either localized or itinerant perspective [@A042252; @A104469]. Nevertheless, the strength of electronic correlations may lie in between the weak and strong correlation limits in the ferrous high $T_C$ superconductors, since electronic band structure measured by ARPES shows certain departures from the LDA band structure [@A062627; @A070398; @A070419; @A072009].
Summary
=======
We have performed a single crystal neutron diffraction study on BaFe$_2$As$_2$. The results are completely consistent with those from our original work using a polycrystalline sample [@A062776]. Despite the appearance of a continuous magnetic order parameter, there is only one combined magnetostructural first-order transition in BaFe$_2$As$_2$. The conflicting results reported by Su using a Sn-flux grown single crystal sample are most likely erroneous to represent intrinsic properties of BaFe$_2$As$_2$.
Work at LANL is supported by U.S. DOE-OS-BES, at USTC by the Natural Science Foundation of China, Ministry of Science and Technology of China (973 Project No: 2006CB601001) and by National Basic Research Program of China (2006CB922005), at UVA by the U.S. DOE through DE- FG02-07ER45384. The SPINS at NIST are partially supported by NSF under Agreement No. DMR-0454672.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The work explores the fundamental limits of coded caching in heterogeneous networks where multiple ($N_0$) senders/antennas, serve different users which are associated (linked) to shared caches, where each such cache helps an arbitrary number of users. Under the assumption of uncoded cache placement, the work derives the exact optimal worst-case delay and DoF, for a broad range of user-to-cache association profiles where each such profile describes how many users are helped by each cache. This is achieved by presenting an information-theoretic converse based on index coding that succinctly captures the impact of the user-to-cache association, as well as by presenting a coded caching scheme that optimally adapts to the association profile by exploiting the benefits of encoding across users that share the same cache.
The work reveals a powerful interplay between shared caches and multiple senders/antennas, where we can now draw the striking conclusion that, as long as each cache serves at least $N_0$ users, adding a single degree of cache-redundancy can yield a DoF increase equal to $N_0$, while at the same time — irrespective of the profile — going from 1 to $N_0$ antennas reduces the delivery time by a factor of $N_0$. Finally some conclusions are also drawn for the related problem of coded caching with multiple file requests.
author:
- 'Emanuele Parrinello, Ayşe Ünsal and Petros Elia[^1] [^2]'
bibliography:
- 'final\_refs2.bib'
nocite:
- '[@MN14; @WanTP15; @YuMA16]'
- '[@lampiris2018lowCSIT; @LampirisZE17]'
title: Fundamental Limits of Caching in Heterogeneous Networks with Uncoded Prefetching
---
Caching networks, coded caching, shared caches, delivery rate, uncoded cache placement, index coding, MISO broadcast channel, multiple file requests, network coding.
Introduction \[sec:intro\]
==========================
In the context of communication networks, the emergence of predictable content, has brought to the fore the use of caching as a fundamental ingredient for handling the exponential growth in data volumes. A recent information theoretic exposition of the cache-aided communication problem [@MN14], has revealed the potential of caching in allowing for the elusive scaling of networks, where a limited amount of (bandwidth and time) resources can conceivably suffice to serve an ever increasing number of users.
Coded Caching
-------------
This exposition in [@MN14] considered a shared-link broadcast channel (BC) scenario where a single-antenna transmitter has access to a library of $N$ files, and serves (via a single bottleneck link) $K$ receivers, each having a cache of size equal to the size of $M$ files.
In a normalized setting where the link has capacity 1 file per unit of time, the work in [@MN14] showed that any set of $K$ simultaneous requests (one file requested per user) can be served with a normalized delay (worst-case completion time) which is at most $T = \frac{K(1-\gamma)}{1+K\gamma}$ where $\gamma {\triangleq}\frac{M}{N} $ denotes the normalized cache size. This implied an ability to treat $K\gamma+1$ users at a time; a number that is often referred to as the cache-aided sum *degrees of freedom* (DoF) $d_{\Sigma} {\triangleq}\frac{K(1-\gamma)}{T}$, corresponding to a caching gain of $K\gamma$ additional served users due to caching.
For this same shared-link setting, this performance was shown to be approximately optimal (cf. [@MN14]), and under the basic assumption of uncoded cache placement where caches store uncoded content from the library, it was shown to be exactly optimal (cf. [@WanTP15] as well as [@YuMA16]).
Such high coded caching gains have been shown to persist in a variety of settings that include uneven popularity distributions [@JiTLC14; @NiesenMtit17Popularity; @ZhangLW:18], uneven topologies [@BidokhtiWT16isit; @ZhangE16b], a variety of channels such as erasure channels [@GhorbelKY:16], MIMO broadcast channels with fading [@ZE:17tit], a variety of networks such as D2D networks [@JiCM16D2D], coded caching under secrecy constraints [@RPK+:16], and in other settings as well [@SenguptaTS15; @CaoTXL16; @RoigTG17a; @CaoTaoMultiAntenna18; @PiovClerckISIT18; @BayatMC:18arxiv]. Recently some progress has also been made in ameliorating the well known subpacketization bottleneck; for this see for example [@ShanmugamJTLD16it; @JiSVLTC15; @YanCTC:17tit; @TangR:17isit; @ShangguanZG:18tit; @ShanmugamTD:17isit; @LampirisEliaJsac18].
{width="0.4\linewidth"} {width="0.4\linewidth"} \[system\_pic\]
Cache-aided Heterogeneous Networks: Coded Caching with Shared Caches
--------------------------------------------------------------------
Another step in further exploiting the use of caches, was to explore coded caching in the context of the so-called heterogeneous networks which better capture aspects of more realistic settings such as larger wireless networks. Here the term *heterogeneous* refers to scenarios where one or more (typically) multi-antenna transmitters (base-stations) communicate to a set of users, with the assistance of smaller nodes. In our setting, these smaller helper nodes will serve as caches that will be shared among the users. This cache-aided heterogeneous topology nicely captures an evolution into denser networks where many wireless access points work in conjunction with bigger base stations, in order to better handle interference and to alleviate the backhaul load by replacing backhaul capacity with storage capacity at the communicating nodes.
The use of caching in such networks was famously explored in the *Femtocaching* work in [@GSDMC:12], where wireless receivers are assisted by helper nodes of a limited cache size, whose main role is to bring content closer to the users. A transition to coded caching can be found in [@Diggavi_IT] which considered a similar shared-cache heterogeneous network as here, where each receiving user can have access to a main single-antenna base station (single-sender) and to different helper caches. In this context, under a uniform user-to-cache association where each cache serves an equal number of users, [@Diggavi_IT] proposes a coded caching scheme which was shown to perform to within a certain constant factor from the optimal. This uniform setting is addressed also in [@MND13], again for the single antenna case. Interesting work can also be found in [@XuGW:18sharedArxiv] which explores the single-stream (shared-link) coded caching scenario with shared caches, where the uniformity condition is lifted, and where emphasis is placed on designing schemes with centralized coded prefetching with small sum-size caches where the total cache size is smaller than the library size (i.e., where $KM < N$).
#### Coded caching with multiple file requests {#coded-caching-with-multiple-file-requests .unnumbered}
Related work can also be found on the problem of coded caching with multiple file requests per receiver, which — for the single-stream, error free case — is closely related to the shared cache problem here. Such work — all in the shared-link case ($N_0=1$) — appears in [@Ji2015CachingaidedCM; @SenguptaT:17TCOMM] in the context of single-layer coded caching. Somewhat related work also appears in [@ZhangWXWL16; @KaramchandaniNMD16IT] in the context of hierarchical coded caching. Recent progress can also be found in [@WeUlukus17] which establishes the exact optimal worst-case delay — under uncoded cache placement, for the shared-link case — for the uniform case where each user requests an equal number of files[^3]. As a byproduct of our results here, in the context of worst-case demands, we establish the exact optimal performance of the multiple file requests problem for any (not necessarily uniform) user-to-file association profile.
#### Current work {#current-work .unnumbered}
In the heterogeneous setting with shared caches, we here explore the effect of user-to-cache association profiles and their non-uniformity, and we characterize how this effect is scaled in the presence of multiple antennas or multiple senders. Such considerations are motivated by realistic constraints in assigning users to caches, where these constraints may be due to topology, cache capacity or other factors. As it turns out, there is an interesting interplay between all these aspects, which is crisply revealed here as a result of a new scheme and an outer bound that jointly provide exact optimality results. Throughout the paper, emphasis will be placed on the shared-cache scenario, but some of the results will be translated directly to the multiple file request problem which will be described later on.
Notation
--------
For $n$ being a positive integer, $[n]$ refers to the following set $[n]\triangleq \{1,2,\dots,n\}$, and $2^{[n]}$ denotes the power set of $[n]$. The expression $\alpha | \beta$ denotes that integer $\alpha$ divides integer $\beta$. Permutation and binomial coefficients are denoted and defined by $P(n,k){\triangleq}\frac{n!}{(n-k)!}$ and $\binom{n}{k}{\triangleq}\frac{n!}{(n-k)!k!}$, respectively. For a set $\mathcal{A}$, $|\mathcal{A}|$ denotes its cardinality. $\mathbb{N}$ represents the natural numbers. We denote the lower convex envelope of the points $\{(i, f(i)) | i \in [n]\cup \{0\}\}$ for some $n\in \mathbb{N}$ by $Conv(f(i))$. The concatenation of a vector $\vv$ with itself $N$ times is denoted by $(\vv \Vert \vv)_{N}$. For $n\in \mathbb{N}$, we denote the symmetric group of all permutations of $[n]$ by $S_n$. To simplify notation, we will also use such permutations $\pi\in S_n$ on vectors $\vv \in \mathbb{R}^n$, where $\pi(\vv)$ will now represent the action of the permutation matrix defined by $\pi$, meaning that the first element of $\pi(\vv)$ is $\vv_{\pi(1)}$ (the $\pi(1)$ entry of $\vv$), the second is $\vv_{\pi(2)}$, and so on. Similarly $\pi^{-1}(\cdot )$ will represent the inverse such function and $\pi_s(\vv)$ will denote the sorted version of a real vector $\vv$ in descending order.
Paper Outline
-------------
In Section \[sec:systemModel\] we give a detailed description of the system model and the problem definition, followed by the main results in Section \[sec:results\], first for the shared-link setting with shared caches[^4], and then for the multi-antenna/multi-sender setting. In Section \[sec:scheme\], we introduce the scheme for a broad range of parameters. The scheme is further explained with an example in this section. We present the information theoretic converse along with an explanatory example for constructing the lower bound in Section \[sec:converse\]. Lastly, in Section \[sec:discussion\] we draw some basic conclusions, while in the Appendix Section \[sec:Appendix\] we present some proof details.
System Model\[sec:systemModel\]
===============================
We consider a basic broadcast configuration with a transmitting server having $N_0$ transmitting antennas and access to a library of $N$ files $W^{1},W^{2},\dots ,W^{N}$, each of size equal to one unit of ‘file’, where this transmitter is connected via a broadcast link to $K$ receiving users and to $\Lambda\leq K$ helper nodes that will serve as caches which store content from the library[^5]. The communication process is split into $a)$ the cache-placement phase, $b)$ the user-to-cache assignment phase during which each user is assigned to a single cache, and $c)$ the delivery phase where each user requests a single file independently and during which the transmitter aims to deliver these requested files, taking into consideration the cached content and the user-to-cache association.
#### Cache placement phase
During this phase, helper nodes store content from the library without having knowledge of the users’ requests. Each helper cache has size $M\leq N$ units of file, and no coding is applied to the content stored at the helper caches; this corresponds to the common case of *uncoded cache placement*. We will denote by $\mathcal{Z}_{\lambda}$ the content stored by helper node $\lambda$ during this phase. The cache-placement algorithm is oblivious of the subsequent user-to-cache association $\mathcal{U}$.
#### User-to-cache association
After the caches are filled, each user is assigned to exactly *one* helper node/cache, from which it can download content at zero cost. Specifically, each cache $
\lambda = 1,2,\dots,\Lambda$, is assigned to a set of users $\mathcal{U}_\lambda$, and all these disjoint sets $$\mathcal{U}{\stackrel{\triangle}{=}}\{\mathcal{U}_1,\mathcal{U}_2,\dots ,\mathcal{U}_\Lambda\}$$ form the partition of the set of users $\{1,2,\dots,K\}$, describing the overall association of the users to the caches.
This cache assignment is independent of the cache content and independent of the file requests to follow. We here consider any arbitrary user-to-cache association $\mathcal{U}$, thus allowing the results to reflect both an ability to choose/design the association, as well as to reflect possible association restrictions due to randomness or topology. Similarly, having the user-to-cache association being independent of the requested files, is meant to reflect the fact that such associations may not be able to vary as quickly as a user changes the requested content.
#### Content delivery
The delivery phase commences when each user $k = 1,\dots,K$ requests from the transmitter, any *one* file $W^{d_{k}}$, $d_{k}\in\{1,\dots,N\}$ out of the $N$ library files. Upon notification of the entire *demand vector* $\dv=(d_1,d_2,\dots,d_{K})\in\{1,\dots,N\}^K$, the transmitter aims to deliver the requested files, each to their intended receiver, and the objective is to design a *caching and delivery scheme $\chi$* that does so with limited (delivery phase) duration $T$, where the delivery algorithm has full knowledge of the user-to-cache association $\mathcal{U}$.
For each transmission, the received signals at user $k$, take the form $$\begin{aligned}
y_{k}=\hv_{k}^{T} \xv + w_{k}, ~~ k = 1, \dots, K\end{aligned}$$ where $\xv\in\mathbb{C}^{N_0\times 1}$ denotes the transmitted vector satisfying a power constraint $\E(||\xv||^2)\leq P$, $\hv_{k}\in\mathbb{C}^{N_0\times 1}$ denotes the channel of user $k$, and $w_{k}$ represents unit-power AWGN noise at receiver $k$. We will assume that the allowable power $P$ is high (i.e., we will assume high signal-to-noise ratio (SNR)), that there exists perfect channel state information throughout the (active) nodes, that fading is statistically symmetric, and that each link (one antenna to one receiver) has ergodic capacity $\log(SNR)+o(log(SNR))$.
#### User-to-cache association profiles, and performance measure
As one can imagine, some user-to-cache association instances $\mathcal{U}$ may allow for higher performance than others; for instance, one can suspect that more uniform profiles may be preferable. Part of the objective of this work is to explore the effect of such associations on the overall performance. Toward this, for any given $\mathcal{U}$, we define the association *profile* (sorted histogram) $$\Lc=(\mathcal{L}_{1},\dots,\mathcal{L}_{\Lambda})$$ where $\mathcal{L}_{\lambda}$ is the number of users assigned to the $\lambda$-th *most populated* helper node/cache[^6]. Naturally, $\sum_{\lambda=1}^\Lambda \mathcal{L}_{\lambda} = K$. Each profile $\Lc$ defines a *class* $\mathcal{U}_{\Lc}$ comprising all the user-to-cache associations $\mathcal{U}$ that share the same[^7] profile $\Lc$.
As in [@MN14], the measure of interest $T$ is the number of time slots, per file served per user, needed to complete delivery of any file-request vector[^8] $\dv$. We use $T(\mathcal{U},\dv,\chi)$ to define the delay required by some generic caching-and-delivery scheme $\chi$ to satisfy demand $\dv$ in the presence of a user-to-cache association described by $\mathcal{U}$. To capture the effect of the user-to-cache association, we will characterize the optimal worst-case delivery time $$T^*(\Lc){\triangleq}\min_{\chi} \max_{(\mathcal{U},\dv) \in (\mathcal{U}_{\Lc},\{1,\dots,N\}^K)} T(\mathcal{U},\dv,\chi) \label{eq:T*_def}$$ for each class. Our interest is in the regime of $N\geq K$ where there are more files than users.
Main Results \[sec:results\]
============================
We first describe the main results for the single antenna case[^9] (shared-link BC), and then generalize to the multi-antenna/multi-sender case.
Shared-Link Coded Caching with Shared Caches\[subsec:single\_antenna\_lower\]
-----------------------------------------------------------------------------
The following theorem presents the main result for the shared-link case ($N_0 = 1$).
\[thm:PerClassSingleAntenna\] In the $K$-user shared-link broadcast channel with $\Lambda$ shared caches of normalized size $\gamma$, the optimal delivery time within any class/profile $\Lc$ is $$\label{eq:TS_L}
T^*(\Lc)=Conv\bigg(\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_r{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}\bigg)$$ at points $\gamma\in \{\frac{1}{\Lambda},\frac{2}{\Lambda},\dots,1\}$.
*Proof.* The achievability part of the proof is given in Section \[sec:scheme\], and the converse is proved in Section \[sec:converse\] after setting $N_0 = 1$.
We note that the converse that supports Theorem \[thm:PerClassSingleAntenna\], encompasses the class of all caching-and-delivery schemes $\chi$ that employ uncoded cache placement under a general sum cache constraint $\frac{1}{\Lambda}\sum_{\lambda=1}^\Lambda |\mathcal{Z}_\lambda | = M$ which does not *necessarily* impose an individual cache size constraint. The converse also encompasses all scenarios that involve a library of size $\sum_{n\in[N]}|W^{n}| = N$ but where the files may be of different size. In the end, even though the designed optimal scheme will consider an individual cache size $M$ and equal file sizes, the converse guarantees that there cannot exist a scheme (even in settings with uneven cache sizes or uneven file sizes) that exceeds the optimal performance identified here.
From Theorem \[thm:PerClassSingleAntenna\], we see that in the uniform case[^10] where $\Lc=(\frac{K}{\Lambda},\frac{K}{\Lambda},\dots,\frac{K}{\Lambda})$, the expression in reduces to $$T^*(\Lc)=\frac{K(1-\gamma)}{\Lambda\gamma+1}$$ matching the achievable delay presented in [@MND13]. It also matches the recent result by [@WeUlukus17] which proved that this performance — in the context of the multiple file request problem — is optimal under the assumption of uncoded cache placement.
The following corollary relates to this uniform case.
\[cor:ressym\] In the uniform user-to-cache association case where $\Lc=(\frac{K}{\Lambda},\frac{K}{\Lambda},\dots,\frac{K}{\Lambda})$, the aforementioned optimal delay $T^*(\Lc)=\frac{K(1-\gamma)}{\Lambda\gamma+1}$ is smaller than the corresponding delay $T^*(\Lc)$ for any other non-uniform class.
The proof that the uniform profile results in the smallest delay among all profiles, follows directly from the fact that in , both $\mathcal{L}_r$ and ${\Lambda-r\choose \Lambda\gamma}$ are non-increasing with $r$.
Multi-antenna/Multi-sender Coded Caching with Shared Caches
-----------------------------------------------------------
The following extends Theorem \[thm:PerClassSingleAntenna\] to the case where the transmitter is equipped with multiple ($N_0>1$) antennas. The results hold for any $\Lc$ as long as any non zero $\mathcal{L}_\lambda$ satisfies $\mathcal{L}_\lambda\geq N_0, ~\forall\lambda\in[\Lambda]$.
\[thm:resmultiant\] In the $N_0$-antenna $K$-user broadcast channel with $\Lambda$ shared caches of normalized size $\gamma$, the optimal delivery time within any class/profile $\Lc$ is $$\label{eq:multi_delay}
T^*(\Lc,N_0)=\frac{1}{N_0}Conv\bigg(\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_{r}{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}\bigg)\\$$ for $\gamma\in \left\{\frac{1}{\Lambda},\frac{2}{\Lambda},\dots,1\right\}$. This reveals a multiplicative gain of $N_0$ with respect to the single antenna case.
*Proof.* The scheme that achieves (\[eq:multi\_delay\]) is presented in Section \[sec:scheme\], and the converse is presented in Section \[sec:converse\].
The following extends Corollary \[cor:ressym\] to the multi-antenna case, and the proof is direct from Theorem \[thm:resmultiant\].
\[cor:ressymMulti\] In the uniform user-to-cache association case of $\Lc=\left(\frac{K}{\Lambda},\frac{K}{\Lambda},\dots,\frac{K}{\Lambda}\right)$ where $N_0\leq \frac{K}{\Lambda}$, the optimal delay is $$\label{delay_unif_multi}
T^*(\Lc)=\frac{K(1-\gamma)}{N_0(\Lambda\gamma+1)}$$ and it is smaller than the corresponding delay $T^*(\Lc)$ for any other non-uniform class.
\[rem:multipleFilerequstsResult\] In the error-free shared-link case ($N_0 = 1$), with file-independence and worst-case demand assumptions, the shared-cache problem here is closely related to the coded caching problem with multiple file requests per user, where now $\Lambda$ users with their own cache, request in total $K\geq \Lambda$ files. In particular, changing a bit the format, now each demand vector $\dv = (d_1,d_2,\dots,d_K)$ would represent the vector of the indices of the $K$ requested files, and each user $\lambda = \{1,2,\dots,\Lambda\}$, would request those files from this vector $\dv$, whose indices[^11] form the set $\mathcal{U}_\lambda \subset [K]$. At this point, as before, the problem is now defined by the user-to-file association $\mathcal{U} = \{\mathcal{U}_1,\mathcal{U}_2,\dots ,\mathcal{U}_\Lambda\}$ which describes — given a fixed demand vector $\dv$ — the files requested by any user. From this point on, the equivalence with the original shared cache problem is complete. As before, each such $\mathcal{U}$ again has a corresponding (sorted) profile $\Lc=(\mathcal{L}_{1},\mathcal{L}_{2},\dots,\mathcal{L}_{\Lambda})$, and belongs to a class $\mathcal{U}_{\Lc}$ with all other associations $\mathcal{U}$ that share the same profile $\Lc$. As we quickly show in the Appendix Section \[sec:AppendixMultipleFileRequests\], our scheme and converse can be adapted to the multiple file request problem, and thus directly from Theorem \[cor:ressym\] we conclude that for this multiple file request problem, the optimal delay $T^*(\Lc){\triangleq}\min_{\chi} \max_{(\mathcal{U},\dv) \in (\mathcal{U}_{\Lc},\{1,\dots,N\}^K)} T(\mathcal{U},\dv,\chi)$ corresponding to any user-to-file association profile $\Lc$, takes the form $T^*(\Lc)= Conv\bigg(\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_{r}{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}\bigg)$. At this point we close the parenthesis regarding multiple file requests, and we refocus exclusively on the problem of shared caches.
Interpretation of Results
-------------------------
### Capturing the effect of the user-to-cache association profile
In a nutshell, Theorems \[thm:PerClassSingleAntenna\],\[thm:resmultiant\] quantify how profile non-uniformities bring about increased delays. What we see is that, the more skewed the profile is, the larger is the delay. This is reflected in Figure \[fig:performance\] which shows — for a setting with $K=30$ users and $\Lambda=6$ caches — the memory-delay trade-off curves for different user-to-cache association profiles. As expected, Figure \[fig:performance\] demonstrates that when all users are connected to the same helper cache, the only gain arising from caching is the well known *local caching gain*. On the other hand, when users are assigned uniformly among the caches (i.e., when $\mathcal{L}_{\lambda}=\frac{K}{\Lambda},\forall\lambda\in[\Lambda]$) the caching gain is maximized and the delay is minimized.
![Optimal delay for different user-to-cache association profiles $\Lc$, for $K=30$ users and $\Lambda=6$ caches.[]{data-label="fig:performance"}](converse_thin.pdf){width="0.95\linewidth"}
### A multiplicative reduction in delay
Theorem \[thm:resmultiant\] states that, as long as each cache is associated to at least $N_0$ users, we can achieve a delay $T(\Lc,N_0)= \frac{1}{N_0}\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_{r}{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}$. The resulting reduction $$\label{eq:ratioT}
\frac{T(\Lc,N_0=1)}{T(\Lc,N_0) }= N_0$$ as compared to the single-stream case, comes in strong contrast to the case of $\Lambda = K$ where, as we know from [@ShariatpanahiMK16it], this same reduction takes the form $$\label{eq:ratioTold}
\frac{T(\Lambda = K,N_0=1)}{T(\Lambda = K,N_0)} = \frac{\frac{K(1-\gamma)}{1+\Lambda\gamma}}{\frac{K(1-\gamma)}{N_0+\Lambda\gamma}} = \frac{N_0+\Lambda\gamma}{1+\Lambda\gamma}$$ which approaches $N_0$ only when $\gamma \rightarrow 0$, and which decreases as $\gamma$ increases.
In the uniform case ($\mathcal{L}_\lambda = \frac{K}{\Lambda}$) with $\Lambda \leq \frac{K}{N_0}$, Corollary \[cor:ressymMulti\] implies a sum-DoF $$d_\Sigma(\gamma)= \frac{K(1-\gamma)}{T} = N_0(1+\Lambda \gamma)$$ which reveals that every time we add a single degree of cache-redundancy (i.e., every time we increase $\Lambda\gamma$ by one), we gain $N_0$ degrees of freedom. This is in direct contrast to the case of $\Lambda = K$ (for which case we recall from [@ShariatpanahiMK16it] that the DoF is $N_0+\Lambda\gamma$) where the same unit increase in the cache redundancy yields only one additional DoF.
### Impact of encoding over users that share the same cache
As we know, both the MN algorithm in [@MN14] and the multi-antenna algorithm in [@ShariatpanahiMK16it], are designed for users with different caches, so — in the uniform case where $\mathcal{L}_\lambda = K/\Lambda$ — one conceivable treatment of the shared-cache problem would have been to apply these algorithms over $\Lambda$ users at a time, all with different caches[^12]. As we see, in the single antenna case, this implementation would treat $1+\Lambda\gamma$ users at a time thus yielding a delay of $T = \frac{K(1-\gamma)}{1+\Lambda\gamma}$, while in the multi-antenna case, this implementation would treat $N_0+\Lambda\gamma$ users at a time (see [@ShariatpanahiMK16it]) thus yielding a delay of $T = \frac{K(1-\gamma)}{N_0+\Lambda\gamma}$. What we see here is that while this direct implementation is optimal (this is what we also do here in the uniform-profile case) in the single antenna case (see [@WeUlukus17], see also Corollary \[cor:ressym\]), in the multi-antenna case, this same approach can have an unbounded performance gap $$\label{eq:gapNaive}
\frac{\frac{K(1-\gamma)}{N_0+\Lambda\gamma}}{\frac{K(1-\gamma)}{N_0(1+\Lambda\gamma)}} = \frac{N_0(1+\Lambda\gamma)}{N_0+\Lambda\gamma}$$ from the derived optimal performance from Corollary \[cor:ressymMulti\]. These conclusions also apply when the user-to-cache association profiles are not uniform; again there would be a direct implementation of existing multi-antenna coded caching algorithms, which would though again have an unbounded performance gap from the optimal performance achieved here.
Coded Caching Scheme\[sec:scheme\]
==================================
This section is dedicated to the description of the placement-and-delivery scheme achieving the performance presented in the general Theorem \[thm:resmultiant\] (and hence also in Theorem \[thm:PerClassSingleAntenna\] and the corollaries). The formal description of the optimal scheme in the upcoming subsection will be followed by a clarifying example in Section \[subsec:example\_scheme\] that demonstrates the main idea behind the design.
Description of the General Scheme
---------------------------------
The placement phase, which uses exactly the algorithm developed in [@MN14] for the case of $(\Lambda=K,M,N)$, is independent of $\mathcal{U},\Lc$, while the delivery phase is designed for any given $\Uc$, and will achieve the optimal worst-case delivery time stated in and . As mentioned, we will assume that any non zero $\mathcal{L}_{\lambda}$ satisfies $\mathcal{L}_{\lambda}\geq N_0, \forall \lambda\in[\Lambda]$.
### Cache Placement Phase \[sec:SchemePlacement\]
The placement phase employs the original cache-placement algorithm of [@MN14] corresponding to the scenario of having only $\Lambda$ users, each with their own cache. Hence — recalling from [@MN14] — first each file $W^n$ is split into $\Lambda \choose \Lambda\gamma$ disjoint subfiles $W^n_\Tau$, for each $\Tau \subset [\Lambda]$, $|\Tau|=\Lambda\gamma$, and then each cache stores a fraction $\gamma$ of each file, as follows $$\Zc_\lambda=\{W^n_\Tau :\Tau\ni\lambda,~ \forall n\in[N]\}.$$
### Delivery Phase\[sec:SchemeDelivery\]
For the purpose of the scheme description only, we will assume without loss of generality that $|\mathcal{U}_1| \geq |\mathcal{U}_2| \geq \dots \geq |\mathcal{U}_{\Lambda}|$ (any other case can be handled by simple relabeling of the caches), and we will use the notation $\mathcal{L}_\lambda {\triangleq}|\Uc_\lambda|$. Furthermore, in a slight abuse of notation, we will consider here each $\mathcal{U}_\lambda$ to be an *ordered vector* describing, in order, the users associated to cache $\lambda$. We will also use $$\label{eq:Alambda}
\boldsymbol{s_{\lambda}} = (\mathcal{U}_\lambda \Vert \mathcal{U}_\lambda )_{N_0}, \lambda\in[\Lambda]$$ to denote the $N_0$-fold concatenation of each $\mathcal{U}_\lambda$. Each such $N_0\mathcal{L}_{\lambda}$-length vector $\boldsymbol{s_{\lambda}}$ can be seen as the concatenation of $\mathcal{L}_\lambda$ different $N_0$-tuples $\boldsymbol{s_{\lambda,j}}$, $j=1,2,\dots, \mathcal{L}_\lambda$, i.e., each $\boldsymbol{s_{\lambda}}$ takes the form[^13] $$\boldsymbol{s_{\lambda}} = \boldsymbol{s_{\lambda,1}} \Vert \boldsymbol{s_{\lambda,2}} \Vert \dots \Vert \underbrace{\boldsymbol{s_{\lambda,\mathcal{L}_\Lambda}}}_{N_0-\text{length}}.$$
The delivery phase commences with the demand vector $\dv$ being revealed to the server. Delivery will consist of $\mathcal{L}_1$ rounds, where each round $j\in[\mathcal{L}_1]$ serves users $$\label{eq:UsersPerRound}
\mathcal{R}_j=\bigcup_{\lambda\in[\Lambda]} \big( \boldsymbol{s_{\lambda,j}}:\mathcal{L}_\lambda \geq j \big).$$
#### Transmission scheme {#transmission-scheme .unnumbered}
Once the demand vector $\dv$ is revealed to the transmitter, each requested subfile $W^{n}_{\Tau}$ (for any $n$ found in $\dv$) is further split into $N_0$ mini-files $\{W^{n}_{\Tau,l}\}_{l\in[N_0]}$. During round $j$, serving users in $\mathcal{R}_j$, we create $\Lambda \choose \Lambda\gamma+1$ sets $\mathcal{Q}\subseteq [\Lambda]$ of size $|\mathcal{Q}|=\Lambda\gamma+1$, and for each set $\mathcal{Q}$, we pick the set of users $$\label{eq:UsersServedPerXOR}
\chi_\mathcal{Q}=\bigcup_{\lambda\in \mathcal{Q}}\big( \boldsymbol{s_{\lambda,j}}:\mathcal{L}_\lambda \geq j \big).$$ If $\chi_\mathcal{Q} = \emptyset$, then there is no transmission, and we move to the next $\mathcal{Q}$. If $\chi_\mathcal{Q}\neq \emptyset$, the server — during this round $j$ — transmits the following vector[^14] $$\label{eq:TransmitSignalGeneral}
\xv_{\chi_{\mathcal{Q}}}=\!\!\!\!\sum_{\lambda\in \mathcal{Q}:\mathcal{L}_\lambda \geq j}\!\!\!\!\mathbf{H}^{-1}_{\boldsymbol{s_{\lambda,j}}}\cdot
\begin{bmatrix}
W^{d_{\boldsymbol{s_{\lambda,j}}(1)}}_{\mathcal{Q}\backslash{\{\lambda\}},l} &\dots &
W^{d_{\boldsymbol{s_{\lambda,j}}(N_0)}}_{\mathcal{Q}\backslash{\{\lambda\}},l}
\end{bmatrix}^T$$ where $W^{d_{\boldsymbol{s_{\lambda,j}}(k)}}_{\mathcal{Q}\backslash{\{\lambda\}},l}$ is a mini-file intended for user $\boldsymbol{s_{\lambda,j}}(k)$, i.e., for the user labelled by the $k$th entry of vector $\boldsymbol{s_{\lambda,j}}$ . The choice of $l$ is sequential, guaranteeing that no subfile $W^{d_{\boldsymbol{s_{\lambda,j}}(k)}}_{\mathcal{Q}\backslash{\{\lambda\}},l}$ is transmitted twice. Since each user appears in $\boldsymbol{s_{\lambda}}$ (and consequently in $\bigcup_{j\in[\mathcal{L}_1]} \mathcal{R}_j$) exactly $N_0$ times, at the end of the $\mathcal{L}_1$ rounds, all the $N_0$ mini-files $W^{d_{\boldsymbol{s_{\lambda,j}}(k)}}_{\mathcal{Q}\backslash{\{\lambda\}},l}$, $l\in[N_0]$ will be sent once. In the above, $\mathbf{H}^{-1}_{\boldsymbol{s_{\lambda,j}}}$ denotes the inverse of the channel matrix between the $N_0$ transmit antennas and the users in vector $\boldsymbol{s_{\lambda,j}}$.
#### Decoding {#decoding .unnumbered}
Directly from , we see that each receiver $\boldsymbol{s_{\lambda,j}}(k)$ obtains a received signal whose noiseless version takes the form $$y_{\boldsymbol{s_{\lambda,j}}(k)}=W^{d_{\boldsymbol{s_{\lambda,j}}(k)}}_{\mathcal{Q}\backslash{\{\lambda\},l}} + \iota_{\boldsymbol{s_{\lambda,j}}(k)}$$ where $\iota_{\boldsymbol{s_{\lambda,j}}(k)}$ is the $k$th entry of the interference vector $$\label{eq:received48}
\sum_{\lambda'\in \mathcal{Q}\setminus{\{\lambda\}}:\mathcal{L}_{\lambda'}\geq j}\!\!\!\mathbf{H}^{-1}_{\boldsymbol{s_{\lambda',j}}}\cdot
\begin{bmatrix}
W^{d_{\boldsymbol{s_{\lambda',j}}(1)}}_{\mathcal{Q}\backslash{\{\lambda'\},l}} &\dots &
W^{d_{\boldsymbol{s_{\lambda',j}}(N_0)}}_{\mathcal{Q}\backslash{\{\lambda'\},l}}
\end{bmatrix}^T .$$ In the above, we see that the entire interference term $\iota_{\boldsymbol{s_{\lambda,j}}(k)}$ experienced by receiver $\boldsymbol{s_{\lambda,j}}(k)$, can be removed (cached-out) because all appearing subfiles $W^{d_{\boldsymbol{s_{\lambda',j}}(1)}}_{\mathcal{Q}\backslash{\{\lambda'\},l}}, \dots, W^{d_{\boldsymbol{s_{\lambda',j}}(N_0)}}_{\mathcal{Q}\backslash{\{\lambda'\},l}}$, for all $\lambda'\in \mathcal{Q}\setminus{\{\lambda\}}, \mathcal{L}_{\lambda'}\geq j$, can be found in cache $\lambda$ associated to this user, simply because $\lambda\in \mathcal{Q}\backslash\{\lambda'\}$.
This completes the proof of the scheme for the multi-antenna case.
### Small modification for the single antenna case\[sec:schemeSingleAntenna\]
For the single-antenna case, the only difference is that now $\boldsymbol{s_{\lambda}} = \mathcal{U}_\lambda$, and that each transmitted vector in during round $j$, becomes a scalar of the form[^15] $$\label{eq:TransmitSignalSingleAntenna}
x_{\chi_{\mathcal{Q}}}=\!\!\!\!\bigoplus_{\lambda\in \mathcal{Q}:\mathcal{L}_\lambda \geq j} W^{d_{\boldsymbol{s_{\lambda,j}}}}_{\mathcal{Q}\backslash{\{\lambda\}},1}.$$ The rest of the details from the general scheme, as well as the subsequent calculation of the delay, follow directly.
Calculation of Delay
--------------------
To first calculate the delay needed to serve the users in $\mathcal{R}_j$ during round $j$, we recall that there are $\Lambda \choose \Lambda\gamma+1$ sets $$\chi_\mathcal{Q}=\bigcup_{\lambda\in \mathcal{Q}}\big( \mathcal{U}_{\lambda}(j):\mathcal{L}_\lambda \geq j \big), \mathcal{Q}\subseteq [\Lambda]$$ of users, and we recall that $|\mathcal{U}_1| \geq |\mathcal{U}_2| \geq \dots \geq |\mathcal{U}_{\Lambda}|$. For each such non-empty set, there is a transmission. Furthermore we see that for $a_j{\stackrel{\triangle}{=}}\Lambda - \frac{|\mathcal{R}_j|}{N_0}$, there are ${a_j \choose \Lambda\gamma+1}$ such sets $\chi_\mathcal{Q}$ which are empty, which means that round $j$ consists of $$\label{eq:totsubround}
{\Lambda \choose \Lambda\gamma+1}-{a_j \choose \Lambda\gamma+1}$$ transmissions.
Since each file is split into ${\Lambda\choose \Lambda\gamma}N_0$ subfiles, the duration of each such transmission is $$\label{eq:dureachtransmission}
\frac{1}{{\Lambda\choose \Lambda\gamma}N_0}$$ and thus summing over all $\mathcal{L}_1$ rounds, the total delay takes the form $$\label{eq:totdelay1}
T=\frac{\sum_{j=1}^{\mathcal{L}_1}{{\Lambda \choose \Lambda\gamma+1}-{a_j \choose \Lambda\gamma+1}}}{{\Lambda\choose \Lambda\gamma}{N_0}}$$ which, after some basic algebraic manipulation (see Appendix \[sec:BinomialChangeProof\] for the details), takes the final form $$\label{eq:totdelay2}
T=\frac{1}{N_0}\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_r{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}$$ which concludes the achievability part of the proof.
Scheme Example: $K=N=15$, $\Lambda=3$, $N_0=2$ and $\Lc=(8,5,2)$\[subsec:example\_scheme\]
------------------------------------------------------------------------------------------
Consider a scenario with $K=15$ users $\{1,2,\dots,15\}$, a server equipped with $N_0=2$ transmitting antennas that stores a library of $N=15$ equally-sized files $W^1,W^2,\dots,W^{15}$, and consider $\Lambda=3$ helper caches, each of size equal to $M=5$ units of file.
In the cache placement phase, we split each file $W^n$ into $3$ equally-sized disjoint subfiles denoted by $W^n_{1},W^n_{2},W^n_{3}$ and as in [@MN14], each cache $\lambda$ stores $W^n_{\lambda}, \forall n\in [15]$.
We assume that in the subsequent cache assignment, users $\mathcal{U}_1=(1,2,3,4,5,6,7,8)$ are assigned to helper node $1$, users $\mathcal{U}_2=(9,10,11,12,13)$ to helper node $2$ and users $\mathcal{U}_3=(14,15)$ to helper node $3$. This corresponds to a profile $\Lc=(8,5,2)$. We also assume without loss of generality that the demand vector is $\dv=(1,2,\dots,15)$.
Delivery takes place in $|\mathcal{U}_1|=8$ rounds, and each round will serve either $N_0=2$ users or no users from each of the following three ordered user groups $$\begin{aligned}
\boldsymbol{s_1}&=\mathcal{U}_1 || \mathcal{U}_1 = (1,2,\dots,7,8,1,2,\dots,7,8),\\
\boldsymbol{s_2}&=\mathcal{U}_2 || \mathcal{U}_2 = (9,10,11,12,13,9,10,11,12,13),\\
\boldsymbol{s_3}&=\mathcal{U}_3 || \mathcal{U}_3 = (14,15,14,15).\end{aligned}$$ Specifically, rounds 1 through 8, will respectively serve the following sets of users $$\begin{aligned}
\mathcal{R}_1&=\{1,2,9,10,14,15\}\\
\mathcal{R}_2&=\{3,4,11,12,14,15\}\\
\mathcal{R}_3&=\{5,6,13,9\}\\
\mathcal{R}_4&=\{7,8,10,11\}\\
\mathcal{R}_5&=\{1,2,12,13\}\\
\mathcal{R}_6&=\{3,4\}\\
\mathcal{R}_7&=\{5,6\}\\
\mathcal{R}_8&=\{7,8\}.\end{aligned}$$ Before transmission, each requested subfile $W^n_\Tau$ is further split into $N_0=2$ mini-files $W^n_{\Tau,1}$ and $W^n_{\Tau,2}$. As noted in the general description of the scheme, the transmitted vector structure within each round, draws from [@LampirisEliaJsac18] as it employs the linear combination of ZF-precoded vectors. In the first round, the server transmits, one after the other, the following $3$ vectors $$\begin{aligned}
\label{eq:ex1}
\xv_{\{1,2,9,10\}}=&\mathbf{H}^{-1}_{\{1,2\}}
\begin{bmatrix}
W^1_{2,1}\\
W^2_{2,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{9,10\}}
\begin{bmatrix}
W^{9}_{1,1}\\
W^{10}_{1,1}
\end{bmatrix}\\
\label{eq:ex2}
\xv_{\{1,2,14,15\}}=&\mathbf{H}^{-1}_{\{1,2\}}
\begin{bmatrix}
W^1_{3,1}\\
W^2_{3,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{14,15\}}
\begin{bmatrix}
W^{14}_{1,1}\\
W^{15}_{1,1}
\end{bmatrix}\\
\label{eq:ex3}
\xv_{\{9,10,14,15\}}=&\mathbf{H}^{-1}_{\{9,10\}}
\begin{bmatrix}
W^9_{3,1}\\
W^{10}_{3,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{14,15\}}
\begin{bmatrix}
W^{14}_{2,1}\\
W^{15}_{2,1}
\end{bmatrix}\end{aligned}$$ where $\mathbf{H}^{-1}_{\{i,j\}}$ is the zero-forcing (ZF) precoder[^16] that inverts the channel $\mathbf{H}_{\{i,j\}}=[\mathbf{h}_i^T \mathbf{h}_j^T]$ from the transmitter to users $i$ and $j$. To see how decoding takes place, let us first focus on users 1 and 2 during the transmission of $\xv_{\{1,2,9,10\}}$, where we see that, due to ZF precoding, the users’ respective received signals take the form $$\begin{aligned}
&y_1=W^1_{2,1}+\underbrace{\mathbf{h}_1^T\mathbf{H}^{-1}_{\{9,10\}}
\begin{bmatrix}
W^{9}_{1,1}\\
W^{10}_{1,1}
\end{bmatrix}}_{\text{interference}}+w_1\\
&y_2=W^2_{2,1}+\underbrace{\mathbf{h}_2^T\mathbf{H}^{-1}_{\{9,10\}}
\begin{bmatrix}
W^{9}_{1,1}\\
W^{10}_{1,1}
\end{bmatrix}}_{\text{interference}}+w_2.\end{aligned}$$ Users 1 and 2 use their cached content in cache node 1, to remove files $W^9_{1,1},W^{10}_{1,1}$, and can thus directly decode their own desired subfiles. The same procedure is applied to the remaining users served in the first round.
Similarly, in the second round, we have $$\begin{aligned}
\xv_{\{3,4,11,12\}}=&\mathbf{H}^{-1}_{\{3,4\}}
\begin{bmatrix}
W^3_{2,1}\\
W^4_{2,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{11,12\}}
\begin{bmatrix}
W^{11}_{1,1}\\
W^{12}_{1,1}
\end{bmatrix}\\
\xv_{\{3,4,14,15\}}=&\mathbf{H}^{-1}_{\{3,4\}}
\begin{bmatrix}
W^3_{3,1}\\
W^4_{3,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{14,15\}}
\begin{bmatrix}
W^{14}_{1,2}\\
W^{15}_{1,2}
\end{bmatrix}\\
\xv_{\{11,12,14,15\}}=&\mathbf{H}^{-1}_{\{11,12\}}
\begin{bmatrix}
W^{11}_{3,1}\\
W^{12}_{3,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{14,15\}}
\begin{bmatrix}
W^{14}_{2,2}\\
W^{15}_{2,2}
\end{bmatrix}\end{aligned}$$ and again in each round, each pair of users can cache-out some of the files, and then decode their own file due to the ZF precoder.
The next three transmissions, corresponding to the third round, are as follows $$\begin{aligned}
&\xv_{\{5,6,13,9\}}=\mathbf{H}^{-1}_{\{5,6\}}
\begin{bmatrix}
W^5_{2,1}\\
W^6_{2,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{13,9\}}
\begin{bmatrix}
W^{13}_{1,1}\\
W^9_{1,2}
\end{bmatrix}
\\[3pt]
&\xv_{\{5,6\}}=\mathbf{H}^{-1}_{\{5,6\}}
\begin{bmatrix}
W^5_{3,1}\\
W^6_{3,1}
\end{bmatrix} \ \
\xv_{\{13,9\}}=\mathbf{H}^{-1}_{\{13,9\}}
\begin{bmatrix}
W^{13}_{3,1}\\
W^{9}_{3,2}
\end{bmatrix}\end{aligned}$$ where the transmitted vectors $\xv_{\{5,6\}}$ and $\xv_{\{13,9\}}$ simply use zero-forcing. Similarly round 4 serves the users in $\mathcal{R}_4$ by sequentially sending $$\begin{aligned}
&\xv_{\{7,8,10,11\}}=\mathbf{H}^{-1}_{\{7,8\}}
\begin{bmatrix}
W^7_{2,1}\\
W^8_{2,1}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{10,11\}}
\begin{bmatrix}
W^{10}_{1,2}\\
W^{11}_{1,2}
\end{bmatrix}
\\[3pt]
&\xv_{\{7,8\}}=\mathbf{H}^{-1}_{\{7,8\}}
\begin{bmatrix}
W^7_{3,1}\\
W^8_{3,1}
\end{bmatrix} \ \
\xv_{\{10,11\}}=\mathbf{H}^{-1}_{\{10,11\}}
\begin{bmatrix}
W^{10}_{3,2}\\
W^{11}_{3,2}
\end{bmatrix}\end{aligned}$$ and round 5 serves the users in $\mathcal{R}_5$ by sequentially sending $$\begin{aligned}
&\xv_{\{1,2,12,13\}}=\mathbf{H}^{-1}_{\{1,2\}}
\begin{bmatrix}
W^1_{2,2}\\
W^2_{2,2}
\end{bmatrix}
+
\mathbf{H}^{-1}_{\{12,13\}}
\begin{bmatrix}
W^{12}_{1,2}\\
W^{13}_{1,2}
\end{bmatrix}
\\[3pt]
&\xv_{\{1,2\}}=\mathbf{H}^{-1}_{\{1,2\}}
\begin{bmatrix}
W^1_{3,2}\\
W^2_{3,2}
\end{bmatrix} \ \
\xv_{\{12,13\}}=\mathbf{H}^{-1}_{\{12,13\}}
\begin{bmatrix}
W^{12}_{3,2}\\
W^{13}_{3,2}
\end{bmatrix}.\end{aligned}$$ Finally, for the remaining rounds $6,7,8$ which respectively involve user sets $\mathcal{R}_6,\mathcal{R}_7$ and $\mathcal{R}_8$ that are connected to the same helper cache 1, data is delivered using the following standard ZF-precoded transmissions $$\begin{aligned}
&\xv_{\{3,4\}}=\mathbf{H}^{-1}_{\{3,4\}}
\begin{bmatrix}
W^3_{2,2}||W^3_{3,2}\\
W^4_{2,2}||W^4_{3,2}
\end{bmatrix}\\[3pt]
&\xv_{\{5,6\}}=\mathbf{H}^{-1}_{\{5,6\}}
\begin{bmatrix}
W^5_{2,2}||W^5_{3,2}\\
W^6_{2,2}||W^6_{3,2}
\end{bmatrix}\\
&\xv_{\{7,8\}}=\mathbf{H}^{-1}_{\{7,8\}}
\begin{bmatrix}
W^{7}_{2,2}||W^{7}_{3,2}\\
W^{8}_{2,2}||W^{8}_{3,2}
\end{bmatrix}.\end{aligned}$$
The overall delivery time required to serve all users is $$T=\frac{1}{6}\cdot 15+\frac{1}{3}\cdot 3 =\frac{21}{6}$$ where the first summand is for rounds 1 through 5, and the second summand is for rounds 6 through 8.
It is very easy to see that this delay remains the same — given again worst-case demand vectors — for any user-to-cache association $\mathcal{U}$ with the same profile $\Lc=(8,5,2)$. Every time, this delay matches the converse $$\begin{aligned}
T^*((8,5,2))&\geq \frac{\sum_{r=1}^{2}\mathcal{L}_r{{3-r}\choose 1}}{2{3\choose 1}}= \frac{8\cdot 2+5\cdot 1}{6}=\frac{21}{6}\end{aligned}$$ of Theorem \[thm:resmultiant\].
Information Theoretic Converse\[sec:converse\]
==============================================
Toward proving Theorems \[thm:PerClassSingleAntenna\] and \[thm:resmultiant\], we develop a lower bound on the normalized delivery time in (\[eq:T\*\_def\]) for each given user-to-cache association profile $\Lc$. The proof technique is based on the breakthrough in [@WanTP15] which — for the case of $\Lambda = K$, where each user has their own cache — employed index coding to bound the performance of coded caching. Part of the challenge here will be to account for having shared caches, and mainly to adapt the index coding approach to reflect non-uniform user-to-cache association classes.
We will begin with lower bounding the normalized delivery time $T(\mathcal{U},\dv,\chi)$, for any user-to-cache association $\mathcal{U}$, demand vector $\dv$ and a generic caching-delivery strategy $\chi$.
#### Identifying the distinct problems {#identifying-the-distinct-problems .unnumbered}
The caching problem is defined when the user-to-cache association $\mathcal{U}=\{\mathcal{U}_\lambda \}_{\lambda=1}^\Lambda$ and demand vector $\dv$ are revealed. What we can easily see is that there are many combinations of $\{\mathcal{U}_\lambda \}_{\lambda=1}^\Lambda$ and $\dv$ that jointly result in the same coded caching problem. After all, any permutation of the file indices requested by users assigned to the same cache, will effectively result in the same coded caching problem. As one can see, every *distinct* coded caching problem is fully defined by $\{\dvlambda\}_{\lambda=1}^\Lambda$, where $\dvlambda$ denotes the vector of file indices requested by the users in $\mathcal{U}_\lambda$, i.e., requested by the $|\mathcal{U}_\lambda|$ users associated to cache $\lambda$. The analysis is facilitated by reordering the demand vector $\dv$ to take the form $$\label{eq:OrderDemand2}
\dv(\Uc){\stackrel{\triangle}{=}}(\boldsymbol{d_1}, \dots, \boldsymbol{d_\lambda}).$$ Based on this, we define the set of worst-case demands associated to a given profile $\Lc$, to be $$\mathcal{D}_{\Lc} = \{\dv(\mathcal{U}): \dv\in \mathcal{D}_{wc}, \mathcal{U} \in \mathcal{U}_{\Lc}
\}$$ where $\mathcal{D}_{wc}$ is the set of demand vectors $\dv$ whose $K$ entries are all different (i.e., where $d_i \neq d_j, ~i,j\in[\Lambda],~i\neq j$, corresponding to the case where all users request different files). We will convert each such coded caching problem into an index coding problem.
#### The corresponding index coding problem {#the-corresponding-index-coding-problem .unnumbered}
To make the transition to the index coding problem, each requested file $W^{\dvlambda(j)}$ is split into $2^\Lambda$ disjoint subfiles $W^{\dvlambda(j)}_\Tau,\Tau\in 2^{[\Lambda]}$ where $\Tau\subset[\Lambda]$ indicates the set of helper nodes in which $W^{\dvlambda(j)}_\Tau$ is cached[^17]. Then — in the context of index coding — each subfile $W^{\dvlambda(j)}_\Tau$ can be seen as being requested by a different user that has as side information all the content $\Zc_\lambda$ of the same helper node $\lambda$. Naturally, no subfile of the form $W^{\dvlambda(j)}_\Tau, \; ~\Tau \ni\lambda$ is requested, because helper node $\lambda$ already has this subfile. Therefore the corresponding index coding problem is defined by $K2^{\Lambda-1}$ requested subfiles, and it is fully represented by the side-information graph $\mathcal{G}=(\mathcal{V}_{\mathcal{G}},\mathcal{E}_{\mathcal{G}})$, where $\mathcal{V}_{\mathcal{G}}$ is the set of vertices (each vertex/node representing a different subfile $W^{\dvlambda(j)}_\Tau, \Tau\not\ni\lambda$) and $\mathcal{E}_{\mathcal{G}}$ is the set of direct edges of the graph. Following standard practice in index coding, a directed edge from node $W^{\dvlambda(j)}_\Tau$ to $W^{\boldsymbol{d_{\lambda'}}(j')}_{\Tau'}$ exists if and only if $\lambda'\in\Tau$. For any given $\mathcal{U}$, $\dv$ (and of course, for any scheme $\chi$) the total delay $T$ required for this index coding problem, is the completion time for the corresponding coded caching problem.
#### Lower bounding $T(\Uc,\dv,\chi)$ {#lower-bounding-tucdvchi .unnumbered}
We are interested in lower bounding $T(\Uc,\dv,\chi)$ which represents the total delay required to serve the users for the index coding problem corresponding to the side-information graph $\mathcal{G}_{\Uc,\dv}$ defined by $\Uc,\dv,\chi$ or equivalently by $\dv(\Uc),\chi$.
In the next lemma, we remind the reader — in the context of our setting — the useful index-coding converse from [@li2017cooperative].
(Cut-set-type converse [@li2017cooperative])\[cor\_dof\] For a given $\Uc,\dv,\chi$, in the corresponding side information graph $\mathcal{G}_{\Uc,\dv}=(\mathcal{V}_{\mathcal{G}},\mathcal{E}_{\mathcal{G}})$ of the $N_0$-antenna MISO broadcast channel with $\mathcal{V}_{\mathcal{G}}$ vertices/nodes and $\mathcal{E}_{\mathcal{G}}$ edges, the following inequality holds $$\label{eq:indexbound}T\geq \frac{1}{N_0}\sum_{{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }\in \mathcal{V_{J}}}|{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }|$$ for every acyclic induced subgraph $\mathcal{J}$ of $\mathcal{G}_{\Uc,\dv}$, where $\mathcal{V}_{\mathcal{J}}$ denotes the set of nodes of the subgraph $\mathcal{J}$, and where $|{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }|$ is the size of the message/subfile/node ${
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }$.
*Proof.* The above lemma draws from [@li2017cooperative Corollary 1] (see also [@Sadeghi:16 Corollary 2] for a simplified version), and is easily proved in the Appendix Section \[proof:cor\_dof\].
#### Creating large acyclic subgraphs {#creating-large-acyclic-subgraphs .unnumbered}
Lemma \[cor\_dof\] suggests the need to create (preferably large) acyclic subgraphs of $\mathcal{G}_{\mathcal{U},\dv}$. The following lemma describes how to properly choose a set of nodes to form a large acyclic subgraph.
\[lem:cons\_acyclic\] An acyclic subgraph $\mathcal{J}$ of $\mathcal{G}_{\Uc,\dv}$ corresponding to the index coding problem defined by $\Uc,\dv,\chi$ for any $\Uc$ with profile $\Lc$, is designed here to consist of all subfiles $W^{\boldsymbol{d_{\sigma_{s}(\lambda)}}(j)}_{\Tau_{\lambda}},~\forall j\in [\mathcal{L}_{\lambda}],~\forall \lambda\in [\Lambda]$ for all $\Tau_{\lambda}\subseteq [\Lambda]\setminus \{\sigma_s(1),\dots,\sigma_s(\lambda)\}$ where $\sigma_s\in S_{\Lambda}$ is the permutation such that $|\mathcal{U}_{\sigma_s(1)}|\geq |\mathcal{U}_{\sigma_s(2)}|\geq\dots\geq |\mathcal{U}_{\sigma_s(\Lambda)}|$.
*Proof.* The proof, which can be found in the Appendix Section \[proof:cons\_acyclic\], is an adaptation of [@WanTP15 Lemma 1] to the current setting.
The choice of the permutation $\sigma_s$ is critical for the development of a tight converse. Any other choice $\sigma\in S_\Lambda$ may result — in some crucial cases — in an acyclic subgraph with a smaller number of nodes and therefore a looser bound. This approach here deviates from the original approach in [@WanTP15 Lemma 1], which instead considered — for each $\dv,\chi$, for the uniform user-to-cache association case of $K = \Lambda$ — the set of *all* possible permutations, that jointly resulted in a certain symmetry that is crucial to that proof. Here in our case, such symmetry would not serve the same purpose as it would dilute the non-uniformity in $\Lc$ that we are trying to capture. Our choice of a single carefully chosen permutation, allows for a bound which — as it turns out — is tight even in non-uniform cases. The reader is also referred to Section \[subsec:example\] for an explanatory example.
Having chosen an acyclic subgraph according to Lemma \[lem:cons\_acyclic\], we return to Lemma \[cor\_dof\] and form — by adding the sizes of all subfiles associated to the chosen acyclic graph — the following lower bound $$T(\Uc,\dv,\chi)\geq T^{LB}(\Uc,\dv,\chi)$$ where $$\begin{aligned}
&T^{LB}(\Uc,\dv,\chi) {\triangleq}\frac{1}{N_0}\Bigg( \sum_{j=1}^{\mathcal{L}_{1}}\sum_{\Tau_{1}\subseteq [\Lambda]\setminus \{\sigma_s(1)\}}|W^{\boldsymbol{d_{\sigma_s(1)}}(j)}_{\Tau_{1}}|\nonumber \\
&+ \sum_{j=1}^{\mathcal{L}_{2}}\sum_{\Tau_{2}\subseteq [\Lambda]\setminus \{\sigma_s(1),\sigma_s(2)\}}|W^{\boldsymbol{d_{\sigma_s(2)}}(j)}_{\Tau_{2}}|+\dots \nonumber \\
&+ \sum_{j=1}^{\mathcal{L}_{\Lambda}}\sum_{\Tau_{\Lambda}\subseteq [\Lambda]\setminus \{\sigma_s(1),\dots,\sigma_s(\Lambda)\}}|W^{\boldsymbol{d_{\sigma_s(\Lambda)}}(j)}_{\Tau_{\Lambda}}|
\Bigg). \label{eq:TLB}\end{aligned}$$
Our interest lies in a lower bound for the worst-case delivery time/delay associated to profile $\Lc$. Such a worst-case naturally corresponds to the scenario where all users request different files, i.e., where all the entries of the demand vector $\dv(\mathcal{U})$ are different. The corresponding lower bound can be developed by averaging over worst-case demands. Recalling our set $\mathcal{D}_{\Lc}$, the worst-case delivery time can thus be written as $$\begin{aligned}
T^*(\Lc)&{\triangleq}\min_{\chi} \max_{(\mathcal{U},\dv) \in (\mathcal{U}_{\Lc},[N]^K)} T(\mathcal{U},\dv,\chi)\\
&\overset{(a)}{\geq} \min_{\chi} \frac{1}{|\mathcal{D}_{\Lc}|} \sum_{\dv(\mathcal{U}) \in \mathcal{D}_{\Lc}} T(\dv(\mathcal{U}),\chi)\label{eq:alternativedefinitionofT}\end{aligned}$$ where in step (a), we used the following change of notation $T(\dv(\mathcal{U}),\chi){\stackrel{\triangle}{=}}T(\mathcal{U},\dv,\chi)$ and averaged over worst-case demands.
With a given class/profile $\Lc$ in mind, in order to construct $\mathcal{D}_{\Lc}$ (so that we can then average over it), we will consider all demand vectors $\dv\in \mathcal{D}_{wc}$ for all permutations $\pi\in S_{\Lambda}$. Then for each $\dv$, we create the following set of $\Lambda$ vectors $$\begin{aligned}
&\boldsymbol{d^{'}_1}= (d_1 : d_{\mathcal{L}_1}),\\
&\boldsymbol{d^{'}_2}= (d_{\mathcal{L}_1+1} : d_{\mathcal{L}_1+\mathcal{L}_2}),\\
& \vdots \\
&\boldsymbol{d^{'}_{\Lambda}}= (d_{\sum_{i=1}^{\Lambda-1}\mathcal{L}_{i}~+1} : d_{K})\end{aligned}$$ and for each permutation $\pi\in S_{\Lambda}$ applied to the set $\{1,2,\dots,\Lambda\}$, a demand vector $\dv(\mathcal{U})$ is constructed as follows $$\begin{aligned}
\dv(\mathcal{U})&{\stackrel{\triangle}{=}}(\boldsymbol{d_1},\boldsymbol{d_2},\dots,\boldsymbol{d_\Lambda})\\
&=(\boldsymbol{d^{'}_{\pi^{-1}(1)}},\boldsymbol{d^{'}_{\pi^{-1}(2)}},\dots,\boldsymbol{d^{'}_{\pi^{-1}(\Lambda)}}).\end{aligned}$$ This procedure is repeated for all $\Lambda!$ permutations ${\pi\in S_{\Lambda}}$ and all $P(N,K)$ worst-case demands $\dv\in \mathcal{D}_{wc}$. This implies that the cardinality of $\mathcal{D}_{\Lc}$ is ${|\mathcal{D}_{\Lc}|=P(N,K)\cdot \Lambda!}$.
Using this designed set $\mathcal{D}_{\Lc}$, now the optimal worst-case delivery time in (\[eq:alternativedefinitionofT\]) is bounded as $$\begin{aligned}
T^{*}(\Lc) &= \min_{\chi}T(\Lc,\chi)\\
& \geq \min_{\chi} \frac{1}{P(N,K)\Lambda!} \sum_{\dv(\mathcal{U}) \in \mathcal{D}_{\Lc}} T^{LB}(\dv(\mathcal{U}),\chi) \label{eq:lowerboundcompact}\end{aligned}$$ where $T^{LB}(\dv(\mathcal{U}),\chi)$ is given by (\[eq:TLB\]) for each reordered demand vector $\dv(\mathcal{U})\in \mathcal{D}_{\Lc}$. Rewriting the summation in (\[eq:lowerboundcompact\]), we get $$\begin{aligned}
\label{eq:longinequality}
&\sum_{\dv(\mathcal{U})\in \mathcal{D}_{\Lc}} T^{LB}(\dv(\mathcal{U}),\chi)= \nonumber \\
&\frac{1}{N_0}\sum_{i=0}^{\Lambda}\sum_{n\in[N]}\sum_{\Tau\subseteq[\Lambda]:|\Tau|=i} |W^n_{\Tau}| \cdot \underbrace{\sum_{\dv(\mathcal{U})\in \mathcal{D}_{\Lc}} \mathds{1}_{\mathcal{V}_{\mathcal{J}_s^{\dv(\mathcal{U})}}}(W^n_{\Tau})}_{{\triangleq}Q_{i}(W^n_\Tau)}\end{aligned}$$ where $\mathcal{V}_{\mathcal{J}_s^{\dv(\mathcal{U})}}$ is the set of vertices in the acyclic subgraph chosen according to Lemma \[lem:cons\_acyclic\] for a given $\dv(\mathcal{U})$. In the above, $\mathds{1}_{\mathcal{V}_{\mathcal{J}_s^{\dv(\mathcal{U})}}}(W^n_{\Tau})$ denotes the indicator function which takes the value of 1 only if $W^n_{\Tau} \subset \mathcal{V}_{\mathcal{J}_s^{\dv(\mathcal{U})}}$, else it is set to zero.
A crucial step toward removing the dependence on $\Tau$, comes from the fact that $$\begin{aligned}
\label{eq:Qi}
Q_{i} &= Q_{i}(W^n_\Tau){\stackrel{\triangle}{=}}\sum_{\dv(\mathcal{U})\in \mathcal{D}_{\Lc}} \mathds{1}_{\mathcal{V}_{\mathcal{J}_s^{\dv(\Uc)}}}(W^n_{\Tau}) \nonumber\\
=&{N-1 \choose K-1}\sum_{r=1}^{\Lambda}P(\Lambda-i-1,r-1)(\Lambda-r)!\mathcal{L}_{r} \nonumber\\
&\times P(K-1,\mathcal{L}_{r}-1) (K-\mathcal{L}_{r})! (\Lambda-i)\end{aligned}$$ where we can see that the total number of times a specific subfile appears — in the summation in , over the set of all possible $\dv(\mathcal{U}) \in \mathcal{D}_{\Lc}$, and given our chosen permutation $\sigma_s$ — is not dependent on the subfile itself but is dependent only on the number of caches $i=|\Tau|$ storing that subfile. The proof of can be found in Section \[proof:lemmaQi\].
In the spirit of [@WanTP15], defining $$x_i{\stackrel{\triangle}{=}}\sum_{n\in[N]}\sum_{\Tau\subseteq[\Lambda]:|\Tau|=i}|W^n_{\Tau}|$$ to be the total amount of data stored in exactly $i$ helper nodes, we see that $$\label{eq:sumfiles}
N=\sum_{i=0}^{\Lambda}x_i=\sum_{i=0}^{\Lambda}\sum_{n\in[N]}\sum_{\Tau\subseteq[\Lambda]:|\Tau|=i}|W^n_{\Tau}|$$ and we see that combining , and , gives $$\label{eq:compacteq}
T(\Lc,\chi)\geq \frac{1}{N_{0}}\sum_{i=0}^{\Lambda}\frac{Q_{i}}{P(N,K)\Lambda!}x_{i}.$$
Now substituting into , after some algebraic manipulations, we get that $$\begin{aligned}
T(\Lc,\chi)&\geq \frac{1}{N_0}\sum_{i=0}^{\Lambda}\frac{\sum_{r=1}^{\Lambda-i}\mathcal{L}_{r} {\Lambda-r\choose i}}{N{\Lambda\choose i}}x_{i} \label{eq:LBwithxi}\\
&=\frac{1}{N_0}\sum_{i=0}^{\Lambda}\frac{x_{i}}{N}c_{i} \label{eq:LBwithxi_2}\end{aligned}$$ where $c_{i}\triangleq \frac{\sum_{r=1}^{\Lambda-i}\mathcal{L}_r{\Lambda-r\choose i}}{{\Lambda\choose i}}$ decreases with $i\in \{0,1,\dots,\Lambda\}$. The proof of the transition from to , as well as the monotonicity proof for the sequence $\{c_i\}_{i\in [\Lambda]\cup \{0\}}$, are given in Appendix Sections \[proof:transition\] and \[sec:monotonicity\] respectively.
Under the file-size constraint given in , and given the following cache-size constraint $$\sum_{i=0}^{\Lambda}i \cdot x_{i}\leq \Lambda M \label{eq:constr2}$$ the expression in serves as a lower bound on the delay of any caching-and-delivery scheme $\chi$ whose caching policy implies a set of $\{x_i\}$.
We then employ the Jensen’s-inequality based technique of [@YuMA16 Proof of Lemma 2] to minimize the expression in , over all admissible $\{x_i\}$. Hence for any integer $\Lambda\gamma$, we have $$\label{eq:optimization1}
T(\Lc,\chi)\geq \frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_r{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}$$ whereas for all other values of $\Lambda\gamma$, this is extended to its convex lower envelop. The detailed derivation of can again be found in Appendix Section \[lastproof\].
This concludes lower bounding $\max_{(\mathcal{U},\dv) \in (\mathcal{U}_{\Lc},[N]^K)} T(\mathcal{U},\dv,\chi)$, and thus — given that the right hand side of is independent of $\chi$ — lower bounds the performance for any scheme $\chi$, which hence concludes the proof of the converse for Theorem \[thm:resmultiant\] (and consequently for Theorem \[thm:PerClassSingleAntenna\] after setting $N_0 = 1$).
Proof of the Converse for Corollary \[cor:ressymMulti\] \[sec:ConverseUniform\]
-------------------------------------------------------------------------------
For the uniform case of $\Lc = [\frac{K}{\Lambda},\frac{K}{\Lambda},\dots,\frac{K}{\Lambda}]$, the lower bound in becomes $$\begin{aligned}
\frac{1}{N_0}\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}\mathcal{L}_r{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}}
& =\frac{1}{N_0}\frac{K}{\Lambda}\frac{\sum_{r=1}^{\Lambda-\Lambda\gamma}{\Lambda-r\choose \Lambda\gamma}}{{\Lambda\choose \Lambda\gamma}} \\
&\overset{(a)}{=} \frac{1}{N_0}\frac{K}{\Lambda}\frac{{\Lambda \choose \Lambda\gamma+1}}{{\Lambda\choose \Lambda\gamma}} \\
& = \frac{K(1-\gamma)}{N_0(\Lambda\gamma+1)}
\end{aligned}$$ where the equality in step (a) is due to Pascal’s triangle.
Example for $N=K=9$, $N_{0}=2$ and $\Lc=(4,3,2)$\[subsec:example\]
------------------------------------------------------------------
We here give an example of deriving the converse for Theorem \[thm:resmultiant\], emphasizing on how to convert the caching problem to the index-coding problem, and how to choose acyclic subgraphs. We consider the case of having $K=9$ receiving users, and a transmitter with $N_{0}=2$ transmit antennas having access to a library of $N=9$ files of unit size. We also assume that there are $\Lambda=3$ caching nodes, of average normalized cache capacity $\gamma$. We will focus on deriving the bound for user-to-cache association profile $\Lc=(4,3,2)$, meaning that we are interested in the setting where one cache is associated to $4$ users, one cache to $3$ users and one cache associated to $2$ users.
Each file $W^n$ is split into $2^{\Lambda}=8$ disjoint subfiles $W^{i}_{\Tau}, \Tau\in 2^{[3]}$ where each $\Tau$ describes the set of helper nodes in which $W^{i}_{\Tau}$ is cached. For instance, $W^1_{13}$ refers to the part of file $W^1$ that is stored in the first and third caching nodes.
As a first step, we present the construction of the set $\mathcal{D}_{\Lc}$. To this end, let us start by considering the demand $\dv=(1,2,3,4,5,6,7,8,9)$ and one of the $6$ permutations $\pi\in S_{3}$; for example, let us start by considering $\pi(1)=2,\pi(2)=3,\pi(3)=1$. Toward reordering $\dv$ to reflect $\Lc$, we construct $$\begin{aligned}
&\boldsymbol{d^{'}_1}=(1,2,3,4),
~~\boldsymbol{d^{'}_2}=(5,6,7),
~~\boldsymbol{d^{'}_3}=(8,9)\end{aligned}$$ to obtain the reordered demand vector $$\begin{aligned}
\dv(\mathcal{U})&=(\boldsymbol{d^{'}_{\pi^{-1}(1)}},\boldsymbol{d^{'}_{\pi^{-1}(2)}},\boldsymbol{d^{'}_{\pi^{-1}(3)}})\\
&=(\boldsymbol{d^{'}_{3}},\boldsymbol{d^{'}_{1}},\boldsymbol{d^{'}_{2}})\end{aligned}$$ which in turn yields $\boldsymbol{d_1}=(8,9),\boldsymbol{d_2}=(1,2,3,4),\boldsymbol{d_3}=(5,6,7)$. Similarly, we can construct the remaining $5$ demands $\dv(\mathcal{U})$ associated to the other $5$ permutations $\pi\in S_{3}$. Finally, the procedure is repeated for all other worst-case demand vectors. These vectors are part of set $\mathcal{D}_{\Lc}$.
With the users demands $\dv(\mathcal{U})$ known to the server, the delivery problem is translated into an index coding problem with a side information graph of $K 2^{\Lambda-1}=9\cdot 2^{2}$ nodes. For each requested file $W^{\boldsymbol{d_\lambda}(j)}$, we write down the $4$ subfiles that the requesting user does not have in its assigned cache. Hence, a given user of the caching problem requiring $4$ subfiles from the main server, is replaced by $4$ different new users in the index coding problem. Each of these users request a different subfile and are connected to the same cache $\lambda$ as the original user.
$$\begin{array}{c@{}c@{}ccc}
\boldsymbol{d_1}=(1,2,3,4),\boldsymbol{d_2}=(5,6,7),
&~~&
\boldsymbol{d_1}=(1,2,3,4),\boldsymbol{d_2}=(8,9),
&
\boldsymbol{d_1}=(5,6,7),\boldsymbol{d_2}=(1,2,3,4),\\
\boldsymbol{d_3}=(8,9)&~~&\boldsymbol{d_3}=(5,6,7)&\boldsymbol{d_3}=(8,9)\\
~~&~~&~~\\
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{2}} & \underline{W^{1}_{3}} & \underline{W^{1}_{23}}\\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{2}} & \underline{W^{2}_{3}} & \underline{W^{2}_{23}}\\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{2}} & \underline{W^{3}_{3}} & \underline{W^{3}_{23}}\\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{2}} & \underline{W^{4}_{3}} & \underline{W^{4}_{23}}\\
\underline{W^{5}_{\emptyset}} & W^{5}_{1} & \underline{W^{5}_{3}} & W^{5}_{13}\\
\underline{W^{6}_{\emptyset}} & W^{6}_{1} & \underline{W^{6}_{3}} & W^{6}_{13}\\
\underline{W^{7}_{\emptyset}} & W^{7}_{1} & \underline{W^{7}_{3}} & W^{7}_{13}\\
\underline{W^{8}_{\emptyset}} & W^{8}_{1} & W^{8}_{2} & W^{8}_{12}\\
\underline{W^{9}_{\emptyset}} & W^{9}_{1} & W^{9}_{2} & W^{9}_{12}\\
\end{array} &~~&
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{2}} & \underline{W^{1}_{3}} & \underline{W^{1}_{23}}\\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{2}} & \underline{W^{2}_{3}} & \underline{W^{2}_{23}}\\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{2}} & \underline{W^{3}_{3}} & \underline{W^{3}_{23}}\\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{2}} & \underline{W^{4}_{3}} & \underline{W^{4}_{23}}\\
\underline{W^{5}_{\emptyset}} & W^{5}_{1} & \underline{W^{5}_{2}} & W^{5}_{12}\\
\underline{W^{6}_{\emptyset}} & W^{6}_{1} & \underline{W^{6}_{2}} & W^{6}_{12}\\
\underline{W^{7}_{\emptyset}} & W^{7}_{1} & \underline{W^{7}_{2}} & W^{7}_{12}\\
\underline{W^{8}_{\emptyset}} & W^{8}_{1} & W^{8}_{3} & W^{8}_{13}\\
\underline{W^{9}_{\emptyset}} & W^{9}_{1} & W^{9}_{3} & W^{9}_{13}\\
\end{array} &
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{1}} & \underline{W^{1}_{3}} & \underline{W^{1}_{13}} \\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{1}} & \underline{W^{1}_{3}} & \underline{W^{2}_{13}} \\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{1}} & \underline{W^{3}_{3}} & \underline{W^{3}_{13}} \\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{1}} & \underline{W^{4}_{3}} & \underline{W^{4}_{13}} \\
\underline{W^{5}_{\emptyset}} & W^{5}_{2} & \underline{W^{5}_{3}} & W^{5}_{23} \\
\underline{W^{6}_{\emptyset}} & W^{6}_{2} & \underline{W^{6}_{3}} & W^{6}_{23} \\
\underline{W^{7}_{\emptyset}} & W^{7}_{2} & \underline{W^{7}_{3}} & W^{7}_{23} \\
\underline{W^{8}_{\emptyset}} & W^{8}_{1} & W^{8}_{2} & W^{8}_{12} \\
\underline{W^{9}_{\emptyset}} & W^{9}_{1} & W^{9}_{2} & W^{9}_{12} \\
\end{array} \\
~~&~~&~~\\
\boldsymbol{d_1}=(5,6,7),\boldsymbol{d_2}=(8,9),
&~~&
\boldsymbol{d_1}=(8,9),\boldsymbol{d_2}=(1,2,3,4),
&
\boldsymbol{d_1}=(8,9),\boldsymbol{d_2}=(5,6,7),\\
\boldsymbol{d_3}=(1,2,3,4)&~~&\boldsymbol{d_3}=(5,6,7)&\boldsymbol{d_3}=(1,2,3,4)\\
~~&~~&~~\\
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{1}} & \underline{W^{1}_{2}} & \underline{W^{1}_{12}} \\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{1}} & \underline{W^{2}_{2}} & \underline{W^{2}_{12}} \\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{1}} & \underline{W^{3}_{2}} & \underline{W^{3}_{12}} \\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{1}} & \underline{W^{4}_{2}} & \underline{W^{4}_{12}} \\
\underline{W^{5}_{\emptyset}} & \underline{W^{5}_{2}} & W^{5}_{3} & W^{5}_{23} \\
\underline{W^{6}_{\emptyset}} & \underline{W^{6}_{2}} & W^{6}_{3} & W^{6}_{23} \\
\underline{W^{7}_{\emptyset}} & \underline{W^{7}_{2}} & W^{7}_{3} & W^{7}_{23} \\
\underline{W^{8}_{\emptyset}} & W^{8}_{1} & W^{8}_{3} & W^{8}_{13} \\
\underline{W^{9}_{\emptyset}} & W^{9}_{1} & W^{9}_{3} & W^{9}_{13} \\
\end{array} &~~&
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{1}} & \underline{W^{1}_{3}} & \underline{W^{1}_{13}} \\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{1}} & \underline{W^{2}_{3}} & \underline{W^{2}_{13}} \\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{1}} & \underline{W^{3}_{3}} & \underline{W^{3}_{13}} \\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{1}} & \underline{W^{4}_{3}} & \underline{W^{4}_{13}} \\
\underline{W^{5}_{\emptyset}} & \underline{W^{5}_{1}} & W^{5}_{2} & W^{5}_{12} \\
\underline{W^{6}_{\emptyset}} & \underline{W^{6}_{1}} & W^{6}_{2} & W^{6}_{12} \\
\underline{W^{7}_{\emptyset}} & \underline{W^{7}_{1}} & W^{7}_{2} & W^{7}_{12} \\
\underline{W^{8}_{\emptyset}} & W^{8}_{2} & W^{8}_{3} & W^{8}_{23} \\
\underline{W^{9}_{\emptyset}} & W^{9}_{2} & W^{9}_{3} & W^{9}_{23} \\
\end{array} &
\begin{array}{cccc}
\underline{W^{1}_{\emptyset}} & \underline{W^{1}_{1}} & \underline{W^{1}_{2}} & \underline{W^{1}_{12}} \\
\underline{W^{2}_{\emptyset}} & \underline{W^{2}_{1}} & \underline{W^{2}_{2}} & \underline{W^{2}_{12}} \\
\underline{W^{3}_{\emptyset}} & \underline{W^{3}_{1}} & \underline{W^{3}_{2}} & \underline{W^{3}_{12}} \\
\underline{W^{4}_{\emptyset}} & \underline{W^{4}_{1}} & \underline{W^{4}_{2}} & \underline{W^{4}_{12}} \\
\underline{W^{5}_{\emptyset}} & \underline{W^{5}_{1}} & W^{5}_{3} & W^{5}_{13} \\
\underline{W^{6}_{\emptyset}} & \underline{W^{6}_{1}} & W^{6}_{3} & W^{6}_{13} \\
\underline{W^{7}_{\emptyset}} & \underline{W^{7}_{1}} & W^{7}_{3} & W^{7}_{13} \\
\underline{W^{8}_{\emptyset}} & W^{8}_{2} & W^{8}_{3} & W^{8}_{23} \\
\underline{W^{9}_{\emptyset}} & W^{9}_{2} & W^{9}_{3} & W^{9}_{23} \
\end{array} \\
\end{array}$$
The nodes of the $6$ side-information graphs corresponding to the aforementioned vectors $\dv(\mathcal{U})$ (one for each permutation $\pi\in S_{3}$) for demand $\dv=(1,2,3,4,5,6,7,8,9)$, are depicted in Figure \[fig:graphs\].
For each side-information graph, we develop a lower bound as in Lemma \[cor\_dof\]. We recall that the lemma applies to acyclic subgraphs, which we create as follows; for each permutation[^18] $\sigma\in S_3$, a set of nodes forming an acyclic subgraph is $$\begin{aligned}
&\{W^{\boldsymbol{d_{\sigma(1)}}(j)}_{\Tau_{1}}\}_{j=1}^{|\mathcal{U}_{\sigma(1)}|}\;\text{ for all}\; \Tau_{1}\subseteq \{1,2,3\}\setminus{\{\sigma(1)\}},\\
&\{W^{\boldsymbol{d_{\sigma(2)}}(j)}_{\Tau_{2}}\}_{j=1}^{|\mathcal{U}_{\sigma(2)}|} \;\text{ for all}\; \Tau_{2}\subseteq \{1,2,3\}\setminus{\{\sigma(1),\sigma(2)\}},\\
&\{W^{\boldsymbol{d_{\sigma(3)}}(j)}_{\Tau_{3}}\}_{j=1}^{|\mathcal{U}_{\sigma(3)}|} \;\text{ for all}\; \Tau_{3}\subseteq \{1,2,3\}\setminus{\{\sigma(1),\sigma(2),\sigma(3)\}}.\end{aligned}$$ Based on this construction of acyclic graphs, our task now is to choose a permutation $\sigma_s\in S_3$ that forms the maximum-sized acyclic subgraph. For the case where $\boldsymbol{d_1}=(8,9),\boldsymbol{d_2}=(1,2,3,4)$ and $\boldsymbol{d_3}=(5,6,7)$, it can be easily verified that such a permutation $\sigma_s$ is the one with $\sigma_s(1)=2$,$\sigma_s(2)=3$ and $\sigma_s(3)=1$. In Figure \[fig:graphs\], for each of the six graphs, we underline the nodes corresponding to the acyclic subgraph that is formed by such permutation $\sigma_s$. The outer bound now involves adding the sizes of these chosen (underlined) nodes. For example, for the demand $\dv(\mathcal{U})=((8,9),(1,2,3,4),(5,6,7))$ (this corresponds to the lower center graph), the lower bound in becomes $$\begin{aligned}
T(\dv(\mathcal{U}))&\geq \frac{1}{2}\left(|W^{1}_{\emptyset}|+|W^{1}_{1}|+|W^{1}_{3}|+|W^{1}_{13}|+|W^{2}_{\emptyset}|\right.\nonumber\\
&+|W^{2}_{1}|+|W^{2}_{3}|+|W^{2}_{13}|+|W^{3}_{\emptyset}|+|W^{3}_{1}|\nonumber\\
&+|W^{3}_{3}|+|W^{3}_{13}|+|W^{4}_{\emptyset}|+|W^{4}_{1}|+|W^{4}_{3}|\nonumber\\
&+|W^{4}_{13}|+|W^{5}_{\emptyset}|+|W^{5}_1|+|W^6_{\emptyset}|+|W^6_1|\nonumber\\
&\left.+|W^7_{\emptyset}|+|W^7_1|+|W^{8}_{\emptyset}|+|W^{9}_{\emptyset}|\right).\end{aligned}$$ The lower bounds for the remaining $5$ vectors $\dv(\mathcal{U})$ for the same $\dv=(1,2,3,4,5,6,7,8,9)$, are given in a similar way, again by adding the (underlined) nodes of the corresponding acyclic subgraphs (again see Figure \[fig:graphs\]).
Subsequently, the procedure is repeated for all $P(N,K)=K!=9!$ worst-case demand vectors $\dv\in \mathcal{D}_{wc}$. Finally, all the $P(N,K)\cdot\Lambda!=9!\cdot 3!$ bounds are averaged to get $$\begin{aligned}
\label{eq:example_final}
&T(\Lc,\chi) \geq \frac{1}{2}\frac{1}{9!\cdot3!}\nonumber \\
&\!\!\! \sum_{\dv(\mathcal{U})\in \mathcal{D}_{\Lc}}\sum_{\lambda\in[3]}\sum_{j=1}^{\mathcal{L}_{\lambda}}\sum_{\Tau_{\lambda}\subseteq [3]\setminus \{\sigma_s(1),\dots,\sigma_s(\lambda)\}}\!\!\!\!\!\!\!\!\!\!|W^{\boldsymbol{d_{\boldsymbol{\sigma_s(\lambda)}}}(j)}_{\Tau_{\lambda}}|\end{aligned}$$ which is rewritten as $$\begin{aligned}
\label{eq:example_step1}
&T(\Lc,\chi) \geq \frac{1}{2}\frac{1}{9!\cdot3!} \nonumber \\
&\sum_{i=0}^{3}\sum_{n\in[9]}\sum_{\Tau\subseteq[3]:|\Tau|=i} |W^n_{\Tau}| \cdot \underbrace{\sum_{\dv(\mathcal{U})\in \mathcal{D}_{\Lc}} \mathds{1}_{\mathcal{V}_{\mathcal{J}_s^{\dv(\mathcal{U})}}}(W^n_{\Tau})}_{Q_{i}(W^n_\Tau)}.\end{aligned}$$ After the evaluation of the term $Q_i(W^n_\Tau)$, the bound in can be written in a more compact form as $$\begin{aligned}
\label{eq:exbound}
T(\Lc,\chi)& \geq \frac{1}{2}\sum_{i=0}^{3}\frac{\sum_{r=1}^{3-i}\mathcal{L}_r{3-r\choose i}}{9{3\choose i}}x_{i}\\
&\geq Conv\Bigg( \frac{1}{2}\frac{\sum_{r=1}^{3-i}\mathcal{L}_r{3-r\choose i}}{{3\choose i}}\Bigg)\label{eq:exbound2}\end{aligned}$$ where the proof of the transition from (\[eq:example\_step1\]) to (\[eq:exbound\]) and from (\[eq:exbound\]) to (\[eq:exbound2\]) can be found in the general proof (Section \[sec:converse\]).
Conclusions\[sec:discussion\]
=============================
We have treated the multi-sender coded caching problem with shared caches which can be seen as an information-theoretically simplified representation of some instances of the so-called cache-aided heterogeneous networks, where one or more transmitters communicate to a set of users, with the assistance of smaller nodes that can serve as caches.
The work is among the first — after the work in [@WanTP15] — to employ index coding as a means of providing (in this case, exact) outer bounds for more involved cache-aided network topologies that better capture aspects of cache-aided wireless networks, such as having shared caches and a variety of user-to-cache association profiles. Dealing with such non uniform profiles, raises interesting challenges in redesigning converse bounds as well as redesigning coded caching which is known to generally thrive on uniformity. Our effort also applied to the related problem of coded caching with multiple file requests. In addition to crisply quantifying the (adverse) effects of user-to-cache association non-uniformity, the work also revealed a multiplicative relationship between multiplexing gain and cache redundancy, thus providing further evidence of the powerful impact of jointly introducing a modest number of antennas and a modest number of helper nodes that serve as caches. We believe that the result can also be useful in providing guiding principles on how to assign shared caches to different users, especially in the presence of multiple senders. Finally we believe that the current presented adaptation of the outer bound technique to non-uniform settings may also be useful in analyzing different applications like distributed computing [@LiAliAvestimehrComputIT18; @PLE:18a; @KonstantinidisRamamoorthyArxiv18; @YanYangWiggerArxiv18; @MingyueJiISIT18] or data shuffling [@AttiaTandon16; @AttiaTandonISIT18; @WanTuninettiShuffling18; @MohajerISIT18] which can naturally entail such non uniformities.
Appendix \[sec:Appendix\]
=========================
Proof of Lemma \[cor\_dof\] \[proof:cor\_dof\]
----------------------------------------------
In the addressed problem, we consider a MISO broadcast channel with $N_0$ antennas at the transmitter serving $K$ receivers with some side information due to caches. In the wired setting this (high-SNR setting) is equivalent to the distributed index coding problem with $N_0$ senders $J_1,\dots,J_{N_0}$, all having knowledge of the entire set of messages, and each being connected via an (independent) broadcast line link of capacity $C_{J_i}=1, i\in[N_0]$ to the $K$ receivers which hold side information. This multi-sender index coding problem is addressed in [@li2017cooperative]. By adapting the achievable rate result in [@li2017cooperative Corollary1] to our problem, we get $$\sum_{{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }\in \mathcal{V}_{\mathcal{J}}}R_{{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }}\leq \sum_{i\in[N_0]}C_{J_i}$$ ($R_{{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }} = \frac{|{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }|}{T}$ is the rate for message ${
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }$), that yields $$\label{eq:lemma_1}
\sum_{{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }\in \mathcal{V}_{\mathcal{J}}}\frac{|{
\mathchoice
{{\scriptstyle\mathcal{V}}} {{\scriptstyle\mathcal{V}}} {{\scriptscriptstyle\mathcal{V}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }|}{T}\leq N_0$$ which, when inverted, gives the bound in Lemma \[cor\_dof\].
Proof of Lemma \[lem:cons\_acyclic\] \[proof:cons\_acyclic\]
------------------------------------------------------------
Consider a permutation $\sigma$ where the subfiles $W^{\boldsymbol{d_{\sigma(\lambda)}}(j)}_{\Tau_\lambda},\forall j\in\mathcal{U}_{\sigma(\lambda)}$ for all $\Tau_\lambda\subseteq[\Lambda]\setminus\{\sigma(1),\dots,\sigma(\lambda)\}$ are all placed in row $\lambda$ of a matrix whose rows are labeled by $\lambda = 1,2,\dots,\Lambda$. The index coding users corresponding to subfiles in row $\lambda$ only know (as side information) subfiles $W^{d_k}_{\Tau}, \ \Tau\ni \sigma(\lambda)$. Consequently each user/node of row $\lambda$ does not know any of the subfiles in the same row[^19] nor in the previous rows. As a result, the proposed set of subfiles chosen according to permutation $\sigma$, forms a subgraph that does not contain any cycle.
A basic counting argument can tell us that the number of subfiles — in the acyclic subgraph formed by any permutation $\sigma\in S_{\Lambda}$ — that are stored in exactly $i$ caches, is $$\label{eq:no_subfiles}
\sum_{r=1}^{\Lambda-i}|\Uc_{\sigma(r)}|{\Lambda-r\choose i}.$$ This means that the total number of subfiles in the acyclic subgraph is simply $$\sum_{i=0}^{\Lambda}\sum_{r=1}^{\Lambda-i}|\Uc_{\sigma(r)}|{\Lambda-r\choose i}.$$ This number is maximized when the permutation $\sigma$ guarantees that the vector $(|\Uc_{\sigma(1)}|,|\Uc_{\sigma(2)}|,\dots,|\Uc_{\sigma(\Lambda)}|)$ is in descending order. This maximization is achieved with our choice of the ordering permutation $\sigma_s$ (as this was defined in the notation part) when constructing the acyclic graphs.
Proof of Equation (\[eq:Qi\]) \[proof:lemmaQi\]
-----------------------------------------------
Here, through a combinatorial argument, we derive $Q_i(W^n_\Tau)$, that is the number of times that a subfile $W^n_\Tau$ with index size $|\Tau|=i$ appears in all the acyclic subgraphs chosen to develop the lower bound.
There are ${N-1\choose K-1}$ subsets $\Upsilon_{m},m\in[{N-1\choose K-1}]$ out of $N\choose K$ unordered subsets of $K$ files from the set $\{W^{j},j\in[N]\}$ that contain file $W^{n}$, and for each $\Upsilon_{m}$ there exists $K!$ different demand vectors $\dv'$. For each $\Upsilon_{m}$, among all possible demand vectors, a subfile $W^{n}_{\Tau}: |\Tau|=i$ appears in the side information graph an equal number of times. For a fixed $\Upsilon_{m}$, file $W^{n}$ is requested by a user connected to any helper node with a certain cardinality $\mathcal{L}_r$. By construction, $Q_{i}(W^n_\Tau)$ can be rewritten as $$\begin{aligned}
Q_{i}(W^n_\Tau)&=\sum_{\dv\in \mathcal{D}_{wc}}\sum_{\pi\in S_{\Lambda}} \mathds{1}_{{\mathcal{V}_{\mathcal{J}_{s}^{\dv_r(\Uc)}}}}(W^{n}_{\Tau})\notag\\
&={N-1 \choose K-1}\sum_{r=1}^{\Lambda}\sum_{\dv'_{r}\in \mathcal{D}_{wc}}\sum_{\pi\in S_{\Lambda}} \mathds{1}_{{\mathcal{V}_{\mathcal{J}_{s}^{\dv'_r(\Uc)}}}}(W^{n}_{\Tau})\label{Qi} \notag\end{aligned}$$ where $\dv'_{r}$ denotes the subset of all demand vectors from $\Upsilon_{m}$ such that $n\in \boldsymbol{d_\lambda}:|\boldsymbol{d_\lambda}|=\mathcal{L}_r$. The number of chosen maximum acyclic subgraphs containing $W^{n}_{\Tau}$ that arise from all the demand vectors $\dv'_{r}(\Uc)$ is evaluated as follows. After fixing the demands such that $n\in\boldsymbol{d_\lambda}:|\boldsymbol{d_\lambda}|=\mathcal{L}_r $, then $W^n_\Tau$ appears in the side information graph only if it is requested by a user connected to helper node $\lambda$ such that $\lambda\notin\Tau$, which corresponds to $(\Lambda-i)$ different *available* positions in the demand vector $\boldsymbol{d'}_{r}(\Uc)$, since $|\Tau|=i$. After fixing one of the $(\Lambda-i)$ positions occupied by $\boldsymbol{d_\lambda}:|\boldsymbol{d_\lambda}|=\mathcal{L}_r$, for the remaining demands ${\boldsymbol{d_\lambda}}:|\boldsymbol{d_\lambda}|=\mathcal{L}_j,\forall j\in[\Lambda]\setminus\{r\}$ there are $P(\Lambda-i-1,r-1)\cdot(\Lambda-r)!$ possible ways to be placed into $\dv$. After fixing the order of $\boldsymbol{d_\lambda},\forall \lambda\in [\Lambda]$ in $\dv$ and $n\in \boldsymbol{d_\lambda}:|\boldsymbol{d_\lambda}|=\mathcal{L}_r$, there are $\mathcal{L}_r$ different positions in which $n$ can be placed in $\boldsymbol{d_\lambda}:|\boldsymbol{d_\lambda}|=\mathcal{L}_r$. This leaves out $\mathcal{L}_{r}-1$ positions with $K-1$ different numbers from the considered set $\Upsilon_{m}\setminus{\{n\}}$, and the remaining $K-\mathcal{L}_r$ positions in $\dv'_{r}$ are filled with $K-\mathcal{L}_r$ numbers. Therefore, there exist $\mathcal{L}_rP(K-1,\mathcal{L}_r-1)(K-\mathcal{L}_r)!$ different demand vectors where the subfile $W^{n}_{\Tau}$ will appear in the associated maximum acyclic subgraphs. Hence, the above jointly tell us that $$\begin{aligned}
&Q_{i}(W^n_\Tau)={N-1 \choose K-1}\sum_{r=1}^{\Lambda}P(\Lambda-i-1,r-1)\nonumber\\
&\times(\Lambda-r)!\mathcal{L}_rP(K-1,\mathcal{L}_r-1)(K-\mathcal{L}_r)!(\Lambda-i)\end{aligned}$$ which concludes the proof.
Transition from Equation (\[eq:compacteq\]) to (\[eq:LBwithxi\]) \[proof:transition\]
-------------------------------------------------------------------------------------
The coefficient of $x_i$ in equation (\[eq:compacteq\]), can be further simplified as follows $$\begin{aligned}
\label{eq:finanumber}
&\frac{Q_{i}}{\Lambda!P(N,K)} \\
=&\frac{(N-1)!(N-K)!}{(K-1)!(N-K)!\Lambda!N!}\sum_{r=1}^{\Lambda}\mathcal{L}_rP(K-1,\mathcal{L}_r-1)
\\ &(K-\mathcal{L}_r)!(\Lambda-i)P(\Lambda-i-1,r-1)(\Lambda-r)!\nonumber\\
=&\frac{1}{(K-1)!\Lambda!N}\sum_{r=1}^{\Lambda}\mathcal{L}_r\nonumber\\ &\frac{(K-1)!(K-\mathcal{L}_r)!(\Lambda-i)(\Lambda-i-1)!(\Lambda-r)!}{(K-\mathcal{L}_r)!(\Lambda-i-r)!}\nonumber\\
=&\frac{1}{\Lambda!N}\sum_{r=1}^{\Lambda}\mathcal{L}_r\frac{(K-1)!(\Lambda-i)!(\Lambda-r)!}{(K-1)!(\Lambda-i-r)!}\nonumber\\
=&\frac{1}{N}\sum_{r=1}^{\Lambda}L_{\pi_s(r)}\frac{(\Lambda-i)!(\Lambda-r)!i!}{\Lambda!(\Lambda-i-r)!i!}\nonumber\\
=&\frac{1}{N}\sum_{r=1}^{\Lambda}\mathcal{L}_r\frac{{\Lambda-r\choose i}}{{\Lambda\choose i}}\end{aligned}$$ which concludes the proof.
Monotonicity of $\{c_i\}$ \[sec:monotonicity\]
----------------------------------------------
Let us define the following sequences $$\begin{aligned}
(a_n)_{n\in[\Lambda-i]}&{\stackrel{\triangle}{=}}& \bigg\{\frac{{{\Lambda-n}\choose i}}{{\Lambda \choose i}}, n\in [\Lambda-i]\bigg\} \\
(b_n)_{n\in[\Lambda-i-1]}&{\stackrel{\triangle}{=}}&\bigg\{\frac{{{\Lambda-n}\choose {i+1}}}{{\Lambda \choose {i+1}}}, n\in [\Lambda-i-1]\bigg\}.\end{aligned}$$ It is easy to verify that $a_n\geq b_n, \; \forall n\in [\Lambda-i]$. Consider now the set of scalar numbers $\{V_j, j\in[\Lambda-i], V_j\in \mathbb{N}\}$. The inequality $a^{*}_n\geq b^{*}_n, \; \forall n\in [\Lambda-i]$ holds for $${(a^{*}_n)_{n\in[\Lambda-i]}{\stackrel{\triangle}{=}}\big\{V_n\cdot a_n, n\in [\Lambda-i]\big\}}$$ and $${(b^{*}_n)_{n\in[\Lambda-i-1]}{\stackrel{\triangle}{=}}\big\{V_n\cdot b_n, n\in [\Lambda-i-1]\big\}}.$$ As a result, we have $$\sum_{n\in[\Lambda-i]}V_n\cdot a_n \geq \sum_{n\in[\Lambda-i]}V_n\cdot b_n \label{ineq:Lnan_Lnbn}$$ which proves that $c_i\geq c_{i+1}$.
Proof of (\[eq:optimization1\]) \[lastproof\]
---------------------------------------------
Through the respective change of variables $t{\stackrel{\triangle}{=}}\Lambda\frac{M}{N}$, $x'_{i}{\stackrel{\triangle}{=}}\frac{x_{i}}{N}$ and $c'_{i}{\stackrel{\triangle}{=}}\frac{c_i}{N_0}$, in equations (\[eq:LBwithxi\]), (\[eq:sumfiles\]) and (\[eq:constr2\]), we obtain $$\begin{aligned}
T(\Lc,\chi)&\geq &\sum_{i=0}^{\Lambda}x'_{i}c'_i \label{eq:compact2}\\
\sum_{i=0}^{\Lambda}x'_{i}&=&1 \label{eq:constr111}\\
\sum_{i=0}^{\Lambda}ix'_{i}&\leq& t .\label{eq:constr22}\end{aligned}$$ Let $X$ denote a discrete integer-valued random variable with probability mass function $f_{X}(x)=\{x'_i ~\text{if}~x=i, \forall i\in \{0,1,\dots,\Lambda\}\}$, where the $x'_i$ are those that satisfy equation . The value $c'_i$ can also be seen as the realization of a random variable $Y{\stackrel{\triangle}{=}}g(X)$, where $g(x)=\frac{\sum_{r=1}^{\Lambda-x}\mathcal{L}_r{\Lambda-r\choose x}}{N_0{\Lambda\choose x}}$, having the same probability mass function as $X$, i.e. $f_{Y}(y)=\{x'_i ~\text{if}~y=c'_i, \forall i\in \{0,1,\dots,\Lambda\}\}$. Due to the equation in (\[eq:constr22\]), the expectation of $X$ is bounded as $\mathbb{E}[X]\leq t$. Similarly, (\[eq:compact2\]) is equivalent to $T(\Lc,\chi)\geq \mathbb{E}[Y]$. From Jensen’s inequality, we have $
T(\Lc,\chi)\geq \mathbb{E}[Y]\geq g(\mathbb{E}[X])
$. Since the sequence $\{c'_i\}$ (and equivalently the function $g(x)$) is monotonically decreasing, the following lower bound holds $$T(\Lc,\chi)\geq g(\mathbb{E}[X])\geq g(t)=\frac{\sum_{r=1}^{\Lambda-t}\mathcal{L}_r{\Lambda-r\choose t}}{N_0{\Lambda\choose t}}.$$ This concludes the proof.
Proof of Equation (\[eq:totdelay2\]) \[sec:BinomialChangeProof\]
----------------------------------------------------------------
We remind the reader that (for brevity of exposition, and without loss of generality) this part assumes that the $|\Uc_{\lambda}|$ are in decreasing order.
We define the following quantity $$b_\lambda{\stackrel{\triangle}{=}}|\Uc_{1}|-|\Uc_{\lambda}|$$ and rewrite the total number of transmissions using the above definition as $$\begin{aligned}
&\sum_{j=1}^{|\Uc_{1}|}{{\Lambda \choose \Lambda\gamma+1}-{a_j \choose \Lambda\gamma+1}}\notag \allowbreak\\
&=|\Uc_{1}|{\Lambda \choose \Lambda\gamma+1}-\sum_{j=1}^{|\Uc_{1}|}{{a_j \choose \Lambda\gamma+1}}\notag \allowbreak\\
&=\sum_{i=1}^{\Lambda-\Lambda\gamma}{(|\Uc_{i}|+b_i){\Lambda-i \choose \Lambda\gamma}}-\sum_{j=1}^{|\Uc_{1}|}\sum_{i=\Lambda\gamma}^{a_j-1}{{i \choose \Lambda\gamma}}\allowbreak \notag \\
&=\sum_{i=1}^{\Lambda-\Lambda\gamma}{|\Uc_{i}|{\Lambda-i \choose \Lambda\gamma}}+{\sum_{i=\Lambda\gamma}^{\Lambda-1}{b_{\Lambda-i}{i \choose \Lambda\gamma}}}\allowbreak \notag\\
&-{\sum_{j=1}^{|\Uc_{1}|}\sum_{i=\Lambda\gamma}^{a_j-1}{{i \choose \Lambda\gamma}}}\allowbreak\notag\end{aligned}$$ $$\begin{aligned}
&\overset{(a)}{=}\sum_{i=1}^{\Lambda-\Lambda\gamma}{|\Uc_{i}|{\Lambda-i \choose \Lambda\gamma}}+{\sum_{i=\Lambda\gamma}^{\Lambda-1}{\sum_{j:a_j\geq i+1}^{|\Uc_{1}|}{i \choose \Lambda\gamma}}}\notag\allowbreak\\
&-{\sum_{j:a_j-1\geq \Lambda\gamma}^{|\Uc_{1}|}\sum_{i=\Lambda\gamma}^{a_j-1}{{i \choose \Lambda\gamma}}}\notag\allowbreak\\
&\overset{(b)}{=}\sum_{i=1}^{\Lambda-\Lambda\gamma}{|\Uc_{i}|{\Lambda-i \choose \Lambda\gamma}}+\sum_{j:a_j\geq \Lambda\gamma+1}^{|\Uc_{1}|}{\sum_{i=\Lambda\gamma}^{a_j-1}{i \choose \Lambda\gamma}}\notag \allowbreak\\
&-\sum_{j:a_j-1\geq \Lambda\gamma}^{|\Uc_{1}|}\sum_{i=\Lambda\gamma}^{a_j-1}{{i \choose \Lambda\gamma}}\notag\\
&=\sum_{i=1}^{\Lambda-\Lambda\gamma}{|\Uc_{i}|{\Lambda-i \choose \Lambda\gamma}}\label{eq:finalform}\end{aligned}$$ where step $(a)$ uses the equality $b_{\Lambda-i}=\sum_{j:a_j\geq i+1}^{|\Uc_{1}|}{1}$, and where step $(b)$ follows by changing the counting order of the double summation in the second summand. Substituting (\[eq:finalform\]) into the numerator of (\[eq:totdelay1\]) yields the overall delivery time given in (\[eq:totdelay2\]). The same performance holds for any $\Uc$ with the same profile $\Lc$.
Transition to the Multiple File Request Problem\[sec:AppendixMultipleFileRequests\]
-----------------------------------------------------------------------------------
We here briefly describe how the converse and the scheme presented in the shared cache problem, can fit the multiple file request problem.
#### Converse {#converse .unnumbered}
In Remark \[rem:multipleFilerequstsResult\] we described the equivalence between the two problems. Based on this equivalence, we will describe how the proof of the converse in Section \[sec:converse\] holds in the multiple file request problem with $N_0=1$, where now simply some terms carry a different meaning. Firstly, each entry $\boldsymbol{d_{\lambda}}$ of the vector defined in equation now denotes the vector of file indices requested by user $\lambda$. Then we see that Lemma \[lem:cons\_acyclic\] (proved in Section \[proof:cons\_acyclic\]) directly applies to the equivalent index coding problem of the multiple file requests problem, where now, for a given permutation $\sigma$ (see Section \[proof:cons\_acyclic\]), all the subfiles placed in row $\lambda$ — i.e., subfiles $W^{\boldsymbol{d_{\sigma(\lambda)}}(j)}_{\Tau_\lambda},\forall j\in\mathcal{U}_{\sigma(\lambda)}$ for all $\Tau_\lambda\subseteq[\Lambda]\setminus\{\sigma(1),\dots,\sigma(\lambda)\}$ — are obtained from different files requested by the same user, and therefore any two of these subfiles/nodes are not connected by any edge in the side information graph. After these two considerations, the rest of the proof of Lemma \[lem:cons\_acyclic\] is exactly the same. The remaining of the converse consists only of mathematical manipulations which remain unchanged and which yield the same lower bound expression.
#### Scheme {#scheme .unnumbered}
The cache placement phase is identical to the one described in Section \[sec:SchemePlacement\], where now each cache $\lambda$ is associated to the single user $\lambda$. In the delivery phase, the scheme now follows directly the steps in Section \[sec:SchemeDelivery\] applied to the shared-link (single antenna) setting, where now $\mathcal{A}_{\lambda}=\mathcal{U}_{\lambda}$ (cf. ). As in the case with shared caches, the scheme consists of $\mathcal{L}_1$ rounds, each serving users $$\label{eq:UsersServerPerRound2}
\mathcal{R}_j=\bigcup_{\lambda\in[\Lambda]} \big( \mathcal{U}_{\lambda}(j):\mathcal{L}_\lambda \geq j \big)$$ where $\mathcal{U}_{\lambda}(j)$ is the $j$-th user in set $\mathcal{U}_{\lambda}$. The expression in now means that the multiple files requested by each user are transmitted in a *time-sharing* manner, and at each round the transmitter serves at most one file per user. Next, equation is replaced by $$\chi_\mathcal{Q}=\bigcup_{\lambda\in \mathcal{Q}}\big( \mathcal{U}_{\lambda}(j):\mathcal{L}_\lambda \geq j \big)$$ and then each transmitted vector described in equation , is substituted by the scalar $$x_{\chi_{\mathcal{Q}}}=\!\!\!\!\bigoplus_{\lambda\in \mathcal{Q}:\mathcal{L}_\lambda \geq j} W^{d_{\mathcal{U}_{\lambda}(j)}}_{\mathcal{Q}\backslash{\{\lambda\}},1}.$$ Finally decoding remains the same, and the calculation of delay follows directly.
[^1]: The authors are with the Communication Systems Department at EURECOM, Sophia Antipolis, 06410, France (email: [email protected], [email protected], [email protected]). The work is supported by the European Research Council under the EU Horizon 2020 research and innovation program / ERC grant agreement no. 725929 (project DUALITY).
[^2]: This work is to appear in part in the proceedings of ITW 2018. A preliminary version of this work, focusing only on the single-stream setting, can be found in [@PEU_arxiv_single].
[^3]: This work explores other cases as well, such as that where the performance measure is the average delivery delay, for which various bounds are presented.
[^4]: We also introduce a very brief parenthetical note that translates these results to the multiple file request scenario.
[^5]: We note that while the representation here is of a wireless model, the results apply directly to the standard wired multi-sender setting. In the high-SNR regime of interest, when $N_0=1$ and $\Lambda = K$ (where each cache is associated to one user), the setting matches identically the original single-stream shared-link setting in [@MN14]. In particular, the file size and log(SNR) are here scaled so that, as in [@MN14], each point-to-point link has (ergodic) capacity of 1 file per unit of time. When $N_0>1 $ and $\Lambda = K$ (again each cache is associated to one user), the setting matches the multi-server *wireline* setting of [@ShariatpanahiMK16it] with a fully connected linear network, which we now explore in the presence of fewer caches serving potentially different numbers of users.
[^6]: Here $\Lc$ is simply the vector of the cardinalities of $\mathcal{U}_\lambda, ~\forall\lambda\in\{1,\dots,\Lambda\}$, sorted in descending order. For example, $\mathcal{L}_{1}=6$ states that the highest number of users served by a single cache, is $6$.
[^7]: An example of a user-to-cache assignment could have that users $\mathcal{U}_1=(14,15)$ are assigned to helper node $1$, users $\mathcal{U}_2=(1,2,3,4,5,6,7,8)$ are assigned to helper node $2$, and users $\mathcal{U}_3=(9,10,11,12,13)$ to helper node $3$. This corresponds to a profile $\Lc=(8,5,2)$. The assignment $\mathcal{U}_1=(1,3,5,7,9,11,13,15)$, $\mathcal{U}_2=(2,4)$, $\mathcal{U}_3=(6,8,10,12,14)$ would have the same profile, and the two resulting $\mathcal{U}$ would belong to the same class labeled by $\Lc=(8,5,2)$.
[^8]: The time scale is normalized such that one time slot corresponds to the optimal amount of time needed to send a single file from the transmitter to the receiver, had there been no caching and no interference.
[^9]: This is also presented in the preliminary version of this work in [@PEU_arxiv_single].
[^10]: Here, this uniform case, naturally implies that $\Lambda|K$.
[^11]: For example, having $\mathcal{U}_2 = \{3,5,7\}$, means that user 2 has requested files $W^{d_{3}},W^{d_{5}},W^{d_{7}}$.
[^12]: This would then require $\frac{K}{\Lambda}$ such rounds in order to cover all $K$ users.
[^13]: Note also that having $\mathcal{L}_\lambda\geq N_0, \forall\lambda\in[\Lambda]$ guarantees that in any given $\boldsymbol{s_{\lambda,j}}, j\in[\mathcal{L}_\lambda]$, a user appears at most once.
[^14]: The transmitted-vector structure below draws from the structure in [@LampirisEliaJsac18], in the sense that it involves the linear combination of one or more *Zero Forcing* precoded (ZF-precoded) vectors of subfiles that are labeled (as we see below) in the spirit of [@MN14].
[^15]: A similar transmission method can be found also in the work of [@JinCaireGlobecom16] for the setting of decentralized coded caching with reduced subpacketization.
[^16]: Instead of ZF, one can naturally use a similar precoder with potentially better performance in different SNR ranges.
[^17]: Notice that by considering a subpacketization based on the power set $2^{[\Lambda]}$, and by allowing for any possible size of these subfiles, the generality of the result is preserved. Naturally, this does not impose any sub-packetization related performance issues because this is done only for the purpose of creating a converse.
[^18]: We caution the reader not to confuse the current permutations ($\sigma$) that are used to construct large-sized acyclic graphs, with the aforementioned permutations $\pi$ which are used to construct $\mathcal{D}_{\Lc}$.
[^19]: Notice that the index coding users/nodes who are associated to the same cache, are not linked by any edge in the corresponding graph.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The structure of topological charge fluctuations in the QCD vacuum is strongly restricted by the spectral negativity of the Euclidean correlator for $x\neq 0$ and the presence of a positive contact term. Some examples are considered which illustrate the physical origin of these properties.'
address:
- 'Dept. of Physics, University of Virginia, Charlottesville, VA 22904'
- 'Department of Physics & Astronomy, University of Kentucky, Lexington, KY 40506, USA'
- 'Center for Nuclear Studies, George Washington University, Washington, DC 20052, USA'
- 'Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA'
- 'CSSM and Dept. of Physics and Math. Physics, University of Adelaide, Adelaide, SA 5005, Australia'
author:
- 'H. Thacker, S.J. Dong, T. Draper, I. Horváth, F.X. Lee, K.F. Liu, J.B. Zhang,'
title: ' Topological Charge Correlators, Spectral Bounds, and Contact Terms [^1]'
---
Introduction
============
Although topological charge plays a fundamental role in our understanding of low-energy hadron physics, the detailed structure of topological charge fluctuations in the QCD vacuum is not well understood. The construction of a local topological charge density operator for QCD [@hasenfratz] in terms of a Dirac operator with GW symmetry [@neuberger] has made it possible not only to study local $q(x)$ distributions in Monte Carlo generated gauge fields, but to analyze these distributions in terms of an eigenmode expansion for the corresponding Dirac operator. As discussed in [@horvath], the resulting “eigenmode filtered” densities provide a physically meaningful way of removing short-wavelength background fluctuations and focusing on whatever longer range structures might appear. The necessity for some such filtering procedure is made clear by a fundamental property of the two-point correlator in [*Euclidean*]{} 4-space[@seiler], namely, that it must be [*negative*]{} at any nonzero separation, $$\label{eq:bound}
G(x)\equiv \langle q(x)q(0)\rangle \leq 0 \;\;{\rm for}\;\; |x|\neq 0$$ This follows from reflection positivity (because $q(x)$ is reflection odd), or equivalently, from spectral positivity in Minkowski space. In the latter derivation, the negative sign of the Euclidean correlator arises from the fact that [**B**]{} fields remain real under Euclidean rotation but [**E**]{} fields acquire a factor of $i$. The bound (\[eq:bound\]) places important restrictions on any realistic picture of topological charge in the QCD vacuum. For any nonzero separation, the positive contributions to the correlator from coherent, finite-size fluctuations of topological charge (e.g. instantons) must necessarily be overwhelmed by anti-correlated background fluctuations. Moreover, the requirement that the topological susceptibility $\chi_t=\int G(x)d^4x$ be positive implies that G(x) must include a positive contact term $\propto \delta^4(x)$ which makes the largest contribution to the $\chi_t$ integral.
The results of a numerical investigation of the topological charge correlator in QCD [@horvath] indicate that it is very short range and consistent with being dominated, in the continuum limit, by an effective delta-function contact term. The fact that the topological charge correlator in QCD is approximately a delta-function can also be inferred from numerical studies of the quenched pseudoscalar hairpin correlator (i.e. the $\eta'$ mass insertion diagram)[@chlogs]. The measured correlator fits extremely well at all time separations to the dipole form $\propto (1+m_{\pi}\tau)\exp(-m_{\pi}\tau)+(\tau\rightarrow T-\tau)$. This implies that the amputated diagram (which, in quenched QCD, is proportional to the topological charge correlator) has very little $q^2$ dependence and is approximately a delta-function in space-time.
In this talk, we will discuss some examples which illustrate how the negativity property (\[eq:bound\]) is satisfied in practice, and also consider the physical origin of the contact term. As a first example, consider the thermodynamics of a nonrelativistic free particle moving in one compact spatial dimension.[@arnold] Denoting the spatial coordinate by $\phi$, the action is $S = \frac{1}{2}\dot{\phi}^2$. The partition function at inverse temperature $\beta$ is given by the Euclidean path integral over all paths satisfying $$\phi(\beta) = \phi(0) + C\nu$$ where $C$ is the circumference of the compact dimension and $\nu$ is the winding number of the path. The winding number is the integral of a local topological charge density, $\nu = \int_{0}^{\beta}q(\tau)\,d\tau$ where $$q(\tau)=C^{-1}\dot{\phi}(\tau)$$ There are classical n-instanton solutions which satisfy the Euclidean equation of motion, $$\phi_n(\tau) = \frac{Cn}{\beta}\tau$$ with action $$S_n = \frac{C^2n^2}{2\beta^2}$$ We decompose any path into the sum of an n-instanton solution and periodic fluctuations around it, $$\phi(\tau) = \phi_n(\tau) + \delta\phi(\tau)$$ where $\delta\phi(\beta) = \delta\phi(0)$. Then it is easy to show that the topological charge correlator separates into a sum over instantons + oscillators, $$\begin{aligned}
\label{eq:correlator}
G(\tau) \equiv \langle \dot{\phi}(\tau)\dot{\phi}(0)\rangle =\nonumber\\
\frac{C^2}{\beta^2}\sum_n n^2e^{-\beta S_n}/\sum_n e^{-\beta S_n} + &\langle\delta\dot{\phi}(\tau)
\delta\dot{\phi}(0)\rangle\end{aligned}$$ The second term, coming from quantum fluctuations around the classical n-instanton solutions, is obtained by differentiating the free propagator $$\langle\delta\dot{\phi}(\tau)\delta\dot{\phi}(0)\rangle =
-\frac{1}{\beta}\frac{\partial^2}{\partial\tau^2} \sum_{q_j}
\frac{e^{-iq_j\tau}}{q_j^2+\lambda^2}$$ where $q_j =\frac{2\pi j}{\beta}$. Here $\lambda\rightarrow 0$ is a small infrared cutoff parameter. Thus, the oscillator contribution is $$\label{eq:oscillator}
\langle\delta\dot{\phi}(\tau)\delta\dot{\phi}(0)\rangle =
\frac{1}{\beta}\sum_{j\neq 0}e^{-iq_j\tau}=\delta(\tau)-\frac{1}{\beta}$$ Now let’s consider how the correlator (\[eq:correlator\]) satisfies the bound (\[eq:bound\]). There are two limiting cases of interest:\
(I) Semiclassical or high temperature limit ($\beta\rightarrow 0$ or $C\rightarrow \infty$). In this limit, the instanton expansion converges, but the terms are exponentially suppressed. The bound (\[eq:bound\]) is satisfied because the negative term $-1/\beta$ from the quantum fluctuations (\[eq:oscillator\]) is always larger than the positive instanton contribution. In this case, if we introduce a $\theta$ term, the instanton expansion gives a good description of $\theta$ dependence (e.g. topological susceptibility).\
(II) Ultra-quantum mechanical or low temperature limit ($\beta\rightarrow \infty$ or $C\rightarrow 0$). In this case, the instanton sum diverges. Instead of expanding in winding number, the instanton series may be resummed by a Poisson transformation $$\sum_ne^{-n^2/\alpha} = \sqrt{\pi\alpha}\sum_m e^{-\alpha\pi^2 m^2}$$ Using this formula, we find, in the large $\beta$ limit, the resummed instanton expansion $\rightarrow +1/\beta$ Thus, in this limit, the $-1/\beta$ from quantum fluctuations exactly cancels the instanton contribution, leaving only the contact term, $$G(\tau) \rightarrow \delta(\tau)$$ Note that, in case II, the expansion of the Poisson-resummed instanton series is in no sense an expansion in number of instantons, but is in fact dual to it. It’s convergence corresponds to a breakdown of the usual instanton expansion. In some respects, this case may be viewed as a greatly oversimplified analog of Witten’s picture of the QCD vacuum, in which topological susceptibility is finite, but is not properly described in terms of an instanton expansion.
Note the origin of the contact term in this 0+1 dimensional example. In momentum space, the two factors of $q$ coming from the derivatives in the definition of the topological charge operators exactly cancel the $1/q^2$ pole of the propagator. This is a manifestation of “vacuum seizing”[@kogut], originally discussed in the Schwinger model as a possible mechanism for resolving the U(1) problem in QCD.
As a second example of a Euclidean topological charge correlator, we consider the CP(N-1) sigma model in two space-time dimensions. We have studied the topological charge correlator via (1) the large N expansion, (2) a lattice strong coupling expansion, and (3) numerical Monte Carlo calculations. A complete discussion of this study will be presented elsewhere [@brelidze]. The results all indicate the dominance of the contact term in the TC correlator. First consider the large N expansion. It is well-known that, to leading order in large N, the auxiliary U(1) gauge field develops a kinetic term and becomes dynamical due to scalar loop effects, thereby generating a long range (confining) Coulomb potential. Thus the gauge field correlator behaves like $$\int d^2x e^{iq\cdot x}\langle A_{\mu}(x)A_{\nu}(0)\rangle
\approx \frac{1}{q^2}\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^2}\right)$$ The corresponding Euclidean correlator for the topological charge operator $q(x)=\epsilon^{\mu\nu}\partial_{\mu}A_{\nu}$ thus produces a contact term, $$\int d^2x e^{iq\cdot x}\langle q(x) q(0)\rangle
\approx const.$$
We have also studied the TC correlator for CP(N-1) in a lattice strong-coupling expansion and by Monte Carlo simulation. For the time-dependent correlator $$G(x_0) = \int dx_1 \langle q(x) q(0) \rangle|_{x_0=\tau} \; ,$$ we find that, for CP(1) in the region $0\leq\beta\leq 2.0$ (correlation length $\leq 35$), $G(\tau)$ is completely dominated by a contact term of the form $G(\tau)= C_0\delta(\tau)
+C_2\delta''(\tau)$, with $C_0\rightarrow \approx 0$ in the weak coupling region. Calculations for larger N models are in progress.
Based on these examples, one might suspect that a short range topological charge correlator dominated by a positive contact term is associated with a strong-coupling vacuum structure for which a description based on classical instanton solutions is inappropriate. In view of the recent QCD results [@horvath] it is interesting to ask whether a theoretical mechanism exists for generating such a contact term in QCD. Using an operator product expansion, it is found that not only does the OPE predict the existence and approximate magnitude of the contact term [@bardeen], but the calculation itself closely resembles the vacuum seizing mechanism encountered in the simpler examples. The key point is that, because $\langle F^2\rangle\neq 0 $ in the QCD vacuum, there is a term in the OPE for the $F\tilde{F}$ correlator where one gluon carries the large momentum $q$, while a pair of soft gluons (one from each source) disappears into the vacuum. Just as in the simpler examples, the $1/q^2$ pole of this gluon propagator is cancelled by the momentum factors which arise from derivatives in the definition of the topological charge operator. A straightforward calculation shows that this effective one-gluon exchange graph gives a contact term $\propto \delta^4(x)$, and also $\propto \langle F^2\rangle$. Using the QCD sum rule estimate of $\langle F^2\rangle$, one finds [@bardeen] that this contact term makes a contribution to the $\eta'$ mass of $m_{\eta'} \approx \sqrt{\alpha_S}\times 400\; MeV$.
[9]{} P. Hasenfratz, V. Laliena, F. Niedermayer, Phys. Lett. B427, 125 (1998). H. Neuberger, Phys. Lett. B417, 141 (1998). I. Horváth et al, [hep-lat/0203027]{}, and I. Horvath, et al, these proceedings. E. Seiler, I.O. Stamatescu, [MPI-PAE/Pth 10/87]{}. W. Bardeen, A. Duncan, E. Eichten, and H. Thacker, Phys. Rev. D62:114505 (2000), and E. Eichten, et al, these proceedings. P. Arnold and L. McLerran, Phys. Rev. D37, 1020 (1988); We thank Peter Arnold for an illuminating discussion of this example. J. Kogut and L. Susskind, Phys. Rev. D11, 1199 (1975). T. Brelidze and H. Thacker (in preparation). W. Bardeen (private communication).
[^1]: Talk presented by H. Thacker
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Some multiple hypergeometric transformation formulas arising from the balanced duality transformation formula are discussed through the symmetry. Derivations of some transformation formulas with different dimensions are given by taking certain limits of the balanced duality transformation. By combining some of them, some transformation formulas for $A_n$ basic hypergeometric series is given. They include some generalizations of Watson, Sears and ${}_8 W_7$ transformations.'
author:
- Yasushi KAJIHARA
title: '**Multiple basic hypergeometric transformation formulas arising from the balanced duality transformation**'
---
[Keywords: basic hypergeometric series, multivariate basic hypergeometric series]{}
Introduction
============
This paper can be considered as a continuation of our paper [@Kaji1]. Namely, we discuss some multiple hypergeometric transformation formulas arising from the balanced duality transformation formula through the symmetry in this paper. We obtain some transformation formulas with different dimensions by taking certain limits of the balanced duality transformation. By combining some of them, we give some transformation formulas for $A_n$ basic hypergeometric series. They include some generalizations of Watson, Sears and nonterminating and terminating ${}_8 W_7$ transformations.
The hypergeometric series ${}_{r+1} F_r$ is defined by $${}_{r+1} F_r
\left[ \
\begin{matrix}
a_0, & a_1, & a_2, & \cdots, & a_r \\
& b_1, & b_2, & \cdots, & b_r \\
\end{matrix}
\ \ ; z \
\right]
:=
\sum_{k \in \Bbb N}
\frac
{[a_0, a_1, \cdots, a_r]_k}
{ k ! \ [b_1, \cdots, b_r]_k} \
z^k,$$ where $[c]_k = c (c+1) \cdots (c+ k -1)$ is Pochhammer symbol and $[d_1, \cdots , d_r]_k = [d_1]_k \cdots [d_r]_k$.
The very well-poised hypergeometric series have nice properties such as the existence of various kinds of summation and transformation formulas which contains more parameters than other hypergeometric series and they have reciprocal structure (For precise see an excellent exposition by G.E. Andrews [@AWP]). In [@B1], W.N.Bailey derived the following transformation formula for terminating balanced and very well-poised ${}_9 F_8$ series which is nowadays called the Bailey transformation formula:
$$\begin{aligned}
\label{CBaileyT1}
&&
{}_{9} F_{8}
\left[ \
\begin{matrix}
a, & a/2 +1,& b, & c, & d, \\
& a/2, & 1 + a -b, & 1 + a-c, & 1 + a -d,
\end{matrix}
\right.
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad
\left.
\begin{matrix}
e, & f, & g, & -N \\
1 + a -e, & 1 +a - f, & 1 + a - g, & 1+ a +N
\end{matrix} \ \ ; \ 1 \
\right]
\nonumber
\\
&=&
\frac
{[1+ a, 1 + \lambda - e,
1 + \lambda - f, 1 + \lambda - g]_N}
{[1+ \lambda, 1 + a - e,
1 + a - f, 1 + a - g]_N}
\nonumber \\
& \times &
{}_{9} F_{8}
\left[ \
\begin{matrix}
\lambda, & \lambda/2 +1,& \lambda + b - a, & \lambda + c - a, & \lambda +d -a, \\
& \lambda /2, & 1 + a -b, & 1 + a-c, & 1 + a -d,
\end{matrix}
\right.
\nonumber
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad
\left.
\begin{matrix}
e, & f, & g, & -N \\
1 + \lambda -e, & 1 + \lambda - f, & 1 + \lambda - g, & 1+ \lambda +N
\end{matrix} \ \ ; \ 1 \
\right],
\nonumber \end{aligned}$$
where $ \lambda = 1 + 2a - b- c- d$, and the parameters are subject to the restriction (called as balancing condition) that $$\label{cbc}
2 + 3 a = b + c + d + e + f + g -N.$$
Among various hypergeometric transformation formulas, the Bailey transformation itself and its special and limiting cases, including their basic and elliptic analogues, have many significant applications in various branches of mathematics and mathematical physics.
On the other hand, Holman, Biedenharn and Louck [@H1], [@HBL] introduced a class of multiple generalization of hypergeometric series in need of the explicit expressions of the Clebsch-Gordan coefficients of the tensor product of certain irreducible representations of the Lie group $SU(n+1)$. In the series of papers, S.Milne [@MilneG] introduced its basic analogue and investigated further. In the course of works (see the expository paper by Milne [@MilneNagoya] and references therein), he and his collaborators succeeded to obtain some multiple hypergeometric transformation and summation formulas by using a certain rational function identity which is nowadays referred as Milne‘s fundamental lemma. After that, many methods of the derivation of multiple hypergeometric identities have been worked out such as ingenious uses of certain matrix techniques by Krattenthaler, Schlosser, Milne, Lilly and Newcomb, see [@BS], [@Milne3], [@LM1], [@MilNew1].
In the previous work [@Kaji1], we derived a certain generalization of the Euler transformation formula for multiple (basic) hypergeometric series with different dimensions by using the techniques in the theory of Macdonald polynomials and Macdonald‘s $q$-difference operators. By interpreting our multiple Euler transformation formula as the generating series, we further obtained several types of multiple hypergeometric summations and transformations.
Among these, we consider the [*(balanced) duality transformation formula* ]{} which generalize the following ${}_9 F_8$ transformation (see Bailey‘s book [@BB]): $$\begin{aligned}
\label{Cmn1BDT1}
&&
{}_{9} F_{8}
\left[ \
\begin{matrix}
a, & a/2 +1,& b, & c, & d, \\
& a/2, & 1 + a -b, & 1 + a-c, & 1 + a -d,
\end{matrix}
\right.
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad
\left.
\begin{matrix}
e, & f, & g, & -N \\
1 + a -e, & 1 +a - f, & 1 + a - g, & 1+ a +N
\end{matrix} \ \ ; \ 1 \
\right]
\nonumber
\\
&=&
\frac
{[1+ a, 1 + a - b-c, 1 + a -b-d, 1+a -b -e, 1 + a - b -f, g]_N}
{[1+ a- b, 1 + a -c, 1 + a -d, 1 + a - e,
1 + a - f, g- b]_N}
\nonumber \\
& \times &
{}_{9} F_{8}
\left[ \
\begin{matrix}
b - g - N, & (b - g - N)/2 +1,& b, & 1 + a - c- g, & 1 + a -d -g, \\
& (b - g -N)/2, & 1 - g - N, & b + c - a- N, & b + d - a- N,
\end{matrix}
\right.
\nonumber
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad
\left.
\begin{matrix}
1 + a - e- g, & 1+ a - f - g, & b - a -N, & -N \\
b + e - a - N, & b + f - a -N, & 1 + a - g, & 1+ b - g
\end{matrix} \ \ ; \ 1 \
\right],
\nonumber \end{aligned}$$ with the balancing condition , as the formula of particular importance. is different from , but can be obtained by duplicating . In the joint work with M.Noumi [@KajiNou], we derived an analogous formula ((3.17) of [@KajiNou]) for elliptic hypergeometric series introduced by Frenkel and Turaev [@FT] by starting from the Frobenius determinant and proposed the notion of [ (balanced) duality transformation]{} there.
In this paper, we present an alternative approach by starting from the balanced duality transformation formula for multiple hypergeometric series of type $A$. In Section 3, we present some transformation formula for basic hypergeometric series of type $A$ with different dimension. They include most of results in our previous work [@Kaji1]. What is remarkable is that from the balanced duality transformation formula, one can obtain multiple Euler transformation formula itself.
By iterating twice a special case of our Sears transformation formula (see section 7 in [@Kaji1]), we verified an $A_n$ Sears transformation formula in [@KajiS]. Later by the same idea as above, we obtained in [@KajiNou] two types of $A_n$ Bailey transformation formulas: one of which is previously known by Milne and Newcomb [@MilNew1] in basic case and Rosengren [@RoseE] in elliptic case and another has appeared to be new in both cases. In Section 4, we further employ this idea to obtain several $A_n$ basic hypergeometric transformations which generalize Watson, Sears and nonterminating ${}_8 W_7$ transformations. They includes known ones due to Milne and his collaborators. We will see here that our class of multiple hypergeometric transformations shed a light to the structure of some $A_n$ hypergeometric transformation formulas.
Preliminaries
=============
In this section, we give some notations for multiple basic hypergeometric series and present the balanced duality transformation formula. We basically refer the notations of $q$-series and basic hypergeometric series from the book by Gasper and Rahman [@GR1]. Throughout of this paper, we assume that $q$ is a complex number under the condition $0 < |q|< 1$. We define $q-$shifted factorial as $$\label{ShiftFact}
(a)_\infty := (a;q)_\infty =\prod_{n \in \Bbb N}(1 - a q^n), \quad
(a)_k := (a;q)_k = \frac{(a)_\infty}{(a q^k)_\infty}
\quad \mbox{for}
\ k \in \Bbb C.$$ where, unless stated otherwise, we omit the basis $q$. In this paper we employ the notation as $$(a_1)_k \cdot (a_2)_k \cdots (a_n)_k
=(a_1, a_2, \cdots , a_n)_k.$$
The basic hypergeometric series ${}_{r+1} \phi_{r}$ is defined by $$\begin{aligned}
\label{BHSdef}
&&
{}_{r+1} \phi_r
\left[
\begin{matrix}
a_1, a_2, \ldots a_{r+1}\\
c_1, \ldots c_r
\end{matrix}
;q; u
\right] =
{}_{r+1} \phi_r
\left[
\begin{matrix}
a_0, \{ a_i \}_{r}\\
\{c_i \}_r
\end{matrix}
;q; u
\right]
\\
&&
\quad \quad \quad \quad \quad \quad \quad
:= \sum_{k \in \Bbb N}
\frac{(a_0, a_1, \ldots ,a_{r+1})_k }
{( c_1, \ldots , c_r, q )_k} u^k
\nonumber\end{aligned}$$ with $r+1$ numerator parameters $ a_0, a_1, \cdots , a_{r+1}$ and $r$ denominator parameters $c_1, \cdots , c_r$. We call ${}_{r+1} \phi_r$ series $k$-balanced if $ q^k a_1 a_2 \cdots a_{r+1} = c_1 \cdots c_r$ and $u=q$: a $1$-balanced series is called balanced (or Saalschützian). An ${}_{r+1} \phi_{r}$ series is called well-poised if $a_0 q = a_1 c_1 = \cdots = a_r c_r$. In addition, if $a_1 = q \sqrt{a_0}$ and $a_2 = - q \sqrt{a_0}$, then the ${}_{r+1} \phi_r $ is called very well-poised. We denote the very well-poised basic hypergeometric series ${}_{r+3} \phi_{r+2} $ as ${}_{r+3} W_{r+2}$ series defined by the following:
$$\begin{aligned}
\label{DefVWP}
&&{}_{r+3} W_{r+2}
\left[
s; \{a_i \}_r;
q; u
\right]
:=
{}_{r+3} \phi_{r+2}
\left[
\begin{matrix}
& s, & q \sqrt{s},& - q \sqrt{s},& \{ a_i \}_r \\
& & \sqrt{s},& - \sqrt{s},& \{ s q / a_i \}_r
\end{matrix};
q; u
\right]
\\
&&
\quad \quad \quad \quad \quad \quad
\ = \
\sum_{k \in {\Bbb N}}
\frac{1 - s q^{2 k} }{1 - s}
\frac{( s, a_1 \cdots ,a_r)_k}
{(q, s q / a_1, \cdots , s q /a_r)_k}
u^k.
\nonumber\end{aligned}$$
Furthermore, all of the very well-poised basic hypergeometric series ${}_{r+3} W_{r+2}$ in this paper, the parameters $s, a_1, \cdots a_r$ and the argument satisfy the very-well-poised-balancing condition $$\label{vwpb}
a_1 \cdots a_r u = \left( \pm (s q)^{\frac{1}{2}} \right)^{r-1}$$ with either the plus and minus sign. We call a ${}_{r+3} W_{r+2}$ series very-well-poised-balanced if holds. Note that a very-well-poised-balanced ${}_{r+3} W_{r+2}$ series is ($1$-)balanced if $$a_1 \cdots a_r = s^{\frac{r-1}{2}} q^{\frac{r-3}{2}}$$ and $u=q$.
Now, we note the conventions for naming series as $A_n$ basic hypergeometric series (or basic hypergeometric series in $SU(n+1)$). Let $\gamma= (\gamma_1, \cdots , \gamma_n) \in \mathbb{N}^n$ be a multi-index. We denote $$\Delta(x) := \prod_{1 \le i < j \le n}
(x_i - x_j)
\quad \mbox{and}
\quad
\Delta(x q^\gamma) := \prod_{1 \le i < j \le n}
(x_i q^{\gamma_i} - x_j q^{\gamma_j}),$$ as the Vandermonde determinant for the sets of variables $x=(x_1, \cdots , x_n)$ and $x q^\gamma = (x_1 q^{\gamma_1}, \cdots , x_n q^{\gamma_n})$ respectively. In this paper we refer multiple series of the form $$\label{DefAnBHS}
\sum_{\gamma \in {\Bbb N}^n}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\
H(\gamma)$$ which reduce to basic hypergeometric series ${}_{r+1} \phi_r$ for a nonnegative integer $r$ when $n=1$ and symmetric with respect to the subscript $1 \le i \le n$ as $A_n$ basic hypergeometric series. We call such a series balanced if it reduces to a balanced series when $n=1$. Very well-poised and so on are defined similarly. The subscript $n$ in the label $A_n$ attached to the series is the dimension of the multiple series .
In our previous work [@Kaji1], we derived a hypergeometric transformation formula for multiple basic hypergeometric series of type $A$ generalizing the following transformation for terminating balanced ${}_{10} W_9$ series in the one dimensional case: $$\begin{aligned}
\label{m1n1BDT1}
&&
{}_{10} W_{9}
\left[
\begin{matrix}
a; b, c, d, e, f, \mu f q^N, q^{-N}
\end{matrix};
q; q
\right]
=
\frac
{(\mu b f/ a, \mu c f / a, \mu d f / a, \mu e f / a,
a q, f)_N }
{(a q / b, a q / c, a q / d, a q/e, \mu q, \mu f / a)_N}
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad \quad
\times
{}_{10} W_{9}
\left[
\begin{matrix}
\mu ; a q / b f, a q / c f, a q / d f, a q / e f,
\mu f / a, \mu f q^{N}, q^{-N}
\end{matrix};
q; q
\right],
\nonumber\end{aligned}$$ where $ \mu = a^3 q^2 /bcde f^2$. Note that is a basic analogue of . The transformation can be obtained by iterating the Bailey transformation formula for ${}_{10} W_9$ series
$$\begin{aligned}
\label{BaileyT1}
&&
{}_{10} W_{9}
\left[
a;b, c, d, e, f,
\lambda a q^{N+1}/e f, q^{-N}
; q; q
\right]
=
\frac{(a q, a q /e f, \lambda q /e, \lambda q /f)_N }
{(a q/ e, a q /f, \lambda q, \lambda q /e f)_N }
\\
&&
\quad \quad \quad
\times
{}_{10} W_{9}
\left[
\lambda; \lambda b /a,
\lambda c/ a, \lambda d / a , e, f,
\lambda a q^{N+1}/e f, q^{-N}
; q; q
\right]
\quad \quad \quad
( \lambda = a^2 q^{} /b c d),
\nonumber\end{aligned}$$
twice. Note also that is a basic analogue of .
To simplify the expressions for multiple very well-poised series, we introduce the notation of $W^{n,m}$ series as the following:
$$\begin{aligned}
\label{DefWSer}
&&
{
W^{n, m}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{a_i\}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{s};{ \{u_k \}_{m}};{\{v_k \}_{m}}; {q; z} \right)
}\\
&&
\quad
:=
\sum_{\gamma \in \mathbb{N}^n}{}\ z^{|\gamma|}
\prod_{1\le i<j\le n}{}\,
\frac{\Delta(x q^\gamma )}{\Delta(x)}\
\prod_{1 \le i \le n}
\frac{1 - sq^{|\gamma|+\gamma_i}x_i}
{1 - s x_i}
\nonumber
\\
&&
\quad \quad
\times
\prod_{1\le j \le n}
\frac{(s x_j)_{|\gamma|}}{( (s q /a_j) x_j) _{|\gamma|}}
\left(
\prod_{1 \le i \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}}{(q x_i / x_j )_{\gamma_i}}
\right)
\nonumber
\\
&&
\quad \quad \quad
\times
\prod_{1 \le k \le m}
\frac{(v_k)_{|\gamma|}}{( s q /u_k)_{|\gamma|}}
\left(
\prod_{1 \le i \le n}
\frac{(u_k x_i)_{\gamma_i}}{( ( s q/ v_k )x_i)_{\gamma_i}}
\right),\nonumber\end{aligned}$$
where $|\gamma| = \gamma_1 + \cdots + \gamma_n$ is the length of a multi-index $\gamma$.
The very starting point of all the discussions of the present paper is the [*balanced duality transformation formula*]{} (Corollary 6.3 of [@Kaji1] with a different notation) between the $W^{n,m+2}$ series ($A_n$ ${}_{2m+8} W_{2m+7}$ series) and $W^{m, n+2}$ series ($A_m$ ${}_{2n+8} W_{2n+7}$ series): $$\begin{aligned}
\label{BDT1}
&&
{
W^{n,m+2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc} \{ b_i\}_n \\ \{ x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{\{ c_k y_k \}_m, d, e};{\{f y_k^{-1}\}_m, \mu f q^N, q^{-N}}; {q; q} \right)
}
\\
&&
\quad
=
\frac{(\mu d f / a, \mu e f / a)_N}
{(a q / d, a q / e)_N}
\prod_{ 1 \le k \le m}
\frac{((\mu c_k f / a) y_k, f y_k^{-1} )_N}
{(\mu q y_k, (a q /c_k) y_k^{-1} )_N}
\prod_{ 1 \le i \le n}
\frac{(a q x_i, ( \mu b_i f / a ) x_i^{-1} )_N}
{((a q / b_i) x_i, (\mu f / a) x_i^{-1} )_N}
\nonumber\\
&&
\quad \quad
\times
{
W^{m,n+2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{ a q / c_k f \}_m \\ \{y_k\}_m\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{\mu};{\{(a q / b_i f) x_i \}_n, a q/ d f, a q/ e f}; \right.
}
\nonumber \\
&&
\quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\quad \quad
{\{ (\mu f / a) x_i^{-1} \}_n, \mu f q^N, q^{-N}};
{q; q}
\Big),\nonumber\end{aligned}$$ where $\mu = a^{m+2} q^{m+1} /BCde f^{m+1}$. Here we denote $B= b_1 \cdots b_n$ and $ C = c_1 \cdots c_m$. In this paper, we frequently use such notations.
In the case when $m=1$ and $y_1= 1$, reduces to $$\begin{aligned}
\label{m1BDT1}
&&
{
W^{n,3}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc} \{ b_i\}_n \\ \{ x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d, e};{f, \mu f q^N, q^{-N}}; {q; q} \right)
} \\
&&
\quad
=
\frac{(\mu c f /a, \mu d f / a, \mu e f / a, f )_N}
{(a q/ c, a q / d, a q / e, \mu q )_N }
\prod_{ 1 \le i \le n}
\frac{(a q x_i, ( \mu b_i f / a ) x_i^{-1} )_N}
{((a q / b_i) x_i, ( \mu f / a) x_i^{-1} )_N}
\nonumber
\\
&&
\quad \quad
\times
{}_{2 n + 8}W_{2 n + 7}
\left[
\mu; \{ (a q / b_i f) x_i \}_n, a q / c f, a q / d f, a q / e f,
\right.
\nonumber
\\
&& \qquad \qquad \qquad \qquad \quad \quad
\left.
\{ (\mu f / a) x_n / x_i\}_n, \mu f q^N, q^{-N}; q; q
\right],
\quad
(\mu = a^{3} q^{2} / B c d e f^{2}).
\nonumber\end{aligned}$$ Note that $m=n=1$ and $x_1 = y_1 = 1$ case of the balanced duality transformation formula is terminating balanced ${}_{10} W_9$ transformation .
In [@Kaji1], was obtained by taking the coefficients of $u^N$ in both sides of “$0$-balanced” case of the multiple Euler transformation formula for multiple basic hypergeometric series of type $A$ with different dimensions (Theorem 1.1 of [@Kaji1])
$$\begin{aligned}
\label{ETG}
&&
\sum_{\gamma \in {\Bbb N}^n}
u^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(b_k x_i y_k)_{\gamma_i}}{(c x_i y_k)_{\gamma_i}}
\\
&&
\quad
=
\
\frac{(A B u/ c^m)_\infty}
{(u)_\infty}
\
\sum_{\delta \in {\Bbb N}^m}
\left( \frac{A B u}{c^m} \right)^{|\delta|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le k, l \le m}
\frac{((c / b_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((c /a_i^{}) x_i y_k )_{\delta_k}}
{(c x_i y_k)_{\delta_k}}.
\nonumber\end{aligned}$$
From this point of view, we can state the balanced duality transformation in more general form: (Proposition 6.2 in [@Kaji1]) $$\begin{aligned}
\label{HBDT1}
&&
\sum_{\gamma \in {\Bbb N}^n, |\gamma| = N}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(b_k x_i y_k)_{\gamma_i}}{(c x_i y_k )_{\gamma_i}}
\\
&&
\quad
= \
\sum_{\delta \in {\Bbb N}^m, |\delta| = N}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\prod_{1 \le k, l \le m}
\frac{((c / b_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((c /a_i^{}) x_i y_k )_{\delta_k}}
{(c x_i y_k )_{\delta_k}},
\nonumber\end{aligned}$$ when $ AB = c^m$.
The balanced duality transformation formula corresponds to the case when $ m,n \ge 2$ of by a rearrangement of parameters in the multiple basic hypergeometric series. Note that, in the case when $m=n=1$, becomes tautological. We shall also remark that the remaining case ($m=1, n \ge 2$) corresponds to the $A_n$ Jackson summation formula for terminating balanced $W^{n,2}$ series $$\begin{aligned}
\label{JS1}
&&
{
W^{n,2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc} \{ b_i\}_n \\ \{ x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{ c, e};{d, q^{-N}}; {q; q} \right)
} \\
&&
\quad \quad \quad \quad \quad \quad
\ = \
\frac{(a q / B c, a q / c d)_N}
{(a q / B c d, a q / c)_N}
\prod_{1 \le i \le n}
\frac{((a q /b_i d) x_i, a q x_i)_N}
{((a q /b_i) x_i, (a q/ d) x_i)_N}
\nonumber\end{aligned}$$ provided $ a^2 q^{N+1} = B cde$, which is originally due to S.Milne [@Milne2] in a different notation. For these facts, the informed readers might see [@Kaji1] for ordinary and basic case and Noumi and the author [@KajiNou] for elliptic case.
In the case when $n=1$ and $x_1= 1$, reduces to the Jackson summation formula for terminating balanced ${}_8 W_7$ series: $$\begin{aligned}
\label{JacksonSum}
{}_8 W_7
[
{a}; b, { c, d, e}
, { q^{-N}};
{q; q}
]
&=&
\frac{(a q, a q / b c, a q / b d, a q / c d)_N}
{(a q / b c d, a q / b, a q / c, a q / d)_N}
, \quad
(a^2 q^{N+1} = b c d e).\end{aligned}$$ We also mention that can be obtained by letting $aq = ef$ in and by relabeling the parameters.
Limit cases
===========
In this section, we shall show that several transformation formulas with different dimension can be obtained from the balanced duality transformation formula by taking certain limits. We see that most of principal transformation formulas in [@Kaji1] which have been obtained by taking a certain coefficient in the multiple Euler transformation formula can be recovered and we find some new transformation formulas. Furthermore we show that itself can be acquired in this manner.
In addition, we shall write down the cases when the dimension of summand in each side of the transformation is one with particular attention. We consider them as particularly significant ones since they have an extra symmetry. We will explore some $A_n$ hypergeometric transformations by using some of them in the next section.
(Non-balanced) Duality transformation formula and its inverse
-------------------------------------------------------------
[**[(Non-balanced) Duality transformation formula]{}**]{}
$$\begin{aligned}
\label{DT1}
&&
{
W^{n, m+1}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{b_i \}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, \{d_k y_k \}_m};{q^{-N}, \{e y_k^{-1} \}_m}; {q; \frac{a^{m+1} q^{N + m + 1} }{B c D e^m}} \right)
}
\\
&&
\quad
=
\
\frac{(a^{m+1} q^{m+1} /B c D e^m)_N}
{(a q /c)_N}
\prod_{ 1 \le i \le n}
\frac
{(a q x_i)_N}{((a q / b_i)x_i)_N}
\prod_{ 1 \le k \le m}
\frac
{(e y_k^{-1})_N}{((a q/ d_i) y_k^{-1} )_N}
\nonumber\\
&&
\quad \quad
\times
\sum_{\delta \in {\Bbb N}^m}
q^{|\gamma|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\frac
{(q^{-N})_{|\gamma|}}
{(a^{m+1 } q^{m+1} /B c D e^m)_{|\gamma|}}
\prod_{1 \le k,l \le m}
\frac{((a q / d_l e ) y_k / y_l)_{\delta_k}} {(q y_k / y_l)_{\delta_k}}
\nonumber
\\
&&
\quad \quad \quad
\times
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{( (a q /b_i e) x_i y_k )_{\delta_k}}
{((a q / e) x_i y_k )_{\delta_k}}
\prod_{1 \le k \le m}
\frac{( (a q /c e) y_k)_{\delta_k}}
{(q^{1-N} e^{-1} y_k)_{\delta_k}},
\nonumber\end{aligned}$$
where $B = b_1 \cdots b_m$ and $D = d_1 \cdots d_n$.
Take the limit $e \to \infty$ in . By rearranging the parameter as $f \to e$, we arrive at the desired identity.
The transformation formula has already appeared in Section 6.1 of our previous work [@Kaji1] with a different notation. Later Rosengren has also obtained in [@RoseKM] by using his reduction formula of Karlsson-Minton type. In contract to that we see here, as is mentioned in [@RoseKM], the balanced duality transformation formula can also be considered as a special case of . Indeed, in [@Kaji1] though we have started the derivation of from the multiple Euler transformation in general case, has been obtained from the “zero”-balanced case of .
In the case when $m=n=1$ and $x_1 = y_1 = 1$, reduces to $$\begin{aligned}
\label{nm1DT1}
&&
{}_{8} W_{7}
\left[
a; b, c, d, e, q^{-N}; q; \frac{a^2 q^{N+2}}{b c d e}
\right]
\\
&& \quad \quad \quad
=
\frac{(a^2 q^2 / b c d e, e, a q)_N}
{(a q / b, a q / c, a q / d)_N}
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, a q / b e, a q / c e, a q / d e \\
q^{1-N} /e, a^2 q^2 / b c d e, a q / e
\end{matrix};
q; q
\right].
\nonumber\end{aligned}$$
In [@Kaji1], is referred as Watson type transformation formula. But, hereafter we shall propose to call (non balanced) duality transformation formula.
In the case when $m=1$ and $y_1$, reduces to
$$\begin{aligned}
\label{m1DT1}
&&
{
W^{n, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{b_i \}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d};{q^{-N}, e }; {q; \frac{a^{2} q^{N + 2} }{B c d e}} \right)
}
=
\frac
{( a^{2} q^{2} / B c d e, e)_N}
{(a q /c, a q / d)_N }
\prod_{ 1 \le i \le n}
\frac
{(a q x_i)_N }
{((a q / b_i) x_i)_N }
\nonumber\\
&& \quad \quad \quad
\times
{}_{n+3} \phi_{n+2}
\left[
\begin{matrix}
q^{-N}, \{(a q /b_i e) x_i \}_{n},
a q / c e, a q / d e \\
q^{1-N} /e, \{(a q /e) x_i \}_{n},
a^2 q^2 / B c d e,
\end{matrix};
q; q
\right].\end{aligned}$$
In the case of $n=1$ and $x_1 =1$, reduces to $$\begin{aligned}
\label{n1DT1}
&&{}_{2 m + 6} W_{2 m + 5}
\left[
a; b, \{c_k y_k \}_{m},
d, \{e y_k^{-1} \}_{m}, q^{-N}; q;
\frac{a^{m+1} q^{N+m+1}}{b C d e^m}
\right]
\\
&&
\quad =
\frac
{( a^{m+1} q^{m+1} / b C d e^{m}, a q)_N}
{(a q / b, a q / d)_N }
\prod_{ 1 \le k \le m}
\frac
{(e y_k^{-1} )_N}
{((a q /c_k) y_k^{-1} )_N}
\nonumber\\
&&
\quad \quad
\times
\sum_{\delta \in {\Bbb N}^{m} }
q^{|\delta|}
\frac{\Delta ({y} q^{\delta})}{\Delta ({y})}
\prod_{1 \le k, l \le m}
\frac{(( a q / c_l e) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\nonumber\\
&&
\quad \quad \quad \times
\frac{(q^{-N})_{|\delta|}}
{( a^{m+1} q^{m+1} / b C d e^{m})_{|\delta|}}
\prod_{1 \le k \le m}
\frac{((a q /b e) y_k, (a q / d e ) y_k)_{\delta_k}}
{(( a q / e) y_k, ( q^{1-N} / e) y_k)_{\delta_k}}.
\nonumber\end{aligned}$$
Further, by letting $aq = de$ in , we obtain Milne‘s $A_n$ generalization of Rogers‘ summation formula for terminating very well-poised ${}_6 W_5$ series:
$$\begin{aligned}
\label{RS1}
{
W^{n, 1}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{b_i \}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c};{q^{-N}}; {q; \frac{a^{} q^{N + 1} }{B c}} \right)
}
&=&
\frac
{( a^{} q^{} / B c)_N}
{(a q /c)_N }
\prod_{ 1 \le i \le n}
\frac
{(a q x_i)_N }
{((a q / b_i) x_i)_N }.
$$
In the case when $n=1$ and $x_1= 1$, reduces to the Rogers‘ summation formula ( (2.4.2) in [@GR1] ): $$\begin{aligned}
\label{RogersSum1}
{}_{6} W_{5}
\left[ a; b, c, q^{-N}; q; \frac{a q^{N+1}}{b c}
\right]
&=&
\frac{(a q, a q / b c)_N}
{(a q / b, a q /c )_N}.\end{aligned}$$
[**[The inverse of the duality transformation]{}**]{}
$$\begin{aligned}
\label{IDT1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}}
{(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac
{(b_k x_i y_k)_{\gamma_i} }
{(e x_i y_k )_{\gamma_i} }
\\
&& \quad \times
\frac{(q^{-N})_{|\gamma|}}
{(d)_{|\gamma|}}
\prod_{1 \le i\le n}
\frac{(c x_i)_{\gamma_i}}
{( (AB c q^{1-N}/ d e^m) x_i)_{\gamma_i}}
\nonumber
\\
&&
\quad \quad
=
\frac{(d e^m /A B)_N}
{(d)_N}
\prod_{ 1 \le k \le m}
\frac
{((d e^m b_k / A B c) y_k )_N}
{((d e^{m+1}/ A B c) y_k )_N}
\prod_{ 1 \le i \le n}
\frac
{((d e^m a_i / A B c) x_i^{-1} )_N}
{((d e^m/ A B c) x_i^{-1} )_N}
\nonumber\\
&& \quad \quad \quad
\times
{
W^{m, n+1}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{e/ b_k \}_m \\ \{y_k\}_m\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{d e^{m+1} q^{-1}/ A B c};{ e/ c, \{ (e / a_i) x_i \}_n}; \right.
}
\nonumber
\\
&&
\quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad
\quad \quad \quad
{q^{-N}, \{ (d e^m / A B c) x_i^{-1} \}_n};
{q; d q^N}
\big).
\nonumber\end{aligned}$$
Substitute the parameters $e$ and $f$ as $aq/e$ and $aq/f$ respectively in . Then take the limit $a \to \infty$. Finally, by rearranging the parameters as $b_i \to a_i \ (1 \le i \le n ), \ c_k \to b_k \ (1 \le k \le m), \ d \to
c, \ e \to d, \ f \to e$, we have the desired result.
In the case when $m=1$ and $y_1 = 1$, reduces to
$$\begin{aligned}
\label{m1IDT1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(b x_i, c x_i)_{\gamma_i}}
{(e x_i, (A b c q^{1-N}/ d e) x_i)_{\gamma_i}}
\\
&&
\quad
\times
\frac{(q^{-N})_{|\gamma|}}
{(d)_{|\gamma|}}
\ = \
\frac
{(d e / A c, d e /A b)_N}
{(d e^{2}/ A b c, d)_N}
\prod_{ 1 \le i \le n}
\frac
{((d e a_i / A b c) x_i^{-1} )_N}
{((d e / A b c) x_i^{-1} )_N}
\nonumber\\
&& \quad \quad
\times \
{}_{2 n +6} W_{2 n + 5}
\left[d e^{2} q^{-1}/ A b c;
\{ (e / a_i) x_i \}_{n},
\{ (d e q^{-1} / A b c ) x_i^{-1} \}_{n},
e/b, e/c, q^{-N};q ; d q^N
\right].
\nonumber\end{aligned}$$
In the case when $n=1$ and $x_1 = 1$, reduces to
$$\begin{aligned}
\label{n1IDT1}
&&
{}_{m+3} \phi_{m+2}
\left[
\begin{matrix}
q^{-N}, a, \{ b_k y_k \}_{m}, c \\
d, \{ e y_k \}_{m}, a B c q^{1-N} / d e^m
\end{matrix}
; q; q
\right]
\\
&&
\quad \quad
\ = \
\frac{(d e^m /a B, d e^m/ B c)_N}
{(d, d e^m/ a B c)_N}
\prod_{ 1 \le k \le m}
\frac
{((d e^m b_k / a B c) y_k)_N}
{((d e^{m+1}/ a B c) y_k )_N}
\nonumber
\\
&& \quad \quad \quad \quad
\times
{
W^{m, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{e/ b_k \}_m \\ \{y_k\}_m\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{d e^{m+1} q^{-1}/ a B c};{ e/ c, e / a };{q^{-N}, d e^m / A B c}; {q; d q^N} \right)
}.
\nonumber\end{aligned}$$
The reason why we call as the inverse of the (non-balanced) duality transformation is that the transformations which one obtain by applying and in both order turn out to be identical as a (multiple) hypergeometric series. Note also that can be obtained by relabeling parameters in appropriately. However, we will use special cases of these transformations to establish an $A_n$ generalization of basic hypergeometric transformation formulas in the next section.
In the case when $m=n=1$, reduces to $$\begin{aligned}
\label{m1n1IDT1}
&&
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, a, b, c \\
d, e , a b c q^{1-N} / d e
\end{matrix}
; q; q
\right]
=
\frac
{(d e / b c, d e /a c, d e/ a b )_N}
{(d e^{2}/ a b c, d e / a b c, d)_N}
\\
&& \quad \quad \quad \quad
\times
{}_{8} W_{7}
\left[d e^{2} q^{-1}/ a b c;
e / a, d e q^{-1} / a b c,
e/b, e/c, q^{-N};q ; d q^N
\right].
\nonumber\end{aligned}$$
We also mention that by letting $e=c$ in and then rearranging the parameter $d$ as $Ab q^{1-N} / c$, we recover an $A_n$ generalization of Pfaff-Saalschütz summation formula for terminating balanced ${}_3 \phi_2$ series due to Milne (Theorem 4.15 in [@Milne3])
$$\begin{aligned}
\label{PSSum1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(b x_i)_{\gamma_i}}
{(c x_i)_{\gamma_i}}
\\
&& \quad \quad \quad \quad
\times
\frac{(q^{-N})_{|\gamma|}}
{(A b q^{1-N} /c )_{|\gamma|}}
=
\frac
{(c/ b)_N}
{(c/ A b)_N}
\prod_{ 1 \le i \le n}
\frac
{((c/ a_i) x_i )_N}
{(c x_i )_N}.
\nonumber\end{aligned}$$
In the case when $n=1$, reduces to Jackson‘s Pfaff-Saalschütz summation formula for terminating balanced ${}_3 \phi_2$ series (the formula (1.7.2) in [@GR1]) $${}_{3} \phi_2 \left[
\begin{matrix}
a, b, q^{-N}\\
c, ab q^{1-N} /c
\end{matrix}
;q; q
\right]
= \frac{(c/a, c/b)_N}{(c, c/ab)_N}.$$
Similarly to the expression for the balanced duality transformation formula , can be stated in more general form:
$$\begin{aligned}
\label{HIDT1}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(b_k x_i y_k)_{\gamma_i}}{(c x_i y_k)_{\gamma_i}}
\\
&&
\quad
\times
\frac{(q^{-N})_{|\gamma|}}
{(A B q^{1-N} /c^m)_{|\gamma|}}
=
\
\frac{(q)_N}{( c^m / A B )_N}
\
\sum_{\delta \in {\Bbb N}^m, \ |\delta| = N}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le k, l \le m}
\frac{((c / b_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((c /a_i^{}) x_i y_k )_{\delta_k}}
{(c x_i y_k)_{\delta_k}}.
\nonumber\end{aligned}$$
Note that can be obtained by taking the coefficient of $u^N$ in the multiple basic Euler transformation formula . We remark that corresponds to the $m \ge 2$ case of by rearrangement of parameters and in the case when $m=1$, reduces to $A_n$ Pfaff-Saalschütz summation formula .
${}_4 \phi_3$ transformation formulas of type $A$
-------------------------------------------------
[**[ Sears transformation of type $A$]{}**]{}
$$\begin{aligned}
\label{ST1}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(c_k x_i y_k )_{\gamma_i}}{(d x_i y_k)_{\gamma_i}}
\\
&& \quad \quad \times
\frac{(q^{-N}, a )_{|\gamma|}}
{(e, a B C q^{1-N}/ d^m e)_{|\gamma|}}
=
\frac{(e / a, d^m e /B C)_N}
{(e, d^m e / a B C)_N}
\sum_{\delta \in {\Bbb N}^m}
q^{|\delta|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\nonumber
\\
&& \quad \quad \quad \times
\frac{(q^{-N}, a )_{|\delta|}}
{(q^{1-N} a / e, d^m e / B C)_{|\delta|}}
\prod_{1 \le k, l \le m}
\frac{((d / c_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((d / b_i^{}) x_i y_k )_{\delta_k}}
{(d x_i y_k )_{\delta_k}}
.
\nonumber\end{aligned}$$
Replace the parameters in $d, e$ and $f$ with $aq/ d, aq/e$ and $aq/f$ respectively. Then put $a=0$. Then rearranging parameters $f \to d$ and $ de f^m q^{N-1} / B C \to a$ leads to the desired identity.
In the case when $m=1$ and $y_1= 1$, reduces to $$\begin{aligned}
\label{m1ST1}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\frac{(q^{-N}, a )_{|\gamma|}}
{(e, a B c q^{1-N}/ d e )_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i )_{\gamma_i}}{(d x_i)_{\gamma_i}}
\\
&& \quad \quad \quad \quad
= \
\frac{(e / a, d e / B c)_N}
{(e, d e / a B c)_N} \
\
{}_{n+3} \phi_{n+2}
\left[
\begin{matrix}
q^{-N}, a, \{ (d / b_i) x_i \}_n, d /c \\
q^{1-N} a / e, \{d x_i\}_n,
d e /B c
\end{matrix}; q, q
\right]
. \nonumber\end{aligned}$$
The multiple Sears transformation formula was originally established in [@Kaji1] (the formula (7.1) in [@Kaji1]). In the case when $m=n=1$ and $x_1 = y_1 = 1$, reduces to the Sears transformation formula for terminating balanced ${}_4 \phi_3 $ series:
$$\begin{aligned}
\label{SearsT1}
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, a, b, c \\
d, e, a b c q^{1-N} / d e
\end{matrix};
q, q
\right]
&=&
\frac{(e/a, d e / b c)_N}
{(e, d e / a b c)_N}
{}_4 \phi_3
\left[
\begin{matrix}
q^{-N}, a, d/b, d/c \\
d, a q^{1-N}/e, d e / b c
\end{matrix};
q, q
\right].\end{aligned}$$
Further informations for multiple Sears transformation can also be found in [@KajiS].
We shall add a few remarks that we have missed in our previous works in [@Kaji1] and [@KajiS].
Let $N$ tend to infinity in , we get the following transformation formula for multiple nonterminating ${}_3 \phi_2$ series of type $A$:
$$\begin{aligned}
\label{NT32}
&&
\sum_{\gamma \in {\Bbb N}^n}
\left( \frac{d^m e}{a B C} \right)^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(c_k x_i y_k )_{\gamma_i}}{(d x_i y_k)_{\gamma_i}}
\\
&&
\quad
\times \
\frac{( a )_{|\gamma|}}
{(e)_{|\gamma|}}
\ = \
\frac{(e / a, d^m e /B C)_\infty}
{(e, d^m e / a B C)_\infty}
\sum_{\delta \in {\Bbb N}^m}
\left( \frac{e}{a} \right)^{|\delta|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\nonumber\\
&&
\quad \quad
\times
\frac{( a )_{|\delta|}}
{(d^m e / B C)_{|\delta|}}
\prod_{1 \le k, l \le m}
\frac{((d / c_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((d / b_i^{}) x_i y_k)_{\delta_k}}
{(d x_i y_k)_{\delta_k}}
\nonumber\end{aligned}$$
holds if it satisfy the convergence condition $\max (|d^m e / a BC|, |e/a| ) < 1$.
was already established as the equation (3.2) in [@KajiS]. In the case when $m=n=1$ and $x_1 = y_1= 1$, reduces to the following nonterminating ${}_3 \phi_2$ transformation formula $$\begin{aligned}
\label{1DNT32}
{}_{3} \phi_{2}
\left[
\begin{matrix}
a, b, c \\
d, e
\end{matrix};
q, \frac{d e}{a b c}
\right]
&=&
\frac{(e/a, d e / b c)_\infty}
{(e, d e / a b c)_\infty}
{}_3 \phi_2
\left[
\begin{matrix}
a, d/b, d/c \\
d, d e / b c
\end{matrix};
q, \frac{e}{a}
\right].\end{aligned}$$
In the case when $m=1$ and $y_1 = 1$, reduces to
$$\begin{aligned}
\label{m1-NT32}
&&
\sum_{\gamma \in {\Bbb N}^n}
\left( \frac{d e}{a B c} \right)^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\frac{( a )_{|\gamma|}}
{(e)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i )_{\gamma_i}}{(d x_i)_{\gamma_i}}
\\
&&
\quad
=
\frac{(e / a, d e /B c)_\infty}
{(e, d e / a B c)_\infty}
{}_{n+2} \phi_{n+1}
\left[
\begin{matrix}
d/c, a, \{ (d / b_i) x_i \}_n \\
d e / B c, \{ d x_i \}_n
\end{matrix}
;
q;
\frac{e}{a}
\right]
\nonumber\end{aligned}$$
holds if it satisfy the convergence condition $\max (|d^m e / a BC|, |e/a| ) < 1$.
Here we give a remark on the convergence of multiple series.
[ **(Convergence of the multiple series)**]{} In the course of obtaining an infinite multiple sum identity such as , one needs to ensure the limiting procedure. To justify the process, one is required to use the dominated convergence theorem. Furthermore, to find the convergence condition of the dominated series such as the series in , one can quote the ratio test for multiple power series in classical work by J.Horn [@Horn]. Since the details of such justifications in this paper are all in the same line as the corresponding discussions in Milne-Newcomb [@MilNew1] (see also Milne [@Milne3] and Milne-Newcomb [@MilNew2]), we shall omit in this paper.
We shall propose the following special case, namely a transformation formula between $A_n$ ${}_{m+2} \phi_{m+1}$ series with integer parameter differences and $A_m$ terminating balanced ${}_{n+2} \phi_{n+1}$ series:
$$\begin{aligned}
\label{MK-PS}
&&
\sum_{\gamma \in {\Bbb N}^n}
\left( q^{1- |M |} /A\right)^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\\
&&
\quad
\times
\frac{( b )_{|\gamma|}}
{( b q )_{|\gamma|}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{( c q^{m_k} x_i y_k )_{\gamma_i}}{(c x_i y_k)_{\gamma_i}}
\nonumber\\
&&
\quad \quad
=
\frac{( q, q^{1-|M|} b/A )_\infty}
{( b, q^{1-|M|} /A )_\infty}
\sum_{\delta \in {\Bbb N}^m}
q^{|\delta|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\prod_{1 \le k, l \le m}
\frac{( q^{- m_l} y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\nonumber\\
&& \quad \quad \quad
\times
\frac{( b)_{|\delta|}}
{( q^{1-|M|} b / A)_{|\delta|}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((c / a_i^{}) x_i y_k)_{\delta_k}}
{(c x_i y_k)_{\delta_k}}.
\nonumber\end{aligned}$$
In the case when $n=1$ and $x_1 = 1$, the multiple series in the right hand side in can be summed by the following $A_n$ generalization of $q$-Pfaff-Saalschütz summation formula $$\begin{aligned}
\label{R-PSSum1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(q^{- m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(b x_i)_{\gamma_i}}
{(c x_i)_{\gamma_i}}
\\
&& \quad \quad \quad \times \
\frac{(a)_{|\gamma|}}
{(a b q^{1-|M|} /c )_{|\gamma|}}
\ = \
\frac
{(c/ b)_{|M|}}
{(c/ a b)_{|M|}}
\prod_{ 1 \le i \le n}
\frac
{((c/ a) x_i )_{m_i}}
{(c x_i )_{m_i}},
\nonumber\end{aligned}$$ which can be obtained from by elementary polynomial argument (see the proof of Corollary 4.1), to recover Gasper’s $q$-analogue of Minton-Karlsson summation formula for ${}_{n+2} \phi_{n+1}$ series [@Gas1]:
$$\begin{aligned}
\label{MK-SumB1}
&&
{}_{n+2} \phi_{n+1}
\left[
\begin{matrix}
a, b, \{ c q^{m_i} x_i \}_n \\
b q, \{ c x_i \}_n
\end{matrix}
;
q;
a^{-1} q^{1-|M|}
\right]
\ = \
b^{|M|}
\
\frac
{(q, b q / a)_\infty}
{(b q, q/a )_\infty}
\prod_{1 \le i \le n}
\frac{((c/b ) x_i)_{m_i}}
{(c x_i)_{m_i}}. \end{aligned}$$
In the case when $m=1$ and $y_1 = 1$, reduces to:
$$\begin{aligned}
\label{m1-MK-PS}
&&
\sum_{\gamma \in {\Bbb N}^n}
\left( q^{1- m} /A\right)^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{( c q^{m} x_i )_{\gamma_i}}{(c x_i)_{\gamma_i}}
\\
&& \quad \quad \quad \times
\frac{( b )_{|\gamma|}}
{( b q )_{|\gamma|}}
\ = \
\frac{( q, q^{1-M} b / A )_\infty}
{( b, q^{1-M} / A )_\infty}
{}_{n+2} \phi_{n+1}
\left[
\begin{matrix}
q^{-M}, b, \{ (c/ a_i) x_i \}_n \\
q^{1-M} b /A, \{ c x_i \}_n
\end{matrix}
;
q;
q
\right].
\nonumber\end{aligned}$$
In the case when $m=n=1$ and $x_1 = y_1 = 1$, reduces to:
$$\begin{aligned}
\label{m1n1-MK-PS}
{}_{3} \phi_{2}
\left[
\begin{matrix}
a, b, c q^M \\
b q, c
\end{matrix}
;
q;
q^{1-M} / a
\right]
&=&
b^{M}
\
\frac
{(q, b q / a, c / b)_\infty}
{(b q, q / a, c )_\infty}.\end{aligned}$$
It may be of worth to note that, furthermore, if we replace $e \to a B C u / d^m$ and take limit $a \to \infty$ in , we recover multiple basic Euler transformation formula of type $A$
$$\begin{aligned}
&&
\sum_{\gamma \in {\Bbb N}^n}
u^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(b_k x_i y_k)_{\gamma_i}}{(c x_i y_k)_{\gamma_i}}
\\
&&
\quad =
\frac{(A Bu/ c^m)_\infty}
{(u)_\infty}
\sum_{\delta \in {\Bbb N}^m}
\left( \frac{A B u}{c^m} \right)^{|\delta|}
\frac{\Delta (y q^{\delta})}{\Delta (y)}
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le k, l \le m}
\frac{((c / b_l^{}) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((c /a_i^{}) x_i y_k )_{\delta_k}}
{(c x_i y_k )_{\delta_k}},
\nonumber\end{aligned}$$
when $ max(|u|, \displaystyle{ \left| AB u / c^m \right|} ) <1 $.
[**[ Reversing version]{}**]{}
Under the balancing condition $$a^{m} B c q^{1-N} = d E f,$$ we have the following:
$$\begin{aligned}
\label{ST2}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i\le n}
\frac{(c x_i)_{\gamma_i}}
{(d x_i)_{\gamma_i}}
\prod_{1 \le k \le m}
\frac{( a y_k)_{|\gamma|}}
{( e_k y_k)_{|\gamma|}}
\\
&&
\quad
\times
\frac{(q^{-N})_{|\gamma|}}
{(f)_{|\gamma|}}
\ = \
\frac
{( E f /a^m B)_N}
{(f )_N}
\prod_{ 1 \le k \le m}
\frac
{( a y_k )_N}
{(e_k y_k )_N}
\prod_{ 1 \le i \le n}
\frac
{(( E f / a^m c ) z_i )_N}
{(( E f / a^m B c) z_i )_N}
\nonumber\\
&&
\quad \quad
\times
\sum_{\delta \in {\Bbb N}^{m} }
q^{|\delta|}
\frac{\Delta ({w} q^{\delta})}{\Delta ({w})}
\prod_{1 \le k, l \le m}
\frac{((e_l/a) w_k / w_l)_{\delta_k}}{(q w_k / w_l)_{\delta_k}}
\prod_{1 \le k \le m}
\frac{(( f/ a) w_k )_{\delta_k}}
{(( d E f / a^{m+1} B c) w_k )_{\delta_k}}
\nonumber\\
&&
\quad \quad \quad
\times
\frac{(q^{-N})_{|\delta|}}{(E f / a^m B )_{|\delta|}}
\prod_{1 \le i \le n}
\frac{( ( E f /a^m b_i c) z_i)_{|\delta|}}
{( ( E f /a^m c) z_i)_{|\delta|}}
\nonumber\end{aligned}$$
where $ z_i = b_i / B x_i, \ ( 1 \le i \le n)$ and $ w_k = y_k^{-1} , \ ( 1 \le k \le m)$ respectively.
Rewrite the parameters $c_k \to aq/ c_k \ ( 1 \le k \le m)$ and $e \to aq/ e$ in . Then put $a=0$. Finally by rearranging the parameters $ f \to a, \ d \to c, \ {B d f^m q^{1-N}}/ {C e} \to d,
\ c_k \to e_k \quad (1 \le k \le m), \ e \to f $, we arrive at the desired identity.
In the case when $m=1$ and $y_1=1$, reduces to
$$\begin{aligned}
\label{m1ST2}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i\le n}
\frac{(c x_i)_{\gamma_i}}
{( d x_i)_{\gamma_i}}
\\
&&
\quad \quad
\times
\frac{(q^{-N}, a)_{|\gamma|}}
{(e, f)_{|\gamma|}}
\ = \
\frac
{(e f / a B, a)_N}
{(e, f)_N}
\prod_{ 1 \le i \le n}
\frac
{(( e f / a c )z_i )_N}
{((e f / a B c) z_i )_N}
\nonumber\\
&&
\quad \quad \quad \quad \quad
\times
{}_{n+3} \phi_{n+2}
\left[
\begin{matrix}
q^{-N}, e/ a, f / a, \{( e f / a b_i c) z_i\}_{n}
\\
def/ a^2 B c, e f / a B, \{( e f / a c) z_i\}_{n}
\end{matrix}
; q; q
\right],
\nonumber\end{aligned}$$
when the following balancing condition holds: $$a B c q^{1-N} = d e f,$$ and where $z_i = b_i / B x_i, i= 1, 2, \cdots n$. By letting $f=a$ in and relabeling the parameters appropriately, we also recover the $A_n$ Pfaff-Saalschütz summation .
has already appeared as the 2nd Sears transformation formula (Proposition 7.2.) in [@Kaji1], up to relabeling parameters. In the case when $m=n=1$ and $x_1 = y_1 =1$, reduces to a transformation formula for terminating balanced ${}_ 4 \phi_3 $ series $$\begin{aligned}
\label{m1n1ST2}
&&
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, a, b, c
\\
d, e, f
\end{matrix}
; q; q
\right]
\ = \
\frac
{(e f / a b, e f/ a c, a)_N}
{(e, f, e f / a b c)_N}
\\
&&
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\times
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N},e f / a b c, e/a, f/a
\\
def/ a^2 b c, e f / a b, e f / a c
\end{matrix}
; q; q
\right]
\nonumber\end{aligned}$$ when $$a b c q^{1-N} = d e f.$$ Note that this is obtained by reversing the order of the summation of the Sears transformation and is also verified by iterating Sears transformation twice in a proper fashion.
Let $N \to \infty$ in . Then we have
$$\begin{aligned}
\label{Hall1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{- \gamma_1} \cdots x_n^{- \gamma_n}
\left( \frac{E f}{a^m B c} \right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i\le n}
{(c x_i)_{\gamma_i}}
\\
&&
\quad
\times
\left[{(f)_{|\gamma|}}\right]^{-1}
\prod_{1 \le k \le m}
\frac{( a y_k)_{|\gamma|}}
{( e_k y_k)_{|\gamma|}}
\nonumber\\
&&
\quad \quad
=
\frac
{( E f /a^m B)_\infty}
{(f )_\infty}
\prod_{ 1 \le k \le m}
\frac
{( a y_k )_\infty}
{(e_k y_k )_\infty}
\prod_{ 1 \le i \le n}
\frac
{(( E f / a^m c ) z_i )_\infty}
{(( E f / a^m B c) z_i )_\infty}
\nonumber\\
&&
\quad \quad \quad
\times
\sum_{\delta \in {\Bbb N}^{m} }
w_1^{-\delta_1} \cdots w_m^{-\delta_m} a^{|\delta|}
q^{ e_2 ( \delta )} \
\frac{\Delta ({w} q^{\delta})}{\Delta ({w})}
\prod_{1 \le k, l \le m}
\frac{((e_l/a) w_k / w_l)_{\delta_k}}{(q w_k / w_l)_{\delta_k}}
\prod_{1 \le k \le m}
{(( f/ a) w_k )_{\delta_k}}
\nonumber\\
&& \quad \quad \quad \quad
\times
\left[{(E f / a^m B )_{|\delta|}}\right]^{-1}
\prod_{1 \le i \le n}
\frac{( ( E f /a^m b_i c) z_i)_{|\delta|}}
{( ( E f /a^m c) z_i)_{|\delta|}},
\nonumber\end{aligned}$$
where $ z_i = b_i / B x_i, \ ( 1 \le i \le n)$, $ w_k = y_k^{-1} , \ ( 1 \le k \le m)$ respectively and $e_2( \gamma )$ is the second elementary symmetric function of the variable $\gamma = ( \gamma_1, \gamma_2, \cdots , \gamma_n )$.
The case $m=1$ and $y_1 = 1$ of is $$\begin{aligned}
\label{m1Hall1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{- \gamma_1} \cdots x_n^{- \gamma_n}
\left( \frac{e f}{a B c}\right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\\
&&
\quad \quad
\times
\frac{( a)_{|\gamma|}}
{(e, f)_{|\gamma|}}
\prod_{1 \le i\le n}
{(c x_i )_{\gamma_i}}
\ = \
\frac
{( e f /a B, a)_\infty}
{(e, f)_\infty}
\prod_{ 1 \le i \le n}
\frac
{((e f / a c)z_i )_\infty}
{((e f /a B c) z_i)_\infty}
\nonumber \\
&&
\quad \quad \quad \quad \quad \quad
\times
{}_{n+2} \phi_{n+1}
\left[
\begin{matrix}
e/ a, f/ a, \{( e f /a B c) z_i \}_{n}
\\
e f / a B, \{( e f/ a c) z_i \}_{n}
\end{matrix}
; q; a
\right],
\nonumber \end{aligned}$$ where $z_i = b_i / B x_i$ for $ i = 1, \cdots, n$. We mention that one yield an $A_n$ Gauss summation theorem by putting $f=a$ and relabeling the parameters: $$\begin{aligned}
\label{2ndGaussSum}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{- \gamma_1} \cdots x_n^{- \gamma_n}
\left( \frac{e f}{a B c}\right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\\
&&
\quad \quad
\times
{(c)_{|\gamma|}}^{-1}
\prod_{1 \le i\le n}
{(a x_i )_{\gamma_i}}
\ = \
\frac
{( c / B)_\infty}
{(c)_\infty}
\prod_{ 1 \le i \le n}
\frac
{((c B / a b_i ) x_i^{-1} )_\infty}
{((c /a b_i) x_i^{-1})_\infty}.
\nonumber \end{aligned}$$
In the case when $m=n=1$ and $x_1 = y_1 = 1$, reduces to ${}_3 \phi_2$ transformation formula $$\begin{aligned}
\label{m1n1Hall1}
&&
{}_{3} \phi_{2}
\left[
\begin{matrix}
a, b, c
\\
e, f
\end{matrix}
; q; \frac{e f}{a b c}
\right]
\ = \
\frac
{(e f / a b, e f/ a c, a )_\infty}
{(e, f, e f / a b c)_\infty}
{}_{3} \phi_{2}
\left[
\begin{matrix}
e/a, f/a, e f / a b c
\\
e f / a b, e f / ac
\end{matrix}
; q; a
\right].
\nonumber\end{aligned}$$ In the case when $mn=1$ and $x_1 = 1$, reduces to the basic Gauss summation formation formula for ${}_2 \phi_1$ $$\begin{aligned}
\label{n1GaussSum}
&&
{}_{2} \phi_{1}
\left[
\begin{matrix}
a, b
\\
c,
\end{matrix}
; q; \frac{c}{a b}
\right]
\ = \
\frac
{(c/ a, c/ b)_\infty}
{(c, c / a b )_\infty}
.
\nonumber\end{aligned}$$ We also remark that and have already appeared in our previous work [@KajiR]. But they contain errors, so we restate them here.
${}_8 W_7$ transformations
---------------------------
[**[ Nonterminating ${}_8 W_7 $ transformation formula]{}**]{}
By taking the limit $N \to \infty$ in , we have the following:
Assume that ${\displaystyle \left| \frac{a^{m+1} q^{m+1} }{B C d e f^{m}} x_i^{-1} \right| <1 }$ for all $i = 1, \cdots , n$ and $|f y_k^{-1}| < 1$ for all $k = 1, \cdots , m$. Then we have $$\begin{aligned}
\label{NT87}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{-\gamma_1} \cdots x_n^{-\gamma_n}
\left(
\frac{a^{m+1} q^{m+1} }{B C d e f^{m}}
\right)^{|\gamma|}
q^{e_2 ( \gamma )} \
\frac{\Delta({x}
q^{\gamma})}
{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i}\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(c_k x_i y_k )_{\gamma_i}}
{(( a q / f ) x_i y_k)_{\gamma_i}}
\nonumber
\\
&&
\quad \quad
\times
\prod_{1 \le i \le n}
\frac{(a x_i )_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\prod_{1 \le k \le m}
\frac{( f y_k^{-1})_{|\gamma|}}
{(( a q / c_k ) y_k^{-1})_{|\gamma|}}
\nonumber\\
&&
\quad \quad \quad
\times
\frac{1}{( a q / d, a q / e )_{|\gamma|}}
\prod_{1 \le i \le n}
\left({(d x_i, e x_i )_{\gamma_i}}
\right)
\nonumber
\\
&=&
\frac
{(\mu d f / a, \mu e f / a)_\infty}
{(a q / d, a q / e)_\infty}
\prod_{ 1 \le k \le m}
\frac
{((\mu c_k f / a) y_k, f y_k^{-1} )_\infty}
{(\mu q y_k, (a q /c_k) y_k^{-1} )_\infty}
\nonumber \\
&&
\quad
\times
\prod_{ 1 \le i \le n}
\frac
{(a q x_i, ( \mu b_i f / a )x_i^{-1} )_\infty}
{((a q / b_i) x_i, (\mu f / a) x_i^{-1} )_\infty}
\nonumber\\
&&
\quad \quad
\times
\sum_{\delta \in {\Bbb N}^{m} }
y_1^{-\delta_1} \cdots y_m^{-\delta_m}
f^{|\delta|}
q^{e_2 ( \delta )} \
\frac{\Delta ({y} q^{\delta})}{\Delta ({y})}
\prod_{1 \le k \le m}
\frac{1 - \mu q^{|\delta| + \delta_k} y_k }
{1 - \mu y_k }
\nonumber\\
&&
\quad \quad \quad
\times
\prod_{1 \le k, l \le m}
\frac{(( a q / c_l^{} f ) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((a q /b_i^{} f) x_i y_k)_{\delta_k}}
{(( a q / f) x_i y_k)_{\delta_k}}
\nonumber
\\
&&
\quad \quad \quad \quad
\times
\prod_{1 \le k \le m}
\frac{( \mu y_k)_{|\delta|}}
{(( \mu c_k f / a ) y_k)_{|\delta|}}
\prod_{1 \le i \le n}
\frac{( (\mu f / a ) x_i^{-1})_{|\delta|}}
{( (\mu b_i f/ a ) x_i^{-1})_{|\delta|}}
\nonumber \\
&&
\quad \quad \quad \quad \quad
\times
\frac{1}{( \mu d f / a, \mu e f / a )_{|\delta|}}
\prod_{1 \le k \le m}
{((a q / d f ) y_k, (a q / e f ) y_k)_{\delta_k}},
\nonumber\end{aligned}$$ where $\mu = a^{m+2} q^{m+1} / B C d e f^{m+1} $.
For the justification of this limiting procedure for , see Remark 3.7.
In the case when $m=1$ and $y_1 = 1$, reduces to
$$\begin{aligned}
\label{m1NT87}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{-\gamma_1} \cdots x_n^{-\gamma_n}
\left(
\frac{a^{2} q^{2} }{B c d e f^{}}
\right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x}
q^{\gamma})}
{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i}
\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i, d x_i, e x_i)_{\gamma_i}}
{(( a q / f ) x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad
\times
\frac{( f)_{|\gamma|}}
{( a q / c, a q / d, a q / e)_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\nonumber\\
&&
\quad
=
\frac
{(\mu c f / a, \mu d f / a, \mu e f / a, f)_\infty}
{(a q / c, a q / d, a q / e, \mu q)_\infty}
\prod_{ 1 \le i \le n}
\frac
{(a q x_i, ( \mu b_i f / a ) x_i^{-1} )_\infty}
{((a q / b_i) x_i, (\mu f / a) x_i^{-1} )_\infty}
\nonumber\\
&&
\quad \quad
\times
{}_{2 n + 4} W_{2 n + 3}
\left[
\mu; \{(a q /b_i^{} f ) x_i \}_{n}
a q / c f,a q / d f,a q / e f,
\{(\mu f / a ) x_i^{-1} \}_{n}
; q; f
\right],\nonumber\end{aligned}$$
where $\mu = a^{3} q^{2} / B c d e f^{2} $.
In the case when $m=n=1$ and $x_1 = y_1 =1$, reduces to the following transformation formula for nonterminating ${}_8 W_7$ series $$\begin{aligned}
\label{mn1NT87}
&&
{}_{8} W_{7}
\left[
\begin{matrix}
a; b, c, d, e, f
\end{matrix};
q; \frac{a^2 q^2}{b c d e f}
\right]
=
\frac
{(\mu b f/ a, \mu c f / a, \mu d f / a, \mu e f / a, a q, f)_\infty }
{(a q / b, a q /c, a q/d, a q/e, \mu q, \mu f / a)_\infty}
\\
&&
\quad \quad \quad \quad \quad
\times
{}_{8} W_{7}
\left[
\begin{matrix}
\mu ; a q / b f, a q / c f, a q / d f, a q / e f,
\mu f / a
\end{matrix};
q; f
\right],\nonumber\end{aligned}$$ where $ \mu = a^3 q^2 /bcde f^2$, under the convergence condition $\max(|{a^2 q^2}/{b c d e f}|,| f|) < 1$.
We mention that by assuming $ e= aq/ f$ in , we get the following:
Assume that ${ \displaystyle \left|\frac{a q }{B c d } x_i^{-1} \right| < 1}$ for all $i=1, \cdots , n$. Then $$\begin{aligned}
\label{NT65Sum}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{-\gamma_1} \cdots x_n^{-\gamma_n}
\left(
\frac{a q }{B c d }
\right)^{|\gamma|}
q^{e_2 ( \gamma )} \
\frac{\Delta({x}
q^{\gamma})}
{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i}
\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
{(c x_i, d x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad
\times
\left[{( a q / c, a q / d)_{|\gamma|}}\right]^{-1}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\nonumber\\
&=&
\frac
{(a q / B c, a q / B d)_\infty}
{(a q / c, a q / d)_\infty}
\prod_{ 1 \le i \le n}
\frac
{(a q x_i, ( a q b_i / B c d ) x_i^{-1} )_\infty}
{((a q / b_i) x_i, (a q / B c d) x_i^{-1} )_\infty}.
\nonumber\end{aligned}$$
In the case when $n=1$ and $x_1 =1$, reduces to the Bailey summation formula for nonterminating ${}_6 W_5$ series (See [@GR1]) $$\begin{aligned}
\label{BaileySum1}
{}_{6} W_{5}
\left[
\begin{matrix}
a; b, c, d
\end{matrix};
q; \frac{a q}{b c d }
\right]
&=&
\frac
{( a q, a q / c d, a q/ b d, a q / b c )_\infty }
{(a q / b c d, a q / b, a q / c, a q/ d)_\infty}.\end{aligned}$$ $A_n$ nonterminating ${}_6 W_5$ summation formula is due to S.C. Milne and has appeared as Theorem 4.27 in [@Milne2] with a different expression (See also Theorem A.4 in Milne-Newcomb [@MilNew2]). Note that can also be obtained by taking the limit $N \to \infty$ in $A_n$ Jackson summation formula .
[**Terminating ${}_8 W_7$ transformation**]{}
$$\begin{aligned}
\label{T87T1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{m+1} q^{m+N + 1}}{B C d^m e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma ) } \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{(c_k x_i y_k)_{\gamma_i}}
{(( a q / d ) x_i y_k )_{\gamma_i}}
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le i\le n}
\frac{1} {( a q^{N+1} x_i, (a q/ e ) x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad \quad
\times
{(q^{-N}, e)_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\prod_{1 \le k \le m}
\frac{( d y_k^{-1})_{|\gamma|}}
{(( a q / c_k ) y_k^{-1})_{|\gamma|}}
\nonumber\\
&=&
\prod_{ 1 \le k \le m}
\frac{((a q /c_k e) y_k^{-1}, d y_k^{-1} )_N}
{((a q /c_k) y_k^{-1}, (d/e) y_k^{-1} )_N}
\prod_{ 1 \le i \le n}
\frac{((a q / b_i e) x_i, a q x_i)_N}
{((a q / b_i) x_i, (a q / e) x_i)_N}
\nonumber\\
&&
\quad
\times
\sum_{\delta \in {\Bbb N}^{m} }
y_1^{\delta_1} \cdots y_m^{\delta_m}
\left( \frac{BC d^{m-1}}{a^m}\right)^{|\delta|}
q^{ - e_2 ( \delta )} \
\frac{\Delta ({y} q^{\delta})}{\Delta ({y})}
\prod_{1 \le k \le m}
\frac{1 - (q^{-N} e/d )q^{|\delta| + \delta_k} y_k}
{1 - (q^{-N} e/ d) y_k }
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le k, l \le m}
\frac{(( a q / c_l^{} d ) y_k / y_l)_{\delta_k}}{(q y_k / y_l)_{\delta_k}}
\prod_{1 \le i \le n, 1 \le k \le m}
\frac{((a q /b_i^{} d) x_i y_k )_{\delta_k}}
{(( a q / d) x_i y_k )_{\delta_k}}
\nonumber\\
&&
\quad \quad \quad
\times
\prod_{1 \le k \le m}
\frac{1}
{((e/d) y_k, ( q^{1-N} / d ) y_k)_{\delta_k}}
\nonumber\\
&&
\quad \quad \quad \quad
\times
{(q^{-N}, e)_{|\delta|}}
\prod_{1 \le k \le m}
\frac{( (q^{-N} e / d) y_k)_{|\delta|}}
{(( q^{-N}c_k e/ a ) y_k )_{|\delta|}}
\prod_{1 \le i \le n}
\frac{( (q^{-N} e/ a ) x_i^{-1})_{|\delta|}}
{( (q^{-N} b_i e / a ) x_i^{-1})_{|\delta|}}.
\nonumber\end{aligned}$$
Relabel $e$ as $\mu f q^N$, i.e. interchange $ e \leftrightarrow \mu f q^N$. Then let $d$ tend to infinity. Finally, by relabeling $f$ as $d$, we arrive at .
In the case when $m=1$ and $y_1= 1$, reduces to
$$\begin{aligned}
\label{m1T87T1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{2} q^{2+N }}{B c d e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{(( a q / d ) x_i, (a q/ e ) x_i, a q^{N+1} x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad
\times
\frac
{(q^{-N}, d, e )_{|\gamma|}}
{(a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i )_{|\gamma|}}
\nonumber\\
&=&
\frac{(a q /c e, d)_N}
{(a q /c, d/e )_N}
\prod_{ 1 \le i \le n}
\frac{((a q / b_i e) x_i, a q x_i)_N}
{((a q / b_i) x_i, (a q / e) x_i)_N}
\nonumber\\
&&
\quad
\times
{}_{2 n + 6} W_{2 n + 5}
\left[ q^{-N} e / d; \{ (a q / b_i d) x_i \}_{n},
\{ (q^{-N } e / a) x_i^{-1} \}_{n}, a q / c d, e, q^{-N}; q;
\frac{B c }{a}
\right].
\nonumber\end{aligned}$$
In the case when $m=n=1$ and $x_1 = y_1 = 1$, reduces to transformation formula for terminating ${}_8 W_7$ series $$\begin{aligned}
\label{mn1T87T1}
&& \quad \quad
{}_{8} W_{7}
\left[ a; b, c, d, e, q^{-N}; q; \frac{a^2 q^{N+2}}{b c d e}
\right]\\
&=&
\frac{(a q / b e, a q /c e, a q, d)_N}
{(a q / b, a q /c, a q / e, d/e )_N}
{}_{8} W_{7}
\left[ q^{-N} e / d; a q / b d, q^{-N } e / a,
a q / c d, e, q^{-N}; q;
\frac{b c }{a}
\right].
\nonumber\end{aligned}$$
Let $aq = cd $ in . Then, by rearranging parameters, we have
$$\begin{aligned}
\label{RS2}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{} q^{1+N }}{B c}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&&
\quad \times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\left[
{(( a q / c) x_i, a q^{1+N} x_i)_{\gamma_i}}
\right]^{-1}
\nonumber\\
&&
\quad \quad
\times
\
{( q^{-N}, c)_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i )_{|\gamma|}}
=
\prod_{ 1 \le i \le n}
\frac{((a q / b_i c) x_i, a q x_i)_N}
{((a q / b_i) x_i, (a q / c) x_i)_N}.
\nonumber\end{aligned}$$
In the case when $n=1$ and $x_1 = 1$, reduces to the Rogers‘ summation formula for terminating ${}_6 W_5$ series .
As we have seen in this section, one can recover our previous results in [@Kaji1] from the balanced duality transformation formula by limiting procedures. Combining with results in [@Kaji1] and Rosengren [@RoseKM], one can consider the master formula for multiple basic hypergeometric transformations of type $A$ with different dimensions presented in this section to either of multiple basic Euler transformation , Sears transformation , (non balanced) duality transformation and balanced duality transformation formula .
$A_n$ hypergeometric transformations
====================================
In this section, we present several hypergeometric transformation formulas with same dimension $n$ (for multiple basic hypergeometric series of type $A_n$) by combining some special cases of hypergeometric transformation with different dimensions which we have obtained in the previous section. It contains new transformation formulas and some of these are previously known by Milne and his collaborators (see [@LM1], [@MilNew1], and [@MilneNagoya]). However, our proofs of them are completely different from theirs and seem to be simpler.
$A_n$ Watson transformations
----------------------------
In this subsection and next, we derive several $A_n$ generalization of the Watson transformation formula between terminating ${}_8 W_7$ series and terminating balanced ${}_4 \phi_3$ series ( (2.5.1) in [@GR1] ): $$\begin{aligned}
\label{WatsonT1}
&& \quad \quad
{}_{8} W_{7}
\left[
a; b, c, d, e, q^{-N}; q;\frac{a^2 q^{2+N}}{b c d e}
\right]
\\
&=& \
\frac{(a q, a q / d e)_N}{(a q / d, a q / e)_N}
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, d, e, a q /b c \\
a q / b, a q / c, d e q^{-N} / a
\end{matrix};
q; q
\right].
\nonumber\end{aligned}$$
Especially, we give two types of $A_n$ Watson transformation formula whose series in the left hand side are expressible in terms of $W^{n, 2}$ series here. We will use a special case ($m=1$ ) of the (non-balanced) duality transformation formula and special cases of ${}_4 \phi_3$ series of type $A$ to the identities below. To be precise, we produce them according to the following diagram.
(100, 30) (0,20)[(15,10)\[r\][$W^{n, 2}$ series]{}]{} (80,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (35, 27)[Watson trans.]{} (45, 0)[(15,10)\[r\][${}_{n+3} \phi_{n+2}$ series]{}]{} (20, 23)[(3,-2)[20]{}]{} (50, 8)[(3,2)[20]{}]{} (20, 14) (65, 14)[ [**(B)**]{} ${}_4 \phi_3$ transformation of type $A$]{}
[**[The 1st one]{}**]{}
$$\begin{aligned}
\label{ltAnWT1}
&&
{
W^{n, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{b_i\}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d};{e, q^{-N}}; {q; \frac{a^2 q^{N+2}}{B c d e}} \right)
}\\
&& \quad \quad
= \
\frac{(a q / B d)_N}
{(a q / d)_N}
\prod_{1 \le i \le n}
\frac
{(a x_i)_N}
{((a q/b_i) x_i)_N}
\
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\nonumber
\\
&& \quad \quad \quad \quad
\times
\frac{(q^{-N}, a q / c e )_{|\gamma|}}
{(B d q^{-N} /a, a q / c)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(d x_i )_{\gamma_i}}{( (a q/ e) x_i)_{\gamma_i}}.
\nonumber\end{aligned}$$
We combine and $n=1$ and $m \to n$ case of
$$\begin{aligned}
\label{n1ST1}
&&
{}_{n+3} \phi_{n+2}
\left[
\begin{matrix}
q^{-N}, a, c, \{u_i \}_{n} \\
e, \{v_i \}_{n}, a c U q^{1-N} / e V
\end{matrix}; q, q
\right]
\\
&&
\quad \quad
=
\frac{(e, e V / a c U)_N}
{(e/a, e V/ c U)_N}
\
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (v q^{\gamma})}{\Delta (v)}
\nonumber
\\
&&
\quad \quad \quad \quad
\times
\frac{(q^{-N}, a )_{|\gamma|}}
{(q^{1-N} a /e, e V / c U)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{( v_i / u_j)_{\gamma_i}} {(q v_i / v_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{( v_i/ c )_{\gamma_i}}{( v_i )_{\gamma_i}}
,
\nonumber\end{aligned}$$
here we give in the useful form for the present proof and latter uses in this section, as [**(B)**]{} in the above diagram. Note that both of the series in the right hand side of and that in the left hand side of satisfy same condition as basic hypergeometric series: namely they are terminating balanced ${}_{n+3} \phi_{n+2}$ series. On the set of variables $$\begin{aligned}
&&
( \{(a q / b_i e) x_i \}_{n}, a q / c e, a q / d e, a^2 q^2 / B c d e, \{(a q / e) x_i\}_{n} ),\end{aligned}$$ we consider the following change of variables: $$\begin{aligned}
\label{covWT1}
\tilde{a} = a q/ c e, & \tilde {c} = a q / d e, & \tilde{e} = a^2 q^2/ B c d e
\nonumber \\
\tilde{u}_i = ( a q / b_i e) x_i, & \tilde{v}_i = (a q/ e) x_i \quad \quad (i= 1, \cdots , n ).&\end{aligned}$$ For given function $\psi$, we denote by $\tilde{\psi} = \psi(a, c, e, \{ u_i \}_n, \{ v_i\}_n)$ the function that is obtained by replacing the variables $(a, c, e, \{ u_i \}_n, \{ v_i\}_n)$ by $( \tilde{a}, \tilde{c}, \cdots).$ In this case, the change of variables is a transposition inside of each sets of numerator parameters in ${}_{n+3} \phi_{n+2} $ series and of denominator parameters. Hence, the right hand side of is invariant under this change of variables. By applying to the series the right hand side in , this invariance implies .
We also give similar transformation formula for multiple series which terminates with respect to a certain multi-index. In this paper, we call such transformation formulas as [*rectangular* ]{} and transformations for the multiple series which terminates with respect to the length of multi-indices as [*triangular*]{}.
$$\begin{aligned}
\label{itAnWT1}
&&
{
W^{n, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{ q^{-m_i} \}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d};{b, e}; {q; \frac{a^2 q^{|M| +2}}{b c d e}} \right)
}\\
&&
\quad \quad =
\frac{(a q / b d)_{|M|}}{(a q / d)_{|M|}}
\prod_{ 1\le i \le n}
\frac{(a q x_i)_{m_i}}{((a q / b ) x_i)_{m_i}}
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\nonumber\\
&& \quad \quad \quad \quad
\times
\frac{(b , a q / c e )_{|\gamma|}}
{(b d q^{-|M|} / a, a q /c)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(q^{- m_j}x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(d x_i )_{\gamma_i}}{((a q / e) x_i)_{\gamma_i}}
.
\nonumber\end{aligned}$$
We first write the product factor in the right hand side of as a quotient of infinite products using . Set $b_i = q^{-m_i}$ in , and notice that is true for $b= q^{-N}$ for all nonnegative integer $N$. Clear the denominators in . Then we find that it is a polynomial identity in $b^{-1}$ with an infinite number of roots. Thus, is true for arbitrary $b$.
All the corollaries in this section can be proved by similar arguments from the formulas in the preceding propositions. So, hereafter we will not repeat this procedure in the rest of this paper.
has appeared in Theorem 6.1 of Milne and Lilly [@LM1].
[**[The 2nd one]{}**]{}
$$\begin{aligned}
\label{ltAnWT2}
&&
{
W^{n, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{b_i\}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d};{e, q^{-N}}; {q; \frac{a^2 q^{N+2}}{B c d e}} \right)
}\\
&& \quad
=
\prod_{1 \le i \le n}
\frac
{(a x_i, (a q / b_i e) x_i)_N}
{((a q / b_i) x_i, ( a q / e ) x_i)_N}
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (z q^{\gamma})}{\Delta (z)}
\nonumber
\\
&&
\quad \quad
\times
\frac{(q^{-N}, e )_{|\gamma|}}
{(a q / c, a q/ d)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(b_j z_i / z_j)_{\gamma_i}} {(q z_i / z_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{((a q / c d ) z_i )_{\gamma_i}}{( (B e q^{-N}/ a) z_i)_{\gamma_i}}
,
\nonumber\end{aligned}$$
where $z_i = b_i / B x_i$ for $1 \le i \le n$.
We combine and $n=1$ and $m \to n$ case of $$\begin{aligned}
\label{n1ST2}
&&
{}_{n+3} \phi_{n+2}
\left[
\begin{matrix}
q^{-N}, a, c, \{u_i \}_{n}\\
e, \{v_i \}_{n}, a c U q^{1-N} /e V
\end{matrix}; q, q
\right]
\\
&&
\quad \quad
=
\frac{(e V /a U, e V /c U)_N}
{(e V / a c U, e)_N}
\prod_{1 \le i \le n}
\frac{(u_i)_N}{(v_i)_N}
\
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (u^{-1} q^{\gamma})}{\Delta (u^{-1})}
\nonumber\\
&&
\quad \quad \quad \quad
\times
\frac{(q^{-N}, e V /a c U )_{|\gamma|}}
{(e V /a U, e V /c U)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(v_j/ u_i)_{\gamma_i}} {(q u_j / u_i)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(e/ u_i)_{\gamma_i}}{(q^{1-N} / u_i)_{\gamma_i}},
\nonumber\end{aligned}$$ here we present in a modified form, as [**(B)**]{}. Notice that both of the series in the right hand side of and that in the left hand side of are terminating balanced ${}_{n+3} \phi_{n+2}$ series. In this case, we consider the following change of variables $$\begin{aligned}
\label{covWT2}
\tilde{a} = a q / c e, & \tilde {c} = a q / d e, & \tilde{e} = a^2 q^2/ B c d e
\nonumber \\
\tilde{u}_i = ( a q / b_i e) x_i, & \tilde{v}_i = ( a q / e) x_i \quad \quad (i= 1, \cdots , n ).&\end{aligned}$$ Since this change of variables is same as in the proof of Proposition 4.1. one can obtain the desired identity by plugging to the series in the right hand side of according to this change of variables.
[**Rectangular version**]{}
$$\begin{aligned}
\label{itAnWT2}
&&
{
W^{n, 2}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{ q^{-m_i} \}_n \\ \{x_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{a};{c, d};{b, e}; {q; \frac{a^2 q^{|M| +2}}{b c d e}} \right)
}\\
&& \quad =
\prod_{ 1\le i \le n}
\frac{(a q x_i, (a q /b e) x_i)_{m_i}}{((a q / b ) x_i, (a q / e) x_i)_{m_i}}
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (z q^{\gamma})}{\Delta (z)}
\nonumber\\
&& \quad \quad \times
\frac{(b , e)_{|\gamma|}}
{(a q /c, a q/ d)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(q^{- m_j} z_i / z_j)_{\gamma_i}} {(q z_i / z_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{( (a q / c e) x_i)_{\gamma_i}}{(( be q^{-|M|} / a) x_i )_{\gamma_i}}
,
\nonumber\end{aligned}$$
where $z_i = q^{-m_i+|M|} x_i^{-1}$ for $i = 1, \cdots , n$.
In the case when $n=1$ and $x_1 = 1$, all of , , and reduce to the Watson transformation formula . can be proved by a similar limiting procedure as the previous section from the following $A_n$ Bailey transformation formula for terminating balanced ${}_{10} W_9$ series:
$$\begin{aligned}
\label{Masatoshi-san}
&&
{
W^{n,3}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc} \{ e_i\}_n \\ \{x_i\}_n \end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{{a}};{b, c, d};{q^{-N}, f, a \lambda q^{1 + N} / E f }; {q} \right)
}
\\
&& \quad =
\prod_{1 \le i \le n}
\frac
{(a q x_i, (a q / e_i f) x_i, (\lambda q / e_i) z_i, (\lambda q / f) z_i)_N}
{((a q / e_i) x_i, (a q / f) x_i, \lambda q z_i, (\lambda q / e_i f) z_i)_N }
\nonumber
\\
&& \quad \quad \quad \times
{
W^{n,3}\left(
\renewcommand{\arraystretch}{0.8}
\begin{array}{cccccccc}\{e_i\}_n \\ \{z_i\}_n\end{array}
\renewcommand{\arraystretch}{1.0}
\Big|\,{\lambda};{a q / c d, a q / b d, a q / b c};{q^{-N}, f, a \lambda q^{1 + N} / E f }; {q} \right)
}
\nonumber\end{aligned}$$
where $\lambda = a^2 q / b c d$ and $z_i = e_i / E x_i$ for $1 \le i \le n$, which has first appeared as (4.36) in [@KajiNou], and by rearranging the parameters.
Another type of $A_n$ Watson transformation
-------------------------------------------
Here, we present a yet another $A_n$ Watson transformation with a different form in the series in both sides from those in the previous subsection. We will use the $m=1$ case of the terminating ${}_8 W_7$ transformation of type $A$ in Section 3.3. and the $n=1$ case of the duality transformation formula in a modified form to produce. We construct it according to the following procedure:
(100, 30) (0,20)[(15,10)\[r\][$W^{n, 2}$ series]{}]{} (80,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (35, 27)[Watson trans.]{} (45, 0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (20, 23)[(3,-2)[20]{}]{} (50, 8)[(3,2)[20]{}]{} (7, 9)[(20, 10)\[r\]]{} (65, 14)[Duality trans. ]{}
$$\begin{aligned}
\label{ltAnWT3}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{2} q^{2+N }}{B c d e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&& \quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{( (a q / d) x_i, (a q/ e) x_i, a q^{N+1} x_i)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad
\times
\frac
{(q^{-N}, d, e )_{|\gamma|}}
{(a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i )_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\nonumber\\
&=&
\frac{(a q / B c )_N}
{(a q /c)_N}
\prod_{ 1 \le i \le n}
\frac{(a q x_i )_N}
{((a q / b_i) x_i)_N}
\
\sum_{\delta \in {\Bbb N}^{n} }
q^{|\delta|}
\frac{\Delta ({x} q^{\delta})}{\Delta ({x})}
\nonumber\\
&& \quad \quad \quad
\times
\frac{(q^{-N})_{|\delta|}}
{(q^{-N} B c / a )_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{( b_j x_i / x_j)_{\delta_i}}{(q x_i / x_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((a q / d e ) x_i, c x_i)_{\delta_i}}
{(( a q / d) x_i, ( a q / e) x_i)_{\delta_i}}
.
\nonumber\end{aligned}$$
We use and with a modified form $$\begin{aligned}
\label{rn1DT1}
&&{}_{2 n + 6} W_{2 n + 5}
\left[
a; b, \{u_i \}_{n},
d, \{ v_i \}_{n}, q^{-N}; q;
\frac{a^{n+1} q^{N+n+1}}{b d U V}
\right]
\\
&&
\quad \quad
=
\frac
{( a^{n+1} q^{n+1} / b d U V, a q)_N}
{(a q / b, a q / d)_N }
\prod_{ 1 \le i \le n}
\frac
{(v_i)_N}
{(a q / u_i )_N}
\
\sum_{\delta \in {\Bbb N}^{n} }
q^{|\delta|}
\frac{\Delta ({v}^{-1} q^{\delta})}{\Delta ({v^{-1}})}
\nonumber\\
&&
\quad \quad \quad \quad
\times
\frac{(q^{-N})_{|\delta|}}
{( a^{n+1} q^{n+1} / b d U V )_{|\delta|}}
\prod_{1 \le i,j \le n}
\frac{(a q / u_j v_i)_{\delta_i}}{(q v_j/ v_i)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{(a q /b v_i, a q / d v_i)_{\delta_i}}
{(a q / v_i, q^{1-N} / v_i)_{\delta_i}}
.
\nonumber\end{aligned}$$ to obtain. It is not hard to see that both of the series in the right hand side of and in the left hand side of satisfy the same condition: they are ${}_{2n+6} W_{2n+5}$ series and very-well-poised-balanced. We consider the following change of variables:
$$\begin{aligned}
\label{covWT3}
\tilde{a} = q^{-N} e / d, & \tilde{b} = a q / c d, &
\tilde{d} = e,
\\
\tilde{u}_i = (a q/ b_i d) x_i, & \tilde{v}_i = (q^{-N} e/ a) x_i^{-1} \quad \quad
(i= 1, \cdots , n), &
\nonumber\end{aligned}$$
which is a transposition of the variables in ${}_{2n + 6} W_{2n+5} $ series in . Note that ${}_{r+3} W_{r+2} $ series is symmetric with respect to the variables $a_1, \cdots , a_r$. So the series in the right hand side of is invariant under this change of variables. This invariance implies by applying according to the change of variables .
[**Rectangular version**]{}
$$\begin{aligned}
\label{itAnWT3}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{2} q^{2+|M| }}{b c d e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i }
{1 - a x_i}
\\
&& \quad
\times
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{( ( a q /b) x_i, (a q / d) x_i, (a q/ e) x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad \quad
\times
\frac
{( b, d, e )_{|\gamma|}}
{(a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q^{m_i} x_i)_{|\gamma|}}
\nonumber\\
&=&
\frac{(a q / b c )_{|M|}}
{(a q /c)_{|M|}}
\prod_{ 1 \le i \le n}
\frac{(a q x_i)_{m_i}}
{((a q / b) x_i)_{m_i}}
\
\sum_{\delta \in {\Bbb N}^{n} }
q^{|\delta|}
\frac{\Delta ({x} q^{\delta})}{\Delta ({x})}
\nonumber
\\
&& \quad \quad
\times
\frac{(b)_{|\delta|}}
{(q^{-|M|} b c / a )_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{( q^{-m_j} x_i / x_j)_{\delta_i}}{(q x_i / x_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((a q / d e ) x_i, c x_i)_{\delta_i}}
{(( a q / d) x_i, ( a q / e) x_i)_{\delta_i}}
\nonumber\end{aligned}$$
In the case when $n=1$ and $x_1 = 1$, and reduce to the Watson transformation formula .
In a similar fashion as we yield , we also obtain the following transformation formula between $A_n$ ${}_8 W_7$ series and $A_n$ terminating balanced ${}_4 \phi_3$ series: $$\begin{aligned}
\label{ltAnWT4}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{2} q^{2+N }}{B c d e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&& \quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{( (a q / d) x_i, (a q/ e) x_i, a q^{N+1} x_i)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad
\times
\frac
{(q^{-N}, d, e )_{|\gamma|}}
{(a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i )_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\nonumber\\
&=& B^N
\frac{(a q / B c )_N}
{(a q /c)_N}
\prod_{ 1 \le i \le n}
\frac{(a q x_i, (a q / b_i d) x_i, (a q / b_i e) x_i)_N}
{((a q / b_i, (a q / d) x_i, (a q / e) x_i) x_i)_N}
\nonumber\\
&&
\quad
\times
\sum_{\delta \in {\Bbb N}^{n} }
q^{|\delta|}
\frac{\Delta ({x^{-1}} q^{\delta})}{\Delta ({x^{-1}})}
\prod_{1 \le i, j \le n}
\frac{( b_j x_j / x_i)_{\delta_i}}{(q x_j / x_i)_{\delta_i}}
\nonumber\\
&& \quad \quad \quad
\times
\frac{(q^{-N})_{|\delta|}}
{(q^{-N} B c / a )_{|\delta|}}
\prod_{1 \le i \le n}
\frac{((q^{-1-N} b_i c d e / a^2 ) x_i^{-1}, (q^{-N} b_i / a ) x_i^{-1})_{\delta_i}}
{(( q^{-N} b_i d / a) x_i^{-1}, ( q^{-N} b_i e / a) x_i^{-1} )_{\delta_i}}
.
\nonumber\end{aligned}$$ We obtain it by applying the change of variables: $$\begin{aligned}
\label{covWT4}
\tilde{a} = q^{-N} e / d, & \tilde{b} = a q / c d,
& \tilde{d} = e,
\\
\tilde{u}_i = (q^{-N} e/ a) x_i^{-1}, \quad &
\tilde{v}_i = (a q/ b_i d) x_i, &
\quad \quad
(i= 1, \cdots , n).
\nonumber\end{aligned}$$ The rectangular version of is given by $$\begin{aligned}
\label{itAnWT4}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^{2} q^{2+|M| }}{b c d e}\right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1 - a q^{ |\gamma|+ \gamma_i} x_i}
{1 - a x_i}
\\
&& \quad
\times
\prod_{1 \le i, j \le n}
\frac{(q^{- m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{( (a q / b) x_i, (a q/ d) x_i, (a q / e) x_i)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad
\times
\frac
{(b , d, e )_{|\gamma|}}
{(a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i )_{|\gamma|}}
{( ( a q^{1+ m~i} ) x_i)_{|\gamma|}}
\nonumber\\
&=&
b^{|M|}
\frac{(a q / b c )_{|M|}}
{(a q /c)_{|M|}}
\prod_{ 1 \le i \le n}
\frac{(a q x_i, (a q / b d) x_i, (a q / b e) x_i)_{m_i}}
{((a q / b) x_i, (a q / d) x_i, (a q / e) x_i) x_i)_{m_i}}
\nonumber\\
&&
\quad
\times
\sum_{\delta \in {\Bbb N}^{n} }
q^{|\delta|}
\frac{\Delta ({x^{-1}} q^{\delta})}{\Delta ({x^{-1}})}
\prod_{1 \le i, j \le n}
\frac{( q^{-m_j} x_j / x_i)_{\delta_i}}{(q x_j / x_i)_{\delta_i}}
\nonumber\\
&& \quad \quad \quad
\times
\frac{(b)_{|\delta|}}
{(q^{-|M|} b c / a )_{|\delta|}}
\prod_{1 \le i \le n}
\frac{((q^{-1- m_i} b c d e / a^2 ) x_i^{-1}, (q^{-m_i} b / a ) x_i^{-1})_{\delta_i}}
{(( q^{-m_i} b d / a) x_i^{-1}, ( q^{-m_i} b e / a) x_i^{-1} )_{\delta_i}}
.
\nonumber\end{aligned}$$ In the case when $n=1$ and $x_1 = 1$, the transformation and reduces to the following transformation formula between terminating ${}_8 W_7$ series and terminating balanced ${}_4 \phi_3$ series: $$\begin{aligned}
\label{WatsonT2}
&& \quad \quad
{}_{8} W_{7}
\left[
a; b, c, d, e, q^{-N}; q;\frac{a^2 q^{2+N}}{b c d e}
\right]
\\
&=& \
b^N
\frac{(a q, a q / b c, a q / b d, a q / b e )_N}{(a q / b, a q / c, a q / d, a q / e)_N}
{}_{4} \phi_{3}
\left[
\begin{matrix}
q^{-N}, b, q^{-1-N} b c d e / a^2, q^{-N} b / a \\
b c q^{-N} / a, b d q^{-N} / a, b e q^{-N} / a,
\end{matrix};
q; q
\right].
\nonumber\end{aligned}$$
$A_n$ Sears transformations
---------------------------
In this and next subsection, we present some $A_n$ generalizations of the Sears transformation formula for terminating balanced ${}_4 \phi_3 $ series . In particular, we will prove two $A_n$ Sears transformations whose form of the series in both sides are same as that in the right hand side of the $A_n$ Watson transformation formulas in Section 4.1. We produce these identities by combining certain special cases of ${}_4 \phi_3$ series of type $A$ in Section 3.2. Our way to prove them is figured as the following diagram:
(100, 30) (0,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (80,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (35, 27)[$A_n$ Sears trans.]{} (45, 0)[(15,10)\[r\][${}_{n+3} \phi_{n+2}$ series]{}]{} (20, 23)[(3,-2)[20]{}]{} (50, 8)[(3,2)[20]{}]{} (-25, 14)[${}_4 \phi_3$ transformation of type $A$ [**(A)**]{}]{} (65, 14)[ [**(B)**]{} ${}_4 \phi_3$ transformation of type $A$]{}
[**The 1st one** ]{}
$$\begin{aligned}
\label{ltAnST1}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i )_{\gamma_i}}{(d x_i)_{\gamma_i}}
\\
&& \quad \times
\frac{(q^{-N}, a)_{|\gamma|}}
{(e, a B c q^{1-N} / d e)_{|\gamma|}}
=
\frac{(e / B, d e /a c)_N}
{(e, d e / a B c)_N}
\
\sum_{\delta \in {\Bbb N}^n}
q^{|\delta|}
\frac{\Delta (x q^{\delta})}{\Delta (x)}
\nonumber\\
&& \quad \quad \quad
\times
\frac{(q^{-N}, d/c)_{|\delta|}}
{(d e/a c, q^{1-N} B / e)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\delta_i}} {(q x_i / x_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((d/a) x_i)_{\delta_i}}{(d x_i)_{\delta_i}}
.
\nonumber\end{aligned}$$
We use as [**(A)**]{} and as [**(B)**]{} in the above diagram. It is not hard to see that both of the series in the right hand side of and in the left hand side in are terminating balanced ${}_{n+3} \phi_{n+2}$ series. We consider the following change of variables:
$$\begin{aligned}
\label{1covST1}
\tilde {a} = e/a, &
\tilde{c} = f/a, & \tilde{e} = d e f /a^2 B c.
\\
\tilde{u_i} = \frac{e f}{a b_i c} z_i, &
{\displaystyle \tilde{v_i} = \frac{e f}{a c}z_i^{}} \qquad (1 \le i \le n). &
\nonumber \end{aligned}$$
Note that the series in the right hand side of is invariant under this change of variables. Applying to the ${}_{n+3} \phi_{n+2}$ series in the right hand side in leads to the desired result .
In [@KajiS], we have already shown that can also be obtained by combining as [**(A)**]{} and as [**(B)**]{}. In this case, The change of variables is given by $$\begin{aligned}
\label{2covST1}
\tilde {a} = d/ c, &
\tilde{c} = a, & \tilde{e} = d e / B c.
\\
\tilde{u_i} = \frac{d}{b_i} x_i, &
{\displaystyle \tilde{v_i} = d x_i^{}} \qquad (1 \le i \le n). &
\nonumber \end{aligned}$$
[**Rectangular version** ]{}
$$\begin{aligned}
\label{itAnST1}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i )_{\gamma_i}}{(d x_i)_{\gamma_i}}
\\
&& \quad
\times
\frac{( a, b )_{|\gamma|}}
{(e, a b c q^{1-|M|} / d e)_{|\gamma|}}
\ = \
\frac{(e / b, d e /a c)_{|M|}}
{(e, d e / a b c)_{|M|}}
\sum_{\delta \in {\Bbb N}^n}
q^{|\delta|}
\frac{\Delta (x q^{\delta})}{\Delta (x)}
\nonumber\\
&& \quad \quad \quad
\times
\frac{(b, d/c)_{|\delta|}}
{(d e/a c, q^{1-|M|} b / e)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} x_i / x_j)_{\delta_i}} {(q x_i / x_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((d/a) x_i )_{\delta_i}}{(d x_i )_{\delta_i}}
.
\nonumber\end{aligned}$$
[**The 2nd one** ]{}
$$\begin{aligned}
\label{ltAnST2}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}{(d x_i)_{\gamma_i}}
\\
&& \quad
\times
\frac{(q^{-N}, a)_{|\gamma|}}
{(e, a B c q^{1-N} / d e)_{|\gamma|}}
\ = \
\frac{(d e /a c)_N}
{( d e / a B c)_N}
\prod_{1 \le i \le n}
\frac{((d/ b_i) x_i)_N}
{(d x_i)_N}
\nonumber
\\
&& \quad \quad \quad
\times \
\sum_{\delta \in {\Bbb N}^n}
q^{|\delta|}
\frac{\Delta (z q^{\delta})}{\Delta (z)}
\frac{(q^{-N}, e/a)_{|\delta|}}
{(d e/a c, e)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(b_j z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((e/c) z_i)_{\delta_i}}{((q^{1-N} B / d) z_i)_{\delta_i}}
,
\nonumber\end{aligned}$$
where $z_i = b_i / B x_i$ for $ 1 \le i \le n$.
We use as [**(A)**]{} and as [**(B)**]{}. In this case, the change of variables is given by $$\begin{aligned}
\tilde{a} = a, &
\tilde{c} = d/ c, & \tilde{e} = d e /B c, \\
\tilde{u_i} = \frac{d}{b_i} x_i , & \tilde{v_i} = d x_i \qquad (1 \le i \le n). &
\nonumber \end{aligned}$$ For the rest, one can easily verify in a similar way as the proof of Proposition 4.4.
Note that can also be obtained by combining as [**(A)**]{} and as [**(B)**]{}. In this case, the change of variables is given by $$\begin{aligned}
\tilde{a} = e/a, &
\tilde{c} = f/ a, & \tilde{e} = e f / a B, \\
\tilde{u_i} = \frac{e f}{a b_i c} z_i , & \tilde{v_i} = {\displaystyle \frac{e f}{a c} z_i}
\qquad (1 \le i \le n). &
\nonumber \end{aligned}$$
[**Rectangular version**]{}
$$\begin{aligned}
\label{itAnST2}
&&
\sum_{\gamma \in {\Bbb N}^n}
q^{|\gamma|}
\frac{\Delta (x q^{\gamma})}{\Delta (x)}
\prod_{1 \le i, j \le n}
\frac{(q^{- m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i )_{\gamma_i}}{(d x_i )_{\gamma_i}}
\\
&&
\quad
\times \
\frac{(a, b)_{|\gamma|}}
{(e, a b c q^{1-|M|} / d e)_{|\gamma|}}
\ = \
\frac{(d e /a c)_{|M|}}
{( d e / a b c)_{|M|}}
\prod_{ 1 \le i \le n}
\frac{((d/b) x_i)_{m_i}}
{(d x_i)_{m_i}}
\nonumber\\
&&
\quad \quad \quad
\times \
\sum_{\delta \in {\Bbb N}^n}
q^{|\delta|}
\frac{\Delta (z q^{\delta})}{\Delta (z)}
\frac{(e/a, b)_{|\delta|}}
{(d e/a c, e)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((e/c) z_i )_{\delta_i}}{((b q^{1-|M|}/d) z_i)_{\delta_i}}
.
\nonumber\end{aligned}$$
where $ z_i = q^{-m_i + |M| } x_i^{-1}$ for $ 1\le i \le n$.
In the case when $n=1$ and $x_1 = 1$, all of $A_n$ terminating balanced ${}_4 \phi_3$ transformation formulas , , and reduce to the Sears transformation . Especially, has already appeared in our previous work [@KajiS] and is originally due to Milne and Lilly (Theorem 6.5 in [@LM1]). Note also that can be proved by duplicating
Two $A_n$ Watson transformations and transpose to one another by $A_n$ Sears transformation formula . Similarly, two $A_n$ Watson transformations and transpose to one another by $A_n$ Sears transformation formula .
Another type of $A_n$ Sears transformation
------------------------------------------
Here, we give a yet another type of $A_n$ Sears transformation formula whose form of the series in both sides are the same as that in the right hand side of $A_n$ Watson transformation formulas in Section 4.2. We use the $m=1$ case of the inversion of the duality transformation and a certain special case of the duality transformation formula . Our road map is as follows:
(100, 30) (0,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (80,20)[(15,10)\[r\][${}_4 \phi_3$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (23, 27)[$A_n$ Sears trans.]{} (0,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (80,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (20, 5)[(1,0)[45]{}]{} (30, 7)[transposition]{} (5, 20)[(0, -1)[10]{}]{} (75, 10)[(0, 1)[10]{}]{} (8, 15) (78, 15)[ ]{}
$$\begin{aligned}
\label{ltAnST3}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(a_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(b x_i, c x_i )_{\gamma_i}}
{(e x_i,
( A b c q^{1-N}/ d e) x_i)_{\gamma_i}}
\\
&& \quad \times
\frac{(q^{-N})_{|\gamma|}}
{(d)_{|\gamma|}}
\ = \
\prod_{ 1 \le i \le n}
\frac
{((d e / b c) z_i, (e/a_i) x_i )_N}
{((d e/ a_i b c) z_i, e x_i )_N}
\sum_{\delta \in {\Bbb N}^{n}}
q^{|\delta|}
\frac{\Delta({z} q^{\delta})}{\Delta({z})}
\nonumber
\\
&& \quad \quad \quad
\times
\frac
{(q^{-N})_{|\delta|}}
{(d)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(a_j z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((d/b) z_i, (d/c) z_i )_{\delta_i}}
{((d e / b c ) z_i, (A q^{1-N}/ e) z_i)_{\delta_i}}
,
\nonumber\end{aligned}$$
where $z_i = a_i / A x_i$ for $i= 1, 2, \cdots , n$.
We use and . Note that both of the series in the right hand side of and in the left hand side of are very-well-poised-balanced ${}_{2n+6} W_{2n+5}$ series. In this case, we consider the following change of variables: $$\begin{aligned}
\label{covST3}
\tilde{a} = \frac{d e^2 q^{-1}}{A b c}, & \tilde{b} = e/b, &
\tilde{d} = e/c,
\\
\tilde{u}_i = (d e q^{-1} /Ab c) x_i^{-1}, & \tilde{v}_i = (e/ a_i) x_i \quad \quad
(i= 1, \cdots , n), &
\nonumber\end{aligned}$$ which is a transposition of the variables in ${}_{2n + 6} W_{2n+5} $ series in . Since the series in the right hand side of is invariant under this change of variables. This invariance implies the desired result by applying according to the change .
[**Rectangular version**]{}
$$\begin{aligned}
\label{itAnST3}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
q^{|\gamma|}
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(b x_i, c x_i )_{\gamma_i}}
{(e x_i,
(a b c q^{1-|M|}/ d e) x_i)_{\gamma_i}}
\\
&&
\quad
\times
\frac{(a)_{|\gamma|}}
{(d)_{|\gamma|}}
\ = \
\prod_{ 1 \le i \le n}
\frac
{((d e / b c) z_i, (e/a) x_i )_{m_i}}
{((d e/ a b c) z_i, e x_i )_{m_i}}
\
\sum_{\delta \in {\Bbb N}^{n}}
q^{|\delta|}
\frac{\Delta({z} q^{\delta})}{\Delta({z})}
\nonumber \\
&& \quad \quad \quad
\times
\frac
{(a)_{|\delta|}}
{(d)_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\prod_{1 \le i \le n}
\frac{((d/b) z_i, (d/c) z_i )_{\delta_i}}
{((d e/ b c) z_i, (a q^{1-|M|}/ e) z_i)_{\delta_i}},
\nonumber\end{aligned}$$
where $z_i = q^{m_i - |M|} x_i^{-1}$ for $i= 1, 2, \cdots , n$.
has originally appeared as Theorem 6.8 in Milne-Lilly [@LM1]. Though they referred as $C_r$ Sears transformation formula there, the sums in both side of are $A_n$ ${}_4 \phi_3$ series. For their point of view, it may be precise to refer it as [ *$A_n$ Sears transformation formula arising from $C_n$ Bailey transform*]{}.
In the case when $n=1$ and $x_1 = 1$, and reduce to the Sears transformation . We also note that can be obtained from $A_n$ Bailey transformation in the following way: First we replace $d \to aq/ d$ and $f \to aq/f$ in . Then let the parameter $a$ tends to infinity in the resulting equation. Finally rearrange the parameters appropriately.
Nonterminating ${}_8 W_7$ transformations
-----------------------------------------
Here we show a $A_n$ nonterminating ${}_8 W_7$ transformation formula. Our tool to produce it is $m=1$ case of of the nonterminating ${}_8 W_7 $ transformation formula in Section 3.3. One can see the way to prove the identity as
(100, 30) (0,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (80,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (23, 27)[$A_n$ nonterminating ${}_8 W_7$ trans.]{} (0,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (80,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (20, 5)[(1,0)[45]{}]{} (30, 7)[transposition]{} (5, 20)[(0, -1)[10]{}]{} (75, 10)[(0, 1)[10]{}]{} (8, 15) (78, 15)
$$\begin{aligned}
\label{AnNT87T1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{- \gamma_1} \cdots x_n^{- \gamma_n}
\left( \frac{a^2 q^2}{b c d E f}\right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i }
\\
&&
\quad
\times
\prod_{1 \le i, j \le n}
\frac{(e_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i\le n}
\frac{(b x_i, c x_i, d x_i)_{\gamma_i}}
{( ( a q / f ) x_i)_{\gamma_i}}
\nonumber\\
&&
\quad \quad
\times
\frac{( f)_{|\gamma|}}
{( a q / b, a q / c, a q / d)_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /e_i ) x_i)_{|\gamma|}}
\nonumber\\
&&
=
\prod_{ 1 \le i \le n}
\frac{
( a q x_i, (a q / e_i f) x_i,
(\lambda q / e_i ) z_i,
(\lambda q / f ) z_i)_\infty}
{( ( a q / e_i) x_i, (a q / f) x_i,
(\lambda q / e_i f ) z_i,
\lambda q z_i)_\infty}
\nonumber\\
&&
\quad
\times
\sum_{\delta \in {\Bbb N}^{n}}
z_1^{- \delta_1} \cdots z_n^{- \delta_n}
\left( \frac{ a q }{E f}\right)^{|\delta|}
q^{ e_2 ( \delta )} \
\frac{\Delta({z} q^{\delta})}{\Delta({z})}
\prod_{1 \le i \le n}
\frac{1- \lambda q^{|\delta| + \delta_i} z_i}
{1- \lambda z_i}
\nonumber\\
&&
\quad \quad
\times
\prod_{1 \le i, j \le n}
\frac{(e_j z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\prod_{1 \le i\le n}
\frac{( ( a q / c d ) z_i, ( a q / b d ) z_i,
(a q / b c) z_i)_{\delta_i}}
{( ( \lambda q / f ) z_i)_{\delta_i}}
\nonumber\\
&&
\quad \quad \quad
\times
\frac
{( f)_{|\delta|}}
{( a q / b, a q / c, a q / d )_{|\delta|}}
\prod_{1 \le i \le n}
\frac{(\lambda z_i)_{|\delta|}}
{( ( \lambda q /e_i ) z_i)_{|\delta|}}
\nonumber\end{aligned}$$
where $ \lambda = a^{2} q^{} / b c d $ and $z_i = \displaystyle{\frac{e_i}{E} x_i^{-1}}$.
We iterate $$\begin{aligned}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_i^{-\gamma_i} \cdots x_n^{-\gamma_n}
\left(
\frac{\mu f}{a}
\right)^{|\gamma|}
q^{ e_2 ( \gamma )} \
\frac{\Delta({x}q^{\gamma})}
{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i }
{1- a x_i}
\\
&& \quad \quad
\times
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\prod_{1 \le i \le n}
\frac{(c x_i, d x_i, e x_i)_{\gamma_i}}
{(( a q / f ) x_i)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad \quad
\times
\frac{( f)_{|\gamma|}}
{( a q / c, a q / d, a q / e)_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\nonumber\\
&=&
\frac
{(\mu c f / a, \mu d f / a, \mu e f / a, f)_\infty}
{(a q / c, a q / d, a q / e, \mu q)_\infty}
\prod_{ 1 \le i \le n}
\frac
{(a q x_i, ( \mu b_i f / a ) x_i^{-1} )_\infty}
{((a q / b_i) x_i, (\mu f / a) x_i^{-1} )_\infty}
\nonumber\\
&&
\quad \quad
\times
{}_{2 n + 4} W_{2 n + 3}
\left[
\mu; \{(a q /b_i^{} f ) x_i \}_{n}
a q / c f,a q / d f,a q / e f,
\{(\mu f / a ) x_i^{-1} \}_{n}
; q; f
\right],\nonumber \\
&& \quad \quad \quad \quad \quad \quad \quad
(\mu = a^{3} q^{2} / B c d e f^{2}),\end{aligned}$$ twice. On the way to obtain , we interchange $ (aq / b_i f) x_i$ and $(\mu f / a) x_i^{-1}$ for all $i=1, \cdots , n$ simultaneously in the ${}_{2n+6} W_{2n+5}$ series.
In the case when $n=1$ and $x_1 = 1$, reduces to the following nonterminating ${}_8 W_7$ transformation: $$\begin{aligned}
\label{NonTermT87}
&& \quad \quad \quad
{}_{8} W_{7}
\left[
a;b, c, d, e, f
; q; \frac{a^2 q^2}{b c d e f}
\right]\\
&=&
\frac{(a q, a q /e f, \lambda q /e, \lambda q /f)_\infty}
{(a q/ e, a q /f, \lambda q, \lambda q /e f)_\infty }
{}_{8} W_{7}
\left[
\lambda; \lambda b /a,
\lambda c/ a, \lambda d / a , e, f
; q; \frac{a q}{e f}
\right],\nonumber\end{aligned}$$ where $\lambda = a^2 q^{} /b c d$. can also be obtained by taking the limit $N \to \infty$ in $A_n$ Bailey transformation formula .
Terminating ${}_8 W_7$ transformations
--------------------------------------
Here we present $A_n$ nonterminating ${}_8 W_7$ transformation formulas. We give a proof by using $m=1$ case of of the nonterminating ${}_8 W_7 $ transformation formula in Section 3.3. The proof is in the same manner as in that of . One can see the way to give the identity as
(100, 30) (0,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (80,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (23, 27)[$A_n$ terminating ${}_8 W_7$ trans.]{} (0,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (80,0)[(15,10)\[r\][${}_{2n+6} W_{2n+5}$ series]{}]{} (20, 5)[(1,0)[45]{}]{} (30, 7)[transposition]{} (5, 20)[(0, -1)[10]{}]{} (75, 10)[(0, 1)[10]{}]{} (8, 15) (78, 15)
$$\begin{aligned}
\label{ltAnTerm8W7-1}
&&
\quad
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^2 q^{N+2}}{B c f g} \right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i}
\\
&& \quad \quad \quad
\times
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( ( a q /b_i ) x_i)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(b_j x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad \quad \quad
\times
\frac{( f, g, q^{-N})_{|\gamma|}}
{( a q / c )_{|\gamma|}}
\prod_{1 \le i\le n}
\frac{(c x_i)_{\gamma_i}}
{( a q^{N+1} x_i, ( a q / f ) x_i, (a q/ g) x_i)_{\gamma_i}}
\nonumber\\
&&
= \
\prod_{ 1 \le i \le n}
\frac{
( a q x_i, (a q / b_i f) x_i, (a q / b_i g) x_i, (a q / f g) x_i)_N}
{( ( a q / b_i) x_i, (a q / f) x_i, (a q / g ) x_i, ( a q / b_i f g) x_i)_N}
\nonumber\\
&&
\quad \quad
\times
\sum_{\delta \in {\Bbb N}^{n}}
z_1^{\delta_1} \cdots z_n^{\delta_n}
\left( \frac{q}{c} \right)^{|\delta|}
q^{ - e_2 ( \delta )} \
\frac{\Delta({z} q^{\delta})}{\Delta({z})}
\prod_{1 \le i \le n}
\frac{1- ( q^{-N-1} B f g / a) q^{|\delta| + \delta_i} z_i }
{1- (q^{-N-1} B f g / a) z_i}
\nonumber\\
&&
\quad \quad \quad \quad
\times
\prod_{1 \le i \le n}
\frac{( (q^{-N} B f g / a) z_i)_{|\delta|}}
{( ( q^{-N} f g / a ) z_i )_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{(b_j z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\nonumber\\
&&
\quad \quad \quad \quad \quad \quad
\times
\frac{( f, g, q^{-N})_{|\delta|}}
{( a q / c)_{|\delta|}}
\prod_{1 \le i\le n}
\frac{( ( q^{-N-1} B c f g / a^2 ) z_i )_{\delta_i}}
{( ( B f g /a ) z_i, ( q^{-N} B g / a ) z_i, ( q^{-N} B f / a) z_i )_{\delta_i}}
.
\nonumber\end{aligned}$$
where $z_i = \displaystyle{\frac{e_i}{E} x_i^{-1}}$.
[**Rectangular version**]{}
$$\begin{aligned}
\label{itAnTerm8W7-1}
&&
\sum_{\gamma \in {\Bbb N}^{n}}
x_1^{\gamma_1} \cdots x_n^{\gamma_n}
\left( \frac{a^2 q^{|M|+2}}{b c f g} \right)^{|\gamma|}
q^{ - e_2 ( \gamma )} \
\frac{\Delta({x} q^{\gamma})}{\Delta({x})}
\prod_{1 \le i \le n}
\frac{1- a q^{|\gamma| + \gamma_i} x_i}
{1- a x_i}
\\
&& \quad \quad
\times
\prod_{1 \le i \le n}
\frac{(a x_i)_{|\gamma|}}
{( a q^{|M|+1 } x_i)_{|\gamma|}}
\prod_{1 \le i, j \le n}
\frac{(q^{-m_j} x_i / x_j)_{\gamma_i}} {(q x_i / x_j)_{\gamma_i}}
\nonumber\\
&& \quad \quad \quad \quad
\times
\frac{( b, f, g)_{|\gamma|}}
{( a q / c )_{|\gamma|}}
\prod_{1 \le i \le n}
\frac{(c x_i)_{\gamma_i}}
{( (a q /b) x_i, ( a q / f ) x_i, (a q/ g) x_i)_{\gamma_i}}
\nonumber\\
&=&
\prod_{ 1 \le i \le n}
\frac{
( a q x_i, (a q / b f) x_i, (a q / b g) x_i, (a q / f g) x_i)_{m_i}}
{( ( a q / b) x_i, (a q / f) x_i, (a q / g ) x_i, ( a q / b f g) x_i)_{m_i}}
\nonumber\\
&& \quad \quad
\times
\sum_{\delta \in {\Bbb N}^{n}}
z_1^{\delta_1} \cdots z_n^{\delta_n}
\left( \frac{q}{c} \right)^{|\delta|}
q^{ - e_2 ( \delta )} \
\frac{\Delta({z} q^{\delta})}{\Delta({z})}
\prod_{1 \le i \le n}
\frac{1- ( q^{-|M|-1} b f g / a) q^{|\delta| + \delta_i} z_i }
{1- (q^{-|M|-1} b f g / a) z_i}
\nonumber\\
&& \quad \quad \quad \quad
\times
\prod_{1 \le i \le n}
\frac{( (q^{-|M|} b f g / a) z_i)_{|\delta|}}
{( ( q^{-|M|} f g / a ) z_i )_{|\delta|}}
\prod_{1 \le i, j \le n}
\frac{( q^{ - m_j} z_i / z_j)_{\delta_i}} {(q z_i / z_j)_{\delta_i}}
\nonumber\\
&& \quad \quad \quad \quad \quad \quad
\times
\frac{( b, f, g)_{|\delta|}}
{( a q / c)_{|\delta|}}
\prod_{1 \le i\le n}
\frac{( ( q^{-|M|-1} b c f g / a^2 ) z_i )_{\delta_i}}
{( ( q^{- |M|} f g /a ) z_i, ( q^{-|M|} b g / a ) z_i, ( q^{-|M|} b f / a) z_i )_{\delta_i}}
.
\nonumber\end{aligned}$$
where $z_i = \displaystyle{q^{m_i - |M|} x_i^{-1}}$.
In the case when $n=1$ and $x_1 = 1$, and reduces to the following terminating ${}_8 W_7$ transformation: $$\begin{aligned}
\label{n1-AnTerm8W7-1}
&&
{}_8 W_7
\left[
a; b, c, f, g, q^{-N}; q;
\frac{a^2 q^{2+N}}{ b c f g}
\right]
=
\frac{
( a q, a q / b f, a q / b g, a q / f g)_N}
{( a q / b, a q / f, a q / g, a q / b f g)_N}
\\
&&
\quad \quad \quad \quad
{}_8 W_7
\left[
q^{-N-1} b f g / a;
b, q^{-N-1} b c f g / a^2, f, g, q^{-N}; q;
\frac{q}{c}
\right].
\nonumber\end{aligned}$$ can also be obtained by taking the limit $d \to \infty$ in .
$A_n$ Watson transformation can be obtained by combining and and by combining and $A_n$ Sears transformations formula .
(100, 30) (0,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (80,20)[(15,10)\[r\][${}_8 W_7$ series in $A_n$]{}]{} (20, 25)[(1,0)[45]{}]{} (37, 27) (0,0)[(15,10)\[r\][ ${}_4 \phi_3$ series in $A_n$]{}]{} (80,0)[(15,10)\[r\][${}_{4} \phi_3$ series in $A_n$ ]{}]{} (20, 5)[(1,0)[45]{}]{} (37, 7) (5, 20)[(0, -1)[10]{}]{} (75, 20)[(0, -1)[10]{}]{} (6, 15) (77, 15) (23, 20)[(4,-1)[40]{}]{} (50, 15)
As we have seen in this section, our discussion here may implies not only that our class of multiple hypergeometric transformations in the previous section are broader class than Milne‘s class of transformation in $A_n$ but also that our class contains more precise informations. In our terminology, one may see that Milne‘s hypergeometric transformations have extra hidden symmetries in the one dimensional (generalized) hypergeometric series.
[**[Acknowledgments]{}**]{}\
I would like to express my sincere thanks to Professors Etsuro Date and Masatoshi Noumi for their encouragements.
[99]{}
G.E. Andrews: [*The well-poised thread: an organized chronicle of some amazing summations and their implications.*]{} Ramanujan J. [**1**]{} (1997), 7–23.
W.N. Bailey: [*Some identities involving generalized hypergeometric series.*]{} Proc. London Math. Soc. (2), [**29**]{}(1929), 503–516.
W.N. Bailey: [*Generalized hypergeometric series.*]{} Cambridge Tracts (1935).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Hall viscosity describes a non-dissipative response to strain in systems with broken time-reversal symmetry. We develop a new method for computing the Hall viscosity of lattice systems in strong magnetic fields based on momentum transport, which we compare to the method of momentum polarization used by Tu et al. \[Phys. Rev. B **88** 195412 (2013)\] and Zaletel et al. \[Phys. Rev. Lett. **110** 236801 (2013)\] for non-interacting systems. We compare the Hall viscosity of square-lattice tight-binding models in magnetic field to the continuum integer quantum Hall effect (IQHE) showing agreement when the magnetic length is much larger than the lattice constant, but deviation as the magnetic field strength increases. We also relate the Hall viscosity of relativistic electrons in magnetic field (the Dirac IQHE) to the conventional IQHE. The Hall viscosity of the lattice Dirac model in magnetic field agrees with the continuum Dirac Hall viscosity when the magnetic length is much larger than the lattice constant. We also show that the Hall viscosity of the lattice model deviates further from the continuum model if the $C_4$ symmetry of the square lattice is broken to $C_2$, but the deviation is again minimized as the magnetic length increases.'
author:
- 'Thomas I. Tuegel'
- 'Taylor L. Hughes'
title: Hall Viscosity and Momentum Transport in Lattice and Continuum Models of the Integer Quantum Hall Effect in Strong Magnetic Fields
---
Introduction
============
The topological response properties of the quantum Hall effect have been intensely studied for more than three decades, and begun with the understanding that the quantized integer/fractional Hall conductance itself is a topological phenomenon.[@laughlin1981; @tknn1982; @niu1985; @wenbook; @fradkinbook] The field has since understood that quantum Hall systems (with and without magnetic fields) also exhibit remarkable responses to changes in geometry.[@wenzee; @avron_viscosity_1995; @levay_berry_1995; @avron1998; @read_non-abelian_2009; @tokatly2009; @haldane_hall_2009; @read_hall_2011; @hughes2011; @kimura_hall_2010; @stone2012; @bradlyn_kubo_2012; @barkeshli2012; @hoyos_hall_2012; @wiegmann2012; @ryu2012; @zaletel_topological_2013; @hughes2013; @hidaka2013; @abanov2013; @biswas_semiclassical_2013; @cho2014; @fremling2014; @bradlyn_low-energy_2014; @abanov_electromagnetic_2014; @parrikar2014; @gromov2014b; @can2014; @park2014; @gromov2015b; @gromov2015; @cho2015b]
One interesting piece of the geometric response is the non-dissipative Hall viscosity.[@avron_viscosity_1995] The Hall viscosity $\eta_H$ is an off-diagonal response coefficient that is only non-vanishing when time-reversal symmetry is broken. Under time-dependent shear strain, the viscosity tensor $\eta$ relates the stress tensor $T$ to the strain rate $\dot{u}$: $$T^{\mu\nu} = -\eta^{\mu\nu\alpha\beta} \dot{u}_{\alpha\beta},
\nonumber$$where the strain tensor is constructed from a symmetrized gradient of the local displacement $u_{\alpha}.$ If only dissipative viscosity coefficients are present, e.g. the bulk and shear viscosities, then $\eta^{\mu\nu\alpha\beta}=\eta^{\alpha\beta\mu\nu}$ and is thus symmetric under exchange of $(\mu\nu)$ with $(\alpha\beta).$ However, being non-dissipative, the Hall viscosity generates an antisymmetric piece satisfying $$\eta_{H}^{\mu\nu\alpha\beta} = -\eta_{H}^{\alpha\beta\mu\nu}.
\nonumber$$ When the system is 2D and isotropic, the antisymmetric part of the viscosity tensor is determined by a single parameter $\eta_H$ that gives, in an isotropic 2D orthonormal frame: [@avron_viscosity_1995; @avron1998] $$\begin{aligned}
\eta_{H}^{1112} = \eta_{H}^{1222} &= \eta_H \nonumber \\
\eta_{H}^{1122} &= 0. \nonumber\end{aligned}$$
The Hall viscosity can be calculated using a variety of different methods. The first calculations were performed via the adiabatic transport of the Hall fluid under shear strain on a torus.[@avron_viscosity_1995; @levay_berry_1995; @read_non-abelian_2009; @tokatly2009] For Schrödinger electrons at integer filling factors, this type of calculation yields $ \eta_H = \hbar\nu\rho/4$ where $\rho$ is the electron number density, and $\nu$ is the integer filling fraction. [@avron_viscosity_1995; @levay_berry_1995; @read_non-abelian_2009; @read_hall_2011] More recently, Ref. developed Kubo formulas for the Hall viscosity which obtain the same result. Also, a new possibility for calculating the Hall viscosity was proposed via the so-called momentum polarization entanglement technique,[@tu_momentum_2013; @zaletel_topological_2013] though there is very little explicit discussion of the results of this method in the literature (see Ref. for a very recent article). Remarkably, from the adiabatic transport calculations it has been shown that for rotationally-invariant integer and fractional quantum Hall systems in large magnetic fields, the viscosity is quantized in units of the density[@read_non-abelian_2009] and takes the form $$\begin{aligned}
\eta_H=\frac{\kappa}{4}\hbar\rho\end{aligned}$$where $\kappa$ is a universal number characterizing the particular integer/fractional quantum Hall phase, and $\rho$ is the uniform electron number density. Generically, the Hall viscosity has units of $[\tfrac{\hbar}{\ell^2}]$ for some length scale $\ell,$ but it need not always retain such a clear quantization in terms of the particle density.
The goal of this article is two-fold: (i) we introduce a new method for the calculation of the Hall viscosity using momentum transport, and compare with the extraction of the Hall viscosity via the momentum polarization entanglement method, and (ii) we study the properties of the viscosity in two different lattice realizations of the Landau-level integer quantum Hall problem (square-lattice Hofstadter, lattice-Dirac model in a magnetic field), and illustrate the competition between contributions of the viscosity from the lattice-length scale and the magnetic-length scale. Our article is organized as follows. In Section \[sec:methods\], we present the momentum transport method used here to compute the Hall viscosity. We also review the momentum polarization method, which allows computation of the Hall viscosity from the entanglement spectrum. In Section \[sec:landau-level\], we describe the application of both methods to the continuum Landau level problem, and compare the results to previous calculations of the Hall viscosity. In Section \[sec:hofstadter\], we present numerical calculations of the Hall viscosity for the Landau levels of a tight-binding model (the Hofstadter model) and discuss the results. In Section \[sec:dirac-landau-level\] we calculate the Hall viscosity of the Landau levels of the continuum Dirac equation, with and without a mass term, building on previous work by @kimura_hall_2010. Finally, in Section \[sec:chern\] we use a lattice analog of the continuum Dirac system in a magnetic field, and study the Hall viscosity for comparison with the continuum results.
Methods {#sec:methods}
=======
We will consider two independent methods for calculating the Hall viscosity in our example systems. The first method considers the transverse flow of momentum when a cylinder is strained with an area-preserving deformation. The second method uses the entanglement spectrum to calculate the phase acquired by the many-body wavefunction when half of a cylinder is sheared; from this phase, one can extract the central charge[@tu_momentum_2013] and Hall viscosity.[@zaletel_topological_2013] Let us introduce and review both of these methods.
#### Momentum Transport
For the first method we study the off-diagonal components of the stress tensor, which represent the momentum flux. We will write the geometric deformation in terms of the strain tensor. To be explicit, if $u_\alpha$ is the displacement vector, then, to lowest order, the strain is [@landau_theory_1986] $$u_{\alpha\beta} = \frac{1}{2}\left(\partial_\alpha u_\beta
+ \partial_\beta u_\alpha\right).$$ In terms of the geometry, if $ds$ and $ds'$ are the original and deformed length elements, respectively, then to lowest order in the deformations, [@landau_theory_1986] $$ds'^2 = ds^2 + 2u_{\alpha\beta} dx^\alpha dx^\beta.$$ Due to the structure of the Hall viscosity terms in the viscosity tensor, shear strain causes momentum transport in the direction of momentum, but pressure/stretching causes momentum transport orthogonal to the direction of momentum. We find the latter is more easily studied in lattice systems when using a cylinder geometry, therefore, we consider deformed metrics of the form $$ds^2 = \frac{1}{\alpha^2} dx^2 + \alpha^2 dy^2
\label{eq:cart-metric}$$ where $\alpha$ can vary. This deformation is area-preserving (shear), so we need not isolate our momentum transport results from effects induced purely by changes to the density.
To calculate the momentum transport under this deformation let us consider a cylinder which is periodic in the $y$-direction with a circumference $L_y$. As $\alpha$ varies, the strain rate is $$\dot{u} = -\frac{\dot{\alpha}}{\alpha^3}dx^2 + \alpha\dot{\alpha}\,dy^2.
\nonumber$$ The corresponding stress tensor components arising from the Hall viscosity contributions are $$T^{xy} = T^{yx} = -2 \eta_H \frac{\dot{\alpha}}{\alpha}.
\nonumber$$ Note that when the system is anisotropic ($\alpha\neq{}1$) we can have $\eta_H^{1122}\neq{}0$; unfortunately that term will not appear in this component of the stress tensor, so it cannot be extracted; but hence, it also will not affect our calculation of the other viscosity coefficients.
As the metric is deformed we want to study the amount of momentum transported from the left-half of the cylinder to the right-half. Consider cutting the cylinder at $x=x_\text{cut}$; if $\mathcal{P}_R$ is the projection operator onto the right side of the cut, then the total momentum on the right of the cut is $$\left\langle P_y \mathcal{P}_R \right\rangle =
\int_{x_\text{cut}}^\infty dx \int_0^{L_y} dy \, \Pi_y$$ where $\Pi_y$ is the momentum density. Typically, we will choose $x_\text{cut}=0$ with the cylinder placed symmetrically around this point. The stress tensor gives the momentum flux across the cut, i.e. $$\frac{d}{dt}\left\langle P_y\mathcal{P}_R \right\rangle =
- \int_0^{L_y}\,dy\,T_y^{\phantom{y}x} = 2 L_y \eta_H \alpha \dot{\alpha}.$$ From this equation we can immediately read-off the important result: $$\eta_H =
\frac{1}{L_y} \frac{d}{d\alpha^2}
\left\langle{}P_y\mathcal{P}_R\right\rangle.
\label{eq:viscosity-momentum-transport}$$ We will use this relationship between $\eta_H$ and the $\alpha$-dependence of the half-cylinder momentum to calculate the viscosity.
#### Momentum Polarization
The second method we discuss uses the entanglement spectrum[@li2008; @peschel2003] to determine the *momentum polarization*.[@tu_momentum_2013] The momentum polarization was initially proposed to calculate the topological spin and central charge of the conformal field theory at the edge of a topological phase. For a system in a cylindrical geometry, these data are extracted from the expectation value of the operator $T_y^L$, which globally translates the left half of the cylinder in the periodic direction. The expectation value can be computed using the reduced density matrix,[@tu_momentum_2013] $$\lambda \equiv \left\langle{}G\right|T_y^L\left|G\right\rangle
= \operatorname{tr}_L \left(\rho_{L}T_y^L\right)$$ where $\left|G\right\rangle$ is the ground state. Ref. shows that $\lambda$ can be easily calculated for free-fermion systems using the entanglement spectrum.
To see that the topological spin and central charge can be extracted from this expectation value, consider that, in the long-wavelength limit, the reduced density matrix of a cylinder cut in half can be written in terms of the Hamiltonians $H_{Ll}$ and $H_{Lr}$ of the respective conformal edge theories of the left and right edges of the left half-cylinder only:[@tu_momentum_2013] $$\rho_{L} = \rho_{Ll} \otimes \rho_{Lr}
= Z^{-1} e^{-\beta_lH_{Ll} -\beta_rH_{Lr}}.$$ The relevant half-cylinder translation operator is $$T_y^L = \exp{\left[\frac{2\pi{}i}{L_y}\left(P_l+P_r\right)\Delta{}y\right]},$$ where $\Delta{}y$ is the distance translated (which we take to be a multiple of the lattice constant for lattice systems), and $P_l$ and $P_r$ are the generators of translations (momentum operators) of the left and right edge theories on the half-cylinder, respectively.[@tu_momentum_2013; @ginsparg_applied_1988] Since the left-most edge is far from the right half, $\beta_l\rightarrow\infty,$ and only the ground state of the left edge contributes. The ground state expectation value of $P_l$ is $h-c/24$ where $h$ is the topological spin and $c$ is the chiral central charge.[@ginsparg_applied_1988] Therefore, the contribution of the left edge is [@tu_momentum_2013] $$\operatorname{tr}_{Ll}{\left(\rho_{Ll}
\exp{\left[\frac{2\pi{}i}{L_y}P_l\Delta{}y\right]}\right)}
= \exp{\left[\frac{2\pi{}i}{L_y}\Delta{}y
\left(h-\frac{c}{24}\right)\right]}.$$ On the other hand, $\beta_r$ takes a finite value because the right edge is entangled with the right half-cylinder. In general, the right edge gives a non-universal contribution [@tu_momentum_2013] $$\operatorname{tr}_{Lr}{\left(\rho_{Lr}
\exp{\left[\frac{2\pi{}i}{L_y}P_r\Delta{}y\right]}\right)}
= \exp{\left[-L_y\alpha\right]}.$$ From this we see that one can extract the central charge and topological spin.
For free fermions, $\lambda$ is easily calculated in terms of the entanglement spectrum for a cylinder by the formula [@tu_momentum_2013] $$\lambda = \prod_{n,k_y}\frac{1}{2}\left[\left(1+e^{ik_y\Delta{}y}\right)
+ \left(1-e^{ik_y\Delta{}y}\right)\tanh\frac{\xi_{k_y,n}}{2}\right]
\label{eq:momentum-polarization-ent-spectrum}$$ where $\prod_{n,k_y}$ is a product over the bands and $y$-momenta, and $\xi_{k_y,n}$ is the entanglement eigenvalue of the state in band $n$ with momentum $\hbar{}k_y$. The entanglement eigenvalues can be expressed in terms of the eigenvalues of the free-electron, equal-time correlation function, [@peschel_calculation_2002] $$\xi_{k_y,n} = \log\frac{1-C^{(L)}_{k_y,n}}{C^{(L)}_{k_y,n}}
\label{eq:free-fermion-ent-eigenvalues}$$where $C^{(L)}_{k_y,n}$ are the eigenvalues of $C^{(L)}_{k_y}=\langle c^{\dagger}_{k_y ia}c_{k_y jb}\rangle$ where $k_y$ are the momenta in the periodic direction, $i,j$ run-over the lattice sites on the left half of the cylinder, and $a,b$ run-over all of the onsite degrees of freedom. Note that this projects states onto the left half of the cylinder, but we will find it more useful to compute this formula in terms of the projections onto the right half, $C_{n,k_y}=1-C^{(L)}_{n,k_y}$. Using these identities, it is convenient to rewrite as $$\lambda = \prod_{n,k_y}\frac{1}{2}\left[\left(1+e^{ik_y\Delta{}y}\right)
+ \left(1-e^{ik_y\Delta{}y}\right)\left(2C_{k_y,n}-1\right)\right].
\label{eq:momentum-polarization-corr-func}$$
In a remarkable extension of this work, Ref. shows that for quantum Hall states one can extract the Hall viscosity from the imaginary part of the “non-universal" coefficient $\alpha.$ Explicitly they find $$\lambda
= \exp{\left[\frac{2\pi{}i}{L_y}\Delta{}y\left(h-\frac{c}{24}\right)
- iL_y\Delta{}y\frac{\eta_H}{\hbar} + \ldots\right]}
\label{eq:viscosity-momentum-polarization}$$where additional non-universal terms that scale differently with $L_y$ have been dropped. In their work they consider a full twist such that $\Delta{}y=L_y,$ but the result carries over for smaller $\Delta{}y$ as well. Thus, the viscosity and central charge can be extracted from a fit of $L_y\mathrm{Arg}\lambda$; the former from the quadratic coefficient, the latter from the constant coefficient.
We can understand how the momentum polarization phase encodes the viscosity by considering the action of the shear strain generators on the ground state. Here, we will show that the Hall viscosity can be extracted by comparing the momentum polarization calculated with a real-space cut to the phase taken with an orbital cut following Ref. . We note that Ref. identified two distinct contributions to the Hall viscosity, and the contribution which interests us here is due to changing the shape of the Landau orbitals under shear strain. The second contribution, the guiding center Hall viscosity, comes from the electron correlations and is absent in the integer quantum Hall models we study here. We will review how the momentum polarization phase calculated with a real-space cut encodes both Hall viscosity contributions. Although we consider only the integer effect, the guiding center Hall viscosity also has a super-extensive term due to the non-zero net momentum in each half of the system.[@park2014] We will show that this background can be subtracted by calculating the momentum polarization phase with an *orbital* cut and comparing the two results.
For most of the remainder of this section we closely follow Ref. . First, let us decompose our physical coordinate $\mathbf{R}$ into a guiding center coordinate $\mathbf{\widetilde{r}}$ and an orbital coordinate $\mathbf{r}$: $$\mathbf{R} = \mathbf{\widetilde{r}} + \mathbf{r}.$$ There is a metric $G_{\mu\nu}$ associated with the physical coordinate $\mathbf{R}$, as well as metrics $\widetilde{g}_{\mu\nu}$ and $g_{\mu\nu}$ associated with each coordinate $\mathbf{\widetilde{r}}$ and $\mathbf{r}$, respectively. Let the operators $\widetilde{\lambda}^{\mu\nu}$ generate shear strain (area-preserving deformations) associated with the metric $\widetilde{g}_{\mu\nu}$; likewise, let $\lambda^{\mu\nu}$ be the shear strain generators associated with $g_{\mu\nu}$. These generators obey commutation relations[@park2014] $$\begin{aligned}
\left[\lambda^{\mu\nu},\,\lambda^{\alpha\beta}\right]
&= -\frac{i}{2} \left(\epsilon^{\mu\alpha}\lambda^{\nu\beta}
+ \epsilon^{\mu\beta}\lambda^{\nu\alpha}
+ \mu\leftrightarrow\nu\right) \nonumber \\
\left[\widetilde{\lambda}^{\mu\nu},\,\widetilde{\lambda}^{\alpha\beta}\right]
&= \frac{i}{2} \left(\epsilon^{\mu\alpha}\widetilde{\lambda}^{\nu\beta}
+ \epsilon^{\mu\beta}\widetilde{\lambda}^{\nu\alpha}
+ \mu\leftrightarrow\nu\right)\nonumber\\
\left[\widetilde{\lambda}^{\mu\nu},\,\lambda^{\alpha\beta}\right]&=0.
\label{eq:mp-strain-commutators}\end{aligned}$$ The strain generator in the physical coordinate is $$\Lambda^{\mu\nu} = \widetilde{\lambda}^{\mu\nu} + \lambda^{\mu\nu}$$ so that the unitary operator implementing strain on quantum states is[@park2014; @bradlyn_kubo_2012] $$U(\alpha) = \exp\left[i \int d^2\mathbf{R}\, \alpha_{\mu\nu}(\mathbf{R}) \Lambda^{\mu\nu} \right]$$ where $\alpha_{\mu\nu}$ is a symmetric matrix parametrizing the strain. Because the strain generators on each coordinate commute, we can also write this as the product of strain transformations on each coordinate: $$\begin{aligned}
U(\alpha) &= u(\alpha)\widetilde{u}(\alpha) \\
u(\alpha) &= \exp\left[i \int d^2\mathbf{R}\,
\alpha_{\mu\nu}(\mathbf{R}) \lambda^{\mu\nu} \right] \\
\widetilde{u}(\alpha) &= \exp\left[i \int d^2\mathbf{R}\, \alpha_{\mu\nu}(\mathbf{R}) \widetilde{\lambda}^{\mu\nu} \right].\end{aligned}$$ To first order in $\alpha_{\mu\nu}$, the variation in the metric under strain is[@park2014] $$\delta G_{\mu\nu}(\mathbf{R})
= -\epsilon^{\alpha\beta}G_{\mu\alpha}(\mathbf{R})\alpha_{\beta\nu}(\mathbf{R})
+ \mu\leftrightarrow\nu.$$ In our particular case, where we shear half the cylinder, this gives $$\alpha_{\mu\nu}(x,\,y) = \left(
\begin{array}{cc}
\delta(x)\Delta{}y & 0 \\
0 & 0
\end{array}\right).
\label{eq:mp-strain}$$ The momentum polarization expectation value $\lambda$ (c.f. Eq. ) is just the ground state expectation value $\left\langle{}U(\alpha)\right\rangle$ under this strain field.
Before we proceed to compute the required expectation values and find the momentum polarization phase, let us see how the Hall viscosity enters the calculation. We can represent the viscosity tensor in terms of the adiabatic curvature of the ground state under shear strain:[@avron_viscosity_1995; @park2014] $$\begin{aligned}
H^{\mu\nu\alpha\beta}(\mathbf{R}) &= 2 \hbar\, \mathrm{Im}
\left\langle\frac{d\Psi(\alpha)}{d\alpha_{\mu\nu}(\mathbf{R})} \Bigg|
\frac{d\Psi(\alpha)}{d\alpha_{\alpha\beta}(\mathbf{R})}\right\rangle
\\
&= -i\hbar\, \left\langle\Psi\right|
\left[\Lambda^{\mu\nu},\,\Lambda^{\alpha\beta}\right] \left|\Psi\right\rangle.\end{aligned}$$ where $\left|\Psi(\alpha)\right\rangle=U(\alpha)\left|\Psi\right\rangle$. Because the strain generator $\Lambda$ is the sum of orbital and guiding center strain generators, we conclude that the viscosity also has contributions due to each strain generator, which we separately denote $$\begin{aligned}
\eta^{\mu\nu\alpha\beta}(\mathbf{R})
&= -i\hbar\, \left\langle\Psi\right|
\left[\lambda^{\mu\nu},\,\lambda^{\alpha\beta}\right]
\left|\Psi\right\rangle \\
\widetilde{\eta}^{\mu\nu\alpha\beta}(\mathbf{R})
&= -i\hbar\, \left\langle\Psi\right|
\left[\widetilde{\lambda}^{\mu\nu},\,\widetilde{\lambda}^{\alpha\beta}\right]
\left|\Psi\right\rangle.\end{aligned}$$ Now, using the strain field in Eq. , we find $$\begin{aligned}
\lambda_{\text{RES}}
&= \left\langle\Psi\right| U(\alpha) \left|\Psi\right\rangle \\
&= \left\langle\Psi\right| \widetilde{u}(\alpha) \left|\Psi\right\rangle
\left\langle\Psi\right| u(\alpha) \left|\Psi\right\rangle,\end{aligned}$$ where $\lambda_{\text{RES}}$ is the momentum polarization phase $\lambda$ in Eq. computed with the real-space entanglement spectrum. Now, the expectation value of $\widetilde{u}$ is the momentum polarization phase computed with the orbital entanglement spectrum.[@park2014], while the expectation value of $u$ is $$\begin{aligned}
\left\langle\Psi\right| u(\alpha) \left|\Psi\right\rangle
&= \left\langle\Psi\right|
\exp\left[i \int d^2\mathbf{R}\, \alpha_{\mu\nu}(\mathbf{R}) \lambda^{\mu\nu} \right]
\left|\Psi\right\rangle \\
&= \left\langle\Psi\right|
\exp\left[i \int d^2\mathbf{R}\, \alpha_{xx}(\mathbf{R}) \lambda^{xx} \right]
\left|\Psi\right\rangle \\
&= \left\langle\Psi\right|
\exp\left[i \int d^2\mathbf{R}\, \delta(x) \Delta{}y \lambda^{xx} \right]
\left|\Psi\right\rangle \\
&= \exp\left[i \int d^2\mathbf{R}\, \delta(x) \Delta{}y
\left\langle\Psi\right|\lambda^{xx}\left|\Psi\right\rangle \right]
\\
&= \exp\left[i L_y \Delta{}y
\left\langle\Psi\right|\lambda^{xx}\left|\Psi\right\rangle \right],\end{aligned}$$ where we have kept terms only to first order in $\Delta{}y$. Using the strain generator commutation relations in Eq. , we substitute $$\left\langle\Psi\right|\lambda^{xx}\left|\Psi\right\rangle
= i\left\langle\Psi\right|\left[\lambda^{xx},\,\lambda^{xy}\right]\left|\Psi\right\rangle
= -\frac{1}{\hbar} \eta^{xxxy} = -\frac{1}{\hbar} \eta_H$$ to find $$\left\langle\Psi\right| u(\alpha) \left|\Psi\right\rangle =
\exp\left[-\frac{i}{\hbar} L_y \Delta{}y \eta_H \right].$$
Returning to our expression for the momentum polarization phase, we have $$\lambda_{\text{RES}} = \lambda_{\text{OES}} \exp\left[-\frac{i}{\hbar}
L_y \Delta{}y \eta_H \right],$$ where $\lambda_{\text{OES}}=
\left\langle\Psi\right| \widetilde{u}(\alpha)
\left|\Psi\right\rangle$ is the momentum polarization phase computed with the orbital entanglement spectrum. Hence, we can determine that an alternate form of the (orbital contribution to the) Hall viscosity is given by $$\eta_H = -\frac{\hbar}{L_y \Delta{}y} \mathrm{Arg}\,
\frac{\lambda_{\text{RES}}}{\lambda_{\text{OES}}}
\label{eq:mp-viscosity}$$for systems in uniform magnetic fields.
Now that we have introduced the two separate methods for calculating the viscosity we will apply them to two different continuum systems, and their matching lattice regularized models.
Continuum Landau Levels {#sec:landau-level}
=======================
![ \[fig:iqhe-convergence\] The Hall viscosity ($\eta_H$) of the specified integer quantum Hall states calculated by the momentum transport (upper panel) and momentum polarization (lower panel) methods. The calculation converges when , i.e. when each half of the cylinder is wider than a single wavefunction. The Hall viscosity is given in units of $\hbar\rho_0$ where $\rho_0=1/2\pi{}\ell_B^2$ is the electron density of the lowest Landau level. ](iqhe-convergence.pdf){width="50.00000%"}
Let us begin with the conventional Landau level problem of 2D electrons in a uniform magnetic field, and consider the possibility of geometric deformations similar to Ref. . The Hamiltonian for electrons in a background electromagnetic field subject to the metric of Equation is $$H = \frac{\alpha^2}{2m}{\left(\hat{p}_x+eA_x\right)}^2 +
\frac{1}{2m\alpha^2}{\left(\hat{p}_y+eA_y\right)}^2.$$ On a cylinder which is periodic in $y$ (with circumference $L_y$), and with a uniform magnetic field normal to the cylinder (in the Landau gauge, $A_x=0$ and $A_y=Bx$), $$H = \frac{\alpha^2}{2m}{\hat{p}_x}^2 +
\frac{1}{2m\alpha^2}{\left(\hat{p}_y+eB\hat{x}\right)}^2.$$ As is conventional, we define the lowering operator $$\hat{a}
= \frac{1}{\sqrt{2\hbar eB}}\left[ \alpha\hat{p}_x
- \frac{i}{\alpha}\left(\hat{p}_y+eBx\right)\right]
\label{eq:iqhe-lowering}$$ and its adjoint, $\hat{a}^\dag$. It is easy to verify that their commutator is $$\left[\hat{a},\,\hat{a}^\dag\right]=1$$ so that these are the usual ladder operators of quantum harmonic oscillator. The Hamiltonian is $$H = \hbar\omega\left(\hat{a}^\dag\hat{a} + \frac{1}{2}\right)$$ where $\omega=eB/m$ is the cyclotron frequency.
The lowest Landau level wavefunction satisfies $$\hat{a} \, \phi_{k,\alpha}^{(0)}=0$$where we are using $p_y=\hbar k.$ The raising operator $\hat{a}^\dag$ generates the higher Landau levels, $$\phi_{k,\alpha}^{(n)}
= \frac{1}{\sqrt{n!}} \left(\hat{a}^\dag\right)^n
\phi_{k,\alpha}^{(0)}.$$ The general formula for the wavefunctions for the $n$-th Landau level is $$\phi^{(n)}_{k,\alpha}(x,\,y) =
\frac{\exp{\left[iky -\frac{{\left(x+k\ell_B^2\right)}^2}{2\alpha^2\ell_B^2}\right]}}
{{\left(2^n n! \alpha \ell_B L_y \sqrt{\pi}\right)}^{\frac{1}{2}}}
H_n\left(\frac{x+k\ell_B^2}{\alpha \ell_B}\right),
\label{eq:iqhe-lll-wavefunction}$$ with $k=2\pi{}n/L_y$ for $n\in{}\mathbb{Z},$ and where the magnetic length $\ell_{B}^2=\tfrac{\hbar}{eB}.$ $H_n$ is the $n$-th Hermite polynomial. When $\alpha=1$, i.e. in the absence of any metric deformation, the wavefunctions assume their well-known isotropic form.
Let us now present the calculations for the Hall viscosity using the two methods we presented in the previous section. To calculate the Hall viscosity by the momentum transport method at a filling $\nu$, we need only compute the derivative (with respect to $\alpha^2$, c.f. Eq. ) of $$\begin{aligned}
\left\langle{}P_y\mathcal{P}_R\right\rangle
&= \sum_{n=0}^{\nu-1} \sum_{k=-K}^K \hbar k \, C_{k,\alpha}^{(n)}
\label{eq:iqhe-momentum-projection} \\
\text{where } C_{k,\alpha}^{(n)}
&= \int_0^\infty dx \int_0^{L_y} dy
\left|\phi_{k,\alpha}^{(n)}\left(x,\,y\right)\right|^2.
\label{eq:iqhe-projection}\end{aligned}$$ We note two things: (i) $C_{k,\alpha}^{(n)}$ is just the probability of finding a particle on the right ($x>0$) half of the cylinder, given that the particle is in the state $\phi_{k,\alpha}^{(n)},$ and (ii) these quantities match the correlation-function eigenvalues $C_{k,n}$ if one calculates the entanglement spectrum of this system by cutting the cylinder at $x=0.$ Thus the projections $C_{k,\alpha}^{(n)}$ of the Landau level wavefunctions onto the right half-cylinder will also used to evaluate the momentum polarization. We list their analytic forms here for the first three Landau levels:
$$\begin{aligned}
C_{k,\alpha}^{(0)}
&= \frac{1}{2} \mathrm{erfc}\left({\frac{k\ell_B}{\alpha}}\right)
\nonumber \\
C_{k,\alpha}^{(1)}
&= {\frac{k\ell_B}{\alpha}}\frac{1}{\sqrt{\pi}}e^{-\left(k\ell_B/\alpha\right)^2}
+ \frac{1}{2} \mathrm{erfc}\left({\frac{k\ell_B}{\alpha}}\right)
\nonumber \\
C_{k,\alpha}^{(2)}
&= \left[\left({\frac{k\ell_B}{\alpha}}\right)^3+\frac{1}{2}{\frac{k\ell_B}{\alpha}}\right]
\frac{1}{\sqrt{\pi}}e^{-\left(k\ell_B/\alpha\right)^2}
\nonumber \\
&\qquad + \frac{1}{2} \mathrm{erfc}\left({\frac{k\ell_B}{\alpha}}\right).
\nonumber\end{aligned}$$
Now, the range of filled $k$ states is determined by the length of the system; the last orbital is centered at . We find that the viscosity derived from the sum over $k$ converges to its expected continuum value when , i.e. when each half of the cylinder is wider than a single wavefunction. We show the result of the viscosity calculation when successively filling up to the first three Landau levels in Fig. \[fig:iqhe-convergence\]. We see that the Hall viscosity contribution from each Landau level converges to the established result [@levay_berry_1995] $$\eta_H^{(n)} = \frac{\hbar}{2\pi{}\ell_B^2}
\frac{1}{2}\left(n+\frac{1}{2}\right),
\label{eq:iqhe-viscosity}$$ which is the Hall viscosity contribution coming from the $n$-th Landau level. The convergence criterion is unsurprising given our treatment of the cylinder’s edges. The edges are not sharp, rather the edge of the cylinder is some region defined by the width of the last occupied wavefunction. If the cylinder is narrower than the width of its edges then it is no surprise that the result does not converge properly.
Using the correlation functions above, and Eq. , nearly the same results are obtained by the momentum polarization method with $$\begin{aligned}
\lambda_{\text{RES}}
&= \prod_{n,k}\frac{1}{2}\left[\left(1+e^{ik\Delta{}y}\right)
+ \left(1-e^{ik\Delta{}y}\right)\left(2C^{(n)}_{k,\alpha}-1\right)\right]
\\
\lambda_{\text{OES}}
&= \prod_{n}\prod_{k>0} e^{ik\Delta{}y} .\end{aligned}$$ The product over $n$ spans the occupied Landau levels. The Hall viscosity can be calculated with Eq. by computing $\lambda_{\text{RES}}$ and $\lambda_{\text{OES}}$ at several values of $L_y$ and extracting the quadratic fit coefficient. The result is shown in the lower panel of Fig. \[fig:iqhe-convergence\], which shows that the calculation converges when . This is the same criterion as for the convergence of the momentum transport calculation: each half of the cylinder must be wider than a single wavefunction.
Hofstadter Model {#sec:hofstadter}
================
![ \[fig:hof-spectrum\] The spectrum of the Hofstadter Hamiltonian in Eq. with $q=20$. The first three Landau levels are highlighted to illustrate the level filling scheme. ](hofstadter-spectrum.pdf)
The utility of the two methods we have introduced is that they can be easily adapted to calculate the Hall viscosity in discrete, lattice systems in magnetic fields as well. The lattice systems have discrete translation and rotation symmetries, and have an additional length scale $a,$ the lattice constant. Since there is not continuous rotation symmetry, then we can no longer appeal to the result that the viscosity is quantized in terms of the density.[@read_non-abelian_2009] Furthermore, when considering momentum transport, we must consider the fact that continuous translation symmetry is broken, and thus we are really considering the transport of quasi-momentum. Additionally, for the momentum polarization technique, there is now a minimal $\Delta{}y$, i.e. the lattice constant in the $y$-direction. In lattice systems we thus might expect that there is a maximum viscosity bound that is physically meaningful, i.e. when twisting the lattice by a single-lattice constant causes the transport of a full reciprocal lattice vector of momentum, then it is as if nothing has been transported. We will save a careful discussion of some of these issues to future work. For now we will compare the results of the two methods to see if they give matching results for the lattice viscosities, and moreover, if they both converge to the continuum limit when the magnetic length becomes much longer than the lattice scale. As an aside, we note that Ref. has also performed some viscosity calculations for the Hofstadter problem in a different context/methodology and recovers the continuum limit of the viscosity for small magnetic fields.
We begin with the Hofstadter model,[@hofstadter1976] which is the tight-binding version of the integer quantum Hall problem. The square lattice tight-binding model with rational flux $\phi=p/q$ per plaquette has a Hamiltonian $$H = \sum_{n,\,k_y} -t_x c_{n+1,k_y}^\dag c_{n,k_y} -t_y
\cos{\left(k_y-2\pi\phi n\right)} c_{n,k_y}^\dag c_{n,k_y}
+ \text{h.c.}$$ on a cylinder, where $c_{n,k_y}$ annihilates an electron in the $y$-momentum mode with wavenumber $k_y$ on the $n$-th site in the $x$-direction.
Although the Hofstadter model in the Landau gauge does not retain the fundamental translation symmetry of the lattice in the $x$ direction, it is symmetric under translation by a whole magnetic cell ($q$ unit cells). To have periodic boundary conditions in a torus geometry, we must respect this symmetry by having an integer number of magnetic cells, $N_x=lq$ for integer $l$. Constructing the cylinder from the properly periodic torus–by setting the wavefunction to zero on all the sites at one $x$ coordinate–preserves the symmetry. Thus, the cylinder has the same number of unit cells as the torus, $lq$, but it has one fewer site, $N_x=lq-1$. [@hatsugai_edge_1993] It is important to note that the consideration of the magnetic translation symmetry not only affects the construction of the lattice, but also the correct scaling of the viscosity (by the lowest Landau level density). We also recall that with commensurate boundary conditions, the spectrum of the Hofstadter Hamiltonian on a cylinder has $q$ nearly flat bands (Landau levels) consisting of $l-1$ states for each momentum mode $k_y$. In the gaps between bands are edge states (one per mode $k_y$) which connect the flat bands. We show an example of the energy spectrum with open boundary conditions and $q=20$ in Fig. \[fig:hof-spectrum\].
![\[fig:hof-viscosity-transport\] The Hall viscosity of the Hofstadter model calculated by Eq. , with $N_y=51$ and $N_x=2q-1$. The Hall viscosity of the continuum model is shown by a dashed line for comparison. ](hofstadter-viscosity-transport.pdf)
Since we are only deforming the diagonal components of the metric, we can the standard nearest-neighbor tight-binding model given above. Hence, the lattice is deformed through the hopping parameters $t_x$ and $t_y$. Absent the magnetic field, the tight-binding model bands are well-known: $$\epsilon(k_x,\,k_y) = -2t_x\cos{(k_xa_x)} - 2t_y\cos{(k_ya_y)}.$$ To see how the metric will enter, we can match parameters to the continuum Hamiltonian through a long wavelength expansion around the bottom of these bands: $$\epsilon(k_x,\,k_y) = \sum_i \left(-2t_i + \frac{t_i}{2}{\left(k_i a_i\right)}^2\right).$$ We can compare this to the Schrödinger equation: $$\epsilon = \epsilon_0 + \sum_{i,j} \frac{\hbar^2}{2m}k_i g^{ij}k_{j}$$ and then equate the coefficients of $k_i.$ Thus, we find the hopping amplitude in each direction is inversely proportional to the lattice constant: $$t_i = \frac{\hbar^2}{2ma_i^2}.$$ Therefore, under the metric deformation in Eq. , we have $t_x\propto\alpha^2$ and $t_y\propto\alpha^{-2}$. Hence, we consider a deformed Hamiltonian: $$\begin{aligned}
H = \sum_{n,\,k_y} &\left(-t_x\alpha^2 c_{n+1,k_y}^\dag c_{n,k_y} \right.\nonumber \\
&\quad-\left.\frac{t_y}{\alpha^2}
\cos{\left(k_y-2\pi\phi n\right)} c_{n,k_y}^\dag c_{n,k_y}
+ \text{h.c.}\right).
\label{eq:hof-hamiltonian}\end{aligned}$$
Now let us explicitly detail how the momentum-transport is calculated. The projected is $$\left\langle{}P_y\mathcal{P}_R\right\rangle
= \sum_{k_y}\sum_{m=1}^{\nu l} \hbar k_y
\left\langle m,\,k_y \right|\mathcal{P}_R\left| m,\,k_y \right\rangle
\label{eq:hof-projected-momentum}$$ where $\mathcal{P}_R$ projects onto the right half of the cylinder: $$\mathcal{P}_R = \sum_{x=0}^{N_x/2}
\left|x\right\rangle\left\langle{}x\right|.$$ The integers $m$ run over energy eigenstates at a given $k_y$ from $1$ (lowest energy) to a value depending on the filling. As a reminder, we point out that if the $\nu>q/2$, the edge states associated with each Landau level above the middle of the spectrum are actually below the flat Landau level, rather than above, so one would need to be careful when choosing which states are filled if the viscosity of those Landau levels is of interest. The filling scheme for the first few Landau levels is illustrated in Fig. \[fig:hof-spectrum\]. We also note that, for our calculations, the site at which the cylinder is cut should fall on the boundary between magnetic cells, i.e. on a site $n=rq$ for $r\in{}\mathbb{Z}$. We need to impose this condition so that the subsystems have commensurate boundary conditions, thus ensuring that the edge states of each half cylinder are the same as the physical edge states of the whole system. [@hatsugai_edge_1993]
The viscosity can be computed directly from the eigenstates of the Hamiltonian using Eqs. and . At large $q$, the magnetic field is weak, and the magnetic length is much larger than the lattice spacing. In this regime we expect that the Hall viscosity should approach the continuum model. In fact, as in Fig. \[fig:hof-viscosity-transport\], we see that it does converge to the continuum result for the fillings we tested. As one increases the magnetic field, the effects of the lattice will become more prominent. This figure also indicates that lattice effects more strongly affect higher Landau levels since the convergence to the continuum limit is slower. Eventually, as the magnetic field strengthens, i.e. as $q\to 0$, the viscosity begins to depend on the lattice scale. From our results in the continuum we expect that, when divided by the density, the viscosity should be a constant, independent of $q.$ Instead we find that the viscosity has contributions that depend on $q$: $$\begin{aligned}
\frac{2\pi\ell_{B}^2}{\hbar} \eta_H^{(1)}
&\sim 0.2499 + \frac{0.0017}{\sqrt{q}} + \frac{0.3865}{q} \\
&= 0.2499 + \frac{0.0017}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{0.3865}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_{B}^2}{\hbar} \eta_H^{(2)}
&\sim 1.0042 - \frac{0.1513}{\sqrt{q}} + \frac{4.3204}{q} \\
&= 1.0042 + \frac{0.1513}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{4.3204}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_{B}^2}{\hbar} \eta_H^{(3)}
&\sim 2.2289 - \frac{0.5938}{\sqrt{q}} + \frac{2.2256}{q} \\
&= 2.2289 + \frac{0.5938}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{2.2256}{2\pi}\frac{a^2}{\ell_B^2},\end{aligned}$$ where we have rewritten the $q$ dependence in terms of the relevant length scales using the fact that $qa^2=2\pi{}\ell_B^2$. We find that the viscosity is unchanged under $B\rightarrow{}-B$, including the $q$-dependent contributions.
The calculations in Fig. \[fig:hof-viscosity-transport\] are performed using derivatives with respect to the metric deformation $\alpha$, but evaluated at the isotropic point $\alpha=1.$ For comparison, in Fig. \[fig:hof-viscosity-aniso\] we fix $q$ large enough $(q=120, 180)$ so that the viscosities for the first three Landau levels have (nearly) saturated at the continuum limit, and then evaluate the momentum transport at different values of $\alpha.$ That is, we see how deforming around an initially anisotropic system affects the calculation. For values of $\alpha\neq 1$ the system only has $180^{\circ}$-rotation symmetry, and is quasi-1D for large deviations. Fig. \[fig:hof-viscosity-aniso\] shows that the viscosity of the Hofstadter model varies as a function of $\alpha$ itself; in comparison, the viscosity of the continuum Landau level is constant as $\alpha$ is varied. If the system is anisotropic we would expect the Hall viscosity to be controlled by more than one coefficient, for example $\eta_{H}^{1112}\neq\eta_{H}^{1222}$ or $\eta_H^{1122}\neq{}0.$ Helpfully, because of the $\alpha\to \alpha^{-1}$ symmetry of the metric when we switch $x$ and $y,$ we should be able to read off both viscosity coefficients from the same figure if we consider both $\alpha$ and $\alpha^{-1}$ simultaneously. However, we do not expect $\eta_H^{1122}$ to enter the momentum transport calculation, and so we cannot extract that coefficient from this figure. Finally, we note that if we directly compare the results at $q=120$ and $q=180$ as shown in Fig. \[fig:hof-viscosity-aniso\], we find that dependence of the Hall viscosity on the anisotropy is weaker for larger $q$, which might be expected since lattice effects will naturally be less important when $\ell_B\gg a.$ In future work, it would be interesting to see if any remnants of lattice anisotropy might survive to affect the thermodynamic limit.
![\[fig:hof-viscosity-aniso\] The Hall viscosity of the Hofstadter model at $q=120$ (red) and $q=180$ (blue) as a function of $\alpha$ in comparison to the continuum quantum Hall model (dashed gray) for filling $\nu=1,2,3$. The system is isotropic when $\alpha=1$ and the unit cells are elongated in the $y$ direction when $\alpha>1$. ](hofstadter-viscosity-aniso.pdf)
![\[fig:hof-polar-viscosity\] The Hall viscosity of the Hofstadter model calculated by Eqs. , , with $N_x=2q-1$. The Hall viscosity of the continuum model is shown by a dashed line for comparison. ](hofstadter-polar-viscosity.pdf)
To close this section, let us compare these results with those obtained from the momentum polarization method. Note that in Eq. , the factor $$C^{(m)}_{k_y,\alpha}
= \left\langle m,\,k_y \right|\mathcal{P}_R\left| m,\,k_y \right\rangle$$ is just the aforementioned correlation function. From Eq. , we can use the correlation function to compute the momentum polarization phases: $$\begin{aligned}
\lambda_{\text{RES}}
&= \prod_{m,k_y} \frac{1}{2}
\left[\left(1+e^{ik_y\Delta{}y}\right)\right.
\nonumber \\
&\qquad + \left.\left(1-e^{ik_y\Delta{}y}\right)
\left(2\left\langle m,\,k_y \right|\mathcal{P}_R\left|
m,\,k_y \right\rangle-1\right)\right]
\nonumber \\
\lambda_{\text{OES}}
&= \prod_{m,k_y} \frac{1}{2}
\left[\left(1+e^{ik_y\Delta{}y}\right)\right.
\nonumber \\
&\qquad + \left.\left(1-e^{ik_y\Delta{}y}\right)
\left(2\Theta(\left\langle m,\,k_y \right|\hat{x}\left|
m,\,k_y \right\rangle)-1\right)\right],
\label{eq:hof-polar}\end{aligned}$$ where $\Theta$ is the Heaviside step function, and $\hat{x}$ is the $x$-coordinate operator. On a lattice, $\Delta{}y$ must be an integer in units of the lattice constant. The resulting viscosity calculation is shown in Fig. \[fig:hof-polar-viscosity\]. The Hall viscosity obtained for the first three Landau levels agrees with the continuum value in the weak field limit. Where the calculation converges, i.e. $q\gtrsim 20,$ it agrees qualitatively with the momentum transport method, although the momentum polarization calculation appears to deviate less from the continuum Hall viscosity at small $q$.
Continuum Dirac Landau Levels {#sec:dirac-landau-level}
=============================
For our second set of examples we focus on 2D Dirac fermions in a magnetic field. The quantum Hall effect in this type of system became fundamentally important with the rise of graphene,[@neto2009] and more recently has become relevant in the study of 3D topological insulators with low-energy surface fermions of Dirac nature.[@hasan2010] We can describe the Landau level problem in the (massive) Dirac Hamiltonian under shear strain with the Hamiltonian $$H = \alpha\left(\hat{p}_x+eA_x\right)\sigma^x
+ \alpha^{-1}\left(\hat{p}_y+eA_y\right)\sigma^y
+ m \sigma^z
\label{eq:dirac-hamiltonian}$$ where we have again chosen the $\alpha$-dependent metric/frame in Eq. , and $\sigma^a$ are the usual Pauli matrices. As above for the Schrödinger equation, let us consider a cylinder which is periodic in the $y$-direction with circumference $L_y,$ and in the Landau gauge where $A_x=0$ and $A_y=Bx$ for a uniform magnetic field normal to the surface. Again the Hamiltonian can be written in terms of the raising and lowering operators : $$H = \left(
\begin{array}{cc}m & \sqrt{2\hbar{}eB}\hat{a} \\
\sqrt{2\hbar{}eB}\hat{a}^\dag & -m\end{array}
\right).$$ The Landau level wavefunctions are, up to normalization, $$\begin{aligned}
\psi^{(0)}
& = \left(\begin{array}{c} 0 \\ \phi_{k,\alpha}^{(0)}\end{array}\right)
\nonumber
\\
\psi^{\pm(n)}
&= \frac{1}{\sqrt{n+{p_{\pm n}(\gamma)}^2}}
\left(\begin{array}{c}
p_{\pm n}(\gamma) \phi_{k,\alpha}^{(n)}
\\
\sqrt{n} \phi_{k,\alpha}^{(n+1)}\end{array}\right)
\nonumber\end{aligned}$$ where $\gamma=m/\sqrt{2\hbar{}eB}$ is the ratio of the two energy scales in the problem, $\phi_{k,\alpha}^{(n)}$ are the Schrödinger Landau level wavefunctions wavefunctions , $p_y=\hbar k,$ and we have denoted for convenience $p_{\pm n}(\gamma)=\gamma\pm\sqrt{\gamma^2+n}$. The energies of each Landau level are $$\begin{aligned}
E_0 & = -\gamma \sqrt{2\hbar{}eB} =-m\nonumber \\
E_{\pm n} &= \pm \sqrt{2\hbar{}eB}\sqrt{\gamma^2+n}. \nonumber\end{aligned}$$
Now that we have the Landau-level wavefunctions, we can calculate the viscosity using Eq. , i.e. by differentiating $$\begin{aligned}
\left\langle{}P_y\mathcal{P}_R\right\rangle^{(0)}
&= \sum_{k=-K}^K \hbar k \, C_{k,\alpha}^{(0)} \nonumber \\
\left\langle{}P_y\mathcal{P}_R\right\rangle^{(n \neq 0)}
&= \sum_{k=-K}^K \hbar k \,
\frac{nC_{k,\alpha}^{(n)}+{p_{\pm n}(\gamma)}^2C_{k,\alpha}^{(n-1)}}{n+{p_{\pm n}(\gamma)}^2}
\label{eq:iqhe-dirac-momentum-projection}\end{aligned}$$ with respect to $\alpha^2$. Recall that $C_{k,\alpha}^{(n)}$ is defined in Eq. , and matches our earlier results since the Dirac Landau-levels are constructed from the Schrödinger Landau-levels.
![ \[fig:iqhe-dirac-transport-convergence\] The Hall viscosity ($\eta_H$) of the specified integer quantum Hall states of the Dirac model calculated by the momentum transport method. As in Fig. \[fig:iqhe-convergence\], the calculation converges when . The derivatives in Eq. were taken numerically with $\left\langle{}P_y\mathcal{P}_R\right\rangle$ given by Eq. . The Hall viscosity is given in units of $\hbar\rho_0$ where $\rho_0=1/2\pi{}\ell_B^2$ is the electron density of the lowest Landau level. ](iqhe-dirac-transport-convergence.pdf){width="50.00000%"}
Because of the connection between the Dirac and Schrödinger Landau-levels we conclude that the Hall viscosity of each Landau level in the continuum Dirac system is given by $$\eta_{H,D}^{(n)} =
\begin{cases}
\eta_{H,S}^{(0)}
& n = 0
\\ & \\
\frac{ n\eta_{H,S}^{(n)}+{p_{\pm n}(\gamma)}^2\eta_{H,S}^{(n-1)}}
{n+{p_{\pm n}(\gamma)}^2}
& n \neq 0
\end{cases}
\label{eq:iqhe-dirac-viscosity}$$ where $\eta_{H,S}^{(n)}$ is the Hall viscosity of the $n$th Landau level of the continuum Schrödinger equation as given in Eq. . In the massless limit, when $m=\gamma=0$ so that $p_{\pm n}=\pm\sqrt{n},$ we find $$\eta_{H,D}^{(n)} =
\begin{cases}
\hbar / \left(8\pi \ell_B^2\right)
& n = 0
\\
\hbar \left|n\right| / \left(4\pi \ell_B^2\right)
& n \neq 0.
\end{cases}$$ This result is in agreement with previous work by @kimura_hall_2010 based on an adiabatic curvature calculation, except for the $n=0$ level for which we have found a value twice as large. We attribute the difference to a probable error in the normalization of the zeroth Landau level in Ref. . We confirm the results numerically in Fig. \[fig:iqhe-dirac-transport-convergence\] using the momentum transport method of calculating the Hall viscosity. Because the Hall viscosity of the Dirac Landau levels is expressed in terms of the Hall viscosity of the Schrödinger Landau levels, the convergence criterion is expected to be the same. Indeed, we find the result converges to the expected value when . The same result is obtained by the momentum polarization method, with similar convergence criteria, though we do not show the figure here.
Let us now test if the Dirac calculation reproduces the Schrödinger result in the large mass limit. Thus, we will consider the $\gamma\rightarrow\pm\infty$ limit. In either limit, $$\begin{aligned}
p_{+n}(\gamma) &\approx 2\gamma + \frac{n}{2\gamma^2} \nonumber \\
p_{-n}(\gamma) &\approx \frac{n}{2\gamma^2} \nonumber\end{aligned}$$ and the resulting wavefunctions are $$\begin{aligned}
\psi^{+(n)}
&\approx \left(\begin{array}{c}0 \\ \phi^{(n+1)}_{k,\alpha}\end{array}\right)
\nonumber \\
\psi^{-(n)}
&\approx \left(\begin{array}{c}\phi^{(n)}_{k,\alpha} \\ 0\end{array}\right).
\nonumber\end{aligned}$$ The limiting values of the wavefunctions can easily be determined by considering the order, with respect to $\gamma,$ of each component of the spinors. Additionally, the $\psi^{(0)}$ wavefunction is completely unmodified in this limit. From this result we can conclude immediately that, in the infinite mass limit, the Dirac Landau levels carry the same set of values of the viscosity as Schrödinger Landau levels, though we still need to see how they are organized. Additionally, in this limit the energy eigenvalues are $$\begin{aligned}
E_0 &= -m \nonumber \\
E_{\pm n}
&\approx \pm\sqrt{2\hbar eB}\left(\gamma+\frac{n}{2\gamma}\right)
= \pm |E_0| \pm\hbar\omega n \nonumber\end{aligned}$$ with $\omega=\left|eB/m\right|$ the usual cyclotron frequency. The spectrum has a gap of width $2|E_0|$ with Landau levels above and below separated from neighboring Landau levels by a gaps of uniform width $\hbar\omega$, much like the Schrödinger spectrum.
The conclusions so far hold generically in the $\gamma\rightarrow\pm\infty$ limits. Let us now consider each limit independently, and furthermore, let us consider taking each limit by fixing $B$ and sending $m\rightarrow\pm\infty$, respectively. In either case, the wavefunction of the $n=0$ Landau level is essentially unchanged from the Schrödinger system. When $m\rightarrow\infty$, the $n=0$ Landau level sits at the top of the valence ($E<0$) band, separated from the $n>0$ Landau levels by the (large) mass gap. On the other hand, when $m\rightarrow-\infty$, the $n=0$ band sits at the bottom of the conduction ($E>0$) band with only the cyclotron gap separating it from the $n>0$ states. It is this configuration, when $m\rightarrow-\infty,$ and with the $E<0$ states filled, which more precisely matches the Schrödinger case. This should not be surprising; the $m\sigma^z$ term of the Hamiltonian attaches a positive mass to the $n=0$ Landau level when $m<0$. Thus, we see that the massive Dirac case matches the Schrodinger case if one focuses on the positive energy levels when $m\to -\infty.$
Now that we have discussed some properties of the continuum Dirac model, let us consider a lattice version.
Lattice Dirac Model {#sec:chern}
===================
Let us consider a lattice regularization of the continuum Dirac model. Despite the fact that the Dirac Landau-level spectrum is celebrated because of its application in graphene, we will not consider such a honeycomb lattice model. The reason is that they honeycomb model presents extra difficulties. For example, there are not only multiple Dirac cones, but the cones are located away from the $\Gamma$-point in the Brillouin zone. The latter issue leads to results which are not easily comparable with the Dirac viscosity calculation in the continuum limit. We have performed cursory calculations on such a system, but we will leave the discussion of lattice viscosity calculations when the low-energy states are near generic points in the Brillouin zone to future work. Instead we will consider a simpler lattice model for a Dirac fermion on a square lattice. When the metric deformation of Eq. is included, the lattice Dirac model Hamiltonian (on a square lattice in a cylinder geometry periodic in the $y$ direction, with rational flux $\phi=p/q$ per plaquette) is $$\begin{aligned}
H = \sum_{n,k_y} &\frac{1}{2}\left(
ic_{n+1,k_y}^\dag\alpha\sigma^xc_{n,k_y}
- c_{n+1,k_y}^\dag\sigma^zc_{n,k_y}
+ \text{h.c.}\right) \nonumber \\
&\quad+ c_{n,k_y}^\dag\sin{(k_y-2\pi\phi n)}\alpha^{-1}\sigma^yc_{n,k_y}
\nonumber \\
&\quad+ c_{n,k_y}^\dag\left[2 - m - \cos{(k_y-2\pi\phi n)}\right]\sigma^zc_{n,k_y},
\label{eq:chern-hamiltonian}\end{aligned}$$ where $c_{n,k_y}$ is a two-component annihilation operator. This model has a single gapless Dirac cone when $m=0$ or $m=4.$ For $m=0$ ($m=4$) the Dirac cone is located near ${\bf{k}}=(0,0) $ (${\bf{k}}=(\pi,\pi))$ in the Brillouin zone. Like the Hofstadter model, since we have included a magnetic field, the system geometry is chosen to preserve the magnetic translation symmetry with an integer number of magnetic cells that are $q$ sites wide in the $x$ direction. Since we are using a cylinder geometry the lattice should have $N_x=lq-1$ so that the boundaries are commensurate. [@hatsugai_edge_1993] The Landau levels at $m=0$ and $m=4$ are shown in Fig \[fig:chern-spectrum\]. We immediately recognize the similarities in the two cases, but should point out a major difference, i.e. that the edge states for $m=0$ ($m=4$) are located near $k_y=0$ ($k_y=\pi$). Below we will discuss how this difference affects the results for these two cases.
![\[fig:chern-viscosity-transport-1\] The Hall viscosity of the lattice Dirac Hamiltonian , calculated by Eq. with $N_y=51$ and $N_x=2q-1$. The indicated individual Landau levels are filled. The dotted grey line indicates the Hall viscosity of the continuum Dirac Landau level given by Eq. . ](chern-viscosity-transport-1.pdf)
![\[fig:chern-viscosity-transport-thru\] The Hall viscosity of the lattice Dirac Hamiltonian , calculated by Eq. with $N_y=51$ and $N_x=2q-1$. The Landau levels are filled from the bottom of the spectrum through the indicated level. Because filling this way causes the number of filled Landau levels to vary with $q$, a linear term $0.011q$ has been subtracted from each series. ](chern-transport-thru-viscosity.pdf)
The viscosity can again be calculated by projecting the total momentum onto the right half of the cylinder, as in Eq. , and then differentiating with respect to $\alpha^2$. The momentum projection is $$\left\langle P_y \mathcal{P}_R \right\rangle
= \sum_{k_y} \sum_{j\,\text{occ.}} \hbar k_y
\left\langle j,\,k_y \right| \mathcal{P}_R \left| j,\,k_y \right\rangle$$ where $\mathcal{P}_R$ projects onto the right half of the cylinder: $$\mathcal{P}_R
= \sum_{x=0}^{\frac{N_x}{2}} \sum_{\sigma=\pm\frac{1}{2}}
\left|x,\sigma\right\rangle\left\langle{}x,\sigma\right|.$$ The integers $j$ run over the occupied energy eigenstates at a given $k_y.$ For most of the cases we consider we only fill the Landau levels near half-filling. We note that near half-filling the $n$-th Landau level consists of the states (see Fig. \[fig:chern-spectrum\]) $$j \in
\begin{cases}
\left(N_x + nl ,\, N_x + (n + 1)l\right] & m = 0 \\
\left(N_x + (n-1)l ,\, N_x + nl\right] & m = 4.
\end{cases}$$ Notice that the $0$-th Landau level moves from the bottom of the conduction band at $m=0$ to the top of the valence band at $m=4$. This is clearly shown in Fig. \[fig:chern-spectrum\].
![\[fig:chern-viscosity-transport-0\] The Hall viscosity of the lattice Dirac model Hamiltonian , calculated by Equation with $N_y=51$ and $N_x=2q-1$. The $n=0$ Landau level Hall viscosity is plotted in each case. The $m=4$ Hall viscosity is plotted twice. The unshifted plot shows the viscosity when the Brillouin zone is unshifted so that the Dirac point is at $k=\pm\pi$. In the shifted plot, the Brillouin zone has been shifted so that the Dirac point is once again at $k=0$, in which case the Hall viscosity agrees exactly with $m=0$. ](chern-viscosity-transport-0.pdf)
The Hall viscosity of the lattice Dirac model was calculated at $m=0$ by the momentum transport method in Eq. to obtain the results in Fig. \[fig:chern-viscosity-transport-1\]. The values here represent the viscosity calculations from individually filling (not successively filling) the $n=0,\pm 1,$ and $\pm 2$ Landau levels (where $n=0$ is referenced to the zeroth Landau level of the Dirac point, not the bottom of the entire bandwidth). To help illustrate, we have shown which Landau levels were filled in Fig. \[fig:chern-spectrum\]. We note that the lattice calculation converges to the continuum value in the large-$q$ (weak magnetic field) limit, i.e. the Hall viscosity of the lattice system approaches the continuum value in the limit where the magnetic length $\ell_B$ is much larger than the spacing between unit cells. As the magnetic field strength increases, so does the effect of the lattice, with the viscosity taking on $q$-dependent terms: $$\begin{aligned}
\frac{2\pi\ell_B^2}{\hbar} \eta_H^{(-2)}
&\sim 0.9868 + \frac{0.4276}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{12.0267}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_B^2}{\hbar} \eta_H^{(-1)}
&\sim 0.5018 + \frac{0.0576}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{1.7067}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_B^2}{\hbar} \eta_H^{(0)}
&\sim 0.2498 + \frac{0.0045}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{0.8290}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_B^2}{\hbar} \eta_H^{(1)}
&\sim 0.5025 + \frac{0.0857}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{1.3075}{2\pi}\frac{a^2}{\ell_B^2} \\
\frac{2\pi\ell_B^2}{\hbar} \eta_H^{(2)}
&\sim 0.9802 + \frac{0.6299}{\sqrt{2\pi}}\frac{a}{\ell_B} + \frac{14.1092}{2\pi}\frac{a^2}{\ell_B^2},\end{aligned}$$ where we have used the relation $qa^2=2\pi{}\ell_B^2$. As predicted by the continuum calculation, the Hall viscosity converges to approximately the same value for positive and negative Landau levels.
Now let us consider what happens if we fill the Landau levels from the absolute bottom of the spectrum, instead of filling individual Landau levels near the Dirac point as was done in Fig. \[fig:chern-viscosity-transport-1\]. We show the results in Fig. \[fig:chern-viscosity-transport-thru\]. Interestingly, we found that the momentum transported scaled linearly in $q$, and we subtracted off this contribution to make Fig. \[fig:chern-viscosity-transport-thru\]. This linear scaling should be expected because the number of filled Landau levels is proportional to $q$ when filling from the bottom of the spectrum. After subtracting off the linear term $0.011q,$ as shown in Fig. \[fig:chern-viscosity-transport-thru\], we find that the momentum transported indeed saturates to a fixed value in the low-field limit. However, the saturation values do not match the continuum results. Instead, it is only the *differences* in the viscosities between filling the $n$-th and $(n+1)$-th Landau levels that exactly matches the continuum value for the viscosity of the added Landau level. This is a surprising result, as it indicates that for lattice systems in magnetic fields the magnitude of the viscosity may be somewhat regularization dependent, but the *difference* in viscosities seems to retain a more universal character. It would be interesting to see if this is a generic feature, an artifact of this model, or can be attributed to finite size effects.
Let us now consider the other massless limit of this model when $m=4.$ We show the viscosity, calculated via momentum transport, in Fig. \[fig:chern-viscosity-transport-0\]. The bare result for the viscosity shows a monotonically decreasing function that does not converge for large $q.$ However, to properly interpret this result, care must be taken to recenter the Brillouin zone. If we keep $m$ fixed, but send $k_y\rightarrow{}k_y-\pi$ in the Hamiltonian, then the momentum transport calculation exactly recovers the result at $m=0.$ If the Brillouin zone is not shifted, extra momentum is transported since the edge states are located near $k_y=\pi,$ and this leads to a different result for the viscosity. We show the $m=0$ case, as well as the shifted and unshifted results for $m=4,$ in Fig. \[fig:chern-viscosity-transport-0\]. Generically, when the low-energy Dirac point(s) are away from ${\bf{k}}=0$ in the Brillouin zone there is extra momentum transport due to the overall momentum shift of the cone. While we have been able to adjust the calculation for this simple case (and consequently any case with a single Dirac cone), the question of how to compare the viscosity of multiple Dirac points at generic momenta to the continuum limit remains a topic for future work.
![\[fig:chern-viscosity-polar\] The Hall viscosity of the lattice Dirac Hamiltonian , calculated by the momentum polarization method with $N_x=2q-1$. The indicated individual Landau levels are filled. Where points are missing it indicates a failure of the fitting required to extract the viscosity. ](chern-viscosity-polar.pdf)
The Hall viscosity of the lattice Dirac model was also calculated by the momentum polarization method as in Equation . As in the Hofstadter model, the correlation function is $$C^{(j)}_{k_y,\alpha}
= \left\langle j,\,k_y \right| \mathcal{P}_R \left| j,\,k_y \right\rangle.$$ Equation allows us to compute $\lambda_{\text{RES}}$ and $\lambda_{\text{OES}}$, $$\begin{aligned}
\lambda_{\text{RES}}
&= \prod_{j,k_y} \frac{1}{2}
\left[\left(1+e^{ik_y\Delta{}y}\right)\right.
\nonumber \\
&\qquad + \left.\left(1-e^{ik_y\Delta{}y}\right)
\left(2\left\langle j,\,k_y \right|\mathcal{P}_R\left|
j,\,k_y \right\rangle-1\right)\right]
\nonumber \\
\lambda_{\text{OES}}
&= \prod_{j,k_y} \frac{1}{2}
\left[\left(1+e^{ik_y\Delta{}y}\right)\right.
\nonumber \\
&\qquad + \left.\left(1-e^{ik_y\Delta{}y}\right)
\left(2\Theta(\left\langle j,\,k_y \right|\hat{x}\left|
j,\,k_y \right\rangle)-1\right)\right].
\nonumber\end{aligned}$$ Again, $\Theta$ is the Heaviside step function, and $\hat{x}$ is the $x$-coordinate operator. As with any lattice model, $\Delta{}y$ must be an integer in units of the lattice constant. The Hall viscosity obtained this way agrees with the momentum transport calculation, showing the same convergence to the continuum value of $\eta_H$ at large $q$ (Figure \[fig:chern-viscosity-polar\]). Note that although both methods show a deviation from the continuum Hall viscosity at small $q$, the momentum polarization method shows a smaller deviation and with opposite sign. Points are missing from these figures where the fitting required for the momentum polarization method has failed.
Discussion and Conclusion
=========================
We have applied two techniques for calculating the Hall viscosity in integer quantum Hall systems. Both methods seem to capture similar results for continuum models, and, more interestingly, both were successfully applied to lattice models. Our original momentum transport method gives results in agreement with the momentum polarization method previously described, especially in the weak magnetic field limit. While we have seen that there are lattice-scale dependent contributions to both the momentum transport and the momentum polarization at strong magnetic field, these corrections seem to be method dependent, at least for the system sizes and parameters we have chosen.
We have demonstrated that either method can determine the Hall viscosity of an isotropic system. However, further work will be required to fully characterize the Hall viscosity of anisotropic systems, though we demonstrated that two of the three non-vanishing Hall viscosity coefficients can already be computed using our methods and geometry. We found that the Hall viscosity coefficients were dependent on the amount of anisotropy, though it is unknown if this variation survives in the thermodynamic limit.
Finally, we have calculated the viscosity for the Dirac Landau-level system, and shown that there is a relationship between the Hall viscosity of continuum Schrödinger and Dirac Landau levels. Namely, that the latter recovers the viscosity of the former in the infinite mass ($m\rightarrow\infty$) limit. Our methods reproduce the known results for these models. Furthermore, we show that both the Hofstadter and lattice Dirac models approach the appropriate continuum Hall viscosity as the magnetic field $B\rightarrow{}0$, with deviations at stronger fields that depend on the lattice scale. Our results for the lattice Dirac model suggest that it may be the viscosity difference between Landau level fillings which is actually quantized (in units of density) in lattice regularized models. We uncovered some difficulties in treating lattice Dirac systems with multiple Dirac points, when they are located at generic points in the Brillouin zone; further work will be required to treat some systems of interest, such as graphene. It is also of interest to understand the competition between time reversal breaking arising from the applied magnetic field, and an intrinsic time reversal breaking coming from a massive Dirac model, i.e. a Chern insulator. The latter is also expected to have a non-vanishing, field-independent, contribution to the viscosity[@hughes2011], though it has yet to be calculated in a lattice regularization. The article Ref. addresses some aspects of this last topic.
We thank S. Ramamurthy and H. Shapourian for discussions. TLH is supported by the US National Science Foundation under grant DMR 1351895-CAR.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Observation of Bloch oscillations and Wannier-Stark localization of charge carriers is typically impossible in single-crystals, because an electric field higher than the breakdown voltage is required. In BaTiO$_3$ however, high intrinsic electric fields are present due to its ferroelectric properties. With angle-resolved photoemission we directly probe the Wannier-Stark localized surface states of the BaTiO$_3$ film-vacuum interface and show that this effect extends to thin SrTiO$_3$ overlayers. The electrons are found to be localized along the in-plane polarization direction of the BaTiO$_3$ film.'
author:
- 'Stefan Muff$^{1,2}$'
- 'Nicolas Pilet$^{2}$'
- 'Mauro Fanciulli$^{1,2}$'
- 'Andrew P. Weber$^{1,2}$'
- 'Christian Wessler$^{2}$'
- 'Zoran Ristić$^{2}$'
- 'Zhiming Wang$^{2,3}$'
- 'Nicholas C. Plumb$^{2}$'
- 'Milan Radović$^{2,4}$'
- 'J. Hugo Dil$^{1,2}$'
title: 'Observation of Wannier-Stark localization at the surface of BaTiO$_3$ films by photoemission'
---
Electric fields are the driving force of electric transport and a variety of electronic properties of semiconductor systems. The ultimate limiting mechanism of conductance in crystals is defect scattering, which prevents ballistic transport. However, in an ideal system, without defect scattering, electrons would perform an oscillating motion for large enough electric fields. These so-called Bloch oscillations form due to the Bragg scattering of the accelerated electrons at the Brillouin zone boundary [@Bloch:1929; @Zener:1934]. The oscillations eventually lead to a Wannier-Stark localization (WSL) of the accelerated electrons as well as the formation of a Wannier-Stark ladder [@Marder:2010; @Hofmann:2014]. These effects set a fundamental limit to coherent transport in crystals and their existence in a real system will provide further insight into its transport mechanisms.
The Bloch oscillation time for one cycle is given by $\tau_B=h/(eFa)$, where $h$ is the Planck constant, $e$ the electron charge, $F$ the electric field present and $a$ the lattice parameter in the direction of the electric field. In real crystals, the critical condition for Bloch oscillations to be possible is a $\tau_B$ smaller than the relaxation time $\tau$ of the system which is determined by the mean free path $\lambda$ and the Fermi velocity $v_F$. In other words the electron has to complete one (or several) periods of the Bloch oscillation before being scattered at random lattice defects. This condition can not be met by applying an external electric field on single crystal semiconductors, because the required fields are orders of magnitudes higher than the breakdown voltages of these systems. This problem was successfully addressed by the engineering of artificial semiconductor lattices of high quality. In artificial superstructures, the lattice parameter $a$ is increased and thus the required electric field is lowered to an achievable value to observe Bloch oscillations and related effects [@Esaki:1970; @Mendez:1988; @voisin:1988; @vonPlessen:1992; @Feldmann:1992; @Waschke:1993]. Alternatively, Bloch oscillations can also be induced for a short time scale using terahertz radiation [@Schubert:2014].
In this Letter we take a new approach, utilizing the electric field present in ferroelectric BaTiO$_3$ (BTO) to directly observe WSL of the two-dimensional states present at its surface by angle-resolved photoelectron spectroscopy (ARPES). Furthermore, it will be shown that this effect can be extended to thin overlayers of SrTiO$_3$ (STO) where the ferroelectric field of the BTO substrate localizes the STO surface states.
BTO is a well known ferroelectric material and closely related to the perovskites SrTiO$_3$, CaTiO$_3$ (CTO) and KTaO$_3$ (KTO). These materials are all known to host a two-dimensional electron gas at their surface [@Santander:2011; @Meevasana:2011; @King:2012; @Santander:2012; @Plumb:2014; @Rodel:2016; @Muff:2017]. Ferroelectric properties due to local lattice relaxations play, together with oxygen vacancies, a key role in the formation of the two-dimensional electron gas on the surfaces of these systems. These three perovskites are all classified as incipient ferroelectrics in which quantum fluctuations prevent a ferroelectric order [@Weaver:1959; @Muller:1979; @Zhong:1996; @Lemanov:1999]. Two-dimensional states can also be expected at the surface of BTO, which presents an excellent opportunity to study the impact of bulk ferroelectric order on these two-dimensional states at the surface and their transport properties [@urakami:2007]. It also offers a means of inducing ferroelectricity in other perovskites through doping or multilayer structure assembly. With the insights into fundamental transport mechanisms by the observation of WSL, new ways to directly manipulate and tailor transport properties of ferroelectric semiconductors become accessible.
Bulk crystalline BTO is ferroelectric below the transition temperature of 120$^{\circ}$, exhibiting three different ferroelectric phases [@Merz:1949; @Potnis:2011]. The phase diagram of BTO thin films is significantly different compared to bulk BTO. In films, the transition temperature is raised for compressive as well as tensile strain [@Li:2006]. For tensile strain, it has been demonstrated that solely an orthorhombic phase exists below the ferroelectric transition temperature [@Tenne:2004; @Li:2006; @Dionot:2014]. For compressive strain the tetragonal phase \[Fig. \[fig:pfm\_xps\](a)\] is the only ferroelectric phase present below the transition temperature, with a preferred polarization along the out-of-plane axis in films with a thickness of around of 5 unit cells (uc) [@Paul:2007; @Dionot:2014]. With increasing film thickness, strain and growth-defect relaxation will be responsible for a mixture of in-plane and out-of-plane domains of the tetragonal phase. In thicker films of BTO grown on STO, a mixture of domains with the size of around 20 nm can be observed as a result of relaxation [@Li:2006; @Dubourdieu:2013]. Furthermore, the formation of domain walls with a 90$^{\circ}$ change in polarization direction are preferred energetically to 180$^{\circ}$ domain walls [@Dionot:2015]. This also favors a mixture of domains with out-of- and in-plane polarization directions.
![(a) Tetragonal BTO uc with indicated polarization axis. (b) AFM topography of a film of 50 uc BTO grown on Nb:STO. (c) and (d) PFM phase and amplitude. Square on the left side is poled by +10V applied to the probe tip, the square on the top with -10V as indicated. (e) XPS measurements with $h\nu=170$ eV photons for normal emission (black) and an emission angle of 45$^{\circ}$ (red).[]{data-label="fig:pfm_xps"}](pfm_xps-01.png){width="50.00000%"}
The films investigated in this work were grown by pulsed laser deposition (PLD), allowing a controlled layer-by-layer growth monitored by reflective high-energy electron diffraction (RHEED). Films with a thickness of 20 uc where grown on commercially available, single-terminated SrTiO$_3$, Nb:SrTiO$_3$ and KTaO$_3$ (001) substrates (SurfaceNet GmbH, see [@SOM:BTO]). The growth was performed at a substrate temperature of 950 K, in a partial oxygen pressure of $1\cdot10^{-5}$ mbar. STO films of 3 and 5 uc thickness where grown on top of this BTO film under similar conditions (see [@SOM:BTO]). The samples were *in-situ* transferred to the high-resolution ARPES endstation and measured with circularly polarized synchrotron light. During the measurements the sample was held at 20 K and kept in ultra high vacuum (UHV) conditions better than $1\cdot 10^{-10}$ mbar. The films were *ex-situ* transferred to the NanoXAS beamline for piezo-response force microscopy (PFM) measurements at room temperature under UHV condition. The sample measured with PFM had a thickness of 50 uc and was grown on a 0.5wt% Nb doped STO substrate under the same conditions as described above. A conductive substrate was chosen in order to have a well-defined back electrode and the higher film thickness in favor of a stronger PFM response signal.
The PFM topography in Fig. \[fig:pfm\_xps\](b) shows a uniform sample surface [@SOM:BTO] according to the vicinality of the substrate, where a 0.2$^{\circ}$ miscut to the (001) surface was chosen to promote a layer-by-layer growth. The PFM phase and amplitude in Fig. \[fig:pfm\_xps\](c,d) of the as-grown sample (bottom part of the field of view) shows no noticeable contrast, indicating that no intrinsic domains of resolvable size ($\gtrsim 20$ nm) are present. After subsequent writing of the surface with +10 V and -10 V applied to the probe tip, a phase and amplitude contrast is noticeable, proving the presence of ferroelectric properties in our films. However, both of the written regions exhibit significant noise, reducing the difference of the mean phase value between the positive and negative poled region to approximately $72^{\circ}$ [@SOM:BTO]. This difference is significantly less than the $180^{\circ}$ phase difference expected for completely opposite polarized regions. The reasons for this observation is a not completely homogeneously polarized surface in the written areas. This indicates a strong locking of the domains in the in-plane direction due to interface strain and relaxation mechanisms.
{width="100.00000%"}
The X-ray photoelectron spectroscopy (XPS) measured at normal emission and at a more surface sensitive emission angle of 45$^{\circ}$ show the Ba 4d, Ti 3p and O 2s core levels \[Fig. \[fig:pfm\_xps\](e)\]. The Ba 4d core level consists of the spin-orbit split Ba 4d$_{3/2}$ and 4d$_{5/2}$ doublet and a lower intensity doublet, shifted by 1.25 eV to higher binding energies. By comparing the peak areas of the two species for the two emission angles we can assign the higher binding energy, chemically-shifted doublet to undercoordinated Ba ions in the BTO surface region [@Jacobi:1987; @Hudson:1993; @Cai:2007]. The Ti 3p core level includes two peaks assigned to the Ti 4$^+$ and Ti 3$^+$ ions whereby the latter is more surface localized. Close to the Fermi energy an in-gap state is located at a binding energy of $0.8$ eV in the bulk band gap [@Rault:2013; @SOM:BTO]. The intensity of the in-gap state and the Ti $3^+$ peak increases under UV irradiation [@SOM:BTO]. Similar effects are observed in the case of STO where the presence of Ti 3$^+$ ions and the in-gap state are explained by surface relaxation and oxygen vacancies [@Plumb:2014]. Comparing the relative Ba 4d and Ti 3p peak areas for the two emission angles, we can conclude that the surface is TiO$_2$ terminated [@Radovic:2009].
The ARPES measurements in Fig. \[fig:arpes80eV\_model\] and Fig. \[fig:STOonBTO\](a-c) show metallic states emerging from the in-gap state. Resonant effects cause strong intensity modulations, but the states show no clear dispersion as a function of photon energy, indicating their two-dimensional nature [@SOM:BTO]. As for STO, matrix element effects are responsible for the suppression of intensity at $k_x=0$ Å$^{-1}$ due to the mainly xy-symmetry of the two-dimensional state [@SOM:BTO]. These states can be attributed to the partially filled Ti 3$d_{xy}$ orbital, that is split from the Ti 3$d_{xz}$ and Ti 3$d_{yz}$ orbitals due to a distortion of the TiO$_6$ octahedron by lattice relaxation [@Santander:2011; @Plumb:2014; @Muff:2017]. However, the Fermi surface around the $\overline{\Gamma}$-points shows no clear bands of the two-dimensional states but features spectral weight, elongated along both $\overline{\Gamma \mbox{X}}$-directions. These elongated states extend over multiple surface Brillouin zones connecting the neighboring $\Gamma$-points as shown in Fig. \[fig:arpes80eV\_model\](a,d). Comparing the Fermi surface with constant energy surfaces at higher binding energies \[Fig. \[fig:arpes80eV\_model\](b,c) and (e,f)\], no dispersion of these states with respect to the binding energy is noticeable. On the other hand, at the binding energy of the in-gap state the spectral weight is smeared out equally in all directions and no pattern is discernible [@SOM:BTO]. As we will explain below, the absence of dispersion in the states around the Fermi level is due to WSL and a direct consequence of the electric field present in the bulk of the film.
The electrons in the two-dimensional state experience an accelerating force in the direction opposite to the electric field present in the ferroelectric domains. Due to the potential barrier at unit cell boundaries, the acceleration is not uniform but is described by Bloch oscillations [@Bloch:1929; @Zener:1934]. This localizes the electron in real space and hence shows smearing in reciprocal space. Considering the lattice parameter of BTO, the condition $\tau>\tau_B$ for Bloch oscillations to exist is fulfilled for an electric field $F \gtrsim 10^9$ V/m assuming a typical relaxation time of $\tau = \lambda / v_F \approx 10^{-14}$ s [@Hofmann:2014], which exceeds the breakdown field strength of known insulators. Due to its ferroelectric properties, the local electric field in the BTO film is several order of magnitudes higher than any possible external electric field. An estimate of the electric field inside BTO films can be obtained from the spontaneous polarization which is $P\approx0.25~C/m^2$ for tetragonal bulk BTO [@Merz:1953] and is predicted to increase for strained films [@ederer:2005]. The resulting local electric field of $1\cdot10^{10}$ V/m in ferroelectric BTO is in the order of the electric field required to meet the condition for Bloch oscillations [@SOM:BTO]. Bloch oscillations are typically accompanied by the formation of a Wannier-Stark ladder, a set of electronic states separated in energy and space [@SOM:BTO]. Due to the surface localization of the Bloch oscillations in the BTO films, the observation of the WS ladder is beyond the compatibility of conventional techniques [@Esaki:1970; @Mendez:1988; @voisin:1988; @vonPlessen:1992; @Feldmann:1992; @Waschke:1993]. For such systems the WSL induced smearing observed by ARPES provides an alternative method to observe these effects.
{width="100.00000%"}
The BTO films grown on a STO substrate have a compressive strain of 2 % at the interface. As a result, the film is expected to stay in a single tetragonal phase below the ferroelectric phase transition [@Paul:2007; @Dionot:2014]. Tetragonal BTO can host a ferroelectric polarization along the $\left\langle 001 \right\rangle$ out-of-plane as well as the $\left\langle 100 \right\rangle$ and $\left\langle 010 \right\rangle$ in-plane directions. While at the interface the polarization direction is preferably along the out-of-plane axis, strain relaxation mediated by growth defects will be responsible for a mixture of domains close to the film surface. The domains with different electric field directions will all contribute differently to the Fermi surface. The in-plane domains, exhibiting an electric field along $\left\langle 100 \right\rangle$ \[Fig. \[fig:arpes80eV\_model\](g)\] or $\left\langle 010 \right\rangle$ \[Fig. \[fig:arpes80eV\_model\](h)\] directions will both give rise to WSL. In ARPES, this WSL becomes visible as one-dimensional states along the $\overline{\Gamma \mbox{X}}$-directions. In domains where the electric field is along the out-of-plane or $\left\langle 001 \right\rangle$ directions \[Fig. \[fig:arpes80eV\_model\](j)\], the electric field will lift the spin-degeneracy of the two-dimensional states. The resulting Rashba-type spin splitting consists of oppositely spin-polarized, concentric rings at the Fermi surface [@Dil:2009R; @Santander:2014; @KrempaskyPRB:2016]. The direction of the spin polarization of the bands will be inverted depending on the sign of the ferroelectric polarization vector.
With a domain size on the order of 20 nm [@Dubourdieu:2013], the synchrotron beam with a size of around 100 $\mu$m will average over several domains with different ferroelectric polarization directions. The resulting model Fermi surface in Fig. \[fig:arpes80eV\_model\](k), formed by an overlay of the contributions from the different domains is in good agreement with the ARPES measurements, especially if further modulation of the ARPES signal by matrix element effects are considered.
For BTO films grown on KTO the compressive strain is reduced to 0.2 % due to the larger lattice constant of KTO compared to STO. With the change in strain also the domain formation is expected to be different for the BTO films on KTO. Furthermore, our KTO substrates have a higher step density as our STO substrates inducing an imbalance between different domains [@SOM:BTO]. When comparing the data of the BTO film on KTO \[Fig. \[fig:arpes80eV\_model\](d-f)\] with the results of the film grown on STO \[Fig. \[fig:arpes80eV\_model\](a-c)\] two differences are noticeable: i) the signature of a circular Fermi surface contribution around $\overline{\Gamma_{00}}$ and ii) the WSL states are more intense along the $k_x$- than the $k_y$-direction. Both of these observations are in agreement with an altered domain configuration. The reduced interface strain and the higher step density are responsible for the formation of larger domains with a higher fraction polarized along the z- and x-directions in the measured BTO films grown on KTO.
The general nature of the WSL is illustrated by its presence in ultrathin STO films grown on top of the BTO layers. The ARPES data for a 3 uc thick STO film in Fig. \[fig:STOonBTO\](d-f) exhibits states very similar to pure BTO Fig. \[fig:STOonBTO\](a-c). The Fermi surface shows stripes extending over several surface Brillouin zones (see [@SOM:BTO]) characteristic for WSL. However, the reduced electric field with increasing STO film thickness results in a lower intensity of the smearing and a shallow electron pocket with polaron replicas [@Moser:2013; @Wang:2016] becomes visible \[see markers in Fig. \[fig:STOonBTO\](f)\]. For the surface of the 5 uc film in Fig. \[fig:STOonBTO\](g-j) the fields of the BTO substrate are so far reduced that no indication of WSL is visible. The ARPES data resembles the electronic structure of bulk STO with the more filled circular $d_{xy}$ states forming the 2DEG and elongated $d_{xz}$ and $d_{yz}$ states that clearly disperse with binding energy (Fig.\[fig:STOonBTO\](h,j)) [@Plumb:2014].
From these results we can conclude that either up to 3 uc of STO on BTO are ferroelectric, or that about 4 uc of STO are needed to let the electric field of BTO decay to a value that no longer influences the electronic properties at the sample surface. Furthermore, the sharp electron-like states at the surface of the STO films verify the high crystalline quality also of our BTO layer. RHEED and XPS data (see [@SOM:BTO]) indicate a layer-by-layer growth of the STO, opening the possibility to study whether a ferroelectric order is induced in the STO by high resolution transition electron microscopy in future work.
To conclude, we have presented combined effects of two different physical properties on BTO film surfaces: the formation of a two-dimensional state and the Wannier-Stark localization of these state. We have further demonstrated that ARPES provides a novel means of probing WSL in reciprocal space. To our knowledge, this is the first time WSL is directly observed in a single crystalline semiconductor. The combined presence of electric fields and two-dimensional states at the surface of a transition metal oxide opens up a rich field to study the interplay of ferroelectricity and interface states. For the study of the macroscopic influence of WSL on the transport properties, BTO films with preferred polarization directions should be prepared by the help of different substrates regarding orientation, lattice parameters and conductivity [@Sinsheimer:2013; @ChenJ:2013; @Guo:2016] and under different growth conditions [@Rault:2013]. Furthermore, our experiments suggest that WSL should be a general effect for ferroelectric materials with surface or interface states, and domains with an in-plane electric field.
This work was financially supported by the Swiss National Science foundation (SNF) Project No. PP00P2\_144742/1 and Project No. 200021-159678.
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**Supplemental Material\
\
Observation of Wannier-Stark localization at the surface of BaTiO$_3$ films by photoemission**
Piezo-Response Force Microscopy
===============================
The piezo-response force microscopy (PFM) data of Fig. \[fig:pfm\] is the full data set of the measurements presented in Fig.1 (b-d) of the main text. The topography \[Fig. \[fig:pfm\](a)\] features a uniform step formation over the whole measured range with step heights \[Fig. \[fig:pfm\](b)\] corresponding to one or multiple unit cells. The position of the step edges are visible in the normal deflection \[Fig. \[fig:pfm\](c)\]. The terrace width of $0.1-0.2~\mu$m is given by the substrate misscut to the (001) surface of 0.1$^{\circ}$-0.2$^{\circ}$. The PFM amplitude and phase \[Fig. \[fig:pfm\](d,f)\] show a visible contrast between the oppositely written areas as well as the unwritten area. The as-grown sample shows no formation of domains larger than the measurement lateral resolution of $\approx$20 nm. Due to the small PFM signal of the BaTiO$_3$ (BTO) film, the measurements where performed at the PFM resonance. This is causing a small cross talk of the PFM phase and amplitude with the topography signal, responsible for the visibility of the step edges in the PFM channels.
{width="90.00000%"}
Two line profiles are taken along the scanning direction in the center of the marked, colored squares for the PFM amplitude \[Fig. \[fig:pfm\](e)\] and phase \[Fig. \[fig:pfm\](g)\] each. The signal in the vicinity of the step edges are excluded from the line profiles. In the amplitude line profiles a clear difference between the positively, the negatively, and the unwritten areas is noticeable. Especially the amplitude of the negative poled region is clearly higher than the positive and the unwritten areas. On the phase signal a clear reduction of noise is noticeable for the negatively poled area, while for the positively poled part the noise is on the level of the unpoled region (see Table \[tab:pfm\]). The phase difference between the oppositely poled area is 6$^{\circ}$ for the top row and 72$^{\circ}$ for the bottom row with an error margin larger than the difference.
The measurements show that writing with a negative potential applied to the probe tip has a more noticeable effect than writing with a positive potential. Especially the strong noise reduction for the phase and the offset in the amplitude is obvious. In the phase, the mean value of the negative written area is, as for the unwritten part, very close to 0$^{\circ}$. This indicates that a larger portion of the intrinsic domains are polarized pointing out of the plane than into the plane. Thus the writing with positive potential applied is less effective. In order to uniformly polarize the two areas by switching all the in-plane domains to the out-of-plane axis, a higher potential than 10 V would be needed. In the lateral deflection \[Fig. \[fig:pfm\](g,h)\] of the probe tips a clear contrast is noticeable between the written regions and the unpoled area. This lowering of the friction at the previously written areas could be due to the reduction of in-plane domains in these regions changing the local polarization fields.
unpoled -10 V +10 V unpoled
-------------------- ---- --------------- ---- ---- --------------- ---- ---- --------------- ---- ---- --------------- ----
Amplitude $0.16\pm0.06$ $0.16\pm0.07$ $0.27\pm0.10$ $0.15\pm0.06$
(arb.units) $0.29\pm0.11$ $0.48\pm0.13$ $0.62\pm0.15$ $0.21\pm0.08$
Phase $2\pm80$ $2\pm86$ $-4\pm29$ $-2\pm78$
(deg.) $-34\pm109$ $-65\pm111$ $7\pm12$ $-25\pm99$
Lateral Deflection $0.45\pm0.02$ $0.44\pm0.03$ $0.44\pm0.02$ $0.47\pm0.02$
(nm) $0.38\pm0.02$ $0.34\pm0.03$ $0.35\pm0.02$ $0.40\pm0.02$
: Table with the average values and corresponding standard deviation for the line profile areas in Fig. \[fig:pfm\](e,g,j).[]{data-label="tab:pfm"}
Topography of the $\mbox{SrTiO}_3$ and $\mbox{KTaO}_3$ substrates
=================================================================
The atomic force microscopy (AFM) topographies of SrTiO$_3$(001) (STO) and KTaO$_3$(001) (KTO) substrates as used for the growth of the BTO films of this study, are presented in Fig. \[fig:afm\_sub\]. These data are taken with a different device than the PFM data in Fig.\[fig:pfm\]. The STO substrate is etched to obtain a TiO$_2$ terminated surface. The etching procedure is described in the SOM of Plumb et al. [@Plumb:2014]. The measured AFM topography in Fig. \[fig:afm\_sub\](a) shows the presence of terraces with a width of approximately 210 nm. The KTO substrate is not etched and accordingly shows a mixed termination in the AFM data \[Fig. \[fig:afm\_sub\](b)\] in the form of higher (dark) patches. The observed terrace width is with approximately 120 nm roughly half the observed size of the STO substrate \[Fig. \[fig:afm\_sub\](c)\].
{width="100.00000%"}
Estimation of the Electric Field
================================
The external electric field of a ferroelectric material is given by its polarization $P$ [@FeynmanV2:1964; @Hofmann:2014]. For bulk, tetragonal BTO the polarization is reported to be $P\approx0.25~C/m^2$ [@Merz:1953] which results in and external electric field of: $$|F|=\frac{P}{(\epsilon_r-1)~\epsilon_0}\approx5\cdot10^{8}~\mbox{V/m}
\label{eq:efield3}$$
Based on this external electric field $F$, the local electric field $F_{loc}$ inside the material is given as $F_{loc}=1/3~(\epsilon_r+2)~F$ [@Hofmann:2014]. With the relation \[eq:efield3\] for the external electric field, the local electric field can be written as a function of the polarization $P$. $$|F_{loc}|=\frac{1}{3}\frac{P\cdot(\epsilon_r+2)}{(\epsilon_r-1)\cdot\epsilon_0}\approx\frac{1}{3}\frac{P}{\epsilon_0}\approx1\cdot10^{10}~\mbox{V/m}
\label{eq:efield}$$
The obtained local electric field is considered relevant for the occurrence of WSL. With a magnitude of $|F_{loc}|\approx1\cdot10^{10}~\mbox{V/m}$ the local electric field of BTO results in a Bloch oscillation time of $\tau_B=\frac{h}{eFa}\approx1\cdot10^{-15}$ s within the unit cell of BTO ($a\approx4$ Å). For a relaxation time of $\tau=10^{-14}$ s the condition $\tau>\tau_B$ is therefore satisfied in a unit cell of ferroelectric BTO and the occurrence of Bloch oscillations and Wannier-Stark localization is expected.
In general, the WSL is accompanied by the formation of a Wannier-Stark ladder, a set of electron states separated in energy and space. In superlattices, where a WSL occurs by the help of an externally applied, tunable electric field, indications of a Wannier-Stark ladder are observed [@Mendez:1988; @voisin:1988]. The energy separation between the steps of the Wannier-Stark ladder is given as $\Delta E=eFa$ [@Marder:2010; @Hofmann:2014] and expected to be between 0.1-6 eV for the films studied, based on an electric field between $5\cdot10^{8}$ V/m to $1.5\cdot10^{10}$ V/m. With the origin of the electric field in the ferroelectric properties of BTO, the local electric field is not expected to be constant due to the variable domain configurations and sizes. Therefore the energy steps of the resulting Wannier-Stark ladder are not isotropic but will vary within the probed area. Therefore, ARPES is not the method of choice to observe these ladders due to the limited coherence length of these states. Local probe techniques using tunneling or optical spectroscopy should be able to address this aspect in future work.
Supplemental ARPES Measurements on BTO
======================================
Constant energy surfaces at higher binding energies
---------------------------------------------------
{width="100.00000%"}
Further constant energy surfaces at higher binding energies, extracted from the same data set shown in Fig. 2(a-c) of the main text, are displayed in Fig. \[fig:cec\_ingap\]. In Fig. \[fig:cec\_ingap\] (b), taken at the Fermi energy we clearly see a checkerboard pattern, with strong intensity at the $\Gamma$-points and the smeared intensity along the $\overline{\Gamma\mbox{X}}$ direction, origination from the WSL of the 2D states. At 400 meV binding energy, the pattern observed at the Fermi energy is still visible, however, also spectral intensity around the $\overline{\mbox{M}}$-point, away from the $\Gamma$-points and the $\overline{\Gamma\mbox{X}}$ direction, becomes apparent. At a binding energy of 800 meV, corresponding to the energy of the in-gap state of BTO as seen in Fig. \[fig:cec\_ingap\](a) the checkerboard pattern observed at the Fermi energy is not distinguishable anymore. There is a constant intensity background with little structure except higher spectral intensities around the $\Gamma$-points due to diffraction effects. This shows that the checkerboard pattern does not originate from the in-gap states.
Measurements along $\overline{\Gamma\mbox{M}}$
----------------------------------------------
The angle-resolved photo electron spectroscopy (ARPES) data presented in the main text are all measured with the same geometry where the entrance slit of the hemispherical analyzer is aligned along the $\overline{\Gamma\mbox{X}}$-direction of the crystal. In the measurements in Fig. \[fig:arpes\_gm\] the crystal is aligned with $\overline{\Gamma\mbox{M}}$ parallel to the analyzer entrance slit \[along $\theta$ see Fig. \[fig:arpes\_gm\](a)\]. The angular scanning direction \[$\psi$ in Fig. \[fig:arpes\_gm\](a)\] is perpendicular to the alignment direction and consequently different for the two cases. Apart from changes in the relative intensities, the altering of the measurement geometry does not affect the data. In particular the WSL states are still visible, smeared along the $\overline{\Gamma\mbox{X}}$-direction. This confirms that our observations are not measurements artifact caused by the probing geometry.
{width="100.00000%"}
The data of Fig. \[fig:arpes\_gm\] are taken from a sample of 10 uc BTO deposited on a SrTiO$_3$ (STO) substrate. In comparison with the data on 20 uc BTO on STO presented in the Fig.3 of the main text, the smearing tend to be more uniform. Possible reasons for this are the combined effect of altered matrix element contributions due to the different measurement geometry and a different domain pattern as a consequence of the reduced film thickness.
Photon Energy Dependency
------------------------
{width="100.00000%"}
Fig. \[fig:arpes67eV\] depicts measured Fermi surfaces, band maps, and photon energy dependence of the two-dimensional states around $k_x=0$ Å$^{-1}$, the $\overline{\Gamma_{00}}$ point. The measured intensity at the Fermi energy as a function of $k_x$ and photon energy \[Fig. \[fig:arpes67eV\](c,e)\] shows bands, forming two parallel lines with photon energy close to $k_x=0$ Å$^{-1}$, that corresponds to $\overline{\Gamma_{00}}$. The different photon energies give access to different $k_z$. Therefore the lack of dispersion of these two parallel bands with photon energy indicates their two-dimensional (or one-dimensional) nature. The observed intensity modulation with photonenergy is given by the excitation to different available final states as well as resonant enhancements. The photon energies of $h\nu=45$ eV and $h\nu=55$ eV \[Fig. \[fig:arpes67eV\](d)\] correspond to the energies of the Ti 3p - Ti 3d and Ti 3p - Ti 4sp resonance, respectively [@Smith:1988; @Tao:2011]. The Ti 3p - 3d resonance has a sharp Fano-like lineshape [@Fano:1961] indicating a low dimensionality of the excited state [@Tao:2011]; i.e. the Ti 3d states close to the Fermi level. On the other hand, the Ti 3p - Ti 4sp resonance is much broader implying that these states, hybridized with oxygen, are more delocalized along the z-direction [@Tao:2011].
Light Polarizations
-------------------
The Fermi surfaces and corresponding band structures for right- and left hand circularly polarized light as well as for s- and p-polarized linear light are depicted in Fig. \[fig:pol\_g0\] and Fig. \[fig:pol\_g2\]. For these data, the analyzer entrance slit is aligned along the $\overline{\Gamma \mbox{X}}$-direction of the sample as in the main text. For s-polarized light the electric field of the synchrotron light is along the $k_y$-direction, for p-polarized light along the $k_x$-direction \[see Fig. \[fig:arpes\_gm\](a)\]. In the data in the vicinity $\overline{\Gamma_{00}}$ \[Fig. \[fig:pol\_g0\]\] measured with a photon energy of $h\nu=67$ eV no differences are noticeable for the two circular polarizations \[Fig. \[fig:pol\_g0\](a,b)\]. The band dispersion along the $k_x$-direction for circularly polarized light shows two features, connected to the in-gap state as discussed in the main text. Along $k_y$ the band structure only hosts a single intensity feature at $k_y=0$ Å$^{-1}$.
{width="80.00000%"}
For s-polarized light \[Fig. \[fig:pol\_g0\](c)\], the Fermi surface consist of two features elongated along the $k_x$-direction, separated by suppressed intensity at $k_x=0$ Å$^{-1}$. Accordingly two features appear in the $k_x$ dispersion, and the $k_y$ dispersion only shows enhanced intensity around $k_y=0$ Å$^{-1}$. The Fermi surface with p-polarized light \[Fig. \[fig:pol\_g0\](d)\] is similar to the one measured with s-polarized light but $90^{\circ}$ rotated, with suppressed intensity along $k_y=0$ Å$^{-1}$.
The suppressed intensity for $k_x=0$ Å$^{-1}$ with s-polarized light and for $k_y=0$ Å$^{-1}$ with p-polarized light is an indication for an xy-symmetry of the probed orbitals. In the case of the other known perovskites hosting a two-dimensional electron gas [@Santander:2011; @Meevasana:2011; @King:2012; @Santander:2012; @Plumb:2014; @Muff:2017], the two-dimensional states are attributed to the Ti 3$d_{xy}$ orbitals and the three-dimensional bands, dispersing with photon energy, with the Ti 3$d_{xz}$ and Ti 3$d_{yz}$ orbitals. It seems likely for BTO to have a similar orbital ordering. However, due to the WSL of the states at the BTO surface, the orbital symmetries of the states present cannot be conclusively assigned.
{width="80.00000%"}
The data in Fig. \[fig:pol\_g2\] of the Fermi surface at $\overline{\Gamma_{10}}$ for the different light polarizations are very similar to the data of $\overline{\Gamma_{00}}$ with respect to the light polarization effects. For right- as well as left-hand circularly polarized light \[Fig. \[fig:pol\_g2\](a,b)\] the intensity of the WSL along $\overline{\Gamma\mbox{X}}$ is visible. However, the intensity distribution along $k_y$ around the $\Gamma$-point is inverted. For the linear polarized light only the WSL states in $k_x$-direction are visible for s-polarized light, while for p-polarized light only intensity elongated in $k_y$-direction are present. In contrast to the data taken at $\overline{\Gamma_{00}}$, the suppresion of intensity at $k_x=0$ Å$^{-1}$ or $k_y=0$ Å$^{-1}$ for linear polarized light is absent due to the emission angle of $\overline{\Gamma_{10}}$ being far off normal emission. The band dispersion along the $k_y$-direction for the four different light polarizations consist of a main feature around $\overline{\Gamma_{10}}$, dispersing from the in-gap state. The band dispersions for the circular polarized lights, show an asymmetry around $k_y=0$ Å$^{-1}$ corresponding with the Fermi surface. For s-polarized light the feature is narrow in the $k_y$-direction while for p-polarized light it is broad, opposite to the band dispersions along the $k_x$-direction.
Time Dependent Behavior {#sec:bto_time}
-----------------------
The BTO films show a time dependent behaviour under UV-irradiation. In order to turn the surface conductive and avoid charging, a path is written by the UV-light starting from the mounting clamp to the center of the sample. This is an established experimental procedure for ARPES measurements of the 2D states of titanates surfaces (see SOM [@Plumb:2014]).
{width="75.00000%"}
Under irradiation, the spectral intensity of the in-gap state at 0.8 eV binding energy and of the Ti $3^+$ shoulder of the Ti 3p core level is increasing with time \[see Fig. \[fig:time\]\]. In case of the Ti $3^+$, its percentage on the total Ti 3p peak area rises from 7% to 18% within one hour \[Fig. \[fig:time\](a)\]. This scales to a free charge carrier density at the surface of 0.18 electrons per uc after one hour, when saturation is reached. Within the same time frame, the intensity of the in-gap state rises by 300% \[Fig. \[fig:time\](b,c)\]. However, the metallic state, visible as a second peak at the Fermi energy \[Fig. \[fig:time\](c)\], only changes by 0.3% in peak area within the same time frame. Thus although the intensity of Ti $3^+$ and the in-gap state seem to be related, the intensity of the metallic states does not directly scale. The surface localized Ti $3^+$ ions are linked to the creation of oxygen vacancies and structural reordering of the surface layers in the titanium based perovskites [@Plumb:2014]. The changes implied in the distortion of the TiO$_6$ octahedra and their respective binding angles due to the reordering will alter the hybridization of titanium and oxygen. Indications of this change in hybridization are observable in the altering peak intensity of the valence band with time. The observed changes under UV-light saturate within 30 min and are persistent with time regardless if the area is further irradiated or not.
Further Characterization of STO thin Films
==========================================
RHEED pattern and Oscillations
------------------------------
The growth process with pulsed laser deposition (PLD), was monitored by reflective high-energy electron diffraction (RHEED) patterns and oscillations. The RHEED pattern and oscillations of a film of 20 uc BTO grown on a STO substrate are depicted in Fig. \[fig:supl\_rheed\](a,d). The RHEED pattern was obtained after the growth and indicates a crystalline two-dimensional surface. Each maxima of the RHEED oscillation corresponds to the formation of a complete BTO layer and therefore allows a precise thickness control of the film while growing. The RHEED pattern of the 3 uc \[Fig. \[fig:supl\_rheed\](b)\] and 5 uc \[Fig. \[fig:supl\_rheed\](c)\] film of STO deposited on a previously grown BTO film of 20 uc shows a good crystalline surface. By the help of the RHEED oscillations of the STO thin film growth \[Fig. \[fig:supl\_rheed\](e)\], a precise termination of the growth process is possible at the oscillation maxima.
{width="70.00000%"}
XPS Measurements
----------------
In Fig. \[fig:xps\_comp\] a comparison of XPS spectra is shown for the clean BTO films and 3 and 5 uc of STO grown on top. The data were normalised to the background after the O 2s core level and the BTO data was offset in Fig. \[fig:xps\_comp\](a) for clarity. As expected the Sr core levels increase with STO coverage whereas the Ba core levels show an exponential decay with coverage and are almost completely suppressed for the 5 uc thick STO film. This indicates a layer-by-layer growth of a closed STO film on top of the BTO substrate.
The small changes of binding energies in the Ba 4d core levels could give insight in the detailed atomic structure of the BTO/STO interface and possible intermixing in the first unit cell. However, this goes far beyond the scope of this work and is best combined with detailed structural investigations.
{width="80.00000%"}
ARPES Measurements
------------------
In Fig. 3(a) of the main text a subsection of the Fermi surface for a 3 uc thick film of STO grown on a 20 uc film of BTO was shown. Although the WSL states are readily discernible their extension becomes more clearly visible in the large range Fermi surface map in Fig. \[fig:arpes\_sto\](a).
{width="80.00000%"}
In Fig \[fig:arpes\_sto\](b-g) a comparison between pure BTO, 3 uc of STO on BTO, and 5 uc of STO on BTO is shown for a different photon energy as in Fig. 3 of the main text. Although the intensity ratio between the 3$d_{xy}$ and 3$d_{xz}$ (or 3$d_{yz}$) bands has changed for the 5 uc data, the general features are independent of photon energy.
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---
abstract: 'Let $R\to S$ be a local ring homomorphism and $N$ a finitely generated $S$-module. We prove that if the Gorenstein injective dimension of $N$ over $R$ is finite, then it equals the depth of $R$.'
address:
- 'Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A.'
- 'Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China'
author:
- Lars Winther Christensen
- Dejun Wu
date: 30 April 2019
title: A Bass equality for Gorenstein injective dimension of modules finite over homomorphisms
---
[^1]
Introduction {#sec:Introduction .unnumbered}
============
The homological theory of modules over commutative noetherian rings comes out particularly elegant for finitely generated modules. One way to relax this finiteness condition—without sacrificing elegance—is to settle for finite generation over some noetherian, but otherwise arbitrary, extension ring. This theme has been systematically explored for at least fifteen years. As part of that effort, this short paper answers an open question in Gorenstein homological algebra.
In this paper a ring means a commutative noetherian ring. Let $R$ and $S$ be local rings with unique maximal ideals ${\mathfrak{m}}$ and ${\mathfrak{n}}$, respectively. A ring homomorphism, $${\nobreak{{\varphi}\colon R\:{\longrightarrow}\:S}}\:,$$ is called *local* if ${\varphi}({\mathfrak{m}}) \subseteq {\mathfrak{n}}$ holds. Given such a homomorphism, every $S$-module is an $R$-module via ${\varphi}$; a finitely generated $S$-module is, when considered as an $R$-module, said to be *finite over ${\varphi}$.* It is evident from the condition ${\varphi}({\mathfrak{m}}) \subseteq {\mathfrak{n}}$ that ${\mathfrak{m}}N \ne N$ holds for every module $N\ne 0$ that is finite over ${\varphi}$. This is an extension of Nakayama’s lemma for finitely generated modules, and the theme that modules finite over ${\varphi}$ behave much like finitely generated $R$-modules was systematically explored by Avramov, Iyengar, and Miller [@AIM-06]. The first theorem in their study is the *Bass Equality,* ${\operatorname{id}_{R}N} = {\operatorname{depth}R}$, which holds if $N$ is finite over ${\varphi}$ and of finite injective dimension over $R$. We extend this result with
Let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a local ring homomorphism and $N\ne 0$ a module finite over ${\varphi}$. If $N$ has finite Gorenstein injective dimension over $R$, then one has $${\operatorname{Gid}_{R}N} {\:=\:}{\operatorname{depth}R}\:.$$
The statement here is a special case of Theorem \[0\]; it provides a positive answer to Question 6.2 in the survey [@CFH-11] by Christensen, Foxby, and Holm. Motivation for this question comes, beyond the Bass Equality [[@AIM-06 Thm. 2.1]]{} cited above, from the similar equality for finitely generated $R$-modules of finite Gorenstein injective dimension, see Khatami, Tousi, and Yassemi [[@KTY-09 Cor. 2.5]]{}, and from the the *Auslander–Bridger Equality,* ${\operatorname{Gfd}_{R}N} = {\operatorname{depth}R}- {\operatorname{depth}_{R}N}$, which by work of Christensen and Iyengar [@LWCSIn07] holds if $N$ is finite over ${\varphi}$ and of finite Gorenstein flat dimension over $R$.
Preliminaries
=============
The proof of the main result uses derived functors on derived categories. Our notation is standard, and to not overload this short paper we refer the reader to the appendix in [@lnm] for unexplained notation.
Let $R$ be a ring, by an $R$-*complex* we mean a complex of $R$-modules. The derived category over $R$ is denoted ${{{\mathsf{D}}(R)}}$. We say that a complex $X$ has *bounded homology* if ${\operatorname{H}_{i}(X)}=0$ holds for $|i|\gg 0$. To capture the homological extent of a complex, set $$\inf{X} {\:=\:}\inf{{\{\mspace{1mu}i\in{\mathbb{Z}}\mid {\operatorname{H}_{i}(X)} \ne 0\mspace{1mu}\}}} {{\quad\text{and}\quad}}\sup{X} {\:=\:}\sup{{\{\mspace{1mu}i\in{\mathbb{Z}}\mid {\operatorname{H}_{i}(X)} \ne 0\mspace{1mu}\}}}\:.$$ We write ${\operatorname{Gid}_{R}X}$ and ${\operatorname{Gfd}_{R}X}$ for the Gorenstein injective dimension and Gorenstein flat dimension of an $R$-complex. For a complex with ${\operatorname{H}_{}(X)} = 0$ it is standard to set $\inf{X} = \infty$, $\sup{X} = -\infty$, and ${\operatorname{Gid}_{R}X} = -\infty = {\operatorname{Gfd}_{R}X}$.
We recall the main results from a paper by Christensen, Frankild, and Holm [@CFH-06].
\[Bass\] Let $R$ be a ring with a dualizing complex $D$. An $R$-complex $X$ with bounded homology has finite Gorenstein injective dimension if and only if it belongs to the *Bass category* ${{{\mathsf{B}}(R)}}$; that is, if and only if the complex ${\operatorname{\mathbf{R}Hom}_{R}(D,X)}$ has bounded homology, and the canonical morphism $${\nobreak{{\beta}_D^X\colon {\nobreak{D\otimes_{R}^{\mathbf{L}}{\operatorname{\mathbf{R}Hom}_{R}(D,X)}}}\:{\longrightarrow}\:X}}$$ is an isomorphism in ${{{\mathsf{D}}(R)}}$.
\[Auslander\] Let $R$ be a ring with a dualizing complex $D$. An $R$-complex $X$ with bounded homology has finite Gorenstein flat dimension if and only if it belongs to the *Auslander category* ${{{\mathsf{A}}(R)}}$; that is, if and only if the complex ${\nobreak{D\otimes_{R}^{\mathbf{L}}X}}$ has bounded homology, and the canonical morphism $${\nobreak{{\alpha}_D^X\colon X\:{\longrightarrow}\:{\operatorname{\mathbf{R}Hom}_{R}(D,{\nobreak{D\otimes_{R}^{\mathbf{L}}X}})}}}$$ is an isomorphism in ${{{\mathsf{D}}(R)}}$.
The next two lemmas slightly improve standard results [[@lnm Lem. (3.2.9)]]{}.
\[lem:A\] Let $Q \to R$ be a ring homomorphism. Assume that $R$ has a dualizing complex and let $I$ be an injective $Q$-module. An $R$-complex $X$ belongs to ${{{\mathsf{A}}(R)}}$ only if ${\operatorname{Hom}_{Q}(X,I)}$ belongs to ${{{\mathsf{B}}(R)}}$, and the converse holds if $I$ is faithfully injective.
Let $D$ be a dualizing complex for $R$. By adjointness there is an isomorphism $$\label{eq:1}
\tag{$*$}
{\operatorname{Hom}_{Q}({\nobreak{D\otimes_{R}^{\mathbf{L}}X}},I)} {\simeq}{\operatorname{\mathbf{R}Hom}_{R}(D,{\operatorname{Hom}_{Q}(X,I)})}$$ in ${{{\mathsf{D}}(R)}}$. It accounts for the horizontal isomorphism in the commutative diagram $$\label{eq:2}
\tag{$\Box$}
\begin{gathered}
\xymatrix{ {\operatorname{Hom}_{Q}({\operatorname{\mathbf{R}Hom}_{R}(D,{\nobreak{D\otimes_{R}^{\mathbf{L}}X}})},I)}
\ar[r]^-{{\operatorname{Hom}_{}({\alpha}_D^X,I)}}
& {\operatorname{Hom}_{Q}(X,I)}\\
{\nobreak{D\otimes_{R}^{\mathbf{L}}{\operatorname{Hom}_{Q}({\nobreak{D\otimes_{R}^{\mathbf{L}}X}},I)}}} \ar[u]^-{\simeq}\ar[r]_-{\simeq}& {\nobreak{D\otimes_{R}^{\mathbf{L}}{\operatorname{\mathbf{R}Hom}_{R}(D,{\operatorname{Hom}_{Q}(X,I)})}}}
\ar[u]^-{{\beta}_D^{{\operatorname{Hom}_{}(X,I)}}} }
\end{gathered}$$ and the vertical isomorphism is Hom evaluation; see Christensen and Holm [[@LWCHHl09 Prop. 2.2(ii)]]{}. If $X$ belongs to ${{{\mathsf{A}}(R)}}$, then ${\operatorname{Hom}_{Q}(X,I)}$ has bounded homology by injectivity of $I$, the complex ${\operatorname{\mathbf{R}Hom}_{R}(D,{\operatorname{Hom}_{Q}(X,I)})}$ has bounded homology by [([\[eq:1\]]{})]{}, and ${\beta}_D^{{\operatorname{Hom}_{}(X,I)}}$ is an isomorphism by [([\[eq:2\]]{})]{}; that is, ${\operatorname{Hom}_{Q}(X,I)}$ belongs to ${{{\mathsf{B}}(R)}}$. Conversely, if $I$ is faithfully injective and ${\operatorname{Hom}_{Q}(X,I)}$ belongs to ${{{\mathsf{B}}(R)}}$, then $X$ has bounded homology, it follows from [([\[eq:1\]]{})]{} that the complex ${\nobreak{D\otimes_{R}^{\mathbf{L}}X}}$ is has bounded homology and from [([\[eq:2\]]{})]{} that ${\alpha}_D^X$ is an isomorphism; that is, $X$ belongs to ${{{\mathsf{A}}(R)}}$.
\[lem:B\] Let $Q \to R$ be a ring homomorphism. Assume that $R$ has a dualizing complex and let $I$ be an injective $Q$-module. An $R$-complex $X$ belongs to ${{{\mathsf{B}}(R)}}$ only if ${\operatorname{Hom}_{Q}(X,I)}$ belongs to ${{{\mathsf{A}}(R)}}$, and the converse holds if $I$ is faithfully injective.
Similar to the proof of [Lemma [\[lem:A\]]{}]{}.
Another key result on Auslander and Bass categories comes from the paper of Avramov and Foxby [@LLAHBF97] in which the categories were introduced.
\[regular\] Let $R$ be local with maximal ideal ${\mathfrak{m}}$ and $R \to R'$ be a flat local homomorphism such that the closed fiber $R'/{\mathfrak{m}}R'$ is regular; such a homomorphism is called *regular*. If $R$ has a dualizing complex, then $R'$ has a dualizing complex, see [@LLAHBF97 (2.11)], and by [[@LLAHBF97 Cor. (7.9)]]{} the next assertions hold:
An $R'$-complex belongs to ${{{\mathsf{A}}(R')}}$ if and only if it belongs to ${{{\mathsf{A}}(R)}}$.
An $R'$-complex belongs to ${{{\mathsf{B}}(R')}}$ if and only if it belongs to ${{{\mathsf{B}}(R)}}$.
The main result
===============
We start by proving the main result in a special case, and then we reduce the general case to the special. Let $R$ be a local ring with maximal ideal ${\mathfrak{m}}$. A local ring homomorphism ${\nobreak{{\varphi}\colon R \rightarrow S}}$ is said to have a *regular factorization* if there is a commutative diagram of local ring homomorphisms $$\xymatrix@=1.5pc{
& R' \ar@{->>}[dr]\\
R \ar[ru]^-{\dot{{\varphi}}} \ar[rr]^{{\varphi}} & & S
}$$ where $\dot{{\varphi}}$ is flat and the closed fiber $R'/{\mathfrak{m}}R'$ is regular.
\[lem:1\] Let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a local ring homomorphism and $N$ an $S$-complex with bounded and degreewise finitely generated homology. Assume that $R$ has a dualizing complex and ${\varphi}$ has a regular factorization. If $N$ has finite Gorenstein injective dimension over $R$, then one has $${\operatorname{Gid}_{R}N} {\:=\:}{\operatorname{depth}R}- \inf{N}\:.$$
Let $D$ be a dualizing complex for $R$ and $R \to R' \to S$ a regular factorization of ${\varphi}$. If ${\operatorname{H}_{}(N)}=0$ the claim is trivial under the conventions from Section 1, so we may assume that ${\operatorname{H}_{}(N)}$ is nonzero. By [\[Bass\]]{} and [\[regular\]]{} the complex $N$ belongs to ${{{\mathsf{B}}(R')}}$, so $g:={\operatorname{Gid}_{R'}N}$ is finite. By [[@CFH-06 Thm. 6.3]]{} one has $$g {\:=\:}{\operatorname{depth}_{}R'} - \inf{N}\:.$$ Let $D'$ be a dualizing complex for $R'$, cf. [\[regular\]]{}, and assume without loss of generality that it is normalized in the sense of [@LLAHBF97]. For every $R'$-complex $X$ with bounded and degreewise finitely generated homology one then has $$\tag{$\dagger$}
\label{eq:dpt}
{\operatorname{depth}_{R'}X} {\:=\:}\inf{{\operatorname{\mathbf{R}Hom}_{R'}(X,D')}}$$ and $$\tag{$\dagger\dagger$}
\label{eq:ddag}
X {\:{\simeq}\:}{\operatorname{\mathbf{R}Hom}_{R'}({\operatorname{\mathbf{R}Hom}_{R'}(X,D')},D')} {\quad\text{in}\quad} {{{\mathsf{D}}(R')}}\:;$$ see [[@LLAHBF97 Lem. (1.5.3), (2.6), and (2.7)]]{}. Moreover, ${\operatorname{Gfd}_{R'}{\operatorname{\mathbf{R}Hom}_{R'}(N,D')}} = g$ holds by [[@CFH-06 Cor. 6.4]]{}. Set $n = -{\operatorname{dim}R}- \inf{N}$; by [[@LWCSIn07 Thm. 3.1]]{} there is a distinguished triangle in ${{{\mathsf{D}}(R')}}$ of complexes with bounded and degreewise finitely generated homology, $$\tag{$\bigtriangleup$}
\label{eq:tri}
{\operatorname{\mathbf{R}Hom}_{R'}(N,D')} {\longrightarrow}P {\longrightarrow}H {\longrightarrow}{\mathsf{\Sigma}^{}{{\operatorname{\mathbf{R}Hom}_{R'}(N,D')}}}\:,$$ where $$\label{eq:H}
\tag{$*$}
{\operatorname{pd}_{R'}P} {\:=\:}g {{\qquad\text{and}\qquad}}\sup{H} {\:\le\:}{\operatorname{Gfd}_{R'}H} {\:\le\:}n\:.$$ By the Auslander–Buchsbaum formula and [[@LWCSIn07 Thm. 4.1]]{} one has $${\operatorname{depth}R}' - {\operatorname{depth}_{R'}P} {\:=\:}g {\:=\:}{\operatorname{depth}R}' - {\operatorname{depth}_{R'}{\operatorname{\mathbf{R}Hom}_{R'}(N,D')}}\:.$$ Combined with [([\[eq:dpt\]]{})]{} and [([\[eq:ddag\]]{})]{} these equalities yield $$\label{eq:di}
\tag{$**$}
{\operatorname{depth}_{R'}P} {\:=\:}\inf{N}\:.$$
Applying the functor ${\operatorname{\mathbf{R}Hom}_{R'}(-,D')}$ to [([\[eq:tri\]]{})]{} one gets via [([\[eq:ddag\]]{})]{} the triangle $$\label{eq:tritri}
\tag{$\bigtriangledown$}
{\mathsf{\Sigma}^{-1}{N}} {\longrightarrow}{\operatorname{\mathbf{R}Hom}_{R'}(H,D')} {\longrightarrow}{\operatorname{\mathbf{R}Hom}_{R'}(P,D')} {\longrightarrow}N\:.$$ One has ${\operatorname{id}_{R'}{\operatorname{\mathbf{R}Hom}_{R'}(P,D')}} = g$; see [[@CFH-06 Cor. 6.4]]{}. As $\dot{{\varphi}}$ is flat, the complex ${\operatorname{\mathbf{R}Hom}_{R'}(P,D')}$ has finite injective dimension over $R$. By [[@AIM-06 Cor. 8.2.2]]{} and [[@DWu15 Thm. 4.4]]{} one has ${\operatorname{id}_{R}{\operatorname{\mathbf{R}Hom}_{R'}(P,D')}} = {\operatorname{depth}R}- \inf{{\operatorname{\mathbf{R}Hom}_{R'}(P,D')}}$, which by [([\[eq:dpt\]]{})]{} and [([\[eq:di\]]{})]{} can be rewritten as $$\label{eq:id}
\tag{$***$}
{\operatorname{id}_{R}{\operatorname{\mathbf{R}Hom}_{R'}(P,D')}} {\:=\:}{\operatorname{depth}R}- \inf{N}\:.$$ The complex ${\operatorname{\mathbf{R}Hom}_{R'}(H,D')}$ has finite Gorenstein injective dimension over $R'$ by [[@CFH-06 Cor. 6.4]]{} and hence over $R$; see [\[regular\]]{}. The first inequality in the next computation holds by [[@CFH-06 Thm. 3.3]]{}; the equality follows from [([\[eq:dpt\]]{})]{}; the second inequality holds by the definition of depth; the final inequality follows from [([\[eq:H\]]{})]{}. $$\begin{aligned}
{\operatorname{Gid}_{R}{\operatorname{\mathbf{R}Hom}_{R'}(H,D')}}
& {\:\le\:}{\operatorname{dim}R}- \inf{{\operatorname{\mathbf{R}Hom}_{R'}(H,D')}}\\
& {\:=\:}{\operatorname{dim}R}- {\operatorname{depth}_{R'}H}\\
& {\:\le\:}{\operatorname{dim}R}+ \sup{H}\\
& {\:\le\:}-\inf{N}\,.
\end{aligned}$$ For every injective $R$-module $I$ and every $i \le \inf{N}$ one gets from [([\[eq:tritri\]]{})]{} an exact sequence in homology $${\operatorname{H}_{i}({\operatorname{\mathbf{R}Hom}_{R}(I,{\operatorname{\mathbf{R}Hom}_{R'}(P,D')})})} {\longrightarrow}{\operatorname{H}_{i}({\operatorname{\mathbf{R}Hom}_{R}(I,N)})} {\longrightarrow}0\:.$$ Thus, per [[@CFH-06 Thm. 3.3]]{} and [([\[eq:id\]]{})]{} one has $${\operatorname{Gid}_{R}N} \le {\operatorname{id}_{R}{\operatorname{\mathbf{R}Hom}_{R'}(P,D')}} = {\operatorname{depth}R}-\inf{N}\:.$$ The opposite inequality ${\operatorname{Gid}_{R}N} \ge {\operatorname{depth}R}-\inf{N}$ holds by [[@CFH-06 Thm. 6.3]]{}.
As is standard, we denote by ${\widehat{R}}$ and ${\widehat{S}}$ the completions of $R$ and $S$ in the topologies induced by their maximal ideals. The homomorphism ${\nobreak{{\varphi}\colon R \rightarrow S}}$ extends to a homomorphism of complete local rings; that is, there is a commutative diagram of local ring homomorphisms $$\xymatrix@=1.5pc{
{\widehat{R}}\ar[r]^-{\hat{{\varphi}}}& {\widehat{S}}\\
R \ar[u] \ar[r]^-{\varphi}& S \ar[u]
}$$ In particular, every ${\widehat{S}}$-complex is an ${\widehat{R}}$-complex.
In the special case $R=S$, and ${\varphi}$ the identity, the next result was proved by Christensen, Frankild, and Iyengar; see Foxby and Frankild [[@HBFAJF07 Thm. 3.6]]{}.
\[lem:2\] Let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a local ring homomorphism and $N$ an $S$-complex with bounded and degreewise finitely generated homology. If $N$ has finite Gorenstein injective dimension over $R$, then ${\nobreak{N\otimes_{S}{\widehat{S}}}}$ has finite Gorenstein injective dimension over ${\widehat{R}}$.
Let $K^S$ be the Koszul complex on a minimal set of generators for ${\mathfrak{n}}$, the maximal ideal of $S$. Since the $S$-complex ${\operatorname{H}_{}(K)}$ has degreewise finite length, one has ${\nobreak{{\widehat{S}}\otimes_{S}K^S}} {\simeq}K^S$ in ${{{\mathsf{D}}(S)}}$. Under the flat map $S \to {\widehat{S}}$ the minimal generators of ${\mathfrak{n}}$ extend to a minimal set of generators for the maximal ideal $\hat{{\mathfrak{n}}}$ of ${\widehat{S}}$, so ${\nobreak{{\widehat{S}}\otimes_{S}K^S}}$ is the Koszul complex $K^{{\widehat{S}}}$ on a minimal set of generators for $\hat{{\mathfrak{n}}}$. Thus one has $$\label{eq:K}
\tag{$\diamond$}
K^S {\simeq}K^{{\widehat{S}}}$$ in ${{{\mathsf{D}}(S)}}$, and we simply denote this complex $K$.
The first step is to notice that ${\nobreak{N\otimes_{S}K}}$ has finite Gorenstein injective dimension over $R$. For every element $x \in {\mathfrak{n}}$ there is an exact sequence of $S$-complexes, $$0 {\longrightarrow}N {\longrightarrow}{\nobreak{\operatorname{Cone}x^N}} {\longrightarrow}{\mathsf{\Sigma}^{}{N}} {\longrightarrow}0\:,$$ where $x^N$ is the homothety. Since $N$ and ${\mathsf{\Sigma}^{}{N}}$ have finite Gorenstein injective dimension over $R$, so has ${\nobreak{\operatorname{Cone}x^N}}$; this folklore fact is dual to a result of Veliche [[@OVl06 Thm. 3.9]]{} for Gorenstein projective dimension. Now, ${\nobreak{\operatorname{Cone}x^N}}$ is isomorphic to ${\nobreak{N\otimes_{S}K(x)}}$, where $K(x)$ denotes the elementary Koszul complex on $x$. Since $K$ is a tensor product of such elementary Koszul complexes, it follows that ${\nobreak{N\otimes_{S}K}}$ has finite Gorenstein injective dimension over $R$.
Set $M = {\nobreak{N\otimes_{S}K}}$; it is an ${\widehat{S}}$-complex via $K$ and, therefore, an ${\widehat{R}}$-complex. The second step is to prove that $M$ belongs to ${{{\mathsf{B}}({\widehat{R}})}}$. The composite $R {\xrightarrow[]{\;{\varphi}\;}} S {\longrightarrow}{\widehat{S}}$, called the semi-completion of ${\varphi}$, has a regular factorization; see Avramov, Foxby, and Herzog [[@AFH-94 Thm. (1.1)]]{}. Let $E'$ denote the injective hull of the residue field of $R'$ and $E$ denote the injective hull ${\operatorname{E}_{R}(k)} {\cong}{\operatorname{E}_{{\widehat{R}}}(k)}$. As ${\operatorname{H}_{}(M)}$ has degreewise finite length over ${\widehat{S}}$ and, therefore, over $R'$ one has $$M {\:{\simeq}\:}{\operatorname{Hom}_{R'}({\operatorname{Hom}_{R'}(M,E')},E')}\:.$$ As ${\operatorname{Gid}_{R}M}$ is finite, it follows from [[@LWCSSW10 Thm. 1.7]]{} that the $R'$-complex $$\begin{aligned}
{\operatorname{\mathbf{R}Hom}_{R}(R',M)} &{\:{\simeq}\:}{\operatorname{\mathbf{R}Hom}_{R}(R',{\operatorname{Hom}_{R'}({\operatorname{Hom}_{R'}(M,E')},E')})}\\
&{\:{\simeq}\:}{\operatorname{Hom}_{R'}({\nobreak{R'\otimes_{R}{\operatorname{Hom}_{R'}(M,E')}}},E')}
\end{aligned}$$ has finite Gorenstein injective dimension. The $R'$-complex ${\nobreak{R'\otimes_{R}{\operatorname{Hom}_{R'}(M,E')}}}$ then has finite Gorenstein flat dimension by [Lemma [\[lem:A\]]{}]{}. As $R'$ is faithfully flat over $R$, it follows from [[@LWCSSW10 Thm. 1.8]]{} that the complex ${\operatorname{Hom}_{R'}(M,E')}$ has finite Gorenstein flat dimension over $R$. By another application of the same result the complex ${\nobreak{{\widehat{R}}\otimes_{R}{\operatorname{Hom}_{R'}(M,E')}}}$ has finite Gorenstein flat dimension over ${\widehat{R}}$, whence it belongs to ${{{\mathsf{A}}({\widehat{R}})}}$. By [Lemma [\[lem:A\]]{}]{} the dual complex $$\begin{aligned}
{\operatorname{Hom}_{{\widehat{R}}}({\nobreak{{\widehat{R}}\otimes_{R}{\operatorname{Hom}_{R'}(M,E')}}},E)} {\:{\cong}\:}{\operatorname{Hom}_{R}({\operatorname{Hom}_{R'}(M,E')},E)}
\end{aligned}$$ belongs to ${{{\mathsf{B}}({\widehat{R}})}}$. As $E$ is faithfully injective, it follows from [Lemma [\[lem:A\]]{}]{} that the complex ${\operatorname{Hom}_{R'}(M,E')}$ belongs to ${{{\mathsf{A}}({\widehat{R}})}}$ and hence to ${{{\mathsf{A}}(R')}}$; see [\[regular\]]{}. By [Lemma [\[lem:B\]]{}]{} the complex $M$ now belongs to ${{{\mathsf{B}}(R')}}$ and hence to ${{{\mathsf{B}}({\widehat{R}})}}$.
To finish the proof we now prove that ${\nobreak{N\otimes_{S}{\widehat{S}}}}$ belongs to ${{{\mathsf{B}}({\widehat{R}})}}$, cf. \[Bass\]. First notice that by [([\[eq:K\]]{})]{} and associativity of the tensor product one has $$\label{eq:4}
\tag{$\diamond\diamond$}
{\nobreak{N\otimes_{S}K}} {\:{\cong}\:}{\nobreak{N\otimes_{S}{({\nobreak{{\widehat{S}}\otimes_{{\widehat{S}}}K}})}}} {\:{\cong}\:}{\nobreak{{({\nobreak{N\otimes_{S}{\widehat{S}}}})}\otimes_{{\widehat{S}}}K}}\:.$$ By [([\[eq:4\]]{})]{} and an application of tensor evaluation [[@LWCHHl09 Prop. 2.2(v)]]{} one gets $$\tag{$\ddag\ddag$}
\begin{aligned}
{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}K}})}
&{\:{\simeq}\:}{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{{({\nobreak{N\otimes_{S}{\widehat{S}}}})}\otimes_{{\widehat{S}}}K}})}\\
&{\:{\simeq}\:}{\nobreak{{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}{\widehat{S}}}})}\otimes_{{\widehat{S}}}K}}\:.
\end{aligned}$$ As ${\nobreak{N\otimes_{S}K}}$ belongs to ${{{\mathsf{B}}({\widehat{R}})}}$, the complex ${\nobreak{{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}{\widehat{S}}}})}\otimes_{{\widehat{S}}}K}}$ has bounded homology, so ${\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}{\widehat{S}}}})}$ has bounded homology. This follows from work of Foxby and Iyengar [@HBFSIn03 1.3]; indeed, as the ${\widehat{R}}$-complex $D$ and the ${\widehat{S}}$-complex ${\nobreak{N\otimes_{S}K}}$ have degreewise finitely generated homology, it follows from [[@AIM-06 Lem. 1.3.2]]{} that the ${\widehat{S}}$-complex ${\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}{\widehat{S}}}})}$ has degreewise finitely generated homology. There is a commutative diagram in ${{{\mathsf{D}}({\widehat{R}})}}$, $$\xymatrix@C=4pc{
{\nobreak{D\otimes_{{\widehat{R}}}^{\mathbf{L}}{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}K}})}}}
\ar[d]_-{\simeq}\ar[r]^-{{\beta}_D^{{\nobreak{N\otimes_{}K}}}}_-{\simeq}& {\nobreak{N\otimes_{S}K}} \ar[d]^-{\simeq}\\
{\nobreak{{({\nobreak{D\otimes_{{\widehat{R}}}^{\mathbf{L}}{\operatorname{\mathbf{R}Hom}_{{\widehat{R}}}(D,{\nobreak{N\otimes_{S}{\widehat{S}}}})}}})}\otimes_{{\widehat{S}}}K}}
\ar[r]^-{{\nobreak{{\beta}_D^{{\nobreak{N\otimes_{}{\widehat{S}}}}}\otimes_{}K}}}
& {\nobreak{{({\nobreak{N\otimes_{S}{\widehat{S}}}})}\otimes_{{\widehat{S}}}K}}
}$$ where the right-hand vertical isomorphism is [([\[eq:4\]]{})]{}, and the left-hand vertical isomorphism follows by tensor evaluation [[@LWCHHl09 Prop. 2.2(v)]]{} and associativity of the tensor product. It follows that ${{\nobreak{{\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}}\otimes_{{\widehat{S}}}K}}}$ is an isomorphism; that is, the mapping cone $${\nobreak{\operatorname{Cone}{{({\nobreak{{\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}}\otimes_{{\widehat{S}}}K}})}}}}
{\:{\simeq}\:}{\nobreak{({\nobreak{\operatorname{Cone}{\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}})}}\otimes_{{\widehat{S}}}K}}$$ is acyclic. As ${\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}}$ per [[@AIM-06 Lem. 1.3.2]]{} is a morphism of ${\widehat{S}}$-complexes with degreewise finitely generated homology, it follows from [@HBFSIn03 1.3] that the complex ${\nobreak{\operatorname{Cone}{\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}}}}$ is acyclic, whence ${\beta}_D^{{\nobreak{N\otimes_{S}{\widehat{S}}}}}$ is an isomorphism in ${{{\mathsf{D}}({\widehat{R}})}}$, and ${\nobreak{N\otimes_{S}{\widehat{S}}}}$ belongs to ${{{\mathsf{B}}({\widehat{R}})}}$.
The main result, which we can now prove, compares to [[@LWCSIn07 Thm. 4.1 and Cor. 4.8]]{}.
\[0\] Let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a local ring homomorphism and $N$ an $S$-complex with bounded and degreewise finitely generated homology. If $N$ has finite Gorenstein injective dimension over $R$, then one has $$\begin{aligned}
{\operatorname{Gid}_{R}N} & {\:=\:}{\operatorname{depth}R}- \inf{N}\\
& {\:=\:}-\inf{{\operatorname{\mathbf{R}Hom}_{R}({\operatorname{E}_{R}(k)},N)}}\\
& {\:=\:}{\operatorname{Gid}_{{\widehat{R}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}}\:.
\end{aligned}$$
The homomorphism ${\widehat{R}}{\xrightarrow[]{\;{\hat{{\varphi}}}\;}} {\widehat{S}}$ has a regular factorization; see [[@AFH-94 Thm. (1.1)]]{}. By [Lemma [\[lem:2\]]{}]{} the ${\widehat{R}}$-complex ${\nobreak{N\otimes_{S}{\widehat{S}}}}$ has finite Gorenstein injective dimension, and it has bounded and degreewise finite homology over ${\widehat{S}}$, so [Lemma [\[lem:1\]]{}]{} yields $${\operatorname{Gid}_{{\widehat{R}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}} {\:=\:}{\operatorname{depth}_{}{\widehat{R}}} - \inf{{({\nobreak{N\otimes_{S}{\widehat{S}}}})}}\:.$$ There are equalities ${\operatorname{depth}_{}{\widehat{R}}} = {\operatorname{depth}R}$ and $\inf{{({\nobreak{N\otimes_{S}{\widehat{S}}}})}} = \inf{N}$; the latter holds by faithful flatness of ${\widehat{S}}$ over $S$. Moreover, one has $${\operatorname{Gid}_{R}N} {\:\ge\:}{\operatorname{depth}R}- \inf{N} {\:=\:}-\inf{{\operatorname{\mathbf{R}Hom}_{R}({\operatorname{E}_{R}(k)},N)}}$$ by [[@CFH-06 Thm. 6.3]]{} and [[@LWCHHl09 Cor. 6.5]]{}, so it is sufficient to prove that the inequality ${\operatorname{Gid}_{R}N} \le {\operatorname{Gid}_{{\widehat{R}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}}$ holds. By [[@LWCSSW10 Thm. 2.2]]{} one has $$\label{eq:ss}
\tag{\S}
\begin{aligned}
{\operatorname{Gid}_{R}N} &{\:=\:}\sup{{\{\mspace{1mu}{\operatorname{depth}_{}R_{\mathfrak{p}}} -
{\operatorname{width}_{R_{\mathfrak{p}}}N_{\mathfrak{p}}} \mid {\mathfrak{p}}\in{\operatorname{Spec}R}\mspace{1mu}\}}}\quad\text{and}\\
{\operatorname{Gid}_{{\widehat{R}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}} &{\:=\:}\sup{{\{\mspace{1mu}{\operatorname{depth}_{}{\widehat{R}}_{\mathfrak{q}}} -
{\operatorname{width}_{{\widehat{R}}_{\mathfrak{q}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}_{\mathfrak{q}}} \mid {\mathfrak{q}}\in{\operatorname{Spec}{\widehat{R}}}\mspace{1mu}\}}}\:.
\end{aligned}$$ For ${\mathfrak{p}}\in {\operatorname{Spec}R}$ choose ${\mathfrak{q}}\in {\operatorname{Spec}{\widehat{R}}}$ minimal over ${\mathfrak{p}}{\widehat{R}}$, the local homomorphism $R_{\mathfrak{p}}\to {\widehat{R}}_{\mathfrak{q}}$ is flat with artinian closed fiber, whence one has ${\operatorname{depth}R}_{\mathfrak{p}}= {\operatorname{depth}_{}{\widehat{R}}_{\mathfrak{q}}}$; see e.g. [[@AFH-94 Prop. (2.8)]]{}. In the next computation the first and fourth equalities hold by the definition of width, the second holds by faithful flatness of ${\widehat{S}}$ over $S$, and the last holds as $R_{\mathfrak{p}}\to {\widehat{R}}_{\mathfrak{q}}$ is a local homomorphism; see Wu and Kong [[@DWuFKn18 Lem. 3.6]]{}. $$\begin{aligned}
{\operatorname{width}_{R_{\mathfrak{p}}}N_{\mathfrak{p}}} & {\:=\:}\inf{{({\nobreak{(R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}})\otimes_{R}N}})}}\\
& {\:=\:}\inf{{\nobreak{{({\nobreak{(R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}})\otimes_{R}N}})}\otimes_{S}{\widehat{S}}}}}\\
& {\:=\:}\inf{{\nobreak{(R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}})\otimes_{R}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}}}}\\
& {\:=\:}{\operatorname{width}_{R_{\mathfrak{p}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}_{\mathfrak{p}}}\\
& {\:=\:}{\operatorname{width}_{{\widehat{R}}_{\mathfrak{q}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}_{\mathfrak{q}}}\:.
\end{aligned}$$ For every prime ideal ${\mathfrak{p}}$ in $R$ there is thus a prime ideal ${\mathfrak{q}}$ in ${\widehat{R}}$ with $${\operatorname{depth}_{}R_{\mathfrak{p}}} - {\operatorname{width}_{R_{\mathfrak{p}}}N_{\mathfrak{p}}} = {\operatorname{depth}_{}{\widehat{R}}_{\mathfrak{q}}} -
{\operatorname{width}_{{\widehat{R}}_{\mathfrak{q}}}{({\nobreak{N\otimes_{S}{\widehat{S}}}})}_{\mathfrak{q}}}$$ so the desired inequality is immediate from [([\[eq:ss\]]{})]{}.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank the anonymous referee for suggestions and comments that helped us improve the exposition.
\#1[0=7=07 by-1ext\#1\#1T\#1 \#1 d\#1\#1 D\#1\#1 l\#1\#1 L\#1\#1\#1]{} \[1\]
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[^1]: L.W.C. was partly supported by Simons Foundation collaboration grant 428308; D.W. was partly supported by NSF of China grants 11761047 and 11861043. The paper was written during D.W.’s year-long visit to Texas Tech University; the hospitality of the TTU Department of Mathematics and Statistics is acknowledged with gratitude.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We discuss the information entropy for a general open pointer-based simultaneous measurement and show how it is bound from below. This entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics. Furthermore, the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results.
[*Keywords*]{}: simultaneous pointer-based measurement, noisy measurement, conjugate observables, entropy, uncertainty relation, quantum mechanics
address: 'Institut f[ü]{}r Quantenphysik and Center for Integrated Quantum Science and Technology (IQ^ST^), Universit[ä]{}t Ulm, D-89069 Ulm, Germany'
author:
- Raoul Heese and Matthias Freyberger
bibliography:
- 'concept.bib'
title: 'Entropic uncertainty bound for open pointer-based simultaneous measurements of conjugate observables'
---
Introduction {#sec:Introduction}
============
The concept of pointer-based simultaneous measurements of conjugate observables is an indirect measurement model, which allows to dynamically describe the properties of a simultaneous quantum mechanical measurement process. Additional to the system to be measured (hereafter just called *system*), the model introduces two additional systems called *pointers*, which are coupled to the system and act as commuting meters from which the initial system observables can be read out after a certain interaction time. In this sense, the pointers represent the measurement devices used to simultaneously determine the system observables. Pointer-based simultaneous measurements date back to Arthurs and Kelly [@arthurs1965] and are based on von Neumann’s idea of indirect observation [@vonneumann1932].
In principle, any pair of conjugate observables like position and momentum or quadratures of the electromagnetic field [@schleich2001], whose commutator is well-defined and proportional to the identity operator, can straightforwardly be measured within the scope of pointer-based simultaneous measurements. We limit ourselves to the measurement of position and momentum in the following. An *open* pointer-based simultaneous measurement [@heese2014] also takes environmental effects into consideration by utilizing an environmental heat bath in the sense of the Caldeira-Leggett model [@caldeira1981; @caldeira1983a; @caldeira1983ae; @caldeira1983b], which leads to a quantum Brownian motion [@chou2008; @fleming2011a; @fleming2011b; @martinez2013] of the system and the pointers, whereas a *closed* pointer-based simultaneous measurement [@arthurs1965; @wodkiewicz1984; @stenholm1992; @buzek1995; @appleby1998a; @appleby1998b; @appleby1998c; @busch2007; @busshardt2010; @busshardt2011; @heese2013] does not involve any environmental effects. A schematic open pointer-based simultaneous measurement procedure is shown in Fig. \[fig:model\].
In this contribution we calculate the information entropy of an open pointer-based simultaneous measurement and discuss its properties as a measurement uncertainty. In particular, we make use of recent results [@heese2013; @heese2014], which we extend and generalize. In Sec. \[sec:generalopenpointer-basedsimultaneousmeasurements\], we present the formal dynamics of open pointer-based simultaneous measurements and then use these results to discuss the entropic uncertainty in Sec. \[sec:entropy\]. In the end, we arrive at a generic lower bound of this entropic uncertainty. Note that we solely use rescaled dimensionless variables [@heese2014] so that $\hbar = 1$.
![Principles of an open pointer-based simultaneous measurement of two conjugate observables, e.g., the simultaneous measurement of position and momentum. The measurement apparatus consists of two quantum mechanical systems, called pointers, which are bilinearly coupled to the quantum mechanical system to be measured. Additionally, an environmental heat bath in the sense of the Caldeira-Leggett model can disturb both the system and the pointers. After the interaction process, one observable of each pointer is directly measured (e.g., the position of each pointer) while the system itself is not subject to any direct measurement. However, from these measurement results, information about the initial system observables can then be inferred. In other words, the final pointer observables act as commuting meters from which the initial non-commuting system observables can be simultaneously read out. The price to be paid for this simultaneity comes in form of fundamental noise terms, which affect the inferred values. The corresponding uncertainties can be described by information entropies, which are bound from below.[]{data-label="fig:model"}](fig1.pdf)
Open pointer-based simultaneous measurements {#sec:generalopenpointer-basedsimultaneousmeasurements}
============================================
As indicated in the introduction, our model of open pointer-based simultaneous measurements consists of a system particle to be measured with mass $M_{\mathrm{S}}$, position observable $\hat{X}_{\mathrm{S}}$ and momentum observable $\hat{P}_{\mathrm{S}}$, which is coupled bilinearly to two pointer particles with masses $M_1$ and $M_2$, position observables $\hat{X}_1$ and $\hat{X}_2$, and momentum observables $\hat{P}_1$ and $\hat{P}_2$, respectively. Both the system and the pointers are bilinearly coupled to an environmental heat bath, which consists of a collection of $N$ harmonic oscillators with masses $m_1,\dots,m_N$, position observables $\hat{q}_1,\dots,\hat{q}_N$, and momentum observables $\hat{k}_1,\dots,\hat{k}_N$. In this section, we first present the general Hamiltonian for this model and then briefly discuss the resulting dynamics.
Hamiltonian {#sec:generalopenpointer-basedsimultaneousmeasurements:hamiltonian}
-----------
The general Hamiltonian for our model reads $$\begin{aligned}
\label{eq:H}
\hat{\mathscr{H}}(t) \equiv \hat{H}_{\mathrm{free}} + \hat{H}_{\mathrm{int}}(t) + \hat{H}_{\mathrm{bath}}(t)\end{aligned}$$ and therefore consists of three parts. First, the free evolution Hamiltonian $$\begin{aligned}
\label{eq:H:free}
\hat{H}_{\mathrm{free}} \equiv \frac{\hat{P}_{\mathrm{S}}^2}{2 M_{\mathrm{S}}} + \frac{\hat{P}_1^2}{2 M_1} + \frac{\hat{P}_2^2}{2 M_2},\end{aligned}$$ which simply describes the dynamics of the undisturbed system and pointers. Second, the interaction Hamiltonian $$\begin{aligned}
\label{eq:H:int}
\hat{H}_{\mathrm{int}}(t) \equiv C_{\mathrm{S}}(t) \hat{X}_{\mathrm{S}}^2 + C_1(t) \hat{X}_1^2 + C_2(t) \hat{X}_2^2 + ( \hat{X}_{\mathrm{S}}, \hat{P}_{\mathrm{S}} ) \mathbf{C}(t) ( \hat{X}_1, \hat{X}_2, \hat{P}_1, \hat{P}_2 )^T,\end{aligned}$$ which describes possible quadratic potentials with the coupling strengths $C_{\mathrm{S}}(t)$, $C_1(t)$, and $C_2(t)$, respectively, as well as bilinear interactions between the system observables and the pointer observables via the $2\times4$ coupling matrix $\mathbf{C}(t)$. These interactions are necessary for an information transfer between system and pointers and are therefore a prerequisite of pointer-based simultaneous measurements. The existence of quadratic potentials is on the other hand not essential, but may be reasonable from a physical point of view when regarding confined particles. One possible interaction Hamiltonian would be the interaction Hamiltonian of the classic Arthurs and Kelly model [@arthurs1965], which can be written as $\hat{H}_{\mathrm{int}} = \kappa ( \hat{X}_{\mathrm{S}} \hat{P}_1 + \hat{P}_{\mathrm{S}} \hat{P}_2 )$ with an arbitrary coupling strength $\kappa \neq 0$.
Lastly, Eq. [(\[eq:H\])]{} contains the bath Hamiltonian [@caldeira1981; @caldeira1983a; @caldeira1983ae; @caldeira1983b] $$\begin{aligned}
\label{eq:H:bath}
\hat{H}_{\mathrm{bath}}(t) \equiv \frac{1}{2} \hat{\mathbf{k}}^{T} \mathbf{m}^{-1} \hat{\mathbf{k}}+ \frac{1}{2} \hat{\mathbf{q}}^{T} \mathbf{c} \hat{\mathbf{q}} + \hat{\mathbf{q}}^{T} \mathbf{g}(t) ( \hat{X}_{\mathrm{S}}, \hat{X}_1, \hat{X}_2 )^T,\end{aligned}$$ which describes the independent dynamics of the bath particles with the $N \times N$ diagonal mass matrix $\mathbf{m}$ containing $m_1,\dots,m_N$, and the $N \times N$ symmetric and positive definite bath-internal coupling matrix $\mathbf{c}$; as well as the coupling of system and pointer positions to the bath positions with the coupling strength $$\begin{aligned}
\label{eq:g}
\mathbf{g}(t) \equiv g(t) \mathbf{g},\end{aligned}$$ which consists of the time-dependent scalar $g(t)$ and the time-independent $N\times 3$ matrix $\mathbf{g}$. To simplify our notation, we make use of the vectorial bath positions $\hat{\mathbf{q}}^{T} \equiv (\hat{q}_1, \dots, \hat{q}_N)$ and the vectorial bath momenta $\hat{\mathbf{k}}^{T} \equiv (\hat{k}_1, \dots, \hat{k}_N)$.
In particular, since all of the following calculations only rely on the bilinear structure of the Hamiltonian, Eq. [(\[eq:H\])]{}, we do not further specify the coupling strengths and therefore consider quite general measurement configurations. Note that the time-dependencies of the coupling strengths in Eqs. [(\[eq:H:int\])]{} and [(\[eq:H:bath\])]{} allow us to design specific coupling pulses for system-pointer interactions [@busshardt2010] or switch-on functions for the bath [@fleming2011a; @fleming2011c].
Dynamics
--------
To determine the complete system and pointer dynamics, it is necessary to solve the coupled Heisenberg equations of motion $$\begin{aligned}
\label{eq:Heisenberg}
& \phantom{=.} \frac{\partial}{\partial t} \left( \hat{X}_{\mathrm{S}}(t), \hat{X}_1(t), \hat{X}_2(t), \hat{P}_{\mathrm{S}}(t), \hat{P}_1(t), \hat{P}_2(t) \right) \nonumber \\
& = i \left[ \hat{\mathscr{H}}(t), \left( \hat{X}_{\mathrm{S}}(t), \hat{X}_1(t), \hat{X}_2(t), \hat{P}_{\mathrm{S}}(t), \hat{P}_1(t), \hat{P}_2(t) \right) \right],\end{aligned}$$ which take on the form of a Volterra integro-differential equation [@burton2005] after explicitly solving the dynamics of the bath particles [@ford1988; @fleming2011a; @fleming2011b]. The general solution of Eq. [(\[eq:Heisenberg\])]{} can formally be expressed by means of a resolvent [@grossmann1968; @becker2006] and leads to system and pointer positions and momenta, which are linearly propagated from their initial values (i.e., the homogeneous solution) and are affected by an additive noise term (i.e., the inhomogeneous solution). Explicit analytical solutions, which are naturally of the same structure as the formal solution, can only be found for specific choices of the Hamiltonian, Eq. [(\[eq:H\])]{}. Otherwise, numerical approaches [@chen1998; @baker2000] might be necessary. Interestingly, since we strive after a general discussion of the dynamics, the ensured existence of a solution and its formal structure is sufficient for all further considerations. We assume in this context that possible unphysical artifacts of the modeling [@weiss1999; @fleming2011a], e.g., renormalization problems, have been treated adequately [@heese2014]. Without loss of generality, we choose $t = 0$ as the initial time.
For the description of a pointer-based simultaneous measurement, we do in fact not need knowledge about the complete system and pointer dynamics. As outlined in Fig. \[fig:model\], the measurement process provides that the two pointers are being measured after an interaction time $t$ in such a way, that we either read out the position or the momentum of each pointer. Consequently, we have knowledge about either $\hat{X}_1(t)$ or $\hat{P}_1(t)$ of the first pointer, and, additionally, about either $\hat{X}_2(t)$ or $\hat{P}_2(t)$ of the second pointer, which is a total of four different possible measurement combinations. To summarize these measurement combinations, we define the measurement vector $\hat{\mathbf{w}}(t)$, which consists of the two pointer observables chosen to be read out, e.g., $\hat{\mathbf{w}}(t) = (\hat{X}_1(t), \hat{X}_2(t))^T$ when measuring both pointer positions. As also outlined in Fig. \[fig:model\], the information gained from measuring the observables in the measurement vector allows us to infer the initial system observables $\hat{X}_{\mathrm{S}}(0)$ and $\hat{P}_{\mathrm{S}}(0)$. For this reason, the connection between the measured observables after a certain interaction time $t$ and the initial system observables builds the framework for a description of pointer-based simultaneous measurements.
With this purpose in mind, we can extract $$\begin{aligned}
\label{eq:inferredsystem}
\begin{pmatrix} \hat{\mathcal{X}}(t) \\ \hat{\mathcal{P}}(t) \end{pmatrix} = \begin{pmatrix} \hat{X}_{\mathrm{S}}(0) \\ \hat{P}_{\mathrm{S}}(0) \end{pmatrix} + \mathbf{B}(t) \hat{\mathbf{J}} + (\mathbf{\Lambda} \star \hat{\boldsymbol{\xi}})(t)\end{aligned}$$ from any (formal) solution of the Heisenberg equations of motion, Eq. [(\[eq:Heisenberg\])]{}. Here we have introduced the so-called generalized inferred observables $$\begin{aligned}
\label{eq:inferredobservables}
\begin{pmatrix} \hat{\mathcal{X}}(t) \\ \hat{\mathcal{P}}(t) \end{pmatrix} \equiv \mathbf{A}(t) \hat{\mathbf{w}}(t),\end{aligned}$$ which are given by a rescaled measurement vector $\hat{\mathbf{w}}(t)$ and can on the other hand be understood as the effectively measured observables from which the system observables can be directly read out. Since the two measured pointer observables in $\hat{\mathbf{w}}(t)$ initially commute and are subject to a unitary evolution, one has $$\begin{aligned}
\label{eq:XPcommute}
[\hat{\mathcal{X}}(t),\hat{\mathcal{P}}(t)] = 0,\end{aligned}$$ which means that the inferred observables can be determined simultaneously.
Furthermore, Eqs. [(\[eq:inferredsystem\])]{} and [(\[eq:inferredobservables\])]{} contain the coefficient matrices $\mathbf{A}(t)$, $\mathbf{B}(t)$, and $\mathbf{\Lambda}(t,s)$ with $0 \leq s \leq t$, for which we do not need to give an explicit expression. In general, they can be straightforwardly calculated from the resolvent of the formal solution of the complete system and pointer dynamics. An exemplary calculation can be found in Ref. [@heese2014]. We also make use of the initial value vector $$\begin{aligned}
\label{eq:J}
\hat{\mathbf{J}} \equiv ( \hat{X}_1(0), \hat{X}_2(0), \hat{P}_1(0), \hat{P}_2(0) )^{T},\end{aligned}$$ and the stochastic force [@weiss1999] $$\begin{aligned}
\label{eq:stochasticforce}
\hat{\boldsymbol{\xi}}(t) \equiv -\mathbf{g}^T(t) \left( \mathbf{m}^{-\frac{1}{2}} \cos( \boldsymbol{\omega} t) \mathbf{m}^{\frac{1}{2}} \hat{\mathbf{q}} + \mathbf{m}^{-\frac{1}{2}} \sin( \boldsymbol{\omega} t) \boldsymbol{\omega}^{-1} \mathbf{m}^{-\frac{1}{2}} \hat{\mathbf{k}} \right)\end{aligned}$$ with the symmetric bath frequency matrix [@fleming2011b] $$\begin{aligned}
\label{eq:omega}
\boldsymbol{\omega} \equiv \sqrt{ \mathbf{m}^{-\frac{1}{2}} \mathbf{c} \mathbf{m}^{-\frac{1}{2}} }.\end{aligned}$$ In particular, the stochastic force results from the homogeneous bath dynamics and describes the noisy influence of the bath on the measurement results, Eq. [(\[eq:H:bath\])]{}. The symbol $\star$ in Eq. [(\[eq:inferredsystem\])]{} represents the integral $$\begin{aligned}
\label{eq:star}
(f \star g)(x) \equiv \int \limits_{0}^{x} \! \mathrm d y f(x,y) g(y)\end{aligned}$$ for two arbitrary functions $f(x,y)$ and $g(x)$. In case of $f(x,y) = f(x-y)$, Eq. [(\[eq:star\])]{} is called a Laplace convolution.
The central aspect of the dynamics contained in Eq. [(\[eq:inferredsystem\])]{} can be understood when considering the expectation value $$\begin{aligned}
\label{eq:inferredexpecation}
\Braket{ \begin{pmatrix} \hat{\mathcal{X}}(t) \\ \hat{\mathcal{P}}(t) \end{pmatrix} } = \Braket{ \begin{pmatrix} \hat{X}_{\mathrm{S}}(0) \\ \hat{P}_{\mathrm{S}}(0) \end{pmatrix} } + \mathbf{s}(t)\end{aligned}$$ of Eq. [(\[eq:inferredsystem\])]{}, where the brackets refer to the mean value with respect to the initial state. Most importantly, Eq. [(\[eq:inferredexpecation\])]{} shows that the knowledge about the first moments of the inferred observables, Eq. [(\[eq:inferredobservables\])]{}, which can be determined by measuring the two pointer observables contained in the measurement vector $\hat{\mathbf{w}}(t)$, allows us to infer the first moments of the initial system observables. The second and third term on the right-hand side of Eq. [(\[eq:inferredsystem\])]{} simply add the shift $$\begin{aligned}
\label{eq:s}
\mathbf{s}(t) \equiv \mathbf{B}(t) \braket{ \hat{\mathbf{J}} } + (\mathbf{\Lambda} \star \braket{ \hat{\boldsymbol{\xi}}})(t)\end{aligned}$$ to this relation. Assuming a suitably separable initial state, this shift is determined by the initial state of the pointers and the bath, but independent of the state of the system. In this sense, our model is only of any practical purpose if the initial pointer and bath states are sufficiently well-known to determine $\mathbf{s}(t)$. Generally, the initial pointer states are at our disposal and can therefore be chosen in such a way that the first term on the right-hand side of Eq. [(\[eq:s\])]{} vanishes. It is furthermore reasonable to suppose that an environmental heat bath in thermal equilibrium leads to a vanishing expectation value of the stochastic force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. Consequently, presuming a vanishing shift $\mathbf{s}(t)$ seems appropriate for a typical measurement configuration. We will confirm this presumption further below for a specifically chosen initial state. Note that the linearity of Eqs. [(\[eq:inferredsystem\])]{} and [(\[eq:inferredobservables\])]{}, which is essential for the inference process, Eq. [(\[eq:inferredexpecation\])]{}, is a direct result of the bilinear structure of the Hamiltonian, Eq. [(\[eq:H\])]{}.
We assume here the non-pathological case that the interaction Hamiltonian $\hat{H}_{\mathrm{int}}(t)$, Eq. [(\[eq:H:int\])]{}, and the measurement vector $\hat{\mathbf{w}}(t)$ are chosen in such a way that inferring system observables from the pointers is possible in the first place, which is defined by the existence of the coefficient matrix $\mathbf{A}(t)$, Eq. [(\[eq:inferredobservables\])]{}. In other words, we require a sufficient information transfer from the system to the pointers. For example, for the classic Arthurs and Kelly model [@arthurs1965] with the interaction Hamiltonian mentioned in Sec. \[sec:generalopenpointer-basedsimultaneousmeasurements:hamiltonian\] and no environmental heat bath, measuring both pointer positions leads to an existing matrix $\mathbf{A}(t)$ for $t>0$, but measuring both pointer momenta does not [@busshardt2010].
Seeing now the role played by the inferred observables, we can quantify the uncertainty of a simultaneous pointer-based measurement with the help of the so-called noise operators [@arthurs1988] $$\begin{aligned}
\label{eq:noiseoperators}
\begin{pmatrix} \hat{N}_{\mathcal{X}}(t) \\ \hat{N}_{\mathcal{P}}(t) \end{pmatrix} \equiv \begin{pmatrix} \hat{\mathcal{X}}(t) - \hat{X}_{\mathrm{S}}(0) \\ \hat{\mathcal{P}}(t) - \hat{P}_{\mathrm{S}}(0) \end{pmatrix} = \mathbf{B}(t) \hat{\mathbf{J}} + (\mathbf{\Lambda} \star \hat{\boldsymbol{\xi}})(t),\end{aligned}$$ which are defined as the difference between the inferred observables $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{P}}(t)$, Eq. [(\[eq:inferredobservables\])]{}, and the respective initial system observables $\hat{X}_{\mathrm{S}}(0)$ and $\hat{P}_{\mathrm{S}}(0)$. The second equal sign in Eq. [(\[eq:noiseoperators\])]{} directly follows from Eq. [(\[eq:inferredsystem\])]{}. The expectation values $$\begin{aligned}
\label{eq:noiseoperators:exp}
\Braket{ \begin{pmatrix} \hat{N}_{\mathcal{X}}(t) \\ \hat{N}_{\mathcal{P}}(t) \end{pmatrix} } = \mathbf{s}(t)\end{aligned}$$ of the noise operators correspond to the shift $\mathbf{s}(t)$, Eq. [(\[eq:s\])]{}. As we have mentioned above, this shift can typically be presumed to vanish. Particularly interesting for our considerations is the symmetrized covariance matrix $$\begin{aligned}
\label{eq:noiseoperators:cov}
\begin{pmatrix} \Braket{ \hat{N}_{\mathcal{X}}^2(t) } - \Braket{ \hat{N}_{\mathcal{X}}(t) }^2 & \Braket{\hat{N}_{\mathcal{X}\mathcal{P}}(t)} - \Braket{ \hat{N}_{\mathcal{X}}(t) }\Braket{ \hat{N}_{\mathcal{P}}(t) } \\ \Braket{\hat{N}_{\mathcal{X}\mathcal{P}}(t)} - \Braket{ \hat{N}_{\mathcal{X}}(t) }\Braket{ \hat{N}_{\mathcal{P}}(t) } & \Braket{ \hat{N}_{\mathcal{P}}^2(t) } - \Braket{ \hat{N}_{\mathcal{P}}(t) }^2 \end{pmatrix} \equiv \begin{pmatrix} \delta_{\mathcal{X}}^2(t) & \delta_{\mathcal{X}\mathcal{P}}(t) \\ \delta_{\mathcal{X}\mathcal{P}}(t) & \delta_{\mathcal{P}}^2(t) \end{pmatrix}\end{aligned}$$ of the noise operators, where $$\begin{aligned}
\label{eq:noiseoperators:NXP}
\hat{N}_{\mathcal{X}\mathcal{P}}(t) \equiv \frac{1}{2} \left( \hat{N}_{\mathcal{X}}(t)\hat{N}_{\mathcal{P}}(t) + \hat{N}_{\mathcal{P}}(t)\hat{N}_{\mathcal{X}}(t) \right)\end{aligned}$$ stands for the symmetrized noise operator. To simplify our notation, we have also introduced the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$, which represent the diagonal elements of the covariance matrix, and the correlation term $\delta_{\mathcal{X}\mathcal{P}}(t)$, which represents the off-diagonal elements. Since a precise knowledge of $\mathbf{s}(t)$, Eqs. [(\[eq:s\])]{} and [(\[eq:noiseoperators:exp\])]{}, is necessary for the success of the inference process, Eq. [(\[eq:inferredexpecation\])]{}, the associated variances given by the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$ can be considered as an intuitive measure for the respective measurement uncertainty. Indeed, in the scope of variance-based uncertainty relations, the noise terms play an important role; see, e.g., Ref. [@ozawa2005] and references therein. Note that the noise terms are also referred to as “errors of retrodiction" [@appleby1998a; @appleby1998b; @appleby1998c]. We will revisit them further below, where we will see that they arise naturally in our entropic description of the measurement uncertainty.
The formal dynamics of the inferred observables $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{P}}(t)$, Eq. [(\[eq:inferredsystem\])]{}, build the framework for all of the following considerations. Most importantly, the intimate connection between $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{P}}(t)$, which result from the measured pointer observables in $\hat{\mathbf{w}}(t)$, Eq. [(\[eq:inferredobservables\])]{}, and the initial system observables $\hat{X}_{\mathrm{S}}(0)$ and $\hat{P}_{\mathrm{S}}(0)$ can clearly be seen from Eq. [(\[eq:inferredsystem\])]{}. It is this connection which lies at the heart of the pointer-based measurement scheme: information about the non-commuting system observables can be gathered from knowledge about the commuting inferred observables. At this point it seems natural to ask: What is the uncertainty of such an indirect measurement process? In the next section, we answer this question with the help of information entropy.
Entropy {#sec:entropy}
=======
The uncertainty principle [@heisenberg1927; @folland1997; @busch2007] is of central importance for quantum mechanical measurements. It manifests itself in the form of uncertainty relations [@busch2014], usually written in terms of variances [@busch2013a; @busch2013b] or information entropies [@buscemi2014; @coles2014]. The concept of information entropies goes back to Ref. [@shannon1948], whereas its general usage in the scope of uncertainty relations has been pioneered by Ref. [@hirschman1957; @beckner1975; @bialynicki1975]. In the context of closed pointer-based simultaneous measurements [@arthurs1965; @stenholm1992], the uncertainty principle leads to well-known variance-based uncertainty relations [@appleby1998a; @appleby1998b; @appleby1998c], which have extensively been discussed, e.g., in the scope of phase-space measurements [@wodkiewicz1984] or energy and timing considerations [@busshardt2010; @busshardt2011]. A respective information entropic form has been derived in Ref. [@buzek1995] and improved in Ref. [@heese2013]. Variances can also be used to describe an uncertainty relation for open pointer-based simultaneous measurements [@heese2014]. However, in comparison with information entropies, variances suffer from two major drawbacks [@hilgevoord1990; @buzek1995; @bialynicki2011]: First, they can become divergent for specific probability distributions and second, they may not reflect what one would intuitively consider as the “width" of a probability distribution. On the other hand, a disadvantage of information entropies is that they are usually much more difficult to calculate than variances. This, however, is only a technical limitation. Therefore, we concentrate on information entropic uncertainties in the following.
In this section, we first introduce the so-called collective entropy as a total measure of uncertainty in the context of open pointer-based simultaneous measurements. Choosing a separable initial state then allows us to calculate the marginal probability distributions this collective entropy is based on. As a result, we can discuss the structural properties of the collective entropy, including an extension of a previously known lower bound [@heese2013].
Collective entropy
------------------
First concepts of information entropies as an uncertainty measure for pointer-based simultaneous measurements can be found in Ref. [@buzek1995], which serves as a foundation for the present section. Since information entropies are based on probability distributions, we first need to define suitable probability distributions before we can deal with the actual entropies.
The probability of measuring an inferred position $\mathcal{X}$ and an inferred momentum $\mathcal{P}$ is given by the joint probability distribution [@wodkiewicz1984; @wodkiewicz1986] $$\begin{aligned}
\label{eq:prXP}
\mathrm{pr}(\mathcal{X},\mathcal{P};t) \equiv \Braket{ \delta(\mathcal{X}-\hat{\mathcal{X}}(t)) \delta(\mathcal{P}-\hat{\mathcal{P}}(t)) }\end{aligned}$$ with the Dirac delta distributions $\delta(\mathcal{X}-\hat{\mathcal{X}}(t))$ and $\delta(\mathcal{P}-\hat{\mathcal{P}}(t))$, which contain the inferred observables $\hat{\mathcal{X}}(t)$ and $\hat{\mathcal{P}}(t)$, Eq. [(\[eq:inferredobservables\])]{}, respectively. Likewise, the probability of measuring either the inferred position $\mathcal{X}$ or the inferred momentum $\mathcal{P}$ is given by the marginal probability distribution of inferred position
\[eq:Pmarginal\] $$\begin{aligned}
\mathrm{pr}_{\mathcal{X}}(\mathcal{X};t) \equiv \int \limits_{- \infty}^{+ \infty} \! \mathrm d \mathcal{P}\ \mathrm{pr}(\mathcal{X},\mathcal{P};t)
\end{aligned}$$ and inferred momentum $$\begin{aligned}
\mathrm{pr}_{\mathcal{P}}(\mathcal{P};t) \equiv \int \limits_{- \infty}^{+ \infty} \! \mathrm d \mathcal{X}\ \mathrm{pr}(\mathcal{X},\mathcal{P};t),
\end{aligned}$$
respectively.
With the help of the marginal probability distributions, Eq. [(\[eq:Pmarginal\])]{}, one can define the collective entropy [@heese2013] $$\begin{aligned}
\label{eq:S}
S(t) \equiv S_{\mathcal{X}}(t) + S_{\mathcal{P}}(t),\end{aligned}$$ which describes the total uncertainty of a simultaneous pointer-based measurement process. It consists of the sum of the marginal entropy of inferred position
\[eq:Smarginal\] $$\begin{aligned}
\label{eq:Smarginal:X}
S_{\mathcal{X}}(t) \equiv - \int \limits_{- \infty}^{+ \infty} \! \mathrm d \mathcal{X}\ \mathrm{pr}_{\mathcal{X}}(\mathcal{X};t) \ln \mathrm{pr}_{\mathcal{X}}(\mathcal{X};t)\end{aligned}$$ and the marginal entropy of inferred momentum $$\begin{aligned}
\label{eq:Smarginal:P}
S_{\mathcal{P}}(t) \equiv - \int \limits_{- \infty}^{+ \infty} \! \mathrm d \mathcal{P}\ \mathrm{pr}_{\mathcal{P}}(\mathcal{P};t) \ln \mathrm{pr}_{\mathcal{P}}(\mathcal{P};t).\end{aligned}$$
So far, our considerations have been completely general.
Separable initial state {#sec:entropy:separable initial state}
-----------------------
A more detailed calculation of the collective entropy, Eq. [(\[eq:S\])]{}, is only possible if we choose a more specific initial state $\hat{\varrho}(0)$ for the measurement configuration. First of all, it is a common approach to assume that the bath is initially in thermal equilibrium and separable from the system and the pointers [^1]. Furthermore, it seems natural to choose initially localized and separable pointer states. These localized pointer states are then propagated by the Hamiltonian, Eq. [(\[eq:H\])]{}, in such a way, Eq. [(\[eq:inferredobservables\])]{}, that their new location can be read out at end of the interaction process to infer the system observables, Eq. [(\[eq:inferredsystem\])]{}. A straightforward realization for localized and separable pointer states are squeezed vacuum states [@barnett1997; @schleich2001], which have already been used in the classic Arthurs and Kelly model [@arthurs1965]. Our system to be measured should, on the other hand, not be subject to any assumptions, so we describe it by a general density matrix.
Thus, for our model of open pointer-based simultaneous measurements, we choose a separable initial state $$\begin{aligned}
\label{eq:InitialState}
\hat{\varrho}(0) \equiv \hat{\varrho}_{\mathrm{S}}(0) \otimes \ket{\sigma_1} \bra{\sigma_1} \otimes \ket{\sigma_2} \bra{\sigma_2} \otimes \hat{\varrho}_{\mathrm{B}}(0).\end{aligned}$$ It consists of the initial state $\hat{\varrho}_{\mathrm{S}}(0)$ of the system to be measured, the squeezed vacuum states $\ket{\sigma_1}$ and $\ket{\sigma_2}$ of the pointers, which can be written as [@schleich2001] $$\begin{aligned}
\label{eq:initialpointerstate}
\braket{x | \sigma_k} \equiv \left( \frac{1}{2 \pi \sigma_k^2} \right)^{\frac{1}{4}} \exp \left[ - \frac{x^2}{4 \sigma_k^2} \right]\end{aligned}$$ in position space with variances $\sigma_k^2$ for $k \in \{1,2\}$, and the thermal state of the bath $$\begin{aligned}
\label{eq:initialbathstate}
\hat{\varrho}_{\mathrm{B}}(0) \equiv \frac{1}{Z} \exp \left[ - \beta \left\{ \frac{1}{2} \hat{\mathbf{k}}^{T} \mathbf{m}^{-1} \hat{\mathbf{k}}+ \frac{1}{2} \hat{\mathbf{q}}^{T} \mathbf{c} \hat{\mathbf{q}} \right\} \right]\end{aligned}$$ with the thermal energy $\beta^{-1}$ and the normalizing partition function $Z$.
The choice of a specific initial state of the bath allows us to determine the statistical properties of the stochastic force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. In particular, one has $$\begin{aligned}
\braket{\hat{\boldsymbol{\xi}}(t)} = 0\end{aligned}$$ and $$\begin{aligned}
\frac{1}{2} \braket{ \hat{\boldsymbol{\xi}}(t_1) \hat{\boldsymbol{\xi}}{}^{T}(t_2) + \hat{\boldsymbol{\xi}}(t_2) \hat{\boldsymbol{\xi}}{}^{T}(t_1) } \equiv g(t_1) g(t_2) \boldsymbol{\nu}(t_1-t_2)\end{aligned}$$ with the noise kernel [@fleming2011b] $$\begin{aligned}
\label{eq:nu}
\boldsymbol{\nu}(t) \equiv \frac{1}{2} \int \limits_{0}^{\infty} \! \mathrm{d} \omega \coth \left( \frac{\beta \omega}{2} \right) \cos ( \omega t ) \mathbf{I}(\omega),\end{aligned}$$ which contains the spectral density [@weiss1999] $$\begin{aligned}
\label{eq:I}
\mathbf{I}(\omega) \equiv \mathbf{g}^{T} \mathbf{m}^{-\frac{1}{2}} \omega^{-1} \delta( \omega \mathds{1} - \boldsymbol{\omega} ) \mathbf{m}^{-\frac{1}{2}} \mathbf{g}.\end{aligned}$$ Here we have made use of the Dirac delta distribution $\delta( \omega \mathds{1} - \boldsymbol{\omega} )$ with the identity matrix $\mathds{1}$ and the bath frequency matrix $\boldsymbol{\omega}$, Eq. [(\[eq:omega\])]{}. In brief, the noise kernel describes the noisy influence of the bath on the measurement process. Note that we do not further specify the structure of the spectral density [^2] and keep it as a general phenomenological expression. Nevertheless, we assume a continuous bath with $N\rightarrow\infty$ for which the definition of a spectral density makes sense in the first place.
We furthermore remark that for our chosen initial state $\hat{\varrho}(0)$, Eq. [(\[eq:InitialState\])]{}, the expectation value shift $\mathbf{s}(t)$, Eq. [(\[eq:s\])]{}, vanishes for all times, i.e., $$\begin{aligned}
\label{eq:s=0}
\mathbf{s}(t) = 0.\end{aligned}$$ This means that the first moments of the inferred observables after a certain interaction time $t$ directly correspond to the first moments of the initial system observables, Eq. [(\[eq:inferredexpecation\])]{}. As we have already stated above, this can be considered as a typical behavior.
Marginal probability distributions {#sec:entropy:marginal probability distributions}
----------------------------------
The choice of the initial state $\hat{\varrho}(0)$, Eq. [(\[eq:InitialState\])]{}, allows us to explicitly calculate the the marginal probability distributions, Eq. [(\[eq:Pmarginal\])]{}. It seems natural to assume that these marginal probability distributions do not directly correspond to the initial position distribution $\braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x}$ or the initial momentum distribution $\braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p}$ of the system, but should in addition also incorporate the disturbance from the indirect measurement via the pointers as well as the noisy effects of the bath. Since pointers and bath are initially in a Gaussian state, Eqs. [(\[eq:initialpointerstate\])]{} and [(\[eq:initialbathstate\])]{}, their influence on the marginal probability distributions can also be expected to have a Gaussian shape. Indeed, the calculations shown in \[sec:appendix:marginal probability distributions\] lead us to the broadened distributions
\[eq:Pmarginal2\] $$\begin{aligned}
\label{eq:Pmarginal2:X}
\mathrm{pr}_{\mathcal{X}}(\mathcal{X};t) = \frac{1}{\sqrt{2 \pi} \delta_{\mathcal{X}}(t)} \int \limits_{- \infty}^{+ \infty} \! \mathrm d x \braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x} \exp \left[ - \frac{(\mathcal{X}-x)^2}{2 \delta_{\mathcal{X}}^2(t)} \right]
\end{aligned}$$ and $$\begin{aligned}
\label{eq:Pmarginal2:P}
\mathrm{pr}_{\mathcal{P}}(\mathcal{P};t) = \frac{1}{\sqrt{2 \pi} \delta_{\mathcal{P}}(t)} \int \limits_{- \infty}^{+ \infty} \! \mathrm d p \braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p} \exp \left[ - \frac{(\mathcal{P}-p)^2}{2 \delta_{\mathcal{P}}^2(t)} \right],
\end{aligned}$$
respectively with the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$, which represent the diagonal elements of the covariance matrix of the noise operators, Eq. [(\[eq:noiseoperators:cov\])]{}. As also shown in \[sec:appendix:marginal probability distributions\], this covariance matrix can be expressed as $$\begin{aligned}
\label{eq:noiseoperators:cov:calc2}
\begin{pmatrix} \delta_{\mathcal{X}}^2(t) & \delta_{\mathcal{X}\mathcal{P}}(t) \\ \delta_{\mathcal{X}\mathcal{P}}(t) & \delta_{\mathcal{P}}^2(t) \end{pmatrix} = & \phantom{+.} \frac{1}{2} \mathbf{B}(t) \left( \braket{\hat{\mathbf{J}} \hat{\mathbf{J}}^{T}} + \braket{\hat{\mathbf{J}} \hat{\mathbf{J}}^{T}}^{T} \right) \mathbf{B}^{T}(t) \nonumber \\
& + \int \limits_{0}^{t} \! \mathrm d t_1 \int \limits_{0}^{t} \! \mathrm d t_2 g(t_1) g(t_2) \mathbf{\Lambda}(t,t_1) \boldsymbol{\nu}(t_1-t_2) \mathbf{\Lambda}^{T} (t,t_2).\end{aligned}$$ Here we have recalled the time-dependent coupling strength of the bath $g(t)$, Eq. [(\[eq:g\])]{}, the coefficient matrices $\mathbf{B}(t)$ and $\mathbf{\Lambda}(t,s)$ from the dynamics of the inferred observables, Eq. [(\[eq:inferredsystem\])]{}, the initial value vector $\hat{\mathbf{J}}$, Eq. [(\[eq:J\])]{}, and the noise kernel $\boldsymbol{\nu}(t)$, Eq. [(\[eq:nu\])]{}. Note that the non-diagonal terms in Eq. [(\[eq:noiseoperators:cov:calc2\])]{} correspond to the correlation term $\delta_{\mathcal{X}\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators:cov\])]{}, which is of no further importance in the following.
Briefly summarized, the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, both consist of a convolution of the system’s initial probability distributions with a Gaussian noise function. This noise function itself is determined by a convolution of a Gaussian pointer noise function, Eq. [(\[eq:FphiResult\])]{}, with a Gaussian bath noise function, Eq. [(\[eq:FbathResult\])]{}. In other words, both the pointers and the bath act as a Gaussian filter [@buzek1995] through which we are forced to look during the measurement process, and which leave us with a distorted image of the initial probability distributions of the system. The noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$ give a description for this combined disturbance of the measurement results from the pointers and the bath. Since we use Gaussian-shaped initial states for the pointers and the bath, Eqs. [(\[eq:initialpointerstate\])]{} and [(\[eq:initialbathstate\])]{}, which are fully characterized by their second moments, the noise terms also contain only second moments: The first term on the right-hand side of Eq. [(\[eq:noiseoperators:cov:calc2\])]{} describes the pointer-based variance contributions to the noise terms, whereas the second term describes the bath-based variance contributions. Both increased pointer-based variance contributions and increased bath-based variance contributions increase the effective disturbance of the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}. In this sense, the noise terms can also be considered as “extrinsic" or “measurement uncertainties" in comparison with the “intrinsic" or “preparation uncertainties" of the system given by $\braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x}$ and $\braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p}$, respectively. This classification of the disturbance is a concept similarly used for variances as uncertainties of pointer-based simultaneous measurements [@appleby1998a; @heese2014].
An optimal measurement configuration for a pointer-based measurement is consequently defined by a minimal noise term product. In \[sec:appendix:minimal noise term product\] we show that the product of the noise terms is bound from below by [^3] $$\begin{aligned}
\label{ineq:dXdP}
\delta_{\mathcal{X}}(t) \delta_{\mathcal{P}}(t) \geq \frac{1}{2},\end{aligned}$$ which defines the best possible accuracy of any measurement apparatus. This statement is closely related to the fact that the variance-based uncertainty of a pointer-based measurement is bound from below by one, see, e.g., Ref. [@arthurs1965; @arthurs1988; @busshardt2010; @heese2014].
Lower bound of the collective entropy
-------------------------------------
In Ref. [@heese2013] we have discussed the collective entropy for closed pointer-based simultaneous measurements with pure initial system states and, with the help of Ref. [@lieb1978], have established a lower bound of this collective entropy. Interestingly, although we have used an open-pointer based measurement and possibly mixed initial system states in the present manuscript, the structure of the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, which determine the collective entropy, Eq. [(\[eq:S\])]{}, is similar to the structure of the marginal probability distribution for closed pointer-based simultaneous measurements. Therefore, with only slight changes, we can adapt the derivation of a lower bound of the collective entropy of closed pointer-based simultaneous measurements to the collective entropy of open pointer-based simultaneous measurements. In the following, we will briefly recapitulate this derivation.
First of all, we recall a theorem from Ref. [@lieb1978], which states that the information entropy $$\begin{aligned}
\label{eq:Lieb:S}
S[ f ] \equiv - \int \limits_{- \infty}^{+ \infty} \! \mathrm d x\ f(x) \ln f(x)\end{aligned}$$ of a Fourier convolution $$\begin{aligned}
\label{eq:ast}
(f \ast g)(x) \equiv \int \limits_{- \infty}^{+ \infty} \! \mathrm d y f(x-y) g(y)\end{aligned}$$ of two probability distributions $f(x)$ and $g(x)$ is bound from below by $$\begin{aligned}
\label{ineq:Lieb:Theorem}
S[ f * g ] \geq \lambda S[ f ] + ( 1 - \lambda ) S[ g ] - \frac{\lambda \ln \lambda + (1 - \lambda) \ln ( 1 - \lambda )}{2}\end{aligned}$$ with an arbitrary weighting parameter $\lambda \in [0,1]$. Equality in Ineq. [(\[ineq:Lieb:Theorem\])]{} holds true if and only if both $f(x)$ and $g(x)$ are Gaussian distributions with variances $\sigma_f^2$ and $\sigma_g^2$, respectively, and the weighting parameter reads $$\begin{aligned}
\label{eqn:GaussianParameter}
\lambda = \frac{\sigma_f^2}{\sigma_f^2+\sigma_g^2}.\end{aligned}$$ It is clear that the marginal entropies, Eq. [(\[eq:Smarginal\])]{}, which contain the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, also represent information entropies of convolutions as given by the left-hand side of Ineq. [(\[ineq:Lieb:Theorem\])]{}. Therefore, we can apply the lower bound of Ineq. [(\[ineq:Lieb:Theorem\])]{} to each of the marginal entropies in the collective entropy $S(t)$, Eq. [(\[eq:S\])]{}. In particular, we insert the initial position and momentum distributions of the system in place of $f(x)$ and their associated Gaussian filter functions in place of $g(x)$. For reasons of simplicity, we use the same weighting parameter $\lambda$ for both of these lower bounds. In a next step, we can eliminate the dependency on the system to be measured by making use of the entropic uncertainty relation [@everett1956; @hirschman1957; @bialynicki1975; @bialynicki2006] $$\begin{aligned}
\label{ineq:Hirsch}
S[\braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x}] + S[\braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p}] \geq 1 + \ln \pi.\end{aligned}$$ It is based on the Babenko-Beckner inequality [@babenko1961; @beckner1975]. Saturation in Ineq. [(\[ineq:Hirsch\])]{} occurs only for pure Gaussian states [@lieb1990; @oezaydin2004]. For a detailed discussion of this and similar entropic uncertainty relations, also see Ref. [@bialynicki2011]. The usage of Ineq. [(\[ineq:Hirsch\])]{}, which contains the density matrix $\hat{\varrho}_{\mathrm{S}}(0)$ of the system to be measured, is the main difference to the derivation from Ref. [@heese2013], where the system to be measured was limited to pure states and a simplified form of Ineq. [(\[ineq:Hirsch\])]{} with pure states had been used.
Accordingly, we arrive at the lower bound $$\begin{aligned}
\label{ineq:Lieb:SingleParam}
S(t) \geq 1 - \lambda \ln \frac{\lambda}{\pi} + (1-\lambda) \ln \left( \frac{2 \pi \delta_{\mathcal{X}}(t) \delta_{\mathcal{P}}(t) }{1-\lambda} \right)\end{aligned}$$ of the collective entropy $S(t)$, Eq. [(\[eq:S\])]{}. In particular, this lower bound depends on the noise terms $\delta_{\mathcal{X}}(t)$ and $\delta_{\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators:cov:calc2\])]{}. A maximization of the right-hand side of Ineq. [(\[ineq:Lieb:SingleParam\])]{} with respect to $\lambda$ reveals the optimal weighting parameter $\lambda = 1/(1+2\delta_{\mathcal{X}}(t) \delta_{\mathcal{P}}(t))$ and finally leads us to the result $$\begin{aligned}
\label{ineq:eur}
S(t) \geq 1 + \ln \bigg[ 2 \pi \Big( \delta_{\mathcal{X}}(t) \delta_{\mathcal{P}}(t) + \frac{1}{2} \Big) \bigg].\end{aligned}$$ This entropic uncertainty bound for open pointer-based simultaneous measurements is of the same form as the entropic uncertainty bound for closed pointer-based simultaneous measurements from Ref. [@heese2013]. Moreover, it is an extension of the more well-known entropic uncertainty bound $S(t) \geq 1 + \ln (2 \pi)$ from Ref. [@buzek1995], to which it can be reduced in case of minimal noise terms, Ineq. [(\[ineq:dXdP\])]{}.
Due to the equality conditions of Ineqs. [(\[ineq:Lieb:Theorem\])]{} and [(\[ineq:Hirsch\])]{}, equality in Ineq. [(\[ineq:eur\])]{} occurs only for initial system states $\hat{\varrho}_{\mathrm{S}}(0)$, which are minimal uncertainty states in the sense of Heisenberg’s uncertainty relation [@ballentine1998], i.e., pure Gaussian states for which the position variance $\sigma_{x}^2$ and the momentum variance $\sigma_{p}^2$ obey $\sigma_{x}^2 \sigma_{p}^2 = 1/4$; and which additionally fulfill $\sigma_{x}^2 = \delta_{\mathcal{X}}(t)/(2 \delta_{\mathcal{P}}(t))$ for a given set of noise terms [@busshardt2011]. Such initial system states can be understood as “minimal entropy states" [@heese2013]. Note that mixed system states with Gaussian density matrices [@mann1993] are not sufficient for equality.
Summarized, the collective entropy of an open pointer-based measurement, Eq. [(\[eq:S\])]{}, behaves exactly like the collective entropy of a closed pointer-based measurement. Particularly, the collective entropy is bound from below by the sharp entropic uncertainty bound given by Ineq. [(\[ineq:eur\])]{}. The only effect of the environmental heat bath is to modify the collective entropy by modifying the noise terms, Eq. [(\[eq:noiseoperators:cov:calc2\])]{}. An optimal measurement accuracy can be achieved for minimal noise terms, Ineq. [(\[ineq:dXdP\])]{}, and requires a pure Gaussian system state with a specific variance which leads to equality in Ineq. [(\[ineq:eur\])]{}.
Conclusion
==========
We have shown that it is possible to determine a formal expression for the collective entropy of a general open pointer-based simultaneous measurement. This collective entropy has a very intuitive structure from which the noisy influence of the pointers and the bath on the measurement result can be understood. Moreover, this structure allows us to show that the collective entropy has a sharp lower bound, which can only be reached for specific pure Gaussian system states. In particular, our results are valid for any bilinear interaction between the system and the pointers and any initially mixed system state and thus extend various previous results on this topic.
Several simplifications have been made to perform our calculations. First of all, the Hamiltonian only includes bilinear terms. However, we expect that terms of higher order would only lead to relatively small correction terms without fundamentally changing our results. Second, ideal single variable measurements of the pointer observables have to be performed in order to realize the measurement procedure, which is an inherent conceptional weakness of pointer-based simultaneous measurements. Yet we hope that with the help of the environmental heat bath, it might be possible to replace this rather theoretical measurement process by a more natural decoherence process [@zurek2003]. Finally, we have used a separable initial state with squeezed states as the initial pointer states and a thermal state as the initial bath state. Although this approach is a clear limitation of our results, we think that our choice is reasonable. It would nevertheless be interesting to discuss different initial states and specifically examine the influence of entanglement and preparation energy on the collective entropy.
Acknowledgement {#acknowledgement .unnumbered}
===============
R. H. gratefully acknowledges a grant from the Landesgraduiertenf[ö]{}rderungsgesetz of the state of Baden-W[ü]{}rttemberg.
Marginal probability distributions {#sec:appendix:marginal probability distributions}
==================================
In this appendix section we briefly describe how to calculate the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, from the definition, Eqs. [(\[eq:prXP\])]{} and [(\[eq:Pmarginal\])]{}, with the help of the chosen initial state, Eq. [(\[eq:InitialState\])]{}. Our calculations are mainly based on characteristic functions and their Fourier transforms.
Inserting the formal dynamics of the inferred observables, Eq. [(\[eq:inferredsystem\])]{}, into the joint probability distribution, Eq. [(\[eq:prXP\])]{}, leads to $$\begin{aligned}
\label{eq:prXP0}
\mathrm{pr}(\mathcal{X},\mathcal{P};t) = \operatorname{tr} \Big\{ & \hat{\varrho}(0) \delta \left[ \mathcal{X} - \hat{X}_{\mathrm{S}}(0) - (1,0) \mathbf{B}(t) \hat{\mathbf{J}} - (1,0) (\mathbf{\Lambda} \star \hat{\boldsymbol{\xi}})(t) \right] \nonumber \\
& \phantom{..} \times \delta \left[ \mathcal{P} - \hat{P}_{\mathrm{S}}(0) - (0,1) \mathbf{B}(t) \hat{\mathbf{J}} - (0,1) (\mathbf{\Lambda} \star \hat{\boldsymbol{\xi}})(t) \right] \Big\}.\end{aligned}$$ The Dirac delta distributions in Eq. [(\[eq:prXP0\])]{} can be expressed as Fourier transforms of unity, i.e., $\delta(x) = \mathcal{F} \left\{ 1 \right\}(x)$. Here we use the notation $$\begin{aligned}
\label{eq:FT}
\mathcal{F} \left\{f\right\}(x) \equiv \frac{1}{2 \pi} \int \limits_{- \infty}^{+ \infty} \! \mathrm d \alpha\ \exp[ \operatorname{i} \alpha x ] f(\alpha)\end{aligned}$$ for the Fourier transform of an arbitrary function $f(x)$, and an analogous notation for Fourier transforms of functions of two variables. Thus, the separability of the initial state $\hat{\varrho}(0)$, Eq. [(\[eq:InitialState\])]{}, allows to rewrite Eq. [(\[eq:prXP0\])]{} as $$\begin{aligned}
\label{eq:prXP1}
\mathrm{pr}(\mathcal{X},\mathcal{P};t) = \mathcal{F}\left\{ F_{\mathrm{S}} F_{\mathrm{P}} F_{\mathrm{B}} \right\}(\mathcal{X},\mathcal{P}) = \left( \mathcal{F}\left\{F_{\mathrm{S}}\right\} \ast \mathcal{F}\left\{F_{\mathrm{P}}\right\} \ast \mathcal{F}\left\{F_{\mathrm{B}}\right\} \right)(\mathcal{X},\mathcal{P})\end{aligned}$$ with the characteristic function of the system $$\begin{aligned}
\label{eq:Fpsi}
F_{\mathrm{S}}(\alpha_1,\alpha_2) \equiv \operatorname{tr} \left\{ \hat{\varrho}_{\mathrm{S}}(0) \exp[ -\operatorname{i} \{ \alpha_1 \hat{X}_{\mathrm{S}}(0) + \alpha_2 \hat{P}_{\mathrm{S}}(0) \} ] \right\},\end{aligned}$$ the characteristic function of the pointers $$\begin{aligned}
\label{eq:Fphi}
F_{\mathrm{P}}(\alpha_1,\alpha_2) \equiv \bra{\sigma_1} \otimes \bra{\sigma_2} \exp[ -\operatorname{i} \{ (\alpha_1,\alpha_2) \mathbf{B}(t) \hat{\mathbf{J}} \} ] \ket{\sigma_1} \otimes \ket{\sigma_2},\end{aligned}$$ and the characteristic function of the bath $$\begin{aligned}
\label{eq:Fbath}
F_{\mathrm{B}}(\alpha_1,\alpha_2) \equiv \operatorname{tr} \left\{ \hat{\varrho}_{\mathrm{B}}(0) \exp[ -\operatorname{i} \{ (\alpha_1,\alpha_2) (\mathbf{\Lambda} \star \hat{\boldsymbol{\xi}})(t) \} ] \right\}.\end{aligned}$$ The symbols $\star$ and $\ast$ are defined in Eqs. [(\[eq:star\])]{} and [(\[eq:ast\])]{}, respectively.
In the following, we calculate the Fourier transforms of the characteristic functions, Eqs. [(\[eq:Fpsi\])]{} to [(\[eq:Fbath\])]{}, one after another with the help of Wigner functions [@schleich2001]. This allows us to explicitly perform the convolution in Eq. [(\[eq:prXP1\])]{}. By straightforward integration of the resulting Gaussian expressions we finally arrive at the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}.
System
------
First of all, the Fourier transform of the characteristic function of the system, Eq. [(\[eq:Fpsi\])]{}, corresponds to the respective Wigner function $$\begin{aligned}
\label{eq:FpsiResult}
W_{\mathrm{S}}(x,p) = \mathcal{F}\left\{F_{\mathrm{S}}\right\}(x,p)\end{aligned}$$ of the system state. We can also use this connection between characteristic function and Wigner function to determine the Fourier transforms of the other two characteristic functions, Eqs. [(\[eq:Fphi\])]{} and [(\[eq:Fbath\])]{}, in the following.
Pointers
--------
The Wigner function $W_{\sigma}$ of a squeezed vacuum state $\ket{\sigma}$ with position variance $\sigma^2$ can be written as $$\begin{aligned}
\label{eq:Wsigma}
W_{\sigma}(x,p) \equiv \mathcal{G}\left(-\frac{1}{2 \sigma^2},-2 \sigma^2,0,\frac{1}{\pi};x,p\right),\end{aligned}$$ where we have introduced the general Gaussian function $$\begin{aligned}
\label{eq:Gaussian}
\mathcal{G}(a,b,c,n;x,p) \equiv n \exp \left[ a x^2 + b p^2 + c x p \right].\end{aligned}$$ Equation [(\[eq:Wsigma\])]{} corresponds to the Fourier transform of the generic characteristic function $$\begin{aligned}
F_{\sigma}'(\alpha_1,\alpha_2) \equiv \braket{ \sigma | \exp[ -\operatorname{i} ( \alpha_1 \hat{x} + \alpha_2 \hat{p} ) ] | \sigma }\end{aligned}$$ of squeezed vacuum states as we have used them for the pointer states, Eq. [(\[eq:initialpointerstate\])]{}, where $\hat{x}$ and $\hat{p}$ stand for the position and momentum observables, respectively, in the Hilbert space of $\ket{\sigma}$. Looking at this relation the other way round yields $$\begin{aligned}
F_{\sigma}'(x,p) = \mathcal{F}^{-1}\left\{ W_{\sigma} \right\}(x,p) = \mathcal{G}\left(-\frac{\sigma^2}{2},-\frac{1}{8 \sigma^2},0,1;x,p\right),\end{aligned}$$ where $\mathcal{F}^{-1}\left\{f\right\}(x)$ denotes the inverse Fourier transform of an arbitrary function $f(x)$ with $\mathcal{F}\left\{\mathcal{F}^{-1}\left\{f\right\}\right\}(x) = f(x)$, Eq. [(\[eq:FT\])]{}. Thus, the characteristic function of the pointers, Eq. [(\[eq:Fphi\])]{}, can be written as $$\begin{aligned}
F_{\mathrm{P}}(\alpha_1,\alpha_2) & = F_{\sigma_1}'( (\alpha_1,\alpha_2) \mathbf{B}(t) (1,0,0,0)^{T} , (\alpha_1,\alpha_2) \mathbf{B}(t) (0,0,1,0)^{T} ) \nonumber \\
& \phantom{=.} \times F_{\sigma_2}'( (\alpha_1,\alpha_2) \mathbf{B}(t) (0,1,0,0)^{T} , (\alpha_1,\alpha_2) \mathbf{B}(t) (0,0,0,1)^{T} )\end{aligned}$$ and one has $$\begin{aligned}
\label{eq:FphiResult}
\mathcal{F}\left\{F_{\mathrm{P}}\right\}(x,p) = \mathcal{G}\left(\frac{a_{\mathrm{P}}(t)}{d_{\mathrm{P}}(t)},\frac{b_{\mathrm{P}}(t)}{d_{\mathrm{P}}(t)},\frac{c_{\mathrm{P}}(t)}{d_{\mathrm{P}}(t)},\frac{1}{2 \pi \sqrt{ d_{\mathrm{P}}(t) }};x,p\right) \end{aligned}$$ with the coefficients
$$\begin{aligned}
\label{eq:abcp}
\begin{pmatrix} -2 b_{\mathrm{P}}(t) & c_{\mathrm{P}}(t) \\ c_{\mathrm{P}}(t) & -2 a_{\mathrm{P}}(t) \end{pmatrix} \equiv \mathbf{B}(t) \mathbf{V} \mathbf{B}^{T}(t)\end{aligned}$$
and $$\begin{aligned}
d_{\mathrm{P}}(t) \equiv 4 a_{\mathrm{P}}(t) b_{\mathrm{P}}(t) - c_{\mathrm{P}}^2(t),\end{aligned}$$
where $$\begin{aligned}
\label{eq:JJ}
\mathbf{V} \equiv \frac{1}{2} \left( \braket{\hat{\mathbf{J}} \hat{\mathbf{J}}^{T}} + \braket{\hat{\mathbf{J}} \hat{\mathbf{J}}^{T}}^{T} \right) = \begin{pmatrix} \sigma_1^2 & 0 & 0 & 0 \\ 0 & \sigma_2^2 & 0 & 0 \\ 0 & 0 & \frac{1}{4 \sigma_1^2} & 0 \\ 0 & 0 & 0 & \frac{1}{4 \sigma_2^2} \end{pmatrix}\end{aligned}$$ denotes the symmetrized pointer covariance matrix, which directly follows from the definition of the initial value vector $\hat{\mathbf{J}}$, Eq. [(\[eq:J\])]{}, and the initial pointer states, Eq. [(\[eq:initialpointerstate\])]{}. In conclusion, Eq. [(\[eq:FphiResult\])]{} can be calculated solely from the coefficient matrix $\mathbf{B}(t)$, Eq. [(\[eq:inferredsystem\])]{}, and the initial pointer position variances $\sigma_1^2$ and $\sigma_2^2$, Eq. [(\[eq:initialpointerstate\])]{}.
Bath
----
The Wigner function of the thermal bath state $\hat{\varrho}_{\mathrm{B}}(0)$, Eq. [(\[eq:initialbathstate\])]{}, reads $$\begin{aligned}
W_{\mathrm{B}}(\mathbf{q},\mathbf{k}) & \equiv \operatorname{det} \left[ \tanh \left( \frac{\boldsymbol{\omega} \beta}{2} \right) \pi^{-1} \right] \nonumber \\
& \phantom{\equiv.} \times \exp \Bigg[ - \mathbf{q}^{T} \mathbf{m}^{\frac{1}{2}} \tanh \left( \frac{\boldsymbol{\omega} \beta}{2} \right) \boldsymbol{\omega} \mathbf{m}^{\frac{1}{2}} \mathbf{q} \nonumber \\
& \phantom{\equiv \times \exp \Bigg[.} - \mathbf{k}^{T} \mathbf{m}^{-\frac{1}{2}} \tanh \left( \frac{\boldsymbol{\omega} \beta}{2} \right) \boldsymbol{\omega}^{-1} \mathbf{m}^{-\frac{1}{2}} \mathbf{k} \Bigg].\end{aligned}$$ Analogously to the calculation of the characteristic function of the pointers, the generic characteristic bath function $$\begin{aligned}
F_{\mathrm{B}}'(\boldsymbol{\alpha}_{\mathbf{1}},\boldsymbol{\alpha}_{\mathbf{2}}) \equiv \operatorname{tr} \left\{ \hat{\varrho}_{\mathrm{B}}(0) \exp[ -\operatorname{i} ( \boldsymbol{\alpha}_{\mathbf{1}}^T \hat{\mathbf{q}} + \boldsymbol{\alpha}_{\mathbf{2}}^T \hat{\mathbf{k}} ) ] \right\}\end{aligned}$$ of baths in thermal equilibrium, which is connected to the characteristic bath function, Eq. [(\[eq:Fbath\])]{}, via $$\begin{aligned}
\label{eq:Fbath1}
F_{\mathrm{B}}(\alpha_1,\alpha_2) = F_{\mathrm{B}}' \left( (\mathbf{\Lambda} \star \mathbf{u_q})^T(t) \left( \alpha_1, \alpha_2 \right)^T, (\mathbf{\Lambda} \star \mathbf{u_k})^T(t) \left( \alpha_1, \alpha_2 \right)^T \right),\end{aligned}$$ can be written as $$\begin{aligned}
F_{\mathrm{B}}'(\mathbf{q},\mathbf{k}) & = \mathcal{F}^{-1}\left\{ W_{\mathrm{B}} \right\}(\mathbf{q},\mathbf{k}) \nonumber \\
& = \exp \Bigg[ - \frac{1}{4} \mathbf{q}^{T} \mathbf{m}^{-\frac{1}{2}} \coth \left( \frac{\boldsymbol{\omega} \beta}{2} \right) \boldsymbol{\omega}^{-1} \mathbf{m}^{-\frac{1}{2}} \mathbf{q} \nonumber \\
& \phantom{= \exp \Bigg[} - \frac{1}{4} \mathbf{k}^{T} \mathbf{m}^{\frac{1}{2}} \coth \left( \frac{\boldsymbol{\omega} \beta}{2} \right) \boldsymbol{\omega} \mathbf{m}^{\frac{1}{2}} \mathbf{k} \Bigg].\end{aligned}$$ Here we have made use of the abbreviations $$\begin{aligned}
\mathbf{u_q}(t) \equiv - \mathbf{g}^{T}(t) \mathbf{m}^{-\frac{1}{2}} \cos(\boldsymbol{\omega} t) \mathbf{m}^{\frac{1}{2}}\end{aligned}$$ and $$\begin{aligned}
\mathbf{u_k}(t) \equiv - \mathbf{g}^{T}(t) \mathbf{m}^{-\frac{1}{2}} \sin(\boldsymbol{\omega} t) \boldsymbol{\omega}^{-1} \mathbf{m}^{-\frac{1}{2}}.\end{aligned}$$ Explicitly performing the Fourier transform of Eq. [(\[eq:Fbath1\])]{} leads to $$\begin{aligned}
\label{eq:FbathResult}
\mathcal{F}\left\{F_{\mathrm{B}}\right\}(x,p) = \mathcal{G}\left(\frac{a_{\mathrm{B}}(t)}{d_{\mathrm{B}}(t)},\frac{b_{\mathrm{B}}(t)}{d_{\mathrm{B}}(t)},\frac{c_{\mathrm{B}}(t)}{d_{\mathrm{B}}(t)},\frac{1}{2 \pi \sqrt{ d_{\mathrm{B}}(t) }};x,p\right) \end{aligned}$$ with the coefficients
$$\begin{aligned}
\label{eq:abcb}
\begin{pmatrix} - 2 b_{\mathrm{B}}(t) & c_{\mathrm{B}}(t) \\ c_{\mathrm{B}}(t) & - 2 a_{\mathrm{B}}(t) \end{pmatrix} \equiv ( \mathbf{\Lambda} \star \mathbf{v} )(t) ( \mathbf{\Lambda} \star \mathbf{v} )^T(t)\end{aligned}$$
and $$\begin{aligned}
d_{\mathrm{B}}(t) \equiv 4 a_{\mathrm{B}}(t) b_{\mathrm{B}}(t) - c_{\mathrm{B}}^2(t),\end{aligned}$$
where $$\begin{aligned}
\label{eq:vv}
\mathbf{v}(t_1) \mathbf{v}^{T}(t_2) \equiv g(t_1) g(t_2) \boldsymbol{\nu}(t_1-t_2)\end{aligned}$$ with the noise kernel $\boldsymbol{\nu}(t)$, Eq. [(\[eq:nu\])]{}.
As a result, Eq. [(\[eq:FbathResult\])]{} is determined by the coefficient matrix $\mathbf{\Lambda}(t,s)$, Eq. [(\[eq:inferredsystem\])]{}, and the noise kernel $\boldsymbol{\nu}(t)$, Eq. [(\[eq:nu\])]{}. Note that for a turned off bath (i.e., $\mathbf{g}(t) = 0$ for all $t\geq0$), one has $d_{\mathrm{B}}(t) = 0$ and thus Eq. [(\[eq:FbathResult\])]{} is not well-defined. However, Eq. [(\[eq:FbathResult\])]{} can in this case be understood as a Dirac delta distribution so that the following considerations are still applicable with $a_{\mathrm{B}}(t) = 0$, $b_{\mathrm{B}}(t) = 0$ and $c_{\mathrm{B}}(t) = 0$.
Coefficients
------------
In a next step, we show how the pointer coefficients $a_{\mathrm{P}}(t)$, $b_{\mathrm{P}}(t)$ and $c_{\mathrm{P}}(t)$, Eq. [(\[eq:abcp\])]{}, and the bath coefficients $a_{\mathrm{B}}(t)$, $b_{\mathrm{B}}(t)$, and $c_{\mathrm{B}}(t)$, Eq. [(\[eq:abcb\])]{}, are related to the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$ and the correlation term $\delta_{\mathcal{X}\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators:cov\])]{}. For this purpose, we recall the noise operators $\hat{N}_{\mathcal{X}}(t)$ and $\hat{N}_{\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators\])]{}. According to Eqs. [(\[eq:noiseoperators:exp\])]{} and [(\[eq:s=0\])]{}, we can simplify their covariance matrix, Eq. [(\[eq:noiseoperators:cov\])]{}, to $$\begin{aligned}
\label{eq:noiseoperators:cov2}
\begin{pmatrix} \delta_{\mathcal{X}}^2(t) & \delta_{\mathcal{X}\mathcal{P}}(t) \\ \delta_{\mathcal{X}\mathcal{P}}(t) & \delta_{\mathcal{P}}^2(t) \end{pmatrix} = \Braket{ \begin{pmatrix} \hat{N}_{\mathcal{X}}^2(t) & \hat{N}_{\mathcal{X}\mathcal{P}}(t) \\ \hat{N}_{\mathcal{X}\mathcal{P}}(t) & \hat{N}_{\mathcal{P}}^2(t) \end{pmatrix} }.\end{aligned}$$ Here we have also recalled the symmetrized noise operator $\hat{N}_{\mathcal{X}\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators:NXP\])]{}. A straightforward calculation using the definition of the noise operators, Eq. [(\[eq:noiseoperators\])]{}, shows that $$\begin{aligned}
\label{eq:noiseoperators:cov:calc}
\begin{pmatrix} \delta_{\mathcal{X}}^2(t) & \delta_{\mathcal{X}\mathcal{P}}(t) \\ \delta_{\mathcal{X}\mathcal{P}}(t) & \delta_{\mathcal{P}}^2(t) \end{pmatrix} = \mathbf{B}(t) \mathbf{V} \mathbf{B}^{T}(t) + ( \mathbf{\Lambda} \star \mathbf{v} )(t) ( \mathbf{\Lambda} \star \mathbf{v} )^T(t)\end{aligned}$$ with the abbreviations introduced in Eqs. [(\[eq:JJ\])]{} and [(\[eq:vv\])]{}. Here we have also made use of the symmetry relation [@fleming2011b] $$\begin{aligned}
\braket{ \hat{\boldsymbol{\xi}}(t_1) \hat{\boldsymbol{\xi}}{}^{T}(t_2) } = \braket{ \hat{\boldsymbol{\xi}}(t_1) \hat{\boldsymbol{\xi}}{}^{T}(t_2) }^{T}\end{aligned}$$ for the stochastic force $\hat{\boldsymbol{\xi}}(t)$, Eq. [(\[eq:stochasticforce\])]{}. A comparison of Eq. [(\[eq:noiseoperators:cov:calc\])]{} with Eqs. [(\[eq:abcp\])]{} and [(\[eq:abcb\])]{} immediately reveals $$\begin{aligned}
\label{eq:noiseterms}
\begin{pmatrix} \delta_{\mathcal{X}}^2(t) & \delta_{\mathcal{X}\mathcal{P}}(t) \\ \delta_{\mathcal{X}\mathcal{P}}(t) & \delta_{\mathcal{P}}^2(t) \end{pmatrix} = \begin{pmatrix} - 2 ( b_{\mathrm{P}}(t) + b_{\mathrm{B}}(t) ) & c_{\mathrm{P}}(t) + c_{\mathrm{B}}(t) \\ c_{\mathrm{P}}(t) + c_{\mathrm{B}}(t) & - 2 ( a_{\mathrm{P}}(t) + a_{\mathrm{B}}(t) ) \end{pmatrix},\end{aligned}$$ which relates the noise and correlation terms on the left-hand side with the pointer and bath coefficients on the right-hand side.
Probability distributions
-------------------------
At this point, we can collect our previous results. By inserting Eqs. [(\[eq:FpsiResult\])]{}, [(\[eq:FphiResult\])]{} and [(\[eq:FbathResult\])]{} into Eq. [(\[eq:prXP1\])]{}, we arrive at the final expression for the joint probability distribution $$\begin{aligned}
\label{eq:prXP2}
\mathrm{pr}(\mathcal{X},\mathcal{P};t) = \left( W_{\mathrm{S}} \ast \mathcal{G}\left(-\frac{1}{2 \Delta_{\mathcal{X}}^2(t)},-\frac{1}{2 \Delta_{\mathcal{P}}^2(t)},\gamma(t),\frac{1}{2 \pi \sqrt{d(t)}}\right) \right) (\mathcal{X},\mathcal{P})\end{aligned}$$ with the coefficients
\[eq:coefficients\] $$\begin{aligned}
\Delta_{\mathcal{X}}^2(t) \equiv \frac{-d(t)}{2 ( a_{\mathrm{P}}(t) + a_{\mathrm{B}}(t) ) },\end{aligned}$$ $$\begin{aligned}
\Delta_{\mathcal{P}}^2(t) \equiv \frac{-d(t)}{2 ( b_{\mathrm{P}}(t) + b_{\mathrm{B}}(t) ) },\end{aligned}$$ $$\begin{aligned}
\gamma(t) \equiv \frac{ c_{\mathrm{P}}(t) + c_{\mathrm{B}}(t) }{ d(t) },\end{aligned}$$ and $$\begin{aligned}
d(t) \equiv 4 ( a_{\mathrm{P}}(t) + a_{\mathrm{B}}(t) ) ( b_{\mathrm{P}}(t) + b_{\mathrm{B}}(t) ) - ( c_{\mathrm{P}}(t) + c_{\mathrm{B}}(t) )^2.\end{aligned}$$
Finally, the marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, follow from Eq. [(\[eq:prXP2\])]{} by straightforward integration as defined in Eq. [(\[eq:Pmarginal\])]{} when we make use of the marginals
$$\begin{aligned}
\int \limits_{- \infty}^{+ \infty} \! \mathrm d p W_{\mathrm{S}}(x,p) = \braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x}\end{aligned}$$
and $$\begin{aligned}
\int \limits_{- \infty}^{+ \infty} \! \mathrm d x W_{\mathrm{S}}(x,p) = \braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p}\end{aligned}$$
of the initial Wigner function of the system $W_{\mathrm{S}}(x,p)$, Eq. [(\[eq:FpsiResult\])]{}, where $\braket{x|\hat{\varrho}_{\mathrm{S}}(0)|x}$ and $\braket{p|\hat{\varrho}_{\mathrm{S}}(0)|p}$ stand for the initial position distribution and the initial momentum distribution of the system state, respectively. In particular, the resulting marginal probability distributions, Eq. [(\[eq:Pmarginal2\])]{}, contain the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{P}}^2(t)$, Eq. [(\[eq:noiseterms\])]{}.
Minimal noise term product {#sec:appendix:minimal noise term product}
==========================
We show in this appendix section that the product of the noise terms $\delta_{\mathcal{X}}^2(t)$ and $\delta_{\mathcal{X}}^2(t)$, Eq. [(\[eq:noiseoperators:cov\])]{}, obeys Ineq. [(\[ineq:dXdP\])]{}. Our proof is closely related to the considerations in Ref. [@arthurs1988]. Due to the definition of the noise terms, Eq. [(\[eq:noiseoperators:cov\])]{}, we can make use of Robertson’s uncertainty relation [@robertson1929], to establish the lower bound $$\begin{aligned}
\label{ineq:dxdpRobertson}
\delta_{\mathcal{X}}(t) \delta_{\mathcal{P}}(t) \geq \frac{1}{2} \left| \Braket{ \left[ \hat{N}_{\mathcal{X}}(t), \hat{N}_{\mathcal{P}}(t) \right] } \right|,\end{aligned}$$ which contains the commutator $$\begin{aligned}
\label{eq:Ncommutator}
\left[ \hat{N}_{\mathcal{X}}(t), \hat{N}_{\mathcal{P}}(t) \right] = \left[ \hat{\mathcal{X}}(t), \hat{\mathcal{P}}(t) \right] - \left[ \hat{\mathcal{X}}(t), \hat{P}_{\mathrm{S}}(0) \right] - \left[ \hat{X}_{\mathrm{S}}(0), \hat{\mathcal{P}}(t) \right] + \left[ \hat{X}_{\mathrm{S}}(0), \hat{P}_{\mathrm{S}}(0) \right]\end{aligned}$$ of the noise operators $\hat{N}_{\mathcal{X}}(t)$ and $\hat{N}_{\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators\])]{}. The first commutator on the right-hand side of Eq. [(\[eq:Ncommutator\])]{} vanishes due to Eq. [(\[eq:XPcommute\])]{}. The second and third commutators read $$\begin{aligned}
\left[ \hat{\mathcal{X}}(t), \hat{P}_{\mathrm{S}}(0) \right] = \left[ \hat{X}_{\mathrm{S}}(0), \hat{\mathcal{P}}(t) \right] = \left[ \hat{X}_{\mathrm{S}}(0), \hat{P}_{\mathrm{S}}(0) \right],\end{aligned}$$ since only the first term on the right-hand side of Eq. [(\[eq:inferredsystem\])]{} acts in the initial system’s Hilbert space. As a result, we have $$\begin{aligned}
\label{eq:Ncommutator:1}
\left[ \hat{N}_{\mathcal{X}}(t), \hat{N}_{\mathcal{P}}(t) \right] = - \left[ \hat{X}_{\mathrm{S}}(0), \hat{P}_{\mathrm{S}}(0) \right] = -i.\end{aligned}$$ Inserting Eq. [(\[eq:Ncommutator:1\])]{} into Ineq. [(\[ineq:dxdpRobertson\])]{} finally leads to Ineq. [(\[ineq:dXdP\])]{}.
We remark that it would also be possible to use Schr[ö]{}dinger’s uncertainty relation [@schroedinger1930] instead of Robertson’s uncertainty relation, Ineq. [(\[ineq:dxdpRobertson\])]{}. This approach would additionally incorporate the correlation term $\delta_{\mathcal{X}\mathcal{P}}(t)$, Eq. [(\[eq:noiseoperators:cov\])]{}, on the right-hand side of Ineq. [(\[ineq:dXdP\])]{}, i.e., $$\begin{aligned}
\delta_{\mathcal{X}}^2(t) \delta_{\mathcal{P}}^2(t) \geq \delta_{\mathcal{X}\mathcal{P}}^2(t) + \frac{1}{4}.\end{aligned}$$ We do not discuss this extension to Ineq. [(\[ineq:dXdP\])]{} in more detail. It could, however, serve as an interesting point of origin for further considerations.
References {#references .unnumbered}
==========
[^1]: For baths which are not initially separable from the system and the pointers, respectively, our method of calculating the entropy in \[sec:appendix:marginal probability distributions\] may fail. However, by choosing an appropriate time-dependency of the Hamiltonian, Eq. [(\[eq:H\])]{}, a smooth switch-on of the bath can be introduced to realize a more physically reasonable model; see, e.g., Ref. [@fleming2011a; @fleming2011c] and references therein.
[^2]: A common choice for Eq. [(\[eq:I\])]{} is an Ohmic spectral density with $\mathbf{I}(\omega) \sim \omega$ and a cut-off term for high frequencies. See, e.g., Ref. [@weiss1999] for a more detailed discussion.
[^3]: We remark that we have assumed in Ref. [@heese2014] (in Eq. (55b)) that the pointer-based variance product, i.e., the product of the diagonal elements of the first term in Eq. [(\[eq:noiseoperators:cov:calc2\])]{}, is bound from below by $1/4$. While this assumption always holds true for a closed pointer-based measurement due to Ineq. [(\[ineq:dXdP\])]{}, it is in fact not generally valid for an open pointer-based measurement.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Safety-Critical Java (SCJ) is a recent technology that restricts the execution and memory model of Java in such a way that applications can be statically analysed and certified for their real-time properties and safe use of memory. Our interest is in the development of comprehensive and sound techniques for the formal specification, refinement, design, and implementation of SCJ programs, using a correct-by-construction approach. As part of this work, we present here an account of laws and patterns that are of general use for the refinement of SCJ mission specifications into designs of parallel handlers used in the SCJ programming paradigm. Our notation is a combination of languages from the family, supporting state-rich reactive models with the addition of class objects and real-time properties. Our work is a first step to elicit laws of programming for SCJ and fits into a refinement strategy that we have developed previously to derive SCJ programs.'
author:
- Frank Zeyda
- Ana Cavalcanti
bibliography:
- 'refine2013\_zeyda.bib'
title: |
Refining SCJ Mission Specifications into\
Parallel Handler Designs
---
[**Keywords:**]{} SCJ, models, refinement, laws, patterns, tactics, .
Introduction {#sec:Introduction}
============
Java is indisputably one of the most popular programming languages. Despite this, its use in the safety-critical industry has been modest due to Java’s generality and rich set of features. Significant issues are, for example, the use of garbage collection and problems related to thread prioritisation [@STR06; @JSR302], which render it inadequate for time-critical applications. Safety-Critical Java (SCJ) [@HHLNSV09], a recent initiative, addresses these issues by introducing a restricted subset of Java; it is based on the Real-time Specification for Java (RTSJ) [@Wel04], but further restricts RTSJ’s execution and memory model. This facilitates the formal analysis of SCJ applications, and thereby enables the application of formal methods to satisfy stringent criteria of certification standards like DO-178C.
SCJ is organised in three levels (Level 0 to Level 2) that define progressively more complex models of execution. Our focus is SCJ Level 1, which roughly corresponds to the Ravenscar profile for Ada [@Bur99]. At Level 1, applications are organised as a sequence of missions, and each mission consists of a set of handlers that are executed in parallel. Handlers can either be periodic, which means they are released at regular time intervals, or aperiodic implying that they are released sporadically by some external event or stimulus. When a handler is released, its `handleAsyncEvent()` method is scheduled for execution.
Our previous work has focused on complementing the informal account of SCJ [@JSR302] with a formal model of SCJ’s mission-based execution paradigm [@ZCW11] and memory model [@CWW11]. Our notation is a combination of languages from the family [@CSW03; @CSW05; @SCJS09], specifically tailored for the specification and development of state-rich reactive systems with the addition of discrete time, object-orientation, and object references. We have also proposed a refinement strategy [@CZWWW12] to transform abstract specifications of SCJ programs into models that directly correspond to SCJ programs. Such a strategy is inherently ambitious and complex, as it simultaneously addresses a multitude of concerns. Therefore, it is not surprising that the existing work [@CZWWW12] only gives a broad description of the top-level approach; details of the application of this strategy to a specific example are available in [@ZCWWW12].
Our contribution in this paper is to examine in detail the refinement of centralised and sequential specifications of missions into parallel handler designs. Our general starting point is a process specification that supports all constructs of , including Z data operations, classes, and Timed CSP constructs, except for parallelism and interleaving. We then show how decomposition at the level of data operations, time budgets, and process actions can be used to transform the model into a uniform shape that determines the structure and behaviour of handlers of an SCJ mission. Refinement laws directly reflect particular program designs that encapsulate the way in which data is shared and how the computational work is divided between the handlers of a mission.
The motivation for our work is to pave the way for automated tool support. Due to the novelty of SCJ, there are not many tools currently available that support the development of critical software in SCJ. The available tools mostly focus on isolated statically-checkable properties [@TPV10; @DHS12; @HL11], but do not address the combination of concerns that characterise the SCJ paradigm. While we do address many concerns of SCJ simultaneously by using a highly expressive language, the practicalities of performing actual refinements are largely an open problem. It is, clearly, unrealistic to carry out such refinements entirely by hand, which is well illustrated by the complexity of the example in [@ZCWWW12]. Some refinement steps are, however, inherently difficult to automate. Our work, most importantly, highlights where automation is feasible, and where human guidance is indispensable to guide the refinement process.
The results in this paper contribute towards elaborating the proposed refinement strategy for SCJ in [@CZWWW12], but they are also useful outside the context of that technique. Decomposition of centralised models is a general issue in refinement-based techniques [@CSW03], and the models we produce can, in principle, serve as a starting point for any form of parallel implementation. As the essence of the SCJ paradigm (its mission-based execution model) can be captured independently of the Java language, our account on mission decomposition is relevant for other languages that adopt a similar execution model, too.
The structure of this paper is as follows. In Section \[sec:Preliminaries\] we review preliminary material: Safety-Critical Java and the family of languages. Section \[sec:Strategy\] then discusses our refinement laws, and Section \[sec:Example\] presents an example of their application. Finally, in Section \[sec:Conclusion\] we conclude and suggest future work.
Preliminaries {#sec:Preliminaries}
=============
We here discuss in more detail Level 1 SCJ and the family of notations.
Level 1 SCJ {#sec:SCJ}
-----------
The execution model for SCJ Level 1 programs is based on four primary conceptual entities: safelet, mission sequencer, missions and handlers. They are realised by classes that derive either from an interface or abstract class of the SCJ API. Namely, these are , , , , and .
[Fig.]{}\[fig:SafeletLifecycle\] illustrates the life-cycle of a Level 1 safelet, the top-level entity of an SCJ application. The SCJ infrastructure[^1] first initialises the safelet. This is followed by a series of mission executions, each involving the initialisation, execution and termination of a particular mission of the safelet. Mission initialisation creates the mission’s event handlers, which are released either periodically or by external events during mission execution. When there are no more missions to execute, the safelet terminates.
![Life-cycle of a safelet during execution of a Level 1 application[]{data-label="fig:SafeletLifecycle"}](SafeletLifecycle.pdf)
In terms of the SCJ API, a class implementing has to provide the methods and , which are called by the SCJ infrastructure to initialise and shutdown the safelet. Another method (not in [Fig.]{}\[fig:SafeletLifecycle\]) is called on the safelet object to obtain the mission sequencer of the application, which defines the sequence of missions to execute. In addition, various methods are called by the infrastructure on the mission sequencer, mission and handler objects during execution of the safelet. Most notably, these are to obtain the next mission to execute, to create the handlers of a mission, and when a handler is released. An SCJ program must provide implementations of these methods, and it thereby defines the architecture of the application in terms of missions and handlers. (We note that although the missions and handlers of a safelet are determined at run-time, we assume in our model that they are *a priori* fixed.)
When a mission terminates, is called on the mission object to perform application-specific clean-up tasks. As already mentioned, the entire safelet terminates when there are no more missions to execute, signalled by returning a reference. In summary, the safelet and the mission sequencer are control components that orchestrate the execution of the missions (and their handlers). The missions and the handlers, on the other hand, are the central components that implement the behaviour of the program, and the main focus of our work here.
The family {#sec:Circus}
----------
[@CSW03] is a language for specification and refinement of state-rich reactive systems. It combines notations from CSP [@Ros97], Z [@WD96], and Morgan’s refinement calculus [@Mor90]. As in CSP, the key elements of models are processes that can interact with their environment through channels. Unlike CSP, processes encapsulate a state that can be modified by actions and data operations of the process. has a denotational semantics defined using the Unifying Theories of Programming [@OCW09].
An example of a process is given in [Fig.]{}\[fig:Target\]. It illustrates the general form of an SCJ handler design, and the laws we discuss in the next section transform (sequential) specifications of safelets into processes of this shape. The name of the process is $SCJDesign$, and its state is defined by the $State$ schema, introducing the components $c_i$ of type $T_i$ ($Inv$ is an optional state invariant). The $T_i$ may be Z schema types or class types, as it is also the case for the types in any of the laws. We then have local action definitions for $Init$, $Mission_i$, $Handler_i$ and $HdlControl$. The actual behaviour of the process is defined by the main action at the bottom after the ‘$\circspot$’; it typically makes use of the local actions.
Local actions can be either specified by Z operation schemas or using a mixture of CSP constructs and guarded commands. Here, $Init$ is a Z operation that initialises the state, and $Mission_i$ and $Handler_i$ are CSP actions that provide models of missions and handlers as they emerge during verification. Each $Mission_i$ action is defined by a parallel composition of a mission-specific set of handler actions. In , parallel composition of two actions $A_1$ and $A_2$ is written as $A_1 \lpar ns_1 | cs | ns_2 \rpar A_2$, where $cs$ is a set of interface channels that require synchronisation of the actions, and $ns_1$ and $ns_2$ are disjoint sets of variables that each action is allowed to write to. Hence, all handlers of a mission write to mutually disjoint parts of the state space, determined by the $ns_i$. This ensures that all constructs (including parallel composition) are monotonic with respect to refinement due to intrinsic non-interference in shared data access by parallel processes and actions.
The action $HdlControl$ is included to incorporate control mechanisms. It controls termination of the mission via the channels $termReq$ (for a termination request raised by one of the handlers) and $termMsn$ (to synchronously terminate the handlers). It also permits the definition of additional control actions (dots) whose design is not a concern for mission decomposition into handlers.
The handler models, captured by the actions $Handler_i$, take different shapes for aperiodic and periodic handlers. Both, however, have the form of a recursion $(\circmu X \circspot A \circseq X \extchoice ~ termMsn \then \Skip)$ that repetitively executes some action $A$ and at the same time enables termination by $HdlControl$. Aperiodic handlers are modelled by an external choice that synchronises on a set of channels $e_{(i, 1)}$, $e_{(i, 2)}$, and so on, which correspond to SCJ events that are bound to the handler $i$ and therefore cause its release. Potentially, each event provides an input $v$, and the method is specified by $A(v)$.
For periodic handlers, the repetitive behaviour is determined by the action $A \circdeadlineterm T ~ \interleave ~ \circwait ~ T$, using additionally constructs from . The $A \circdeadlineterm T$ operator imposes a termination deadline $T$ on $A$, $\circwait ~ T$ corresponds to a delay of $T$ time units, and the interleaving with $\circwait ~ T$ prevents the action from terminating *before* $T$ time units have elapsed. Hence, we obtain a cyclic behaviour that executes the method $A$ once every $T$ time units. For clarification, we point out that all constructs take relative times as their arguments.
We note that interleaving ($A_1 \interleave A_2$) is a special case of parallelism where the synchronisation set $cs$ is empty; termination only occurs when both parallel actions have terminated. Later on, we also make use of the $\circwait ~ t_1 \upto t_2$ statement, which corresponds to a nondeterministic delay between $t_1$ and $t_2$ time units, and $A \circdeadlinesync ~ T$ which is a deadline on $A$ to interact via a visible event, such as a communication or synchronisation.
The main action of $SCJDesign$ at the end first initialises the state and then executes all missions in sequence (operator $A_1 \circseq A_2$). In [Fig.]{}\[fig:Target\], we use notations from both and [@SCJS09], and generally also support the use of constructs from [@CSW05] for class objects. The UTP [@HH98] enables us to give a sound semantic foundation to this combination of languages.
Refinement Laws {#sec:Strategy}
===============
Our starting point is a centralised mission specification that defines communication patterns, data operations, and timing restrictions using sequential actions. We deal with three aspects of the verification of a mission implementation with respect to such a centralised specification. The first aspect is decomposition of data operations to introduce functional models of handlers. The second is distribution of time budgets between the handlers. And the third is parallelisation of handlers to match the architecture of Level 1 SCJ; this also addresses data flow and control mechanisms via communications. We present here collections of refinement laws and tactics to address each of these verification issues. Although some of these laws have already been given in [@ZRC] and [@CSW03], the parallelisation laws in [Fig.]{}\[fig:ParPattn2\] and [Fig.]{}\[fig:conj-to-par-2\] are to our knowledge novel, and so are the laws in Section \[sec:Stage2\].
Decomposition of data operations {#sec:Stage1}
--------------------------------
Here we target data operations. We note that we do not generally require that the specification of a mission involves a single data operation. For missions with simple interaction patterns, such as reading an input, performing a computation, and writing an output, it is possible to capture the functional aspects of the mission in a single data operation. In the general case, however, where inputs and outputs may occur sporadically during mission execution, a functional mission model may be split into more than one data operation. We assume, on the other hand, that all data operations specify mission behaviour at a suitably high level of abstraction: this means they are centralised models of functionality, and hence do not already encapsulate any form of computational design.
Our goal is to decompose data operations so that the (functional) specifications of individual handlers emerge. We employ schema composition to model sequential execution of handlers, and schema conjunction to model parallel execution of handlers. All refinement is carried out at the level of Z. The Z Refinement Calculus [@ZRC; @Gro02], whose laws are valid in [@OCW09], provides the foundation for our laws here. The laws we present are therefore applicable and relevant for Z refinement in general.
Though [@ZRC; @Gro02], for example, present a collection of laws that address issues of decomposition too, it is well understood that decomposition of data operations is overall difficult to automate. We propose a number of specialised laws that cover a broad spectrum of mission designs. Each law encapsulates either a sequential or parallel design that carries out a centralised computation by two or more handlers.
#### Laws for sequential decomposition of data operations
We distinguish two fundamental cases. The first one assumes no dependency between the data operations in terms of the computed results. The corresponding law is presented in [Fig.]{}\[fig:SeqPattn1\]. We assume the existence of a $State$ schema that specifies the state on which the operations act. It is partitioned into two disjoint lists of variables, $x$ and $y$, which are respectively constrained by the state invariants $I_1(x)$ and $I_2(y)$. The law decomposes $Op$ into a sequence $Op_1 \semi Op_2$, where $Op_1$ only modifies the components in $x$, and $Op_2$ only modifies the components in $y$ and does not depend on $x$. Application of this law entails transforming the predicate of an operation schema into a form $P(x, y, x') \land Q(y, y')$. This may in general require intelligent decision making, but in some cases ought to be automatable using elementary laws and syntax-driven rewriting realised by generic tactics of proof for the Z mathematical notation.
\[law:SeqPattn1\]
The second case is where there exists a dependency between the data operations in terms of the result. That is, the second operation uses data that is computed by the first one. Here, we have the general law in [Fig.]{}\[fig:SeqPattn2\]. The crucial difference is in the shape of the predicate of the refined operation $Op$, where $Q(x', y, y')$ refers to the final value of $x$. The state invariant is decomposed as well, namely into a conjunct $I_1(x)$ that only considers constraints on $x$, and another conjunct $I_2(x, y)$ that relates $x$ and $y$. The invariant $I_2(x, y)$ is not enforced by the $\Delta$ and $\Xi$ schemas but moved into the predicates for technical reasons: we observe that $I_2$ may in fact not hold for the intermediate state of the decomposition.
\[law:SeqPattn2\]
The propagation of invariants proves to be especially important to facilitate further decomposition and later algorithmic refinement. Invariant decomposition once again requires guidance. It involves the transformation of a single invariant $I(x, y)$ into the conjunction $I_1(x) \land I_2(x, y)$ so that all relevant knowledge about the components in $x$ is encoded by $I_1(x)$.
We have defined several variations of the previous two laws that moreover deal with inputs and outputs of operations. We omit their discussion as they are straightforward generalisations. They can, however, be found in the appendix of [@CZWWW12]. Next, we take a look at parallel decomposition.
#### Laws for parallel decomposition of data operations
As before, we have a pair of laws that consider the case of independent and dependent data operations. Dependency here means that the operations cumulatively participate in the computation of some result. For independent data operations, the law is similar to that in [Fig.]{}\[fig:SeqPattn1\] with a small modification of the right-hand side: firstly, the sequence $Op_1 \semi Op_2$ is replaced by a conjunction $Op_1 \land Op_2$, and secondly, we remove the $\Xi$ schemas in the declaration part of $Op_1$ and $Op_2$. The fact that both laws have the same left-hand side illustrates that there is often more than one possible handler design, giving rise to different degrees of parallelisation.
A more interesting parallelisation law is presented in [Fig.]{}\[fig:ParPattn2\]. There, we have $n$ handlers participating in the computation of the result $r$ and using the components $x$. The behaviour of the handlers is specified by the predicate $Q(r_i, i, x)$ for $1 \le i \le n$. Decomposition here yields a conjunction that includes a conjunct $POp$ for each handler, as well as a merge operation $MOp$ that collects the partial results $r_i$ to compute the overall result of the refined operation. Following the Z convention, the symbols ‘$?$’ and ‘$!$’ in the declaration part of the schemas $POp$ and $MOp$ are used to identify input and output parameters. The merge operation is parametrised by a bag to enforce syntactically that the order in which the results are delivered is irrelevant. Hence, we require that the binary operation used in the merge is associative and commutative; the merge basically consists of folding this operation over the list of partial results.
\[law:ParPattn2\]
It turns out that the application of the above decomposition laws, in comparison to subsequent sets of laws, is the most challenging to automate. The developer needs to determine the target of each law application, that is, the schema predicates on the right-hand side of the laws. With that, a verification condition can be generated to establish that the predicate of the schema being refined can be written in the form required for the application of the law. Specialised proof tactics will be useful in this context.
Distribution of time budgets {#sec:Stage2}
----------------------------
Data operations in are atomic and instantaneous. Hence, all timing behaviour has to be specified explicitly using timed action operators. Time budgets specify the permissible amount of time that an implementation may take to execute a data operation; in , they can be captured by nondeterministic wait statements of the form $\circwait 0 \upto t$ that precede a data operation. The laws in this section are hence essentially about $\circwait$ statements modelling time budgets, and, therefore, are useful in any context where we want to reason about the timing of Z data operations in ().
Our general assumption is that the specification of mission behaviour may utilise $\circwait$ statements in arbitrary places. The laws in this section decompose and distribute those $\circwait$s in order to attach them to the data operations emerging from decomposition in the previous section. Using these laws, we can equip each decomposed data operation $Op$ with an operation-specific time budget $\circwait 0 \upto TB_{Op}$, where $TB_{Op}$ determines the amount of time the operation may take to execute in the SCJ program.
The refinement laws needed can be divided into two classes. In the first class, we have two key laws (given in [Fig.]{}\[fig:SplitDistrTB\]) for the decomposition and narrowing of time budgets. Whereas the first [Law ]{}\[law:SplitBudget\] replaces a single time budget by a sequence of two time budgets, the second [Law ]{}\[law:NarrowBudget\] incurs a reduction of nondeterminism that narrows a time budget. A point of design in applying these laws is to decide on the values of $t_1$ and $t_2$, which subsequently determine the amount of time available to the underlying data operations. Decomposition may, of course, be applied iteratively, so that a single time budget can be split into several time budgets for any given number of operations.
The second class of laws addresses the issue of moving the decomposed time budgets to suitable locations to attach them to their respective data operations. For this, we first transform all Z schema compositions into action sequences. The standard law for this is recaptured below from [@ZRC].
$Op_1 \semi Op_2 \; \equiv Op_1 \circseq \! Op_2 \;\; \mbox{provided} \;\; \pre(Op_1 \semi Op_2) \land Op_1 \implies \pre'(Op_2)$
As usual, $\pre(Op)$ yields the precondition (domain) of a Z operation $Op$, and we use $\pre'(Op)$ to indicate that the variables in the result are primed. We note that the semicolon ‘$\semi$’ is used for composition of Z operations, as opposed to ‘$\circseq\!$’ which is used for composition of actions.
We further require the specialised distribution [Law ]{}\[law:DistrBudget\] in [Fig.]{}\[fig:SplitDistrTB\]. This law is in fact non-compositional: it is a law about processes rather than actions. Hence, it only holds if the underlying action $\circwait t_1 \upto t_2 \circseq \! Op$ is embedded in a process $P$. The justification for the law comes from the structure and semantics of processes that prevents observation of the precise time at which an (internal) state change takes place. A proof is possible by induction over the structure of processes.
We note that no distribution laws exist to move time budgets across prefixes, since such transformations would not be correct as they alter the observable behaviour. Consider, for example, $c \then \circwait ~ t \circseq A$. Refining this action by $\circwait ~ t \circseq c \then A$ would be wrong since the refining action refuses communication on the channel $c$ for $t$ time units, whereas the refined action offers it immediately. Some general laws for refinement in [@Oli05] are useful, too, namely to distribute time budgets into and out of internal and external choice. Lastly, we have a fusion law for nondeterministic choice of time budgets:
$\circwait t_1 \upto t_2 \; \intchoice \; \circwait t_1' \upto t_2' \; \equiv \; \circwait \, \min(t_1, t_1') \upto \max(t_2, t_2')$
This law is useful as it enables the combination of two budgets.
The laws we present here are evidently complete for mission specifications in which each abstract data operation is already associated with an (abstract) time budget. Automation of the refinement can be envisaged by annotating each data operation with the intended time budget, and using tactics to mechanically perform the decomposition and distribution steps. An overall caveat for the transformation is that we cannot distribute time budgets into parallel data operations which are represented by Z schema conjunctions. This is because the conjunction operator only applies to schemas and not to actions, and the schema calculus does not support timing constructs such as $\circwait ~ t_1 \upto t_2$. (In our strategy, we, therefore, distribute the budgets of parallel operations after the parallel operators are introduced.)
The next section examines the refinement of sequential actions and schema conjunctions, as they emerge from the laws discussed so far, into parallel actions.
Introduction of parallel handler actions {#sec:Stage3}
----------------------------------------
In Section \[sec:Stage1\], we have presented laws to parallelise data operations using schema conjunction, but considered no laws to parallelise actions. The laws we discuss next can be used to parallelise mission actions. Like in Section \[sec:Stage1\], we divide the necessary laws into two classes: laws that account for sequential designs and laws that cater for parallel designs. The shapes we target are precisely those produced by earlier decomposition of data operations, which makes this aspect of the verification more susceptible to automation. In the sequel, we discuss both classes of laws.
#### Laws for sequential handler designs
Two central laws for parallelisation of handlers are given in [Fig.]{}\[fig:seq-to-par-1\] and [Fig.]{}\[fig:seq-to-par-2\]. The first one assumes that there exists no data dependency between the sequential handler actions $A_1$ and $A_2$, hence we have the proviso $\wrt(A_1) \, \cap \, \used(A_2) = \emptyset$, which states that the state components written by $A_1$ are disjoint from those read by $A_2$. A fresh typeless channel $c$ is introduced to control the order of execution of the parallel actions: they both have to synchronise on it, so that the right parallel action $c \then A_2$ blocks until the left parallel action is ready to execute the prefix $c \then \Skip$. The channel $c$ models an SCJ event that is bound to the second handler and fired by the first handler.
The second law ([Fig.]{}\[fig:seq-to-par-2\]) assumes that there is a data dependency between the sequential handlers. In that case, the channel $c$ is parametrised by the type of the data that is passed between $A_1$ and $A_2$. Multiple data items can be passed by using product types, and, as mentioned earlier, class types are permissible, too. An interesting observation at this point is that the channel $c$ fulfils a dual purpose: it controls both the order of execution of handlers and makes available shared data. Further refinement is hence required to untangle these concerns, namely by way of encapsulating the shared data independently of the control aspect. This is, however, beyond the scope of parallelisation of handlers and a separate and orthogonal design issue, so we do not discuss it further here. The report [@ZCWWW12] examines it in detail though.
We emphasise that the parallelisations performed by [Law ]{}\[law:seq-to-par-1\] and [Law ]{}\[law:seq-to-par-2\] are to align the model with the SCJ paradigm and architecture. In other words, they do not parallelise the computations of the respective handlers, which are still performed in sequence here. This reflects that any sequentialism in an SCJ design needs to be explicitly enforced, while parallel execution (of handlers) is the default.
For multiple applications of the two laws, we also require the application of several elementary laws between each application of [Law ]{}\[law:seq-to-par-1\] or [Law ]{}\[law:seq-to-par-2\]. Their purpose is firstly to extract the newly introduced channel $c$ to the outer level of the mission action in which the targeted (refined) action is embedded, and secondly to distribute prefixes $c~[?~x] \then A_2$ introduced in the right-hand parallel action into $A_2$, namely if $A_2$ is itself an action sequence or parallelism.
We conclude by observing that the first parallelisation [Law ]{}\[law:seq-to-par-1\] targets precisely the shape of models generated by earlier application of [Law ]{}\[law:SeqPattn1\] ([Fig.]{}\[fig:SeqPattn1\]), and the second parallelisation [Law ]{}\[law:seq-to-par-2\] precisely the shape of models generated by earlier application of [Law ]{}\[law:SeqPattn2\] ([Fig.]{}\[fig:SeqPattn2\]), subsequent to replacing Z compositions by action sequences, which is done collaterally as part of the distribution of time budgets.
#### Laws for parallel handler designs
A key law for transforming parallel data operations modelled by conjunctions into parallel actions is presented in [Fig.]{}\[fig:conj-to-par-1\]. It applies to data operations $Op_1$ and $Op_2$ that write to disjoint sets of variables, which is what we usually expect from a parallelism at that level.
Although this law permits us to replace parallel data operations by parallel actions, this might not immediately yield a top-level parallelism of handlers as present in our refinement target in [Fig.]{}\[fig:Target\]. It shows, in general, that due to the fact that the conjunction might be embedded into action sequences (see [Law ]{}\[law:ParPattn2\]), there is still a considerable number of refinement steps and specialised laws required to arrive at the desired shape. In particular, these refinements involved further decomposition of time budgets related to the particular parallel design adopted. Applying [Law ]{}\[law:conj-to-par-1\], for instance, to the result of [Law ]{}\[law:ParPattn2\] ([Fig.]{}\[fig:ParPattn2\]), we observe that there still remains a sequential composition with $MOp$. We can parallelise it using [Law ]{}\[law:seq-to-par-2\] in the previous section, but this does not completely eliminate it due to a prefix emerging in the left parallel action. In [@ZCWWW12], we precisely detail the basic refinement steps that are needed prior and subsequent to application of [Law ]{}\[fig:conj-to-par-1\]; they involve two specialised laws: one for channel decomposition and one for distribution of an interleaving of basic communications into a preceding parallelism.
We also consider high-level parallelisation laws. Namely, [Law ]{}\[law:conj-to-par-2\] in [Fig.]{}\[fig:conj-to-par-2\] directly targets shapes emerging from parallelising data operation via [Law ]{}\[law:ParPattn2\] and at the same time caters for further decomposition of time budgets. This shows in the time budgets $POp_{TB}$, $Rec_{TB}$ and $Merge_{TB}$ replacing the global time budget $Op_{TB}$. We hence have a proviso $POp_{TB} + n*Rec_{TB} + Merge_{TB} \leq Op_{TB}$ that considers the time allowance of the parallelised operations to compute the partial results, the time to record them, and the time needed to merge them. The concrete value of these budgets has to be determined by the developer as part of the verification process.
A design artifact of [Law ]{}\[law:conj-to-par-2\] is that it introduces a fresh typed channel $rec$ that is used to communicate the partial results to a parallel operation that receives and merges them into the final result. From this, a control fragment emerges that is later refined into shared data to hold the partial result(s); it contributes to the $HdlControl$ action in [Fig.]{}\[fig:Target\] and its refinement gives rise to further design of how partial results are stored and processed; this relies on its own set of laws which are omitted here.
To conclude this aspect of the refinement, we observe that we can either tackle it by way of applying the more general [Law ]{}\[law:conj-to-par-1\], or use specialised high-level laws like [Law ]{}\[law:conj-to-par-2\] that encapsulate particular designs. Since it is still an open issue how the general case can profit from further elementary laws and their automation, we recommend the use of high-level laws. Therefore, we assume that for every decomposition law into a parallel data operation, there exists at least one specialised action law that directly targets the emerging shape. So far, this appears to be the case, however, further experience needs to be gained to ascertain this. In [@ZCWWW12], we sketch a proof of [Law ]{}\[law:conj-to-par-2\] which uses a few novel and interesting elementary laws. Beyond this, future work may propose alternative parallelisation laws with more sophisticated merge operations that can, for instance, deal with partial results of heterogeneous type. We next look at an example that illustrates the refinement of a realistic SCJ program.
Example {#sec:Example}
=======
As an example, we consider the refinement of an action that models the behaviour of the collision detector ( benchmark) in [@KHPPTV09]. The SCJ program consists of a single mission that periodically carries out the following tasks: reading a set of aircraft positions from a radar device, calculating their predicted motions, and identifying the number of aircraft at risk of colliding due to their distances decreasing below a certain threshold. Whereas [@KHPPTV09] provides a sequential implementation using a single handler, we have developed a parallel program by breaking down the mission design into seven handlers: (1) a cyclic input handler that reads the next radar frame; (2) a reducer handler that performs a voxel-hashing algorithm, which partitions the space; (3) four parallel detector handlers that carry out the detection work; and (4) an output handler that communicates the result.
Our starting point is the abstract operation $ComputeCycle$ in [Fig.]{}\[fig:CDxCycle\]. It is embedded into an action that defines the cyclic mission behaviour, as specified below.
CDxMission X\
1 (next\_frame ? frame ComputeCycle) INP\_DL\
0 (FRAME\_PERIOD - INP\_DL - OUT\_DL)\
(output\_collisions ! collisions ) OUT\_DL\
FRAME\_PERIOD X
The channel $next\_frame$ (of a type $Frame$ encoding radar frames) is used to read the next frame of aircraft positions, and $output\_collisions$ (of type $\num$) to output the detected number of collisions. Collisions are computed by $ComputeCycle$ and stored in a state component $collisions$. The constant $FRAME\_PERIOD$ determines the length of a cycle, and $INP\_DL$ and $OUT\_DL$ are deadlines on external communications. We observe that $ComputeCycle$ is equipped with a time budget $FRAME\_PERIOD - INP\_DL - OUT\_DL$, obtained by subtracting from the cycle time the maximal amount of time that the communications are permitted to take. Besides, $RawFrame$, $StateTable$ and $Partition$ are classes.
We start by decomposing $ComputeCycle$ into sequences and conjunctions of data operations. This is done by applying [Law ]{}\[law:SeqPattn2\] three times, followed by an application of [Law ]{}\[law:ParPattn2\]. This is not trivial, however, since the $ComputeCycle$ operation contains further existentially quantified variables that either correspond to abstract model variables ($posns$ and $motions$) here arising from earlier data refinement, or local variables like $voxel\_map$, capturing the result of the voxel-hashing algorithm. These quantifiers either have to be eliminated using the one-point rule, or localised to predicates corresponding to single handlers.
Another issue that needs to be addressed is that the data flow is not always explicit in abstract operations specifying missions. In our SCJ program, for example, data is transmitted between the reducer handler that carries out the voxel-hashing, and the detector handlers that perform the detection. That is, the reducer handler writes to the component $work$ which determines how the computational work is split, and this variable is also read by the detector handlers. In the operation, the last existential conjunct $$\exists collset : \finset ~ (Aircraft \cross Aircraft) | collset = CalcCollisionSet(posns', motions') @
\\
\t1 (\# collset = 0 \land collisions' = 0) \lor (\# collset > 0 \land collisions' \geq (\# collset) \div 2)$$ models the detector handlers, and we notice that the new value of $collisions$ is determined by the function $CalcCollisions(posns', motions')$ in terms of the abstract model variables. To reformulate it in terms of $work$ requires some *ad hoc* refinements that appear to be difficult to automate by a machine. We skip further details and merely present the result of the decomposition.
CDxMission X\
1 next\_frame ? frame \
RecordFrame\
ReduceAndPartitionWork\
DetectCollisions INP\_DL\
0 (FRAME\_PERIOD - INP\_DL - OUT\_DL)\
(output\_collisions ! collisions ) OUT\_DL\
FRAME\_PERIOD X
where the decomposed Z operations $CalcPartCollisions$, $SetCollisionsFromParts$ and $DetectCollisions$ can be found in Appendix \[sec:DecompExample\]. We next decompose and distribute the time budget between the newly introduced sequential operations. For this, we introduce the handler-specific time budgets $RF_{TB}$, $RPW_{TB}$ and $DC_{TB}$. This yields the following refinement.
CDxMission X\
1 next\_frame ? frame \
0 RF\_[TB]{} RecordFrame\
0 RPW\_[TB]{}\
ReduceAndPartitionWork\
0 DC\_[TB]{} DetectCollisions INP\_DL\
… X
The time budget $DC_{TB}$ is further decomposed during the parallelisation of actions. For the last part of the refinement parallelising the action above, we refer to the detailed description in our technical report [@ZCWWW12], which is available from <http://www.cs.york.ac.uk/circus/publications/techreports/>. It entails applying [Law ]{}\[law:seq-to-par-2\] and [Law ]{}\[law:conj-to-par-2\], and after a finalising transformation that uses elementary laws and can be automated, too, we obtain an action that has the shape in [Fig.]{}\[fig:Target\].
Conclusion {#sec:Conclusion}
==========
We have presented a collection of refinement laws that can be used to refine sequential specifications of SCJ mission behaviour into parallel designs that match the SCJ Level 1 programming model. Our refined models are a suitable starting point for further refinement of shared data and control mechanisms. We have also highlighted challenges for automation: they are, primarily, in the decomposition of sequential and parallel data operations, and to provide a repository of parallelisation laws, both at the level of data operations and actions, that deal with a wide spectrum of recurring program designs. Due to the novelty of SCJ, there are still open issues related to the designs that ought to be supported, and hence we do not claim completeness at this stage. On the other hand, our results showed that the decomposition of time budgets can largely be automated, and so can (the intermediate steps in) the refinement of data operations into parallel handler actions, which ultimately creates a positive outlook. Like in SCJ, our model and strategy also supports data being shared between missions. But this is less of an issue for the refinement laws because no write conflicts or race conditions can arise. The mission design in fact emerges where sequential actions of an abstract centralised model are retained during refinement.
In practical terms, we propose to facilitate the decomposition of data operations, the more difficult aspect of a refinement, by asking the developer to identify intermediate target models that permit the application of one of the decomposition laws. Each intermediate model generates a refinement proof obligation which can be tackled in isolation, and, as we hope, its resolution will be able to take some advantage of automatic refinement tactics. The development of useful tactics is still work in progress, however, their mechanisation may use a tool like [@ZOC12] to ensure soundness of refinements and laws alike.
An open issue is the validation of our laws against a semantics for the particular combination of languages that we use. Our recent work explores in detail the semantics of , and this shall provide a platform to prove, for instance, the laws about time budgets in Section \[sec:Stage2\]. Further work is, however, required to integrate that semantics with that of . And importantly, we require a proof that the laws from either language ( and ) hold within the combined language. The Unifying Theories of Programming (UTP) [@HH98], the common semantic foundation for all dialects, ought to facilitate such a proof. It is an issue that is high on our agenda of research.
Related work includes action systems and their refinement [@Back90; @BK83]. Action systems combine state and behaviour by away of atomic *actions* that operate on the state and that can be executed concurrently if there are no write conflicts to variables. Like , action systems come with an extensive refinement calculus, supporting the refinement of centralised sequential specifications into distributed implementations [@Back90; @BW03]. The computational paradigm is, however, more restrictive since actions have to adhere to a specific form, whereas actions can, for instance, use all of CSP’s constructs.
Event-B [@Abr11] is a practically-oriented formalism closely-related to action systems; it has been successfully used in the formal development of distributed systems in academia and industry. Research has been prompted to overcome initial restrictions of the method to deal with decomposition [@But09] and time [@CMR06]. It would be interesting to see whether Event-B would be expressive enough for SCJ handler models, and whether the refinement laws we propose can be formulated and perhaps validated.
SCJ is still a very recent technology, and, as far as we know, this is the first work that looks at refinement more specifically in the context of the SCJ programming model. Our results though contribute to a wider objective of proposing and proving refinement laws for *all* aspects of the verification of SCJ programs. These are, among others, data refinements in and the introduction of class objects, the refinement of shared data and use of object references, and the transformation of models into , a new language sufficiently concrete to be directly translatable into code. They are all immediate areas for future work, each bringing its own set of challenges for refinement and automation.
#### Acknowledgements
This work was funded by the EPSRC grant EP/H017461/1. We are grateful to Andy Wellings for many useful clarifications of SCJ, and we also thank the anonymous reviewers for their pertinent and useful suggestions.
Decomposed data operations of the example {#sec:DecompExample}
=========================================
[CalcPartCollisions]{} \[currentFrame : RawFrame; state : StateTable; work : Partition; collisions : \]\
i? : 1 4\
pcolls! : pcolls! = \# {a\_1 : Aircraft; a\_2 : Aircraft | l : work . getDetectorWork(i?). elems () @ …} 2
[SetCollisionsFromParts]{} \[currentFrame : RawFrame; state : StateTable; work : Partition; collisions : \]\
collsbag? : int currentFrame’ = currentFrame state’ = state voxel\_map’ = voxel\_map work’ = work\
s : int | s = items collsbag? @ collisions’ = s
DetectCollisions \
1 colls1, colls2, colls3, colls4 :\
1 (i? : @ CalcPartCollisions\[colls1 / pcolls!\] i? = 1)\
(i? : @ CalcPartCollisions\[colls2 / pcolls!\] i? = 2)\
(i? : @ CalcPartCollisions\[colls3 / pcolls!\] i? = 3)\
(i? : @ CalcPartCollisions\[colls4 / pcolls!\] i? = 4)\
1 SetCollisionsFromParts(colls1, colls2, colls3, colls4 )
[^1]: By ‘SCJ infrastructure’ we mean an SCJ-compliant virtual machine.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A major point of interest in cometary plasma physics has been the diamagnetic cavity, an unmagnetised region in the inner-most part of the coma. Here, we combine Langmuir and Mutual Impedance Probe measurements to investigate ion velocities and electron temperatures in the diamagnetic cavity of comet 67P, probed by the Rosetta spacecraft. We find ion velocities generally in the range 2-4 km/s, significantly above the expected neutral velocity $\lesssim$1 km/s, showing that the ions are (partially) decoupled from the neutrals, indicating that ion-neutral drag was not responsible for balancing the outside magnetic pressure. Observations of clear wake effects on one of the Langmuir probes showed that the ion flow was close to radial and supersonic, at least w.r.t. the perpendicular temperature, inside the cavity and possibly in the surrounding region as well. We observed spacecraft potentials $\lesssim$-5 V throughout the cavity, showing that a population of warm ($\sim$5 eV) electrons was present throughout the parts of the cavity reached by Rosetta. Also, a population of cold ($\lesssim0.1$ eV) electrons was consistently observed throughout the cavity, but less consistently in the surrounding region, suggesting that while Rosetta never entered a region of collisionally coupled electrons, such a region was possibly not far away during the cavity crossings.'
bibliography:
- 'references.bib'
title: Ion velocity and electron temperature inside and around the diamagnetic cavity of comet 67P
---
The ion velocity exceeded the neutral velocity, showing that the ions were not strongly collisionally coupled to the neutral gas.
A population of warm electrons was present throughout the parts of the cavity reached by Rosetta, driving the spacecraft potential negative.
A population of cold electrons was consistently observed inside the cavity, and intermittently also in the surrounding region.
This is the pre-peer reviewed version of the following article: *Ion velocity and electron temperature inside and around the diamagnetic cavity of comet 67P* \[Odelstad et al., Journal of Geophysical Research: Space Physics, 123, 2018\], which has been published in final form at DOI:[10.1029/2018JA025542](https://doi.org/10.1029/2018JA025542). This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
Introduction {#sec:intro}
============
The Rosetta mission
-------------------
Between August 2014 and September 2016, the European Space Agency’s Rosetta spacecraft followed the short-period, Jupiter Family comet 67P/Churyumov-Gerasimenko in its orbit around the Sun [@Glassmeier2007; @Taylor2017]. The comet heliocentric distance ranged from 3.5 au at arrival of the spacecraft, to 1.26 au at perihelion in September 2015, to 3.83 au at the end of mission (EOM). The cometocentric distance of the spacecraft was typically on the order of a few tens to a few hundreds of kilometers, providing unprecedented access to the inner coma of a comet. The relative speed of the spacecraft w.r.t. the nucleus was generally on the order of one meter per second or less. Previous cometary space missions (e.g. ICE at 21P/Giacobini-Zinner [@Rosenvinge1986], Giotto, Sakigake/Suisei and VEGA (1&2) at 1P/Halley [@Reinhard1986; @Hirao1987; @Sagdeev1987] and Giotto at 26P/Grigg-Skjellerup [@Grensemann1993], which all carried plasma instruments) have been short flybys at distances of at least a few hundred km (and relative speeds of tens of kilometers per second). Thus, the Rosetta mission was unprecedented also with regard to its prolonged stay at the target comet, for the first time allowing the long-term evolution of a comet to be observed by in-situ measurements.
The diamagnetic cavity
----------------------
A major point of interest in cometary plasma physics has been the existence, extent and formation mechanism of the diamagnetic cavity, a region in the inner-most part of the coma into which the interplanetary magnetic field cannot reach and which, in the absence of an intrinsic magnetic field of the nucleus [@Auster2015], will be magnetic-field-free. First predicted theoretically by @Biermann1967, it has since been observed in situ by the Giotto spacecraft at comet 1P/Halley [@Neubauer1986]. @Biermann1967 proposed a pressure balance between the magnetic pressure on the outside of the cavity and the ion dynamic pressure on the inside to account for its formation and extent. However, @Ip1987 and @Cravens1986 [@Cravens1987] found this to be insufficient to explain the extent of the cavity observed at Halley and instead invoked the ion-neutral drag force inside the cavity to balance the outside magnetic pressure. This was supported by observations of near-equal ion and neutral velocities inside the cavity ($\sim$1 km/s and $\sim$0.9 km/s, respectively), consistent with strong ion-neutral collisional coupling, and clear stagnation of the ion flow in the region just outside the cavity boundary [@Balsiger1986; @Krankowsky1986].
A diamagnetic cavity was first detected around comet 67P in magnetometer (RPC-MAG [@Glassmeier2007], hereafter MAG) data from July 26, 2015 [@Goetz2016a], near perihelion at a distance of 170 km from the nucleus (in the terminator plane). The spacecraft remained inside the cavity for about 25 min during this event. Subsequent analysis has identified a total of 665 cavity crossings in MAG data [@Goetz2016b], between April 2015 and February 2016 (i.e. some preceding the original detection). They ranged in duration from 8 s up to 40 min, in distance to the nucleus from 40 to 380 km and in heliocentric distance from 1.25-2.4 au. The low velocity of Rosetta ($\lesssim 1$ m/s) implies that these highly transient events were the result of the cavity expanding and contracting over Rosetta’s position, rather than resulting from the spacecraft moving into and out of a stationary cavity. Another possibility is blobs of unmagnetized plasma detaching from the main cavity structure and convecting past the spacecraft. The distance to the nucleus of the cavity crossings exhibited a strong statistical dependence on the long-term production rate, but was unaffected by diurnal variations and short-duration events such as outbursts or varying solar wind consitions. @Goetz2016a therefore suggested a Kelvin-Helmholtz type instability, driven by a presumed velocity shear at the cavity boundary, to account for its short-term dynamics. This was also proposed to explain the fact that cavity distances were generally found to be larger than predicted for a steady-state cavity sustained by the above pressure balance, as in hybrid simulations by @Koenders2015 and @Rubin2012. The existence of instabilities at the cavity boundary was indeed confirmed in these simulations.
Density measurements by the Mutual Impedance Probe (RPC-MIP [@Trotignon2007], hereafter MIP) inside the diamagnetic cavity showed densities ranging from $\sim$100 to $\sim$1500 cm$^{-3}$ on longer time scales, but that were almost constant inside any given cavity or between closely successive events [@Henri2017]. The surrounding regions of magnetized plasma were in contrast characterized by large density variations, predominantly in the form of large-amplitude compressible structures with relative density fluctuations $\delta n/n \sim$ 1 [@Harja2018]. These generally matched similar structures observed in the magnetic field near the cavity by @Goetz2016a. The plasma density inside the cavity was found to be entirely determined by the ionization of the cometary neutral atmosphere and the cavity boundary generally located close to the electron exobase. Hence, @Henri2017 suggested that the cavity formation and extent was the result of electron-neutral collisionality rather than the ion-neutral collisionality previously invoked. They also proposed a Rayleigh-Taylor type instability of the cavity boundary, driven by the electron-neutral drag force acting as an “effective gravity”, instead of the Kelvin-Helmholtz type suggested by @Goetz2016a. @Timar2017 obtained good fits of observed cavity distance values to the ion-neutral drag model of @Cravens1986 [@Cravens1987], using the cometary neutral production rate and solar wind dynamic pressure estimated from magnetic field data as well as several different solar wind propagation models. This possibly eliminates the need for an instability at the cavity boundary to account for the intermittent nature of the cavity crossings, in favour of variations in the solar wind pressure.
@Nemeth2016 found that accelerated electrons in the 100 eV range, typically present in the inner coma, were absent inside the diamagnetic cavity, suggesting that these electrons were bound to the field lines and therefore excluded from the cavity.
Neutral gas velocity
--------------------
For the first few months at the comet, through fall of 2014 up to early 2015, empirical or semi-empirical estimates of the expansion velocity of the neutral coma gas are available from many different sources: doppler shift of the spectral lines of water observed by the Microwave Instrument on the Rosetta Orbiter (MIRO, @Gulkis2007) [@Lee2015; @Biver2015; @Gulkis2015], simulation outputs of DSMC models [@Bieler2015] constrained by ROSINA-DFMS data [@Fougere2016a; @Fougere2016b] and direct measurements by ROSINA-COPS ram and nude gauges [@Tzou2017]. They all typically give terminal velocities at a few kilometers from the nucleus surface in the range 400 - 800 m/s, with generally a positive correlation between velocity and local outgassing intensity. However, from the period between April 2015 and February 2016 considered in this Paper, published measurements are scarce. @Marshall2017 used a range of 400 m/s to 1 km/s, with a preferred value of 700 m/s, to obtain local effective production rates of H$_2$O from H$_2^{16}$O/H$_2^{18}$O line area ratios obtained by MIRO for the entire period from August 2014 to April 2016, though no more specific values are given from within this period. @Heritier2017 used a one-dimensional model for the neutral gas based on an adiabatic fluid expansion around the nucleus driven by inner boundary conditions on gas outflow velocity from @Huebner2000 and temperature from the thermophysical model of @Davidsson2005 to find terminal velocities of about 800 m/s. For the purpose of comparison to observed ion velocities in this study, we note simply that the neutral outgassing velocity is on the order of 1 km/s, and that exact values are likely to be lower than this estimate rather than higher.
Ion velocity
------------
The primary ionization processes in the cometary coma, photoionization and electron impact ionization , produce ions that are initially cold and flowing with the neutral gas. This is an effect of conservation of momentum: the momentum of the ionizing particle is minuscule compared to that of the much heavier neutral molecule and therefore does not affect its motion in any significant way. The excess energy from the ionization instead goes to the electrons, which are therefore born warm ($T_{\textnormal{e}}$ $\sim$ 10 eV) [@Häberli1996; @Galand2016]. If there is no electric field, or if the ions are strongly collisionally coupled to the neutrals, the ions can thus be expected to be cold and flowing with the neutral gas. In the presence of a magnetic field of solar wind origin, the assumption of no electric field fails because of the existence of a convective electric field, which will cause the ions to gyrate, $\mathbf{E} \times \mathbf{B}$ drift and eventually become dynamically part of the solar wind flow (which will be decelerated and deflected by mass-loading) [@Coates2004; @Szego2000]. This so called *ion pick-up* process takes place over spatial scales on the order of the ion gyro-radius, which for singly charged water group ions ($m_{\textnormal{i}} \approx 18$) in a cometary plasma with typical magnetic field strength $\lesssim$20 nT [@Goetz2017] is $\gtrsim$10 km for ion velocities $\gtrsim$1 km/s. Inside the diamagnetic cavity, or at distances outside of it smaller than about 10 km, this process is therefore unimportant for the ion motion. However, the presence of warm electrons (to be further discussed below) suggests the existence of an ambipolar electric field (at least inside the diamagnetic cavity) of a strength on the order of $k_{\textnormal{B}} T_{\textnormal{e}}/q_{\textnormal{e}}r$ to maintain quasi-neutrality of the radially expanding cometary plasma. In such a case, strong collisional coupling to the neutrals is necessary if the ions are to remain at the neutral velocity. Estimates of the location of the ion-neutral collisionopause by @Mandt2016 suggested that Rosetta was generally in a region where ion-neutral collisions were important. However, these estimates did not take into account the reduced collisionality of accelerated ions due to the cross-section for ion-neutral collisions decreasing with energy. @Vigren2017 used a 1D model to simulate the radial acceleration of water group ions interrupted by collisions (primarily charge transfer processes) with neutral water molecules, taking into account the energy-dependence of the cross-sections. They found that for an outgassing rate $\sim$2$\cdot 10^{28}$ s$^{-1}$, typical of 67P near perihelion, even a weak electric field of 0.03 mV/m, typical of what would be expected for an ambipolar field, is sufficient to partially decouple the ions from the neutrals, giving a bulk ion velocity of about 4 km/s at distances $\sim$200 km from the nucleus, typical of the Rosetta spacecraft around perihelion. For an outgassing rate $\sim$2$\cdot 10^{29}$ s$^{-1}$, typical of Halley during the Giotto encounter, collisional coupling was found to prevail. @Vigren2017b combined Langmuir probe (RPC-LAP [@Eriksson2007], hereafter LAP) and MIP measurements to produce estimates of the ion velocity for a three-day period near perihelion, including one diamagnetic cavity crossing. They obtained values typically in the range 2-8 km/s, roughly in line with the predictions of @Vigren2017, lending further support to the supposition that ions are collisionally decoupled from the neutrals at 67P. The presence of an ambipolar electric field, the velocity of the ions and the formation and dynamics of the diamagnetic cavity at 67P are at present poorly understood. In this Paper we attempt to shed some light on these issues by using the method of combined LAP and MIP measurements to produce estimates of the ion velocity throughout the diamagnetic cavity and compare to the surrounding region.
Electron temperature
--------------------
@Odelstad2015 [@Odelstad2017] presented measurements of the spacecraft potential ($V_{\textnormal{S/C}}$) by LAP, showing that $V_{\textnormal{S/C}}$ was mostly negative throughout Rosetta’s stay at the comet, often below -10 V and sometimes below -20 V. This was attributed to a population of warm ($\sim$5-10 eV) coma photoelectrons, whose presence was explained by the neutral gas not being dense enough to effectively cool these electrons (which are born warm, as mentioned above) by collisions. Positive spacecraft potentials ($\sim$0-5 V) were only observed in regions of very low electron density ($\lesssim$10 cm$^{-3}$), typically far from the nucleus or above the more inactive areas on it, where the positive $V_{\textnormal{S/C}}$ could be explained by low density rather than temperature and where significant electron cooling by neutrals was not possible. Thus, it was concluded that such warm electrons were persistently present in the parts of the coma reached by Rosetta, most notably also around perihelion, where the elevated neutral density would perhaps have been expected to effectively cool the electrons. The statistical nature of this study could not rule out the existence of some brief events of low spacecraft potential hiding in the data set, which would indicate the near-absence of warm electrons. In this paper, we examine this in detail for the diamagnetic cavity crossings and discuss the implications for the physics of the cavity.
In addition to these warm electrons, clear signatures of cold ($\lesssim$0.1 eV) electrons have also been observed by LAP [@Eriksson2017] and MIP [@Gilet2017]. In LAP, these show up in high-time-resolution current measurements at fixed bias voltage in the form pulses of typical duration between a few seconds and a few minutes, and in bias voltage sweeps in the form of very steep slopes in the current-voltage curve at high positive bias voltages (to be discussed further below). In MIP, they produce a second resonance in the mutual impedance spectra below the total plasma frequency. Since local electron cooling was negligible as evidenced by the presence of warm electrons, this cold plasma was inferred to have formed in a region closer to the nucleus than reached by Rosetta. Together with the intermittent nature of the signatures in LAP data, this was taken as evidence for strong filamentation of the cold plasma close to the nucleus, with individual filaments extending far outside the collisionally dominated region, perhaps even detaching from it entirely. Similar structures were observed to develop in connection to the diamagnetic cavity in global 3D hybrid simulations by [@Koenders2015] and have been proposed to result from the instability of the cavity boundary (e.g. @Henri2017). However, observations of cold electrons are not limited to the diamagnetic cavity so the relationship (if any) between the structure and dynamics of cold and unmagnetized plasma, respectively, is not clear. In this paper, we investigate this by examining in detail the presence of cold electrons in the diamagnetic cavity and the surrounding region.
Instrumentation and measurements {#sec:instruments}
================================
The Rosetta spacecraft carried a suite of 5 plasma instruments, the Rosetta Plasma Consortium (RPC) [@Carr2007], including a Langmuir probe instrument (LAP) [@Eriksson2007] and a Mutual Impedance Probe (MIP) [@Trotignon2007] which are of primary importance in this paper.
LAP {#sec:LAP}
---
### Physical characteristics
LAP comprises two spherical Langmuir probes, which we denote LAP1 and LAP2. They are both 5 cm in diameter and have a surface coating of titanium nitride (TiN). Each probe is mounted at the end of a stiff boom protruding from the spacecraft main body, see Figure \[fig:SCgeo\]a.
![a) Geometrical configuration of the LAP (yellow), MIP (red) and COPS (light blue) sensors. b) Angles used to describe the spacecraft attitude.[]{data-label="fig:SCgeo"}](SC_geometry.pdf){width="\textwidth"}
The LAP1 boom is 2.24 m in length and extends from near one of the nominally comet-pointing corners of the shadow ($-x$) side of the spacecraft (the side on which the lander was originally mounted) at an angle of 45$^\circ$ from the nominal comet-pointing direction. The LAP2 boom (also known as the MAG boom, since it also hosts the magnetometer sensors) is 1.62 m in length and extends from near the corner “downstream” of LAP1 on the shadow side, at an angle of about 120$^\circ$ from the nominal comet-pointing direction. Thus, LAP2 is likely to be much more susceptible to wake effects than LAP1.
For future reference, Figure \[fig:SCgeo\]b introduces the Comet Elevation Angle (CEA) and Comet Aspect Angle (CAA) to describe the spacecraft attitude w.r.t. the comet nucleus. These are the elevation and azimuth angles, respectively, of the comet position vector in a spherical coordinate system with zenith direction along the spacecraft y-axis and azimuth reference direction along the spacecraft x-axis. Thus, CAA is the angle of the projection of the nucleus position vector in the x-z plane from the z-axis (in the range \[-180$^\circ$,180$^\circ$\], positive for positive $x$) and CEA is the angle between the comet position vector and this projection (in the range \[-90$^\circ$,90$^\circ$\], positive for positive $y$). The corresponding angles for the Sun are denoted SEA and SAA, respectively.
### Operational modes
LAP supports three main operating modes: current measurements at fixed bias potential, potential measurements at fixed bias current (or with a floating probe, i.e. disconnected from the biasing circuitry) and Langmuir probe bias potential sweeps. In the latter mode, which is the one used in this paper, the bias voltage is sequentially stepped through a range of values and the current sampled at each step. The resulting $I$-$V$ curve can the be used to derive various parameters of the surrounding plasma by comparison to theoretical models for the probe current-voltage relationship. LAP supports bias potentials between -30 and +30 V, with typical voltage steps between 0.25 and 1 V. Full sweeps typically take between 1 and 4 s and are performed at a cadence of 64-160 s.
### Theoretical models
The theory of current collection by spherical (and cylindrical) Langmuir probes immersed in a plasma was pioneered by @Mott-Smith1926 for the case of a stationary maxwellian plasma in the *orbit-motion limited* (OML) regime, where the Debye length $\lambda_\textrm{D}$ is much smaller than the radius of the probe. For an ideal isolated spherical probe, the expression reads
$$I_{j} = \left\{ \begin{array}{ll}
I_{j\textnormal{0}} ( 1 - \chi_j), & \chi_j \leq 0, \\
I_{j\textnormal{0}} \exp \left\{-\chi_j \right\}, & \chi_j \geq 0. \end{array} \right.
\label{eq:OML_I}$$
where $j$ denotes the particle species (i for ions and e for electrons, respectively) and the random thermal current $I_{j\textnormal{0}}$ and normalized potential $\chi_j$ are given by
$$\begin{aligned}
I_{j\textnormal{0}} & = & -4\pi a^2 n_j q_j \sqrt{\frac{k_{\textnormal{B}} T_j}{2\pi m_j}}, \label{eq:OML_I0} \\
\chi_j & = & \frac{q_j V_{\textnormal{p}}}{k_{\textnormal{B}} T_j}. \label{eq:OML_Xi}\end{aligned}$$
Here, $a$ is the probe radius, $V_{\textnormal{p}}$ the probe potential w.r.t. the ambient plasma and $n_j$, $q_j$, $T_j$ and $m_j$ are, respectively, the number density, charge, temperature (in Kelvin) and mass of particle species $j$. LAP uses the spacecraft as electrical ground, thus $V_{\textnormal{p}}$ is related to the controlled bias potential $U_{\textnormal{B}}$ as $V_{\textnormal{p}} = U_{\textnormal{B}} + V_{\textnormal{S/C}}$, where $V_{\textnormal{S/C}}$ is the spacecraft potential. In the case of Langmuir probe bias potential sweeps where $U_{\textnormal{B}}$, and thus $V_{\textnormal{p}}$, is incrementally varied on timescales short enough that all other parameters can be assumed to be constant, the probe current due to species $j$ will be linearly proportional to $V_{\textnormal{p}}$, and thus $U_{\textnormal{B}}$, for attractive bias potentials $\chi_j \leq 0$ (i.e. electrons and a positively charged probe or positive ions and a negative probe). The proportionality constant, hereafter referred to as the ion or electron slope, is given by
$$\frac{\partial I_j}{\partial U_{\textnormal{B}}} = \frac{\partial I_j}{\partial V_{\textnormal{p}}} = a^2 n_j q_j^2 \sqrt{\frac{8\pi}{m_j k_{\textnormal{B}} T_j}}.
\label{eq:eslope}$$
Equation (\[eq:eslope\]) for the electron slope is used in this paper together with empirical electron slopes from LAP sweeps and density measurements from MIP to constrain the electron temperature and investigate the respective prevalences of warm and cold electrons in the diamagnetic cavity and the surrounding region (cf. Section \[sec:res\]).
For repulsive bias potentials $\chi_j \geq 0$ (i.e. electrons and a negatively charged probe or positive ions and a positive probe), the probe current falls off exponentially with increasing $|V_{\textnormal{p}}|$. Due to the larger ion mass, the random thermal ion current $I_{\textnormal{i0}}$ is much smaller than the random thermal electron current $I_{\textnormal{e0}}$, thus the ion current is typically negligible at repulsive bias potentials. The electron current on the other hand is generally not negligible unless $-V_{\textnormal{p}} \gtrsim k_{\textnormal{B}} T_{\textnormal{e}}/q_{\textnormal{e}}$. Probe currents at repulsive bias potentials are not utilized in this paper other than to note that ion and electron slopes have to be determined from sweep regions at sufficiently large $|V_{\textnormal{p}}|$ that the current due to the oppositely charged species is effectively suppressed.
In addition to currents due to collection of ambient plasma particles, particle emission from the probe surface in the form of photoelectrons and secondary electrons in response to solar EUV and impacting plasma electrons, respectively, may also contribute to the probe current. Attraction or repulsion of photoelectrons is determined by the probe potential w.r.t. its immediate surroundings, which may differ from $V_{\textnormal{p}}$ if part of the spacecraft potential field persists at the position of the probe. For a sunlit probe the photoelectron current will be independent of the bias potential at $U_{\textnormal{B}} < -\alpha V_{\textnormal{S/C}}$, where $\alpha$ is the fraction of $V_{\textnormal{S/C}}$ remaining at the probe position, since all emitted electrons are repelled and escape from the probe, contributing to the current. At $U_{\textnormal{B}} > -\alpha V_{\textnormal{S/C}}$, the photoelectron current falls off exponentially with increasing $U_{\textnormal{B}}$, with a characteristic e-folding typically on the order of 1-2 V. This regime change at $U_{\textnormal{B}} = -\alpha V_{\textnormal{S/C}}$ is typically identifiable as a sharp knee in the sweeps, hereafter denoted $V_{\textnormal{ph}}$, from which an estimate of the spacecraft potential can be obtained as $V_{\textnormal{S/C}} = -V_{\textnormal{ph}}/\alpha$. $\alpha$ has been shown to generally be in the range 0.7-1 by @Odelstad2017. Such spacecraft potential measurements have previously been used by @Odelstad2015 [@Odelstad2017] to demonstrate the overall pervasiveness of warm ($\sim$5 eV) electrons in the coma of 67P; in this paper we more carefully examine this during the times when the spacecraft is inside the diamagnetic cavity.
The OML theory for spherical probes was extended to the case of a drifting maxwellian plasma by @Medicus1961. An analytically simpler semi-empirical approximation to the rather cumbersome expressions of @Medicus1961 was presented by @Fahleson1967 for attractive probe potentials (e.g. positive ions and a negatively charged probe). Here, the attracted-ion current is still given by Equation (\[eq:OML\_I\]) but with modified expressions for $I_{\textnormal{i0}}$ and $\chi_{\textnormal{i}}$:
$$\begin{aligned}
I_{\textnormal{i0}} & = & -4\pi a^2 n_{\textnormal{i}} q_{\textnormal{i}} \sqrt{\frac{k_{\textnormal{B}} T_{\textnormal{i}}}{2\pi m_{\textnormal{i}}} + \frac{v_{\textnormal{D}}^2}{16}} \label{eq:FahlesonI0}\\
\chi_{\textnormal{i}} & = & q_{\textnormal{i}} U_{\textnormal{B}} \Big/ \left( k_{\textnormal{B}} T_{\textnormal{i}} + \frac{m_{\textnormal{i}} v_{\textnormal{D}}^2}{2} \right), \label{eq:FahlesonX}\end{aligned}$$
where $v_{\textnormal{D}}$ is the drift velocity of the ions. This gives for the ion slope:
$$\frac{\partial I_{\textnormal{i}}}{\partial U_{\textnormal{B}}} = \frac{2\pi a^2 n_{\textnormal{i}} q_{\textnormal{i}}^2}{m_{\textnormal{i}}} \underbrace{\frac{\sqrt{\frac{8k_{\textnormal{B}} T_{\textnormal{i}}}{\pi m_{\textnormal{i}}} + v_{\textnormal{D}}^2}}{\frac{2k_{\textnormal{B}} T_{\textnormal{i}}}{m_{\textnormal{i}}} + v_{\textnormal{D}}^2}}_{1/v_{\textnormal{i}}},
\label{eq:ionslope}$$
where for convenience, the factor containing the dependence on ion motion ($T_{\textnormal{i}}$ and $v_{\textnormal{D}}$) has been denoted $v_{\textnormal{i}}$. It should be noted at this point that in the case of a stationary plasma ($v_{\textnormal{D}} \rightarrow 0$) Equations (\[eq:FahlesonI0\]) - (\[eq:FahlesonX\]) reduce to the expressions of Equations (\[eq:OML\_I0\]) - (\[eq:OML\_Xi\]) and $v_{\textnormal{i}}$ reduces to $\frac{\sqrt{\pi}}{2} v_{\textnormal{th}} \approx v_{\textnormal{th}}$, where $v_{\textnormal{th}} = \sqrt{2 k_{\textnormal{B}} T_{\textnormal{i}} / m_{\textnormal{i}}}$ is the thermal velocity of the ions, defined as their most probable speed. In the case of cold drifting ions ($T_{\textnormal{i}} \rightarrow 0$), $v_{\textnormal{i}}$ reduces to the drift velocity $v_{\textnormal{D}}$. Thus, $v_{\textnormal{i}}$ represents an effective ion velocity that combines the effects of thermal and drift motions of the ions on the probe current collection, the contributions of which cannot be separated by Langmuir probe measurements alone.
### Practical aspects
While the theory of spherical probes in a (possibly drifting) single-component maxwellian plasma is well developed, the behavior of such probes in the highly variable and dynamic multi-component (at least in terms of the electron temperature [@Eriksson2017]) cometary plasma and in close proximity to a large, negatively charged spacecraft is not well understood. Space-charge sheath and wake effects are to be expected, complicating the analysis and interpretation of probe measurements [@Sjogren2012; @Johansson2016sctc]. Among the more robust parameters that can reliably be determined from LAP measurements alone are the spacecraft potential [@Odelstad2015; @Odelstad2017] and the photosaturation current [@Johansson2017]. In addition to these, the aforementioned ion slope can also most often be reliably identified in LAP sweeps. While this curve parameter alone cannot determine any parameter of the plasma, if the ion density is known from some other source, Equation (\[eq:ionslope\]) can be used to obtain the effective ion velocity $v_{\textnormal{i}}$ if assumptions are made on the values of $m_{\textnormal{i}}$ and $q_{\textnormal{i}}$. The cometary plasma is most often dominated by singly charged H$_2$O$^+$ and H$_3$O$^+$ ions , giving $m_{\textnormal{i}} \approx 18$ u and $q_{\textnormal{i}} = 1$ e. The total plasma density can often be reliably obtained from MIP (cf. Section \[sec:MIP\]). Thus, combining MIP density measurements with the ion slope from LAP sweeps provides a means of measuring the ion velocity in the cometary plasma. This method was used by @Vigren2017b to obtain ion velocities for a three-day period near perihelion. It must be noted here that this method presumes that the LAP ion slope is unaffected by any sheath or wake effects of the spacecraft. The validity of this assumption, and possible consequences on obtained results when it fails, will be discussed in detail in Section \[sec:disc\].
A more detailed discussion of the general appearance and interpretation of LAP1 sweeps can be found in Appendix A, along with descriptions of how the various sweep parameters (e.g. ion and electron slopes) are obtained from the sweeps.
MIP {#sec:MIP}
---
The MIP antenna consists of four cylindrical electrodes (1 cm in diameter) arranged in a linear array along the LAP1 boom, see Figure \[fig:SCgeo\]. The two middle electrodes, 20 cm apart, are transmitting monopoles while the two outer ones, each at a 40 cm distance from the nearest transmitter, make up a receiving electric dipole with a total separation distance of 1 m. In the most commonly used active mode, an sinusoidal current is fed to the transmitting electrodes, either in phase or anti-phase, and the mutual impedance between the transmitting and receiving electrodes is computed from the induced voltage difference across the receiving dipole. This is repeated for a number of different frequencies of the driving sinusoidal current, producing a mutual impedance spectrum from which properties of the surrounding plasma can be deduced. In particular, the modulus of the mutual impedance spectrum exhibits either a peak or a cut-off (depending on the plasma properties) at the electron plasma frequency $f_{\textnormal{p}} = \frac{1}{2\pi}\sqrt{n_{\textnormal{e}} q_{\textnormal{e}}^2 / \varepsilon_{\textnormal{0}} m_{\textnormal{e}}} \approx 9 \sqrt{n_{\textnormal{e}}}$ , where $n_{\textnormal{e}}$ in cm$^{-3}$ gives $f_{\textnormal{p}}$ in kHz, from which the total electron density can thus be obtained. This signature at the plasma frequency appears in the mutual impedance spectra only when the Debye length is small enough compared to the emitter-receiver distance. For longer-Debye-length plasmas, typically when it exceeds about half a meter, LAP2 was used to transmit the MIP oscillating signal into the plasma, still received on the MIP receivers, in order to provide a larger transmitter-receiver distance, close to 5 m, and extend the usable range of the mutual impedance technique to plasmas characterized by a Debye length up to a few meters. This operational mode is called the Long Debye Length (hereafter LDL) mode, while the previously described mode is referred to as the Short Debye Length (hereafter SDL) mode.
While the theory of mutual impedance probes in a homogeneous maxwellian plasma is well developed, the behavior of such probes in close proximity to a large, negatively charged spacecraft is not well understood. In particular, questions may be raised regarding what density is actually measured by MIP from its location inside the plasma sheath created by the potential field of the charged spacecraft. The local electron density at the position of the probes is expected to be much suppressed by this potential field, especially since the boom on which the electrodes are mounted is conductive and grounded to the spacecraft. In passive mode the picture is clearer: conservation of energy requires the frequency of externally generated waves to be constant during propagation through the density gradient surrounding the spacecraft. The spectral peak has been observed not to change much when switching between active and passive modes, when electrostatic waves not generated by the MIP experiments but by other plasma processes are observed at the plasma frequency (not shown here). Thus, we surmise that also active mode measurements are reliable for obtaining the density of the ambient plasma, unperturbed by the presence of the spacecraft. In fact, all MIP density measurements used in this paper have been obtained from active modes, since MIP has a better power resolution in active than in passive mode. The power resolution in passive mode is generally too low to observe variations of $f_{\textnormal{p}}$, except on some occasions when it confirms the active signature.
@Gilet2017 showed that in the presence of two distinct electron populations at different temperatures, the resonance peak at the total electron plasma frequency observed in the modulus of the mutual impedance spectrum is supplemented by a second resonance peak at lower frequency, close to the plasma frequency of the cold population, corresponding to electron acoustic waves excited in the plasma by the MIP experiment. This allows MIP to detect the presence of such a cold electron population in the cometary plasma, as has indeed been the case for parts of the mission [@Gilet2017]. In such cases, the position of the resonance at the total plasma frequency gives the total electron density; this is the density estimate used throughout this paper. In principle, MIP could be used also to investigate the prevalence of cold electrons. However, the technique for doing this systematically with MIP is still under development and in this paper we primarily use LAP for that purpose.
Combining LAP and MIP data {#sec:interpolation}
--------------------------
Both MIP and LAP working modes are organized in synchronized 32-s long sequences composed of elementary working modes. The 64-160-s cadence of LAP sweeps implies that sweeps are performed every other to every fifth such 32-s sequence, always at the very beginning of the sequence. MIP begins each 32-s sequence with a full active-mode spectrum, which is then repeated with a cadence of typically just over 4 s, when MIP is in burst mode. Thus, each LAP sweep is concurrent with a MIP spectrum, which in the most common case of a $\sim$3-s sweep from negative to positive bias voltage is generally acquired during the first half of the sweep, i.e. simultaneously with the ion side of the sweep. Thus the timing between MIP density measurements and LAP ion slope measurements is generally very good. However, not all spectra are well-behaved enough to allow automated identification of the plasma frequency signature so some sweeps will not have a concurrent density estimate from MIP. In the quiescent plasma inside the diamagnetic cavity, such sweeps are instead combined with density estimates from adjacent spectra, provided that the mid-points are less than 4 s apart. This is justified by the fact that the plasma density has been observed to be almost constant during the diamagnetic cavity crossings [@Henri2017]. In the more variable plasma outside the cavity, such sweeps are omitted when producing ion velocity estimates by combining LAP1 ion slopes with MIP densities.
Results {#sec:res}
=======
Case study of the Nov 19-21 2015 cluster of diamagnetic cavity observations {#sec:res1}
---------------------------------------------------------------------------
In order to facilitate comparative statistics between the cavity and the surrounding region, we seek an interval with a comparable number of measurements inside and outside the cavity. It should be short enough for secularly varying parameters such as comet outgassing, latitude and radial distance of the spacecraft to be constant but still long enough to contain sufficient measurements for good statistics. The diamagnetic cavity events came in the form of single events and clusters; the most prominent clusters occurred on 30 July 2015 and 19-21 November 2015 [@Goetz2016b]. We found that the 50 h interval between 08:00 on 19 November 2015 and 10:00 on 21 November 2015 fulfilled the above criteria. During this interval the spacecraft spent about one third of the time inside the cavity and both MIP density measurements and LAP sweeps from both probes are available with sufficient quality to perform our study. Latitude and radial distance varied from -40$^\circ$ to -57$^\circ$ and 126 km to 150 km, respectively.
Figure \[fig:timeseries\] shows a multi-instrument time series plot of this interval.
![Multi-instrument time series plot of the interval from 08:00 19 November 2015 to 10:00 21 November 2015. See text for description.[]{data-label="fig:timeseries"}](timeseries_plot.pdf){width="140.00000%"}
Figures \[fig:timeseries\]a and \[fig:timeseries\]b show sweeps from LAP1 and LAP2, respectively, with probe bias potential on the (left-hand) y-axis and measured probe current color-coded on the surface plot. The color scale is set quite narrow in order to bring out at least the most prominent features on the ion (negative-voltage) side of the sweeps. As a consequence, the much larger electron currents on the positive-voltage side often saturate the color scale. For LAP1, the photoelectron knee, roughly corresponding to the negative of the spacecraft potential, is over-plotted as a white line, to be read off the same y-axis. To more clearly make out the behavior of the sweep ion current, the ion slope calculated from each sweep is plotted as an orange line on top of the negative-voltage side of the sweeps, to be read off the respective right-hand y-axes. Figure \[fig:timeseries\]c contains scatter plots of all the MIP plasma density measurements acquired during the interval (grey points) and those for which the timing coincide with the timing of LAP sweeps (black points). Here, and in all the remaining panels, time intervals when @Goetz2016b have identified the spacecraft to be inside the diamagnetic cavity are shaded purple. Figure \[fig:timeseries\]d shows the ion velocities derived from combining the LAP ion slopes with the simultaneous MIP densities through Equation (\[eq:ionslope\]). Figure \[fig:timeseries\]e shows the neutral density measurements by COPS during the plotted interval. Finally, Figure \[fig:timeseries\]f displays the spacecraft attitude through the angles defined in Figure \[fig:SCgeo\]b, with dark shaded regions denoting the angular intervals where each of the probes is obscured from view of the sun (SAA) or comet (CAA, CEA) and a lighter shaded region where Probe 2 is possibly shaded behind the High Gain Antenna (HGA), depending on the HGA orientation which we do not look further into here.
Turning first our attention to Figure \[fig:timeseries\]c, we recall the observation of @Henri2017 that the density is remarkably stable inside the cavity. This is true both during each individual cavity event, but also when comparing densities from different events in this interval. Outside the cavity on the other hand, the density is highly variable, with scatter primarily towards higher densities. This is shown more clearly in Figure \[fig:hist\]a,
![Histograms of data in the interval from 08:00 19 November 2015 to 10:00 21 November 2015.[]{data-label="fig:hist"}](histograms.pdf){width="\textwidth"}
where we show a histogram of the density measurements in the plotted time interval. Here, we also see that the density is quite narrowly and almost symmetrically distributed around 600-700 cm$^{-3}$ inside the cavity, with most measurements falling in the range 500-800 cm$^{-3}$. The measurements from outside the cavity exhibit much larger spread and distribute less symmetrically.
The LAP1 ion slopes in Figure \[fig:timeseries\]a exhibit similar behavior, as shown more clearly in the histogram in Figure \[fig:hist\]b. Here too the spread is much larger in the region surrounding the cavity than inside of it. The distribution is less asymmetric outside the cavity than it was for the densities, though some preponderance of lower values may be discerned.
The velocities in Figure \[fig:timeseries\]d, a histogram of which is shown in Figure \[fig:hist\]c, display an even larger difference between the cavity and the surrounding region. Inside the cavity, the ion velocities are very narrowly distributed around 3.5-4 km/s, with almost all measurements falling in the range 3-4.5 km/s. In fact, the spread here is so small that it may just be an effect of measurement noise, there being no statistically significant variation of the derived ion velocity at all inside the cavity during this interval. We note specifically that the ion velocity is significantly larger than the neutral velocity of $\lesssim$1 km/s at all times, contrary to what would be expected if the ion-neutral drag force [@Cravens1986; @Cravens1987] was responsible for balancing the outside magnetic pressure at the cavity boundary. Also, we note from Figure \[fig:timeseries\]e that the neutral density varies by $\sim$50% during this time interval (excluding the spacecraft slew between 12:00 and 16:00 on November 19, to be discussed below); this does not come through at all in the ion velocity measurements inside the cavity. The velocity outside the cavity is on the contrary highly variable, with a spread entirely towards higher velocities, up to and above 10 km/s. Here too there is no sign of correlation with the neutral density. We see no sign of a decrease of the ion velocities at the cavity boundary or in the surrounding region. However, our ion velocity estimate cannot distinguish between bulk drift and thermal speeds or different flow directions, so this does not preclude a decrease in the outward radial bulk flow outside the cavity.
A histogram of the spacecraft potential inside and outside the cavity is shown in Figure \[fig:hist\]d. $V_{\textnormal{S/C}}$ is consistently $< -5$ V both inside and outside the cavity, attesting to the persisting presence of warm ($\sim$5 eV) electrons in both these regions. The measurements distribute narrowly around -13 V inside the cavity, with most measurements falling in the range between -14 and -12 V. Outside the cavity the distribution is much wider, with significant spread primarily towards more negative potentials. In this sense, the spacecraft potential measurements mirror the density measurements in Figure \[fig:hist\]a, consistent with $V_{\textnormal{S/C}}$ being governed mainly by the electron density.
Figure \[fig:hist\]e shows a histogram of the LAP1 electron slopes inside and outside the cavity. Outside the cavity, the measurements distribute into two distinct groups, $\lesssim$20 nA/V and $\gtrsim$300 nA/V, respectively. Inside the cavity, only the population $\gtrsim$300 nA/V is observed. The electron slope can be related to the electron temperature and density through Equation (\[eq:eslope\]). Solving Equation (\[eq:eslope\]) for $T_{\textnormal{e}}$ in units of eV gives
$$T_{\textnormal{e}} \textnormal{ [eV]} = 0.04 \cdot n_{\textnormal{e}} \textnormal{ [cm$^{-3}$]} \Bigg/ \frac{\partial I_{\textnormal{e}}}{\partial U_{\textnormal{B}}} \textnormal{ [nA/V]},
\label{eq:eslope2}$$
where $n_{\textnormal{e}}$ \[cm$^{-3}$\] and $\partial I_{\textnormal{e}} / \partial U_{\textnormal{B}}$ \[nA/V\] are the density in units of cm$^{-3}$ and electron slope in nA/V, respectively. For $n_{\textnormal{e}} \lesssim 800$ cm$^{-3}$ and slopes $\gtrsim$300 nA/V, typical inside the diamagnetic cavity, this gives $T_{\textnormal{e}} \lesssim 0.3$ eV. Thus, such cold electrons are pervasive throughout the cavity. For slopes $\lesssim$20 nA/V, $T_{\textnormal{e}}$ comes out $\gtrsim$2.5 eV for $n_{\textnormal{e}} \gtrsim 150$ cm$^{-3}$, which is the very lowest density ever observed throughout this time interval. This corresponds well to the temperature range inferred for the warm electron population from the spacecraft potential measurements above. Thus, these slopes correspond to sweeps where the cold electron population is not observed and the warm population prevails. For larger densities on the order of several hundreds or thousands of cm$^{-3}$, more typical of what is generally observed in the region surrounding the cavity, Equation (\[eq:eslope2\]) gives electron temperatures on the order of a few tens to several hundreds of eV, entirely inconsistent with the spacecraft potential measurements. The electron slopes, especially the low ones corresponding to warm-only electrons, are most likely heavily influenced by spacecraft sheath effects. Thus, the temperature estimates obtained from them, especially for the warm ones, should not be taken as a quantitative measure of $T_{\textnormal{e}}$, but rather as an indication of the presence or absence of the cold electron population in the LAP sweeps.
Several spacecraft slews were performed during the plotted interval, as can be seen in the attitude angles in Figure \[fig:timeseries\]f (Slew 1, Slew 2 and Slew 3). These are also graphically illustrated in the inset next to Figure \[fig:timeseries\]d. We focus here on Slew 1, between about 14:00 and 17:40 on November 20, during which the spacecraft was inside the cavity most of the time. Looking at panel b in Figure \[fig:timeseries\], we see a clear regime change in the current collection of LAP2 coinciding with this slew. The LAP2 ion slope, generally on the order of 0.2 nA/V during nominal pointing conditions, effectively doubles to $\sim$0.4 nA/V during the slew, bringing it much closer to typical LAP1 values $\gtrsim$0.5 nA/V inside the cavity. This regime change also comes through clearly on the positive-voltage side of the LAP2 sweeps, with a substantial increase of the electron current at bias potentials $\gtrsim$10 V due to collection of cold electrons. Cold electrons are only very rarely seen in LAP2 sweeps before and after this slew, but are clearly and consistently present during the slew, at least inside the cavity. These observations strongly point to LAP2 being in or near a wake created by the spacecraft in the ion flow during nominal pointing, from which it comes out, at least partially, during the slew as the pointing changes and the probe becomes better exposed to the flowing plasma. This explains the lower ion slopes in LAP2 during nominal pointing (compared to both LAP1 and LAP2 during the slew) since the ion flow into a wake would be weakened. It also explains the increase of the cold electron current, since in the presence of warm electrons such a wake in the ion flow would become negatively charged, thereby prohibiting cold electrons from entering. In order for a wake to form, the ion flow must be supersonic, at least w.r.t. the perpendicular ion temperature. The observations during this slew thus constitute clear evidence of a directed ion flow inside the cavity. They are consistent with a flow direction radially outward from the nucleus, at least inside the cavity.
A zoom-in of Slew 1 is shown in Figure \[fig:slew\], where panels a, b and c are zoom-ins of panels b, c and f in Figure \[fig:timeseries\], respectively, and the spacecraft potential ($-V_{\textnormal{ph}}$) is over-plotted on the MIP densities in Figure \[fig:slew\]b in blue, to be read off the right-hand y-axis.
![Zoomed-in plot of Slew 1 in Figure \[fig:timeseries\]. Panels a, b and c are zoom-ins of panels b, c and f in Figure \[fig:timeseries\], respectively.[]{data-label="fig:slew"}](zoomed_slew.pdf){width="\textwidth"}
We note clear decreases of the LAP2 ion slope and drop-outs of the cold electrons on at least a few occasions when the spacecraft briefly leaves the cavity. This could possibly be taken as an indication that LAP2 goes back into the wake when leaving the cavity, suggesting the existence of a wake and implying directed supersonic flow also outside the cavity. The fact that this happens even though the spacecraft attitude does not change could potentially be taken as an indication of a difference in the ion flow direction. However, we note that many other aspects of the environment also change upon leaving the cavity, most notably the spacecraft potential and density (and consequently also the Debye length), which can be expected to impact the extent and dynamics of the wake.
Looking at a similar slew a few hours later (Slew 2), between about 00:30 and 06:00 on November 21 (cf. Figure \[fig:timeseries\]), when the spacecraft was mostly outside the cavity, we indeed see a more complicated picture. While there was definitely an increased proclivity towards lager ion slopes and electron currents in LAP2 sweeps (Figure \[fig:timeseries\]b) during this slew, these effects were less stable and consistent than inside the cavity, indicating that LAP2 was only intermittently out of the wake during this slew. Changes in the ion flow direction would have to be both rapid and erratic to account for such volatile wake effects. This could instead be attributed this to a more dynamic and variable wake outside the cavity. Finally, we note the major slew between 11:30 and 16:00 on November 19 (Slew 3), also mostly outside the cavity, when the attitude change was so large as to bring LAP2 to a position w.r.t. the flow comparable to that of LAP1 during nominal pointing (and actually prompting some rather prominent wake effects in COPS, as evidenced by the drop-out in observed neutral density). LAP2 sweeps here exhibited the largest ion and electron currents observed during the plotted time interval. The variability was large here too, even though LAP2 should be consistently out of the wake at these attitudes. This likely reflects the high variability of the plasma observed outside the cavity as discussed previously in the context of LAP1 measurements.
Statistical survey of all cavity crossings {#sec:res2}
------------------------------------------
In this Section, we broaden our scope to look at statistics of all cavity crossings throughout the mission. Figure \[fig:overview\] shows an overview of the 300-day period from April 2015 to February 2016 during which all of the cavity crossings occurred.
![Overview of the 300-day period from April 2015 to February 2016 during which all of the cavity crossings occurred. See text for description.[]{data-label="fig:overview"}](overview_plot_Q_loc.pdf){width="145.00000%"}
Figure \[fig:overview\]a shows a bar chart of the number of cavity crossings each day during this period (the conspicuous coloring will be explained in connection with Figure \[fig:global\_vi\_hist\] below). The distribution is very uneven, with most cavity events occurring in clusters in the end of July, early August, and the end of November 2015, as previously noted by @Goetz2016b and @Henri2017. This is even more clear in Figure \[fig:overview\]b, which shows a bar chart of the total time spent inside the cavity during each day. Here, the Nov 19-21 2015 cluster, examined in detail in the previous Section, really stands out, showing that most of the available data from inside the cavity actually comes from that brief interval. This observational bias is further aggravated by the fact that one of the most prominent days before perihelion, 26 July 2015, is useless for ion velocity measurements since MIP was run in LDL mode throughout most of that day, while the plasma density was likely above the maximum plasma density measurable in LDL mode (about 300 cm-3), so that the actual plasma density was missed by MIP during that period. Figure \[fig:overview\]c shows neutral density measurements by COPS, filtered by a 10 point moving median filter to remove spurious outliers (grey dots). We have also excluded data collected at spacecraft attitudes corresponding to CAA $>$ 42$^\circ$ or CEA $>$ 90$^\circ$, since COPS appears to be subject to wake-related density drop-outs at these attitudes (cf. Section \[sec:res1\]). In order to bring out the medium to long-term evolution of the neutral density, we also plot the median density for each synodic rotation ($\sim$12 h) of the comet nucleus w.r.t. the spacecraft (black line in Figure \[fig:overview\]c). Figure \[fig:overview\]d shows the radial distance of the spacecraft in terms of the nominal electron exobase [@Henri2017], denoted $R^*$ and calculated as
$$R^* = \frac{r}{L_\textnormal{e-n}} = \frac{1}{\sigma_\textnormal{en} n_\textnormal{n} r} \quad ,
\label{eq:exobase}$$
where $\sigma_\textnormal{en}$ is the electron-neutral cross-section, which we have set to $5 \cdot 10^{-16}$ cm$^{-3}$ in accordance with , $n_\textnormal{n}$ is the neutral density observed by COPS onboard the spacecraft and $r$ is the radial distance to the nucleus. The most prominent clusters of cavity crossings occur at small $R^*$, while many of the minor ones actually at larger $R^*$, as previously noted also by @Henri2017. We point out here that $R^*$ primarily follows the (reciprocal of the) neutral density, which in turn largely varies in response to changes in the spacecraft latitude, shown in Figure \[fig:overview\]e (black line), as a consequence of the seasonal variations in outgassing over the comet nucleus. We also note that the conspicuous lack of cavity events at small $R^*$ around the turn of the months Aug-Sep and Oct-Nov can possibly be explained by the low solar zenith angle (SZA, also known as phase angle), shown in Figure \[fig:overview\]e (red line) at these times; the extent of the cavity is expected to be smaller closer to the Sun-comet line. Figure \[fig:overview\]f shows the total H$_2$O production rate model by @Hansen2016 (black line) and the local production rate calculated as $4\pi n_{\textnormal{n}}r^2\cdot u$, for a neutral outflow velocity $u$ of 1 km/s (grey dots). (@Hansen2016 actually gives two distinct models for the inbound and outbound passages, hence the discontinuity at perihelion). The most prominent clusters of cavity crossings occur at total production rates between about $3-8 \cdot 10^{27}$ s$^{-1}$. The lack of cavity crossings at higher production rates is likely due to the increased radial distance of the spacecraft during that period. Finally, for context, we also show radial and heliocentric distances in Figures \[fig:overview\]g and \[fig:overview\]h, respectively.
### Ion observations
Figure \[fig:global\_vi\_hist\]a shows a histogram of ion velocity measurements during all diamagnetic cavity crossings throughout the mission, color-coded by time to bring out any long-term temporal variation.
![Histogram of ion velocity measurements during all diamagnetic cavity crossings throughout the mission, color-coded by time.[]{data-label="fig:global_vi_hist"}](global_vi_hist.pdf){width="67.00000%"}
The distribution peaks at velocities around 3.5-4 km/s, although it is clear that this peak derives entirely from the Nov 19-21 2015 cluster. In fact, the rest of the ion velocity measurements distribute more widely between 0-6 km/s, with a central maximum between 2-3 km/s, i.e. somewhat lower than the Nov 19-21 cluster. In Figure \[fig:global\_stats\] we examine the data in detail for possible causes of this, by scatter-plotting the ion velocity measurements versus a range of different parameters of interest.
![Scatter plots of ion velocity vs. a) radial distance b) in situ neutral density c) radial distance relative to electron exobase d) total production rate e) local production rate f) plasma density g) LAP1 ion slopes and i) spacecraft potential, color-coded by time as in Figure \[fig:global\_vi\_hist\]. Panel h shows a scatter plot of ion slopes vs. plasma density.[]{data-label="fig:global_stats"}](global_stats.pdf){width="144.00000%"}
Figure \[fig:global\_stats\]a shows the ion velocity measurements scatter-plotted versus radial distance to the nucleus with the same temporal color-coding as in Figure \[fig:global\_vi\_hist\]. This shows an apparent inverse relationship between radial distance and ion velocity. The radial distance generally decreases with time for the cavity crossings as Rosetta could go closer when activity decreased after perihelion, so it is possible that this just reflects an underlying dependence on some other temporally varying parameter. The most prominent deviation from monotonically decreasing distance is the interval 3-18 Aug 2015, when the radial distance was larger than the preceding interval 19 July - 3 Aug 2015. The ion velocity estimates from this period (cyan points just above 200 km in Figure \[fig:global\_stats\]a) come out lower than from the preceding interval (light blue points just below 200 km) in accordance with the general inverse radial dependence but in opposition to the general temporal trend, suggesting that the data is indeed better organized by radial distance than time. On the other hand, the red points just below 100 km, which come from the interval 16-31 Dec 2015, instead deviate from the radial trend, so the picture is not clear.
Figures \[fig:global\_stats\]b-e show similar scatter plots of $v_{\textnormal{i}}$ versus in situ neutral density and $R^*$, total (H$_2$O) production rate and local production rate. The in situ neutral density does not organize the ion velocity measurements very well, suggesting that the ion velocity is not determined by a local collisional equilibrium. $R^*$ organizes the data somewhat better; at least the major blue and orange point clouds separate roughly in agreement with a trend towards higher ion velocities at larger $R^*$, qualitatively consistent with the ions being accelerated radially outward by an electric field outside the exobase. However, this is not very convincing in light of the large scatter in Figure \[fig:global\_stats\]c and the fact that velocity observations from the other time intervals are not well organized by this.
The ion velocity does not appear to correlate much with the local production rate, whereas the total production rate organizes data slightly better, although not better than $R^*$ in Figure \[fig:global\_stats\]c. $R^*$ and the local production rate are really just different ways of combining the radial distance and neutral density; the fact that these combinations do not yield any significant improvement in data organization over the radial distance alone suggests that either the radial distance is the main determining factor for the ion velocity or we have not yet found the real underlying cause. Systematic measurement errors are also a possibility to be considered, as will be further discussed in Section \[sec:disc\]. In Figures \[fig:global\_stats\]f-g, we plot the ion velocity versus the plasma density observed by MIP and LAP1 ion slope, respectively, since these are the underlying parameters from which $v_{\textnormal{i}}$ is obtained. The main blue and orange point clouds separate nicely in Figure \[fig:global\_stats\]f, showing that the plasma density was generally higher during the 19 July - 18 Aug 2015 interval than for the rest of the data set. However, there is a lot of scatter and no clear correlation between $v_{\textnormal{i}}$ and $n_{\textnormal{e}}$ within the different subgroups. Also, the red and dark blue points, obtained at the very beginning and end of the 300-day period of cavity crossings, respectively, do not follow the general trend. We note here that the plasma density distributes similarly to the neutral density in Figure \[fig:global\_stats\]c, a result of the close relationship between $n_{\textnormal{e}}$ and $n_{\textnormal{n}}$ inside the cavity observed by @Henri2017, though the plasma density seems to have less variance within each subinterval. Thus, the plasma density does not really provide any new information by which to organize the data. The ion slopes in Figure \[fig:global\_stats\]g groups velocity measurements from the different clusters similarly as the plasma density, but organize them better within each subgroup, especially for the 19 July - 18 Aug 2015 (light blue) interval. In Figure \[fig:global\_stats\]h, we plot the ion slope versus the plasma density. The 19 July - 18 Aug 2015 interval stands out here as being well off the general linear trend that organizes the rest of the measurements. Such a linear trend between ion velocity and plasma density would be expected in the case of a constant ion velocity, in which case the slope of this trend can be obtained from the partial derivative of Equation (\[eq:ionslope\]) w.r.t. $n_{\textnormal{i}}$:
$$\frac{\partial^2 I_{\textnormal{i}}}{\partial n_{\textnormal{i}} \partial U_{\textnormal{B}}} = \frac{2\pi a^2 q_{\textnormal{i}}^2}{m_{\textnormal{i}} v_{\textnormal{i}}}.
\label{eq:ionslopeslope}$$
Solving Equation (\[eq:ionslopeslope\]) for $v_{\textnormal{i}}$ provides an additional way of obtaining the ion velocity:
$$v_{\textnormal{i}} = \frac{2\pi a^2 q_{\textnormal{i}}^2}{m_{\textnormal{i}}} \Biggm/ \frac{\partial^2 I_{\textnormal{i}}}{\partial n_{\textnormal{i}} \partial U_{\textnormal{B}}}.
\label{eq:vi2}$$
The slope of the linear trend in Figure \[fig:global\_stats\]h is roughly $8\cdot 10^{-4}$ nA/Vcm$^{-3}$, giving $v_{\textnormal{i}} \approx 4$ km/s, i.e. in good agreement with the point-wise determination for these time intervals. This gives some extra credibility to these measurements, since if the ion slopes were significantly perturbed by the presence of the charged spacecraft, this effect would be expected to vary with spacecraft potential and Debye length and hence with the plasma density, and the linear relationship would likely be distorted. For the 19 July - 18 Aug 2015 interval it appears that there is a substantial amount of scatter towards higher ion slopes that is not matched by similar increases in MIP density as required for a constant ion velocity. However, we cannot say for sure whether this is due to actual variations in ion velocity or measurement inaccuracies.
Finally, Figure \[fig:global\_stats\]i shows a similar scatter plot of ion velocity versus spacecraft potential. The magnitude of the spacecraft potential, which can be observed to roughly follow the plasma density as expected, should be important for the formation and extent of a spacecraft sheath and pre-acceleration of ions before collection by the probe. We note here that spacecraft potential measurements above -4 V are generally unreliable, as will be discussed below. A general trend towards lower ion velocities for lower-magnitude spacecraft potentials can be observed, possibly indicative of a systematic overestimation of the ion velocities due to pre-acceleration of the ions in the potential field of the spacecraft. However, the scatter is large and, as before, the strong dependence of the spacecraft potential on plasma density, which in turn depends on the neutral density, means that effects of any or all of these parameters cannot be disentangled. In addition, $V_{\textnormal{S/C}}$ also depends on the temperature of the warm electrons, which determines the magnitude of the ambipolar electric field and hence the acceleration of the ions, providing another possible explanation of the observed trend in Figure \[fig:global\_stats\]i.
### Spacecraft potential and warm electrons
Figure \[fig:global\_hists\]a shows a histogram of spacecraft potential estimates from all 1138 LAP1 sweeps obtained inside the diamagnetic cavity throughout the mission, color-coded by observation time as before. Virtually all of the spacecraft potential values were negative, by at least a few volts. The automated algorithm used to find the location of the photoemission knee in the sweeps is not always very precise, errors of several volts are not uncommon (cf. Appendix A). In previous studies [@Odelstad2015; @Odelstad2017] this was not a major concern due to the long-term, statistical nature of these studies. The more limited and detailed study of the diamagnetic cavity in this paper however calls for a more detailed analysis. Consequently, all sweeps with $V_{\textnormal{ph}} \leq 4$ V in Figure \[fig:global\_hists\]a have been manually double-checked, to find that in all these cases, $V_{\textnormal{ph}}$ was indeed imprecisely determined and that none of these sweeps really had $V_{\textnormal{ph}} < 2$ V, and all but a handful conclusively $\gtrsim$4 V. In addition, $V_{\textnormal{ph}}$ is expected to underestimate the magnitude of $V_{\textnormal{S/C}}$ by a factor 0.7-1 [@Odelstad2017], where the lower end of the interval can be expected to hold for low-magnitude $V_{\textnormal{S/C}}$, as those just discussed, which presumably correspond to low densities and consequently long Debye lengths and inefficient screening of the spacecraft potential. Thus, we conclude that $V_{\textnormal{S/C}}$ was persistently $\lesssim -5$ V throughout all Rosetta’s passages through the cavity, attesting to the persistent presence of a warm ($\sim$5 eV) population of electrons all through the parts of the cavity probed by Rosetta.
![Histograms of a) spacecraft potential measurements during all diamagnetic cavity crossings throughout the mission, color-coded by time, b) electron slopes from LAP1 sweeps during all diamagnetic cavity crossings throughout the mission, color-coded by time, c) electron slopes from LAP1 sweeps during all diamagnetic cavity crossings throughout the mission, color-coded by neutral density, and d) effective electron temperatures (see text), color-coded by time, for all LAP1 sweeps obtained during a diamagnetic cavity crossing and for which concurrent MIP density estimates are available.[]{data-label="fig:global_hists"}](global_hists.pdf){width="144.00000%"}
### Cold electrons
Figure \[fig:global\_hists\]b similarly shows a histogram of electron slopes from all LAP1 sweeps inside the cavity. Sweeps with clear signatures of cold electrons (electron slope $\gtrsim 100$ nA/V) dominate almost entirely inside the cavity. However, unlike the Nov 19-21 interval showed in Section \[sec:res1\], there is a non-negligible number (49) of sweeps lacking the steep-slope signature of cold electrons. These originate predominantly, but not exclusively, from the Dec 2015 time period. Non-detection by LAP does not preclude the presence of cold electrons, since their reaching the probe hinges on the probe bias potential being positive enough to overcome the potential barrier caused by the negatively charged spacecraft (cf. Appendix A). Indeed, manual analysis reveals that a handful of the sweeps lacking signatures of cold electrons occur at unusually strong spacecraft potentials, where this could possibly be the case. Also, if the steep-slope part commences very close to the upper edge of the sweep, the electron slope as derived here may not capture it (cf. Appendix A); a few cases have been found manually where this was the case. However, some 40 sweeps remain inside the cavity where a cold electron signature is conspicuously absent. About three quarters of these sweeps have concurrent MIP spectra; a manual analysis of these spectra has revealed one case with a clear cold-electron signature and about ten cases where cold the cold-electron signature was clearly absent. The remaining cases were inconclusive due to instrumental effects. Figure \[fig:global\_hists\]c shows the electron slope histogram color-coded by neutral density. All the sweeps lacking a cold-electron signature occur at very low neutral densities (in fact all but one of the points with cyan to green coloring and electron slopes $\lesssim 20$ nA/V in Figure \[fig:global\_hists\]c occur at unusually strong spacecraft potentials or have steep-slope parts that commence too close to the sweep edge to be captured by the electron slope estimate used here, as described above, and the last one can probably be attributed to a spurious COPS measurement), where inefficient cooling might be expected.
Using Equation (\[eq:eslope2\]), effective electron temperatures can be calculated from the electron slopes for all LAP1 sweeps inside the diamagnetic cavity for which simultaneous MIP density measurements are available. The electron slopes being a combination of the slopes due to the cold and warm electron populations, this gives only an effective temperature representing their combined contribution to the probe current, although this will be heavily weighted towards the temperature of the cold population, even if its relative abundance is not large. Thus, the effective temperature $T_{\textnormal{e,eff}}$ should be a reasonably accurate estimate of the temperature of the cold electrons, such that $T_{\textnormal{e,cold}} \lesssim T_{\textnormal{e,eff}}$. Figure \[fig:global\_hists\]d shows a histogram of the resulting effective electron temperatures, color-coded by time as in many of the previous figures. They distribute almost entirely in the range 0.02 - 0.2 eV ($\sim 200 - 2000$ K) with a peak between 0.04 - 0.06 eV ($\sim 500 - 700$ K), expected for cometary electrons collisionally cooled by the neutral gas. Both LAP electron slopes and MIP density measurements enter squared in Equation (\[eq:eslope2\]), thus $T_{\textnormal{e,eff}}$ can be expected to be rather sensitive to errors in either of these quantities. Measurements outside the 0.02 - 0.2 eV range should likely not be trusted. Values $\gtrsim 1$ eV additionally often derive from sweeps without the steep slopes associated with cold electrons, in which case the resulting effective electron temperatures should rather be associated with the warm population. However, these slopes are likely to be affected by potential barrier and spacecraft sheath effects and should not be trusted, though we note that the cluster of values around 5 - 10 eV in Figure \[fig:global\_hists\]d does fall squarely in the expected range for this population. The steep slopes associated with the cold electron population should be less sensitive to potential barrier and spacecraft sheath effects, as suggested by @Olson2010. From the color-coding in Figure \[fig:global\_hists\]d, a clear temporal trend can be observed, with $T_{\textnormal{e,eff}}$ generally decreasing over time. An effort has been made to determine the cause of this, but the results are as yet inconclusive and will not be further discussed here. This issue should be addressed in more detail in future works.
Discussion {#sec:disc}
==========
Uncertainties
-------------
### MIP uncertainties
In this study, the plasma frequency is identified as the frequency of peak power in MIP spectra. While this is consistent with mutual impedance probe models in plasma characterized by small enough Debye lengths (compared to the transmitter-receiver distance), the plasma frequency is expected to be located below the mutual impedance spectral power peak for larger Debye length (closer to the transmitter-receiver distance) [@Geiswiller2001; @Gilet2017]. The MIP density estimates may therefore be slightly overestimated. The magnitude of this over-estimation varies from spectrum to spectrum, depending on the Debye length, being virtually negligible for high densities (small Debye lengths) and up to several tens of percent for low densities (larger Debye length), depending on the actual operational mode used for each measurement. A detailed investigation into the interpretation of MIP spectra is beyond the scope of this paper; we note simply that by Equation (\[eq:ionslope\]), observed ion velocities will be over-estimated if MIP densities are, with the same relative errors ($v_{\textnormal{i}}$ being directly proportional to $n_{\textnormal{i}}$ in Equation (\[eq:ionslope\])). However, such errors on the order of several tens of percent do not affect the general statistics and conclusions of this paper.
### LAP uncertainties
LAP1 ion currents are subject to distortions due to the proximity of the large negatively charged spacecraft. First of all, the bias potential $U_{\textnormal{B}}$ w.r.t. the ambient plasma is shifted by the spacecraft potential, the probe being grounded to the spacecraft. This effect does not affect the ion (or electron) slopes, since the slope of a linear curve is invariant under translation. (Its effect on other sweep parameters can also largely be compensated for if the spacecraft potential is known, e.g. from $-V_{\textnormal{ph}}$). Secondly, ion trajectories are likely strongly perturbed in the potential field of the spacecraft. The spacecraft potential ($\lesssim$-5 V) is generally greater in magnitude than typical ion energies ($\lesssim$ 2 eV) so effects can be expected also upstream of the spacecraft, where LAP1 is located, even though the ion flow is supersonic. Investigations of this are ongoing, but preliminary results from PIC simulations using the SPIS (Spacecraft Plasma Interaction System, @Spis-Sci) software package indicate that $v_{\textnormal{i}}$ obtained from Equation (\[eq:ionslope\]) is overestimated as a result of these effects. The magnitude of the overestimation depends on the spacecraft potential, possibly becoming as large as a factor of 2.5 for spacecraft potentials around $-20$ V ($n_{\textnormal{e}} \sim 10^3$ cm$^{-3}$). However, this study is ongoing and possibly overestimates this effect due to uncertainties regarding the ability of the simulations to accurately reproduce the dynamics of flowing low-energy ions, so this should be taken as an upper limit of this error. We note that even an error of this magnitude is insufficient to bring our ion velocity estimates down to presumed neutral velocities $\lesssim$1 km/s.
### Ion mass uncertainty
The ion velocities in this paper have been derived from Equation (\[eq:ionslope\]) with the assumption of an ion mass $m_{\textnormal{i}} = 18$ u, typical of H$_2$O$^+$ and H$_3$O$^+$ ions presumed to dominate the ion composition. The main other candidate species is CO$_2^+$ with a mass of 44 u, i.e. about a factor 2.4 higher. An underestimation of the ion mass in Equation (\[eq:ionslope\]) would lead to an overestimation of the ion velocity derived from observed ion slopes by the same factor. In the presence of multiple species, the harmonic mean of the individual masses, weighted by their relative abundances, is the relevant quantity for ion velocity determinations by Equation (\[eq:ionslope\]). The CO$_2$/H$_2$O abundance ratio of the neutral gas did not exceed 1 until after the northward equinox in March 2016 [@Gasc2017; @Fougere2016b], i.e. after the time period studied here. This does not necessarily constrain the relative abundance of CO$_2^+$ in the plasma since the photoionization frequency of CO$_2$ is about twice that of H$_2$O . On the other hand, CO$_2^+$ has quite a large cross section for electron charge exchange with water which should help suppress the CO$_2^+$ abundance. We have adapted the analytical model of @Vigren2018 to estimate the fractional abundance of CO$_2^+$ ions as a function of the total outgassing rate, an assumed CO$_2$/H$_2$O ratio and an assumed expansion velocity. The relative abundance of CO$_2^+$ at Rosetta’s position increases with CO$_2$/H$_2$O ratio, outgassing velocity, and the spacecraft radial distance from the nucleus, but decreases with the outgassing rate. For the two intervals contributing most of the measurements in this study, 19 July - 3 Aug 2015 and 19-21 Nov 2015, CO$_2$/H$_2$O ratios were $\lesssim 0.1$ and $\lesssim 0.5$, respectively [@Fougere2016b]. Global production rates from @Hansen2016 (cf. Figure \[fig:overview\]) were $\gtrsim 1\cdot 10^{27}$ s$^{-1}$. Assuming outgassing velocities (of the neutrals) of $\lesssim 1$ km/s gives (harmonic) mean ion masses of $\lesssim 22$ u and $\lesssim 20$ u, respectively, corresponding to overestimations of $v_{\textnormal{i}}$ by $\lesssim 22$% and $\lesssim 11$%, respectively. This is too small to affect the overall statistics, although we cannot rule out that there could be single events at lower outgassing rate and larger CO$_2$/H$_2$O ratio where errors could be larger, though errors in excess of 50% are hard to achieve by any reasonable parameters using this model, even for single events.
Ion drift speed from a simple flux conservation model
-----------------------------------------------------
@Vigren2017b proposed a method for obtaining an estimate of the effective ion drift speed from a simple flux conservation model assuming radial outflow. In steady state, the total production of plasma inside the position of the spacecraft must then equal the radial flux of plasma past the spacecraft. Assuming production through photoionization only and neglecting recombination loss, flux conservation gives
$$n_{\textnormal{i}} u_{\textnormal{i}} = n_{\textnormal{n}} \nu (r - r_{\textnormal{c}}) \quad \Rightarrow \quad u_{\textnormal{i}} = \frac{n_{\textnormal{n}} \nu (r - r_{\textnormal{c}})}{n_{\textnormal{i}}},
\label{eq:flux_conservation}$$
where $\nu$ is the photoionization frequency and $r_{\textnormal{c}}$ is the nucleus radius, which is small compared to the spacecraft radial distance throughout the time period covered in this study and can therefore be neglected. Assuming quasi-neutrality, $n_{\textnormal{i}} \approx n_{\textnormal{e}}$, Equation (\[eq:flux\_conservation\]) can be used to obtain estimates of the effective ion drift speed from COPS neutral density and MIP electron density measurements. This provides a different ion velocity estimate, completely independent of LAP measurements and any errors therein, that captures specifically the radial outflow velocity, without any thermal or non-radial contributions. The assumptions of steady state, non-radial outflow and neglect of electron-impact ionization by supra-thermal electrons are well-founded inside the diamagnetic cavity, while the neglect of recombination loss is less clear, leading to a possible overestimation of the ion velocity on the order of a few tens of percent [@Vigren2017b; @Vigren2015]. Outside the cavity, where the plasma is much more variable and dynamic and the ion motion is affected by the presence of a magnetic field, this velocity estimate is more dubious. We use it here for comparison to and validation of the ion velocities previously obtained inside the diamagnetic cavity by combining MIP densities with LAP ion slopes. We use daily averaged photo-ionization frequencies for H$_2$O, computed at the location of 67P from the Timed SEE L3 v12 database, corrected for phase shift and heliocentric distance, from @Heritier2018. Figure \[fig:global\_vi\_ui\_hists\] shows a histogram of the resulting effective ion drift speeds inside the cavity, together with the $v_{\textnormal{i}}$ values derived from LAP ion slopes.
![Comparison of effective ion drift speeds obtained from the flux conservation model of @Vigren2017b and the LAP ion slope method during all diamagnetic cavity crossings throughout the mission.[]{data-label="fig:global_vi_ui_hists"}](global_vi_ui_hists2.pdf){width="67.00000%"}
The effective ion drift speeds from the flux conservation model are in good agreement with the ion-slope derived values throughout the cavity. Some shift of the peak of the distribution towards lower velocities can be observed, but our main result that the ion drift velocity is significantly elevated w.r.t. the neutral velocity is supported by these measurements.
Implications for the pressure balance at the cavity boundary
------------------------------------------------------------
The fact that we observe ion velocities well in excess of the neutral velocity both inside the cavity and in the surrounding region shows that the ions were not collisionally coupled to the neutrals, implying that the ion-neutral drag force was not responsible for balancing the outside magnetic pressure at the cavity boundary. The poor collisional coupling is a result of the lower outgassing rate of 67P compared to comet 1P/Halley [@Vigren2017], where ion-neutral drag was indeed found to prevail [@Ip1987]. In this context it is interesting that @Timar2017 managed to achieve such good fit between observed cavity distance values and the ion-neutral drag model of @Cravens1986 [@Cravens1987], which assumes that the ions are collisionally coupled to the neutrals and stagnant at (or just outside) the cavity boundary. The good fit may essentially express the strong dependence of observed cavity boundary distances on the neutral production rate and the varying solar wind pressure. It cannot be ruled out that some other process, driven by the same inputs, might be able to produce an equally good fit, if the resulting dependence of the cavity boundary distance on outgassing and solar wind pressure is similar to that of the ion-neutral drag model.
The elevated ion velocities observed here produce an ion dynamic pressure $n_{\textnormal{i}} m_{\textnormal{i}} v_{\textnormal{i}}^2$ of $\sim$5$\cdot 10^{-10}$ kgm$^{-1}$s$^{-2}$ ($n_{\textnormal{i}} = 1000$ cm$^{-3}$, $m_{\textnormal{i}} = 18$ u and $v_{\textnormal{i}} = 4$ km/s). Typical magnetic field strengths outside the cavity are $\sim$20 nT, giving a magnetic pressure $B^2/(2\mu_0) \sim 2 \cdot 10^{-10}$ kgm$^{-1}$s$^{-2}$. Thus the ion dynamic pressure is of the right order of magnitude to balance the magnetic pressure on the outside. If this was the case, the ion dynamic pressure would be expected to decrease outside the cavity, by an amount comparable to the magnitude of the magnetic pressure. We see no such decrease in ion velocity outside the cavity, rather there is some tendency toward increased velocity. While the ion velocity measurements presented in this paper are incapable of distinguishing between bulk and thermal motion and don’t give a flow direction, the apparent persistence of a spacecraft wake outside the cavity suggests that no significant thermalization of the ions take place. Since the geometry of the slews during which LAP2 comes out of the wake appear to be generally consistent with radial flow from the nucleus, even outside the cavity, there does not seem to be any clear change in flow direction of the ions either. However, the plasma density generally increases immediately outside the cavity [@Henri2017; @Harja2018] thus the flux conservation model of @Vigren2017b, if applied there, would give a decrease in the effective radial ion drift speed and of the ion dynamic pressure. This observation is hard to reconcile with the ion slope based velocity measurements without invoking thermalization or non-radial flow of the ions, which in turn is hard to reconcile with the apparent persistence of a spacecraft wake outside the cavity. The dynamics of the plasma on the outside of the cavity boundary and the role of the ion dynamic pressure in balancing the magnetic pressure should be further investigated in future works.
We have seen that there is always a substantial population of warm ($\sim$5 eV) electrons present throughout the parts of the inner coma probed by Rosetta, so also inside the diamagnetic cavity. These contribute a thermal pressure $n_{\textnormal{e}} k_{\textnormal{B}} T_{\textnormal{e}} \sim 8 \cdot 10^{-10}$ kgm$^{-1}$s$^{-2}$ ($n_{\textnormal{e}} = 1000$ cm$^{-3}$, $T_{\textnormal{e}} = 5$ eV), i.e. of the same order of magnitude as the ion dynamic pressure and well sufficient to maintain the pressure balance at the cavity boundary. However, the fact that the electron density increases immediately outside the cavity means that a substantial decrease in electron temperature would be required there. It is hard to conceive of a mechanism that can achieve this in the region immediately outside the cavity boundary. Direct measurements of the electron temperature have not yet been obtained from the data at sufficient accuracy to detect such changes, so the role of the electron thermal pressure for the pressure balance at the cavity boundary cannot be conclusively determined at this time.
Summary and conclusions {#sec:conclusions}
=======================
We have combined density measurements by MIP with ion slopes from LAP sweeps to produce measurements of the ion velocity in and around the diamagnetic cavity of 67P. As noted before by @Henri2017, the density inside the cavity was remarkably stable, while the surrounding region was characterized by high density variations. This turned out to hold also for the ion slopes; they were much more variable outside than inside the cavity. The resulting ion velocity measurements inherited this feature, with stable velocities narrowly distributed around 3.5-4 km/s inside the cavity and scattered toward higher velocities ($\lesssim$8-10 km/s, though still heavily weighted towards 4 km/s) in the surrounding region. The ion velocity inside the cavity was clearly elevated w.r.t. the neutral velocity of $\lesssim$1 km/s and in good agreement with the model predictions of @Vigren2017. This indicates that the ions were not collisionally coupled to the neutrals and implies that the ion-neutral drag force was not responsible for balancing the outside magnetic pressure at the cavity boundary. It also suggests the existence of an ambipolar electric field to accelerate the ions, at least inside the cavity.
Clear evidence of wake effects on LAP2 has been identified inside the cavity during a spacecraft slew. The geometry of this slew was such that LAP2 was brought forward from a position close to being obscured from view of the nucleus to a position more exposed to radial flow from the comet. Inside the cavity, the effect of this on LAP2 was to increase its collection of ions and cold electrons and generally make its current-voltage characteristics more similar to that of LAP1, which was consistently well-exposed to any radial flows. This shows that the flow of cometary ions inside the cavity was indeed close to radial and supersonic, at least w.r.t. the perpendicular temperature. Thus, the observed ion velocities inside the cavity are to be taken as a radial velocity of the ions, possibly a cold radial drift, although a non-negligible radial temperature of the ions cannot be rule out. Outside the cavity, the effects of these slews on the LAP2 current were less clear and consistent, indicative of a more dynamic wake, or possibly a more variable ion flow direction.
We also made a detailed examination of the spacecraft potential measurements inside the cavity, finding it to be consistently $\lesssim$-5 V. This proves that a population of warm ($\sim$5-10 eV) electrons was present throughout the parts of the cavity probed by Rosetta. This was shown to hold on larger time scales throughout the mission already by @Odelstad2017, but has now been confirmed on a sweep-by-sweep level for all cavity events. This shows that Rosetta never entered the region of collisionally coupled electrons presumed to exist in the innermost part of the coma, not even during any of the passes through the diamagnetic cavity.
Finally, we have found that a population of cold ($\lesssim$ 0.1 eV) electrons, first shown by @Eriksson2017 to be intermittently present at Rosetta, is in fact observed consistently throughout the diamagnetic cavity. Already immediately outside the cavity, sweeps lacking a signature of cold electrons begin to turn up intermittently. This reinforces the notion of @Henri2017 that the formation and extent of the cavity is closely related to electron-neutral collisionality and suggests that the the region of collisionally coupled electrons, while never entered by Rosetta, was possibly not that far away during the cavity crossings. It also suggests that the filamentation of the cold electrons begins at or near the cavity boundary and is possibly related to an instability of this boundary.
Sweep analysis {#sec:sweep_an}
==============
Figure \[fig:ex\_sweeps\] shows four examples of LAP1 sweeps from inside the diamagnetic cavity.
![Four examples of LAP1 sweeps from inside the diamagnetic cavity, illustrating the effect of cold electrons and how the electron and ion slopes and the photoelectron knee ($V_{\textnormal{ph}}$) are obtained.[]{data-label="fig:ex_sweeps"}](ex_sweeps.pdf){width="\textwidth"}
Together, these four sweeps contain examples of all the major sweep features of relevance to the present analysis and of which account needs to be taken in deriving relevant parameters (e.g. electron and ion slopes) from the sweeps. Due to there often being a large difference in scales between various sweep features, the complete sweeps, plotted in the outer axes (Figure \[fig:ex\_sweeps\]a), are complemented by an inset panel (Figure \[fig:ex\_sweeps\]b) displaying zoomed versions of the low-current parts of the sweeps. Here, sweeps 3 and 4 exhibit, at least qualitatively, the behavior expected for spherical probes in a single-temperature maxwellian plasma: The sweeps are linear at negative bias potentials, corresponding to attracted-ion current only, have a sharp inflection point at the photoelectron knee potential ($V_{\textnormal{ph}}$), corresponding to the local plasma potential w.r.t. the spacecraft (which may differ from the plasma potential at infinity due to some fraction of the spacecraft potential field remaining at the position of the probe), and then become linear again at more positive potentials, corresponding to attracted-electron current only. In sweeps 1 and 2 on the other hand, the linear parts above $V_{\textnormal{ph}}$ are interrupted by secondary inflection points at about 16 and 12 V, respectively, followed by a much steeper slope. This is attributed to a second population of cold ($\lesssim$0.1 eV) electrons [@Eriksson2017], required to produce such steep slopes in the sweeps. The fact that they don’t appear in the sweeps until at least a few volts above $V_{\textnormal{ph}}$ we ascribe to the existence of a potential barrier between the probe and the ambient plasma due to some fraction of the potential field of the negatively charge spacecraft remaining outside the position of the probe [@Olson2010]. The probe would then have to go to somewhat more positive potentials than the local plasma potential in its immediate vicinity in the potential well of the spacecraft for this barrier potential to be fully suppressed, allowing the cold electrons to reach the probe. Assuming this barrier potential to be much lower in magnitude than typical temperatures of the uncooled thermal electron population, which must exist to explain the persistently negative spacecraft potentials [@Odelstad2017], the bulk of the thermal electrons will have sufficient energy to overcome it and will appear in the sweeps already at the photoelectron knee potential.
The photoelectron knee is identified in the sweeps primarily by seeking peaks in the sweep second derivative. This requires considerable filtering and pre-processing of the sweeps to eliminate noise and spurious effects, without thereby smoothing the knee so much that it avoids subsequent detection. Further analysis includes conditioning on various supplementary sweep parameters, such as the sign of the current at candidate knees, to single out real from spurious knees and distinguish between the photoelectron knee and the secondary knee associated with collection of cold electrons. The algorithm currently in use is complex and will not be further described here. For the purpose of this study, it suffices to note that the resulting $V_{\textnormal{ph}}$ estimates have typical errors up a few volts (as can be observed for sweeps 1 and 2 in Figure \[fig:ex\_sweeps\]b).
The electron slopes are obtained from an ordinary least squares (OLS) linear fit to the last 10 points of each sweep that have currents below 9 $\mu$A. This cap is used since in some operational modes, the LAP electronics saturate at currents above this value, as for example sweep 4 in Figure \[fig:ex\_sweeps\]a. The limited range of only 10 points for the fit is used to assure that in the presence of cold electrons the fit is performed well above the second inflection point, so that the two very different slopes below and above this point are not convoluted. This generally works well, but for some rare cases where the second inflection point appears within 10 points of the sweep upper edge and the electron slope is consequently underestimated. As the linear electron part is the steepest on the sweeps, we can never overestimate the slope (except for small random errors). The resulting linear fits are shown in cyan for the sweeps in Figure \[fig:ex\_sweeps\].
The ion slopes are obtained from similar linear fits to the part of each sweep lying between 40% and 80% of the distance up from the first (most negative) bias potential to the photoelectron knee ($V_{\textnormal{ph}}$). In previous work where sweep ion slopes have been used, either directly [@Odelstad2017; @Vigren2017b], or indirectly, they have been obtained from the first 40% of each sweep below $V_{\textnormal{ph}}$, i.e. just below the part used here. This choice was made to ensure that the repelled electron current had fully decayed and did not skew the ion slopes. In this study, we have found numerous examples of spurious features in these parts of the sweeps, one example of which is shown in sweep 4 in Figure \[fig:ex\_sweeps\]b, where there is a clear bend in the sweeps at the most negative bias potentials. Therefore, the ion slopes in this paper are derived from further up the sweep, where such effects have not been observed and which are more consistently linear. We surmise that the electron current is well suppressed in this interval and that the previous selection was unnecessarily draconian in this regard.
In LDL mode, interference from MIP produces spurious outlying points in the part of the sweeps used to obtain the ion slope. An example of this can be seen in sweep 3 in Figure \[fig:ex\_sweeps\]b. To remedy this, a generalized extreme Studentized deviate (ESD) test [@Rosner1983] is iteratively performed on sequential fits, testing for and removing one outlier at a time, until no outliers (with respect to a normal distribution and at a significance level of 0.001) are found in the fit residuals. (This approach to outlier removal was used together with the total linear least squares (TLS) algorithm by @Odelstad2017; we use it here together with the OLS algorithm instead and with a factor 10 lower significance level.) This procedure has been found to generally produce accurate and robust linear fits to the aforementioned parts of the sweeps, even in the presence of such strong interferences from MIP. It is used blindly on all sweeps in this paper; no effort has been made to selectively apply it only to operational modes where MIP interferences can be expected (LDL modes, cf. Section \[sec:MIP\]). The resulting linear fits are shown in green for the sweeps in Figure \[fig:ex\_sweeps\].
Rosetta is a European Space Agency (ESA) mission with contributions from its member states and the National Aeronautics and Space Administration (NASA). The work on RPC-LAP data was funded by the Swedish National Space Board under contracts 109/12, 168/15 and 166/14 and Vetenskapsr[å]{}det under contract 621-2013-4191. Work at LPC2E was supported by CNES, by ESEP and by ANR under the financial agreement ANR-15-CE31-0009-01. This work has made use of the AMDA and RPC Quicklook database, provided by a collaboration between the Centre de Donn[é]{}es de la Physique des Plasmas (CDPP) (supported by CNRS, CNES, Observatoire de Paris and Universit[é]{} Paul Sabatier, Toulouse) and Imperial College London (supported by the UK Science and Technology Facilities Council). Work at the University of Bern on ROSINA COPS was funded by the State of Bern, the Swiss National Science Foundation, and the European Space Agency PRODEX Program. The data used in this paper will soon be made available on the ESA Planetary Science Archive and is available upon request until that time.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The Orion Nebula Cluster (ONC) is the nearest region of massive star formation and thus a crucial testing ground for theoretical models. Of particular interest amongst the ONC’s $\sim 1000$ members are: $\theta^1$ Ori C, the most massive binary in the cluster with stars of masses $38$ and $9\,{\rm{M_\odot}}$ ; the Becklin-Neugebauer (BN) object, a $30\,{\rm km\:s^{-1}}$ runaway star of $\sim
8\,{\rm{M_\odot}}$ [@2004ApJ...607L..47T]; and the Kleinmann-Low (KL) nebula protostar, a highly-obscured, $\sim 15\,{\rm{M_\odot}}$ object still accreting gas while also driving a powerful, apparently “explosive” outflow [@1993Natur.363...54A]. The unusual behavior of BN and KL is much debated: How did BN acquire its high velocity? How is this related to massive star formation in the KL nebula? Here we report the results of a systematic survey using $\sim 10^7$ numerical experiments of gravitational interactions of the [$\rm \theta^1C$]{} and BN stars. We show that dynamical ejection of BN from this triple system at its observed velocity leaves behind a binary with total energy and eccentricity matching those observed for [$\rm \theta^1C$]{}. Five other observed properties of [$\rm \theta^1C$]{} are also consistent with it having ejected BN and altogether we estimate there is only a $\lesssim 10^{-5}$ probability that [$\rm \theta^1C$]{} has these properties by chance. We conclude that BN was dynamically ejected from the [$\rm \theta^1C$]{} system about 4,500 years ago. BN has then plowed through the KL massive-star-forming core within the last 1,000 years causing its recently-enhanced accretion and outflow activity.
author:
- SouravChatterjee
- 'Jonathan C. Tan'
title: Gravitational slingshot of young massive stars in Orion
---
Introduction {#sec:intro}
============
Massive stars impact many areas of astrophysics. In most galactic environments they dominate the radiative, mechanical and chemical feedback on the interstellar medium, thus regulating the evolution of galaxies. Many low-mass stars form in clusters near massive stars, and their protoplanetary disks can be affected by this feedback also. There is some evidence that our own solar system was influenced in this way . Despite this importance, there is no consensus on the basic formation mechanism of massive stars. Theories range from scaled-up versions of low-mass star formation [@2003ApJ...585..850M], to competitive Bondi-Hoyle accretion at the center of forming star clusters [@2001MNRAS.323..785B; @2010ApJ...709...27W], to stellar collisions [@1998MNRAS.298...93B]. The Orion Nebula Cluster (ONC) is the nearest region of massive star formation and thus a crucial testing ground for theoretical models.
Of particular interest amongst the ONC’s $\sim 1000$ members in this regard are: $\theta^1$ Ori C, the most massive binary in the cluster with stars of masses $\approx 38$ and $9\,{\rm{M_\odot}}$ ; the Becklin-Neugebauer (BN) object, a $30\,{\rm km\:s^{-1}}$ runaway star of $\approx 8 M_\odot$ [@2004ApJ...607L..47T]; and the Kleinmann-Low (KL) nebula protostar, a highly-obscured, about $15\,{\rm{M_\odot}}$ object still accreting gas while also driving a powerful, apparently “explosive” outflow [@1993Natur.363...54A]. The unusual behavior of BN and KL is much debated bearing implications towards massive-star formation theories: How did BN acquire its high velocity? How is this related to massive star formation in the KL nebula?
BN, like KL, is heavily obscured by dust so its luminosity of $\sim
(5\pm3)\times 10^3\,{\rm{L_\odot}}$ mostly emerges in the infrared [@1998ApJ...509..283G]. The above luminosity constrains BN’s mass to be $m_{\rm BN} \simeq 9.3\pm2.0{\rm{M_\odot}}$, assuming it is on the zero age main sequence [@2004ApJ...607L..47T]. For this estimate and throughout the paper we have adopted $414\pm7$ pc for the distance to the cluster . Astrometry based on mm and radio observations indicate that BN is a runaway star [@1995ApJ...455L.189P; @2004ApJ...607L..47T], with some recent measurements of its motion in the ONC frame of $\mu_{\rm BN}=13.2\pm
1.1\:{\rm mas\:yr^{-1}}$ towards P.A.$_{\rm
BN}=-27^\circ.5\pm4^\circ$[@2008ApJ...685..333G] and $\mu_{\rm
BN}=13.4\pm 1.1\:{\rm mas\:yr^{-1}}$ towards P.A.$_{\rm
BN}=-18^\circ.8\pm4.6^\circ$ [@2011ApJ...728...15G] (Figure 1). This corresponds to a velocity $v_{\rm
2D,BN}=25.9\pm2.2\:{\rm km\:s^{-1}}$ [@2008ApJ...685..333G]. BN has an observed radial (LSR) velocity of $+21\pm\sim 1\:{\rm km\:s^{-1}}$ [@1983ApJ...275..201S], while the ONC mean is $+8.0\:{\rm km\:s^{-1}}$ [based on a mean heliocentric velocity of about 1000 ONC stars of $+26.1\:{\rm km\:s^{-1}}$; @2008ApJ...676.1109F]. Including this $+13\pm\sim
1\:{\rm km\:s^{-1}}$ radial velocity with respect to the ONC mean, the 3D ONC-frame velocity of BN is $v_{\rm 3D,BN}=29\pm3\:{\rm
km\:s^{-1}}$. This is much greater than the velocity dispersion of ONC stars, variously inferred to be $\sigma_{\rm 3D} = 2.4 \:{\rm km\:s^{-1}}$ based on the proper motions of $\sim 50$ bright ($V\lesssim 12.5$) stars within 30$^\prime$ of the ONC center [@1988AJ.....95.1744V], $\sigma_{\rm 3D} = 3.8 \:{\rm km\:s^{-1}}$ based on proper motions of $\sim 900$ fainter stars within 15$^\prime$ of the ONC center [@1988AJ.....95.1755J], and $\sigma_{\rm 3D} = 5.4 \:{\rm km\:s^{-1}}$ based on radial velocity measurements (potentially affected by motion induced by binarity) of $1100$ stars within $\sim$60$^\prime$ of the ONC center [@2008ApJ...676.1109F]. Thus there is little doubt that BN is a runaway star, which formed and was then accelerated in the ONC.
Supernova explosion of one member of a binary can lead to the other being ejected at high speeds [@1957moas.book.....Z]. The ONC is too young [most stars are $<3$ Myr old; @2010ApJ...722.1092D] for a supernova to have occurred. Nor is there any evidence for a recent supernova. Alternatively, runaway stars can be produced via dynamical ejection — a gravitational slingshot — from a triple or higher multiple system [@1967BOTT....4...86P; @1983ApJ...268..319H; @1986ApJS...61..419G], in which the lowest mass member tends to be ejected. Indeed, such dynamical ejection of stars naturally happen in dense and young clusters [e.g., @2004MNRAS.350..615G; @2011Sci...334.1380F]. Similar ejections have also been discussed in the context of some other ONC stars [@2004MNRAS.350..615G]. Thus BN, having formed in the ONC, should have been accelerated via dynamical ejection. The predictions of this scenario are very specific: somewhere along BN’s past trajectory should be a massive binary (or higher order multiple), with two components likely more massive than BN, recoiling in the opposite direction, and, as we shall see, with specific orbital properties.
Two scenarios for the dynamical ejection of BN have been proposed. [(1)]{} Ejection from the [$\rm \theta^1C$]{} binary [@2004ApJ...607L..47T]: In this scenario the BN star is ejected via a strong gravitational scattering interaction [e.g., @1983ApJ...268..319H; @2011MNRAS.410..304G] and later plows, [*by chance*]{}, through the KL star-forming core to drive tidally-enhanced accretion and thus outflow activity. If so, a model of formation of the KL massive protostar via an ordered collapse of a gas core to a central disk[@2003ApJ...585..850M], similar to how low-mass stars are thought to form, is still broadly applicable, though subject to the tidal perturbation from BN’s fly-by. [(2)]{} Ejection from the KL (source [*I*]{}) protostar [@2005AJ....129.2281B; @2008ApJ...685..333G]: Here it is proposed that the KL outflow is related to the disintegration of a [ *forming*]{} triple system, which ejected BN and produced a binary suggested to be the radio source I [@1995ApJ...445L.157M]. This binary has recoiled southwards from the original formation site and is now hidden, [*by chance*]{}, behind or in the dense gas core near the center of the KL nebula. In this scenario the gas from the original formation site, like the stars, has also been expelled in this event to form the outflow, and the core that formed these massive stars has been destroyed. If true, this is a very different formation process, and would indicate that chaotic gravitational interactions between multiple protostars followed by complete ejection of both stars and gas are intrinsic features of massive star formation [@2005AJ....129.2281B; @2011ApJ...727..113B], at least in this case in Orion.
Figure 1 shows a near-IR image of the central region of the ONC, including BN, the KL protostar (marked by radio source [ *I*]{}) and the famous Trapezium stars, of which [$\rm \theta^1C$]{} is the brightest. The past trajectory of BN is indicated based on its present motion and assuming no acceleration. It goes near KL [*and*]{} the Trapezium stars. The high obscuration to KL means that there is little direct constraint on the properties of the star(s): for example there is no evidence that it is even a binary. In contrast, the properties of [$\rm \theta^1C$]{} have been measured much more precisely and so the scenario of ejection of BN via a binary-single strong scattering can be tested much more rigorously and is the goal of this study.
In §\[sec:method\] we summarize our methods and numerical calculations. In §\[sec:results\] we present our key results, which show that [$\rm \theta^1C$]{} has orbital properties expected if it ejected BN. In §\[sec:probability\] we estimate the probability that [$\rm \theta^1C$]{} has not been responsible for BN’s ejection and has these and other observed properties simply by chance. We summarize and conclude in §\[sec:conclude\].
Methods {#sec:method}
=======
We investigate the scenario of ejection of BN from the [$\rm \theta^1C$]{} binary by carrying out calculations of gravitational scattering between the three stars. We adopt the central values of the observed masses of the [$\rm \theta^1C$]{} binary members , $m_{\rm
\theta^1C_1}=38.2\pm3.6\,{\rm{M_\odot}}$ and $m_{\rm \theta^1C_2}=8.8\pm
1.7\,{\rm{M_\odot}}$. For BN we adopt the central value of the mass estimate $m_{\rm BN}=8.2\pm2.8\,{\rm{M_\odot}}$, which is based on the observed cluster-frame proper motion of [$\rm \theta^1C$]{} of $\mu_{\rm
\theta^1C}=2.3\pm0.2\:{\rm mas\:yr^{-1}}$ [@1988AJ.....95.1744V], i.e., $v_{\rm 2D,\theta^1C} =
4.5\pm0.4\:{\rm km\:s^{-1}}$, and assuming it is due to recoil from ejecting BN. Note that this mass is consistent with that inferred from the luminosity of BN, discussed above. The error range includes an assumed pre-ejection motion of the center of mass of the 3 stars along the ejection axis of 0.7 ${\rm mas\:yr^{-1}}$, i.e., similar to that observed for other bright ONC stars [@1988AJ.....95.1744V].
We carry out a systematic investigation of the three possible strong scattering interactions between a binary and a single star. Depending on the initial perturber and the binary members these interactions can be divided into three types. [**Type 1 - BN star is the perturber:**]{} Here BN is initially a single star that interacts with [$\rm \theta^1C$]{} members [$\rm \theta^1C_1$]{} and [$\rm \theta^1C_2$]{}. We denote this initial configuration as \[[$\rm \theta^1C_1$]{}, [$\rm \theta^1C_2$]{}\] BN. The square brackets denote a bound binary and the star outside the brackets is a single stellar perturber. [**Type 2 - [$\rm \theta^1C_2$]{} is the perturber:**]{} This can be denoted as \[[$\rm \theta^1C_1$]{}, BN\] [$\rm \theta^1C_2$]{}. [**Type 3 - [$\rm \theta^1C_1$]{} is the perturber:**]{} This can be denoted as \[[$\rm \theta^1C_2$]{}, BN\] [$\rm \theta^1C_1$]{}.
Out of all outcomes of our numerical experiments we focus on the ones that could lead to ejection of BN. [**Case 1 - BN fly-by:**]{} This is a Type 1 interaction and the outcome of interest is “preservation" where the initial binary members remain unchanged and BN flies by after interaction with the initial binary. We write this interaction as [$\rm [\theta^1C_1,\theta^1C_2]\:BN \longrightarrow [\theta^1C_1,\theta^1C_2]\: BN$]{}. The arrow points from the initial to the final configuration. [**Case 2 - ejection of BN from a binary via exchange with $\bf \theta^1{\rm
C}_2$:**]{} [$\rm [\theta^1C_1,BN]\:\theta^1C_2 \longrightarrow [\theta^1C_1,\theta^1C_2]\: BN$]{}. [**Case 3 - ejection of BN from a binary via exchange with $\bf \theta^1{\rm C}_1$:**]{} [$\rm [\theta^1C_2 , BN]\:\theta^1C_1 \longrightarrow [\theta^1C_1,\theta^1C_2]\: BN$]{}. The “Types" and “Cases" are different in the fact that for the Types we only take into account the initial conditions and allow all outcomes, whereas, the Cases are more restrictive and only considers the outcomes where BN is ejected leaving behind a bound binary.
There are 7 parameters that describe the initial conditions of each binary-single star interaction and since the 3-body problem is chaotic we must sample over the expected distributions of these parameters: (1) Eccentricity of the initial binary, $e_i$. For each case we investigate two extreme distributions: (A) Circular, $e_i=0$, for all systems, which may be expected if the binaries have recently formed from a gas disk that has damped out noncircular motions; (B) Thermal [@2003gmbp.book.....H], $dF_b/de_i= 2 e_i$, where $F_b$ is the fraction of the binary population. This is an extreme scenario that would result if binaries have had time to thermalize via stellar interactions with other cluster stars. The actual situation for ONC binaries should be between these limits. (2) Semi-major axis of the initial binary, $a_i$. We assume a flat distribution [@1989ApJ...347..998E] $dF_b/d {\rm log} a_i = 0.208$ from $0.1\,{\rm{AU}}$ (approximately the limit resulting from physical contact) to $6300\,{\rm{AU}}$ [the hard-soft boundary beyond which binaries are expected to be disrupted by interactions with other cluster stars; @2003gmbp.book.....H]. (3) Initial impact parameter, $b_i$. For each Case and each sampling of $a_i$ we investigate the full range of impact parameters that can lead to scattering events strong enough to eject BN with its observed high velocity. This is achieved by increasing $b_i$ from small values until the regime where all interactions are weak fly-bys incapable of increasing BN’s velocity to the observed large value. (4) Initial relative velocity at infinity, $v_i$, in the frame of the center of mass of the binary. We assume that the stars have velocities following a Maxwellian distribution with a dispersion of $\sigma_{\rm{3D}}=3\:\rm{km\:s^{-1}}$ [@2008ApJ...676.1109F]. We have repeated the numerical experiments with $\sigma_{\rm{3D}}=2\:{\rm \rm
km\:s^{-1}}$, finding qualitatively similar results. (5) The initial angle of the orbital angular momentum vector ($\vec L_i$) of the binary with respect to the velocity vector ($\vec v_i$) of the approaching single star, which is assumed to be randomly oriented. (6) The initial angle between the major axis of the binary orbit and the velocity vector of the approaching single star, which is assumed to be randomly oriented. (7) The initial orbital phase of the binary, which is assumed to be random.
We find cross-sections ($\Sigma$) of the various outcomes of the Cases 1, 2, & 3 binary-single interactions numerically using the [[Fewbody]{}]{} software [@2004MNRAS.352....1F], which uses an order $8$ Runge-Kutta integrator, by performing $\sim 10^7$ numerical scattering experiments to sample the $7$ dimensional parameter space that is needed to describe the possible interactions. This large number of numerical scattering experiments gives us rigorous sampling of all properties in the dynamical scattering problem, including the initial semimajor axis of the binary ($a_i$), initial eccentricity ($e_i$), binary orbital phases, initial velocity at infinity ($v_i$) of the single star, the initial impact parameter of the encounter ($b_i$), the angle between the initial major axis relative to $\vec v_i$ and the initial orientation of the binary ($\vec L_i . \vec v_i$). For example, we sample $b_i$ as fine as $10^{-4}b_0$, where $b_0$ is the impact parameter at infinity that results in a closest approach within $2a_i$. Starting from a small value, $b_i$ is sampled with the above-mentioned resolution up to at least $b_{i,{\rm max}} = b_0$. Within this interval smaller intervals of $b_i$, $\delta b_i = 10^{-4} b_0$ are chosen. The impact parameter is chosen from each of these intervals uniform in the area of the annulus between $b^{'}_{i}$ and $b^{'}_{i} + \delta b_i$. If a particular final outcome of interest or “event” is achieved (in particular [[BN-Velocity]{}]{} or [[BN-True]{}]{} events, defined below), then contribution of that event to the total $\Sigma$ is simply $\delta\Sigma = 2\pi b^{'}_{i} \delta b_i$ (@1996ApJ...467..348M; but see more recently @2006ApJ...640.1086F). Afterwards, to ascertain that all energetic encounters are sampled, we increase the maximum $b_i$ geometrically until $b_{i,{\rm max}} = 100 b_0$. For our case, this large value of $b_{i,{\rm max}}$ corresponds to as large a physical distance as the cluster size making sure that all possible energetic encounters are captured in the determination of the cross-sections $\Sigma$.
Using the large ensemble of numerical gravitational scattering experiments we evaluate the $\Sigma$s for outcomes where BN is ejected leaving behind [$\rm \theta^1C_1$]{} and [$\rm \theta^1C_2$]{} in a bound binary (henceforth, “[[BN-Ejection]{}]{}" events). A subset of the [[BN-Ejection]{}]{} events where BN is ejected with the observed velocity of $29\pm3\,{\rm km\:s^{-1}}$ are called “[[BN-Velocity]{}]{}” events. We do not put any constraints on the binary properties that is left behind for the [[BN-Velocity]{}]{} events. We further calculate the $\Sigma$s of a subset of [[BN-Velocity]{}]{} events where the final binary is left with orbital properties similar to those observed of [$\rm \theta^1C$]{}, namely, $a =
18.13\pm1.28\,\rm{AU}$, and $e = 0.592\pm0.07$ . The [[BN-Ejection]{}]{} events are used to explore the velocity distribution of BN if it is ejected via a strong binary-single interaction. [[BN-Velocity]{}]{} events, a subset of the [[BN-Ejection]{}]{} events, show us all possible interactions over a range of $a_i$ where BN could have an energy compatible with the observed energy. A further subset, the [[BN-True]{}]{} events, give us stronger constraints and indicates the range of initial binary properties most likely to create the observed [$\rm \theta^1C$]{} binary as well as the runaway BN star.
Results {#sec:results}
=======
In this section we present the key results of our numerical experiments. We start with overall outcomes of all our numerical experiments for all Cases and $e_i$ distributions and then increasingly focus our attention towards the observed BN-[$\rm \theta^1C$]{} system and compare its various properties with those predicted from our simulations.
Figure \[fig:branching\] shows the branching ratios of all outcomes in general from our numerical experiments. A handful of interesting aspects are evident in the branching ratios for the given masses of the 3 stars involved in these interactions.
Disruption (or ionization) of the initial binary happens only when the binary is dynamically soft [@1983ApJ...268..319H], i.e., the value of the binary binding energy is $\lesssim$ the kinetic energy of the perturber. This is achieved at large $a_i \gtrsim 300\,{\rm{AU}}$, for Types 1 and 2. For Type 3 ionization can happen at relatively smaller $a_i \sim 50\,{\rm{AU}}$ due to the higher mass of the perturber and relatively lower binding energy of the initial binary. Nevertheless, even for Type 3, branching ratio for ionization becomes comparable or greater than exchange outcomes only at $a_i \sim 10^3\,{\rm{AU}}$.
For Types 1 and 2, exchange with the primary is very unlikely, since here the primary is significantly more massive ($38.2\,{\rm{M_\odot}}$) than the secondary ($8.8$ and $8.2\,{\rm{M_\odot}}$ for interactions of Types 1 and 2, respectively). In interaction of Type 3, since the initial binary consists of two stars with comparable masses, both exchanges are almost equally likely. For Type 3 the fraction of exchange outcomes is comparable to the fraction of fly-by events for a large range of $a_i$ taking into account sufficiently strong encounters (see §\[sec:method\] for the value of the maximum impact parameter). The fraction of fly-by outcomes is of course formally infinite since one can always use a sufficiently large impact parameter where nothing but a weak fly-by is the outcome.
Collisional outcomes are comparable with exchanges only for sufficiently small $a_i$ values. In interactions of Type 1, preservation is the channel that can produce the observed BN-[$\rm \theta^1C$]{} system. For Type 1 for both $e_i$ distributions collisions become important for $a_i \lesssim 1\,{\rm{AU}}$. For interactions of Types 2 and 3, exchange of the perturber with the primary is the channel that can produce the observed BN-[$\rm \theta^1C$]{} system. For Type 2, collisions become comparable with the BN-[$\rm \theta^1C$]{} producing channel for $a_i \lesssim$ a few AU. Whereas, for interactions of Type 3, collisions remain comparable to the BN-[$\rm \theta^1C$]{} producing channel for $a_i \lesssim 1\,{\rm{AU}}$. In interactions of all Types collisions happen more often for the Thermal $e_i$ distribution since for the Thermal $e_i$ distribution the pericenter distances for the stars in binary can be much smaller than that for the Circular $e_i$ distribution for any given $a_i$. We later show (§\[sec:cross-sections\]) that [[BN-True]{}]{} events happen in a given range of $a_i$ for a given interaction Type. For all interaction Types and $e_i$ distributions collisions have much lower branching ratios compared to the branching ratios for the BN-[$\rm \theta^1C$]{} production channels for the ranges of $a_i$ where [[BN-True]{}]{} events can occur.
In the following sections we increasingly focus on outcomes that are similar to the observed BN-[$\rm \theta^1C$]{} system. We first present results for all events where the BN star is ejected ([[BN-Ejection]{}]{}, §\[sec:vdist\]). Then we present results for all outcomes where the BN star is ejected with a velocity within the observed range of $29 \pm 3\,{\rm km\:s^{-1}}$ ([[BN-Velocity]{}]{}, §\[sec:eratio\]). We then restrict our attention to only a subset of the [[BN-Velocity]{}]{} events where the final binary has properties similar to the observed [$\rm \theta^1C$]{} binary ([[BN-True]{}]{}, §\[sec:cross-sections\] and §\[sec:angle\]).
Velocity Distribution of the Ejected BN Star {#sec:vdist}
--------------------------------------------
Increasing the kinetic energy of BN by about two orders of magnitude compared to the value expected given the ONC’s velocity dispersion is at the heart of the problem. Hence, we focus on the velocity distribution of BN following ejection. In addition, we focus on energy considerations for the scattering problem, especially the ratio of the kinetic energy of BN’s ejection to the total energy of the binary left behind.
First we explore given the masses of the three stars in the interaction, and given that BN is ejected leaving the other stars in a binary, how likely it is for BN to acquire a velocity significantly higher than the velocity dispersion ($v_\sigma = 3\,{\rm km\:s^{-1}}$) in the ONC. We calculate the cross-section $\Sigma_{{{\tt BN-Ejection}}}$ for [[BN-Ejection]{}]{} events for a given $a_i$, for each Cases 1–3, and each $e_i$ distribution. The overall cross-section for [[BN-Ejection]{}]{} events for [*any*]{} $a_i$ is calculated by multiplying $\Sigma_{{{\tt BN-Ejection}}}$ with the probability of finding an initial binary with that $a_i$ assuming the semimajor axis distribution for binaries is flat in logarithmic intervals within the physical limits discussed in §\[sec:method\]. Thus, the normalized $\Sigma_{{{\tt BN-Ejection}}}$ is $\int P(a_i) \Sigma_{{{\tt BN-Ejection}}} (a_i, v_{\rm{BN}}, e) d a_i$, where, $P(a_i)da_i = d {\rm log} (a_i) / {\rm log} (6310/0.1)$.
Figure \[fig:vdist\] shows the cumulative distribution of $\Sigma_{{{\tt BN-Ejection}}} (v_{\rm{BN}})$ as a function of BN’s velocity ($v_{\rm{BN}}$) calculated using $\int^{v}_{v_{\rm{BN}} = 2v_\sigma} dv_{\rm{BN}} \times d\Sigma_{{{\tt BN-Ejection}}}/dv_{\rm{BN}}$. We find that binary-single interactions involving the three stars in question can eject the BN star with velocities that can exceed $v_\sigma$ by more than two orders of magnitude. However, the cross-sections for such events reduce as $v_{\rm{BN}}$ increases. For both $e_i$ distributions Case 3 shows a higher fraction of high-velocity ejection events. This is because in Case 3 the perturber is the most massive star in the triplet, the one that finally becomes the [$\rm \theta^1C_1$]{} star. Hence, the total available energy is higher in Case 3 events.
The problem of binary-single scattering can be understood by comparing the kinetic energy and the potential energy of the systems since the outcomes differ qualitatively depending on the relative values of these quantities [e.g., @1983ApJ...268..319H]. We explore if BN is ejected with $v_{\rm{BN}} \geq 2v_\sigma$, then what is the distribution of cross-section for the various Cases and $e_i$-distributions as a function of the $E_{\rm{ratio}}$ and the velocity of ejection, $v_{\rm{BN}}$. Here, $E_{\rm{ratio}}\equiv T_{\rm ejection}/|E_{\rm binary}|$ is the ratio of the final kinetic energy ($T_{\rm ejection}$) of [*both*]{} the single star and the binary star system (based on the motion of its center of mass) to the total energy (gravitational energy plus kinetic energy of orbital motion) of the binary ($E_{\rm{binary}}$). Figure \[fig:eratio\_v\] shows a $2D$ distribution of the overall $\Sigma_{{{\tt BN-Ejection}}}$ for any $a_i$ as a function of $v_{\rm{BN}}$ and $E_{\rm{ratio}}$ for all Cases 1–3, and all $e_i$ distributions. The highest ejection velocities for the BN star happens for $E_{\rm{ratio}}$ between 0.1 and 1 for all Cases. Note that the distributions for Cases 1 and 2 are very similar. This is due to the similarity in masses of BN and [$\rm \theta^1C_2$]{}. However, in Case 3 the most massive star is the perturber. In addition, the most massive star exchanges into the final binary increasing the binding energy of the final binary star system. Hence, there is a tail for high $E_{\rm{ratio}}$ values where BN can still be ejected with a high velocity. Even for the more energetic Case 3 events, the maximum velocity for ejections are achieved within a narrow range of $0.1 < E_{\rm{ratio}} < 1$.
A high value ($\sim 10$) of the ratio between the velocities of the incoming single star (expected to be near $v_\sigma$) and the ejected single star is needed to create runaway stars by definition. This is possible for strong encounters involving a binary and a single star where a fraction of binding energy of the binary is converted into the kinetic energy ($T$) of the ejected star. If the final outcome is again a binary and the runaway ejected star (the binary members are not required to remain the initial ones, e.g., for Cases 2 and 3) then the kinetic energy of the stars undergoing dynamical ejection, $T_{\rm ejection}= (1/2) m_{\rm BN} v_{\rm BN}^2 + (1/2) (m_{\rm
\theta^1C_1}+ m_{\rm \theta^1C_2}) v_{\rm \theta^1C}^2\rightarrow
(6.9\pm2.7)\times 10^{46}\:{\rm erg} + (1.2\pm0.5)\times 10^{46}\:{\rm
erg} = (8.1\pm2.8)\times 10^{46}\:{\rm erg}$ (here $\rightarrow$ indicates the observed values) is expected to be less than the magnitude of the total energy of the resulting binary, $|E_{\rm binary}|= G m_{\rm
\theta^1C_1} m_{\rm \theta^1C_2}/(2a)\rightarrow (16.4\pm4.9)\times
10^{46}\:{\rm erg}$. In addition to requiring $E_{\rm ratio}<1$, one also expects it to achieve a value of order unity, i.e. not too much less than one. For $E_{\rm {ratio}}\ll 1$, collisional outcomes dominate [Figure \[fig:branching\]; also see e.g., @1983ApJ...268..319H; @2004MNRAS.352....1F]. On the other hand, for $E_{\rm {ratio}}\gg 1$ outcomes with a disruption of the binary dominates creating 3 single stars (Figure \[fig:branching\]). The observed value for the BN-[$\rm \theta^1C$]{} system is $E_{\rm
ratio} \rightarrow 0.49\pm 0.22$, consistent with it being a result of a binary-single interaction.
Kinetic Energy of Ejection and Eccentricity of Binary {#sec:eratio}
-----------------------------------------------------
We now focus on the subset of [[BN-Ejection]{}]{} events ([[BN-Velocity]{}]{}) where BN is ejected with the observed velocity of $29\pm3$ (§\[sec:method\]) via any one of the Cases 1–3 (a small strip in the vertical axis in Figure \[fig:eratio\_v\]). The [$\rm \theta^1C$]{} orbit is eccentric ($e \approx 0.6$). Indeed, strong encounters are expected to leave behind binaries with generally high eccentricities. We now want to see the $2D$ distribution of overall cross-section for all [[BN-Velocity]{}]{} events for any $a_i$ as a function of $E_{\rm{ratio}}$ and the final eccentricity.
In Figure \[fig:eratio\_e\] we plot the 2D distributions of $\Sigma$ for the [[BN-Velocity]{}]{} events in the $E_{\rm ratio}$ versus final $e$ plane for Cases 1, 2, & 3 for both the Circular and Thermal $e_i$ distributions. The observed values of the BN-[$\rm \theta^1C$]{} system are also shown. These overlap within the $1\sigma$ contours for all cases. The energy of the [$\rm \theta^1C$]{} binary is just what we would expect if it had ejected BN at the observed velocity. Its high eccentricity, $\approx 0.6$, is also naturally explained by the recent ejection of BN since during ejection the potential of the system is changing rapidly.
Note that if the [$\rm \theta^1C$]{} binary was unrelated to BN, then $E_{\rm{ratio}}$ could have values in a large range spanning many orders of magnitude. For example, the range in $a$ for the binary is determined by contact ($\sim$ a few stellar radii) to the hard-soft boundary in ONC ($\sim
6000\,{\rm{AU}}$ for the velocity dispersion in ONC). Thus $E_{\rm{binary}}$ for [$\rm \theta^1C$]{} if unrelated to BN’s ejection, could be expected to have values anywhere between $\sim 10^{44}$ and $10^{49}\,\rm{ergs}$. It is thus very interesting to find the observed BN-[$\rm \theta^1C$]{} system with energies so close to the expected energies if they have had a binary-single scattering encounter in the past. The likelihood of this occuring simply by chance rather than by being caused by interaction with BN is explored in §\[sec:probability\].
Cross-Sections of [[BN-Velocity]{}]{} and [[BN-True]{}]{} Events: {#sec:cross-sections}
-----------------------------------------------------------------
We evaluate the cross-sections for outcomes where BN is ejected with the observed velocity of $29\pm3\,{\rm km\:s^{-1}}$ (“[[BN-Velocity]{}]{}” events). We further calculate the cross-section of a subset of [[BN-Velocity]{}]{} events where the final binary is left with orbital properties similar to those observed of [$\rm \theta^1C$]{}, namely, $a = 18.13\pm1.28\,\rm{AU}$, and $e
= 0.592\pm0.07$ (; “[[BN-True]{}]{}” events).
For Cases 1, 2, 3 with Circular initial $e_i$ distribution, $\Sigma_{\rm BN-Velocity} = (2.7, 0.94, 2.9)\times 10^3\:{\rm AU}^2$, while for Thermal initial $e_i$ distribution, $\Sigma_{\rm BN-Velocity} = (2.9, 1.4, 2.5)\times 10^3\:{\rm AU}^2$. For Cases 1, 2, 3 with Circular $e_i$-distribution, $\Sigma_{\rm BN-True}
= 56, 24, 105\:{\rm AU}^2$, while for thermal $e_i$ distribution, $\Sigma_{\rm BN-True} = 58, 34, 84\:{\rm AU}^2$. The three cases have similar cross-sections, with Case 3, i.e. [$\rm [\theta^1C_2 , BN]\:\theta^1C_1 \longrightarrow [\theta^1C_1,\theta^1C_2]\: BN$]{}, somewhat more preferred.
The cross-sections for [[BN-Velocity]{}]{} and [[BN-True]{}]{} events as a function of $b_i$, are shown in Figure \[fig:b\_sigma\]. The cross-sections for a given $b_i$ and for all explored $a_i$ are normalized using the probability, $P(a_i)$, of finding a binary with semimajor axis $a_i$, assuming a semimajor axis distribution flat in log intervals. For small values of $b_i\lesssim300\,{\rm{AU}}$, the cross-sections grow geometrically as $b_i^2$ and then decline. This decline, seen in all cases, is simply due to the fact that the interactions are happening at larger and larger impact parameters and at some point no interactions are expected to be strong enough to increase the energy of the ejected star to the observed value of BN. The areas under the histograms are the total cross-sections for the [[BN-Velocity]{}]{} and [[BN-True]{}]{} events. Figure \[fig:b\_sigma\] also includes a table summarizing the total cross-sections.
The cross-sections for [[BN-Velocity]{}]{} and [[BN-True]{}]{} events as a function of $a_i$ are shown in Figure \[fig:a\_sigma\]. We find that [[BN-Velocity]{}]{} events happen via binary-single scattering encounters for a large range of $a_i$, but [[BN-True]{}]{} events place much tighter constraints on $a_i$. For example, Case 3, which is the most favored, requires the initial binary \[$\theta^1{\rm C}_2$, BN\], i.e. two approximately equal-mass stars of $\sim 9\,{\rm{M_\odot}}$, to have originally had $a_i\simeq 8\pm 2$ AU. Note that depending on the case, different ranges of $a_i$ contribute towards creating systems similar to the observed BN-[$\rm \theta^1C$]{} system. Moreover, note that the $a_i$ ranges where the [[BN-True]{}]{} events occur via the three Cases 1 – 3, ejection of BN is the most likely outcome among all other possible outcomes (except of course weak fly-bys; Figure \[fig:branching\]). This gives us further confidence in the scenario that the BN-[$\rm \theta^1C$]{} system has been created via a strong binary-single interaction involving these three stars.
Throughout this study we have used the central values of the estimated masses for the three stars. Dynamically there should be no qualitative difference in the outcomes if the masses are changed within the mass errors. However, note that the masses of $\theta^1{\rm C}_2$ and BN are comparable. In fact the error ranges actually overlap. Due to the comparable masses, dynamically there is only a small difference between ejection of BN and ejection of $\theta^1{\rm C}_2$, as named here. This is reflected in our results to some degree. Cases 1 and 2 contribute towards creating the BN-[$\rm \theta^1C$]{} system over very similar $a_i$ ranges. Their contributions are also comparable. The small differences in the $a_i$ range and $\Sigma_{{\tt BN-True}}$ come from the small difference between $\theta^1{\rm C}_2$ and BN’s assumed masses and also to some extent the scenario, a fly-by being more likely compared to an exchange if all else is kept unchanged. In fact, similar results will be recovered if BN and [$\rm \theta^1C_2$]{} are interchanged among themselves. However, in that case, the definitions of Cases 1 and 2 will also be interchanged.
Orientation of the Orbital Plane of [$\rm \theta^1C$]{} Binary Relative to the Direction of BN’s Motion {#sec:angle}
-------------------------------------------------------------------------------------------------------
One additional variable in the dynamical ejection problem is the angle $\alpha$ between the angular momentum vector, $\vec{L}_{\theta^1{\rm C}}$, of the [$\rm \theta^1C$]{} binary and the $3D$ velocity vector of BN, $\vec{v}_{\rm{BN}}$. Figure \[fig:alpha\_sigma\] shows the distribution of $\Sigma_{{{\tt BN-True}}}$ for all [[BN-True]{}]{} events for [*any*]{} $a_i$ as a function of $\alpha$ for all Cases 1–3 and all $e_i$ distributions. The distributions of $\alpha$ are quite broad for Cases 1 and 2. In comparison, for Case 3 there is a strong peak near $\alpha=90^\circ$.
To calculate the observed value of $\alpha$ for comparison with the predictions of our numerical results we adopt different measured values in existing literature. Orientation of $\vec{L}_{\theta^1{\rm C}}$ is calculated using data given in . The LSR velocity of BN is obtained from @1983ApJ...275..201S. There are two independent proper-motion measurements for BN. Adopting the values given in @2008ApJ...685..333G we find $\alpha = 95^\circ$. Adopting the values given in @2011ApJ...728...15G we find a slightly different value of $\alpha = 87^\circ$. If the direction of BN’s motion in the sky-plane is obtained by simply joining the expected position of the binary-single encounter and BN’s current position, then $\alpha = 99^\circ$. (this may be a more accurate value, since we expect BN to have suffered a recent change in its proper motion vector via interaction with source [*I*]{}, see below) Note all of the above values of $\alpha$ for the observed BN-[$\rm \theta^1C$]{} system are consistent with the predicted distribution of $\alpha$ from our numerical experiments, and give some support for the ejection having resulted via Case 3.
Tests for the Ejection Scenario of BN from [$\rm \theta^1C$]{} and Probability of Chance Agreement {#sec:probability}
==================================================================================================
The system which ejected BN must be located along BN’s past trajectory and have a total mass $\gtrsim 2 m_{\rm BN}$. These conditions are potentially satisfied for [$\rm \theta^1C$]{}, the 3 other Trapezium stars $\rm
\theta^1A$, $\rm \theta^1B$, $\rm \theta^1D$, another ONC member $\theta^2A$, and probably for source [*I*]{} (assuming it is the main source of luminosity in the KL nebula). Indeed, a number of authors have argued BN was launched from source [ *I*]{} [@2005AJ....129.2281B; @2008ApJ...685..333G]. However, as we now discuss, there are 6 independent properties of [$\rm \theta^1C$]{} that have the values expected if it were the binary left behind after ejecting BN (7 if we assume ejection via Case 3 and include the angle $\alpha$ between the angular momentum vector, $\vec{L}_{\theta^1{\rm C}}$, of the [$\rm \theta^1C$]{} binary and the $3D$ velocity vector of BN, $\vec{v}_{\rm{BN}}$). To consider the likelihood that all of these properties of the BN-[$\rm \theta^1C$]{} system are as observed [*by chance*]{}, we take that to be our null hypothesis. Given that BN has the runaway velocity, for each of these properties we assign a probability that [$\rm \theta^1C$]{} has its values by chance to finally calculate the composite probability of chance agreement of [$\rm \theta^1C$]{}’s properties with those expected from a binary-single scattering scenario. We discuss these properties and our estimates of the chance-agreement probabilities below. These probabilities are summarized in Table \[tab:properties\] along with the values predicted by the binary-single ejection scenario, and the observed values of the BN-[$\rm \theta^1C$]{} properties.
\(1) [*ONC-Frame Proper Motion in Declination ($\mu_{\rm
\delta,ONC}({\rm \theta^1C})$):*]{} If [$\rm \theta^1C$]{} ejected BN, then, in the frame of the center of mass of the pre-ejection triple, the predicted value of $\mu_{\rm
\delta,T}({\rm \theta^1C}) = -(m_{\rm BN}/m_{\rm \theta^1C})
\mu_{\rm \delta,T}({\rm BN}) \rightarrow
-([9.3\pm2.0\,{\rm{M_\odot}}]/[47\pm4\,{\rm{M_\odot}}]) 11.7\pm1.3\:{\rm mas\:yr^{-1}}
\rightarrow -2.3\pm0.6\:{\rm mas\:yr^{-1}}$. Here we have used the luminosity-based mass estimate for BN [@2004ApJ...607L..47T] and the proper motion measurements of @2008ApJ...685..333G for BN, including a 0.70 mas/yr uncertainty of the motion in declination of the pre-ejection triple with respect to the ONC frame. The predicted value of the ONC-frame motion of [$\rm \theta^1C$]{} is then $\mu_{\rm \delta,ONC}({\rm \theta^1C})[{\rm predicted}] = -2.3 \pm 0.9\:{\rm mas\:yr^{-1}}$, with the error increasing again because of the uncertain motion of the pre-ejection triple[^1]. The observed value [@1988AJ.....95.1744V] is $\mu_{\rm
\delta,ONC}(\theta^1C)[{\rm observed}] = -1.8\pm 0.2 \:{\rm mas\:yr^{-1}}$. Given the observed [@1988AJ.....95.1744V] 1D proper motion dispersion of the bright ONC stars of 0.7 mas/yr, the probability for [$\rm \theta^1C$]{} to be in the predicted range is 0.023. Indeed, @1988AJ.....95.1744V already noted that [$\rm \theta^1C$]{} has an abnormally large proper motion.
\(2) [*ONC-Frame Proper Motion in Right Ascension ($\mu_{\rm \alpha,ONC}({\rm \theta^1C}) {\rm cos}\delta$):*]{} Similarly, in the frame of the center of mass of the pre-ejection triple: $\mu_{\rm \alpha,T}({\rm \theta^1C}) {\rm cos}\delta =
-(m_{\rm BN}/m_{\rm \theta^1C}) \mu_{\rm \alpha,T}({\rm BN}){\rm
cos}\delta \rightarrow -([9.3\pm2.0\,{\rm{M_\odot}}]/[47\pm4\,{\rm{M_\odot}}])
(-6.1\pm1.2)\:{\rm mas\:yr^{-1}} \rightarrow +1.21 \pm 0.36\:{\rm
mas\:yr^{-1}}$. The predicted value of the ONC-frame motion of [$\rm \theta^1C$]{} is then $\mu_{\rm \alpha,ONC}({\rm \theta^1C}) {\rm cos}\delta [{\rm predicted}]
=+1.2\pm 0.8 \:{\rm mas\:yr^{-1}}$. The observed value [@1988AJ.....95.1744V] is $\mu_{\rm \alpha,\theta^1C} {\rm
cos} \delta [{\rm observed}] = +1.4\pm 0.2\:{\rm mas\:yr^{-1}}$ and the probability that [$\rm \theta^1C$]{} is in the predicted range by chance is 0.27.
\(3) [*ONC-Frame Radial Velocity ($v_{\rm r,ONC}({\rm \theta^1C})$):*]{} Similarly, the radial recoil in the frame of the pre-ejection triple should satisfy: $v_{\rm r,T}({\rm \theta^1C})= -(m_{\rm
BN}/m_{\rm \theta^1C}) v_{\rm r,T}({\rm BN})\rightarrow
-([9.3\pm2.0\,{\rm{M_\odot}}]/[47\pm4\,{\rm{M_\odot}}])(+13\pm 1.8\:{\rm km\:s^{-1}})
\rightarrow -2.57\pm0.69 \:{\rm km\:s^{-1}}$. The predicted value of the ONC-frame motion of [$\rm \theta^1C$]{} is then $v_{\rm r,ONC}({\rm \theta^1C})[{\rm predicted}]
= -2.6\pm 1.6\:{\rm km\:s^{-1}}$, with the error range dominated by the assumption that the pre-ejection triple had a motion similar to the other bright ONC stars [@1988AJ.....95.1744V] with $\sigma_{\rm 1D}=1.4\:{\rm km\:s^{-1}}$. [$\rm \theta^1C$]{} has an observed heliocentric velocity of $+23.6\:{\rm km\:s^{-1}}$ i.e. an LSR velocity of $5.5\:{\rm km\:s^{-1}}$, i.e. an ONC frame velocity of $v_{\rm r,ONC}({\rm \theta^1C})[{\rm observed}] = -2.5\:{\rm km\:s^{-1}}$. For a Gaussian distribution with $\sigma_{\rm 1D}=1.4\:{\rm km\:s^{-1}}$, i.e. based on the proper motion dispersion of bright stars [@1988AJ.....95.1744V] the probability of being in the predicted velocity range by chance is 0.24.
\(4) [*Mass of Secondary ($m_{\theta^1{\rm C}_2}$):*]{} Given a [$\rm \theta^1C$]{} primary mass of $38.2\,{\rm{M_\odot}}$, what is the probability of having a secondary star with mass $\gtrsim m_{\rm BN}$? We estimate this probability using the low value ($m_{\rm{BN}} = 7.3\,{\rm{M_\odot}}$) of the luminosity-based mass estimate for BN of $9.3\pm2.0\,{\rm{M_\odot}}$. If the secondary star is drawn from a Salpeter power-law mass function, $dF/dm_*\propto m_*^{-2.35}$, where $F$ is the fraction of the stellar population, with maximum mass equal to the primary mass and lower mass limit equal to $1.0\,{\rm{M_\odot}}$ (a relatively top-heavy IMF, with average mass of $2.8\:M_\odot$, compared to the global ONC IMF, which has a broad peak around $0.5\:M_\odot$, e.g., @2002ApJ...573..366M), then the probability of having a secondary with mass $>7.3\,{\rm{M_\odot}}$ is 0.061. @1998ApJ...492..540H find evidence for mass segregation in the IMF ([*of primary stars*]{}) in the center of the ONC, with average stellar mass reaching peak values of $\sim
1.3-2\:M_\odot$ in the vicinity of [$\rm \theta^1C$]{} (excluding [$\rm \theta^1C$]{} from the average). If we raise the lower limit of the above Salpeter IMF to $2\,{\rm{M_\odot}}$ (i.e. an average mass of $5.1\:M_\odot$), this raises the probability of obtaining a sufficiently massive secondary to 0.16 and we adopt this number as a conservative estimate. We note that none of the other Trapezium stars has a secondary mass that satisfies this condition [@2004ApJ...607L..47T].
\(5) [*Ratio of Ejection Kinetic Energy to Binary Total Energy ($E_{\rm ratio}$(BN-[$\rm \theta^1C$]{})):*]{} From our numerical experiments we find that in order for [$\rm \theta^1C$]{} to have ejected BN at its observed velocity, $0.23<E_{\rm ratio}[{\rm predicted}]<0.72$. This range contains about 70% of the [[BN-Velocity]{}]{} events. Recall, $E_{\rm
ratio}[{\rm observed}]=0.49\pm 0.22$. Given the primary and secondary masses of [$\rm \theta^1C$]{}, what is the probability the total energy of the binary, $E_{\rm binary}$, falls in the predicted range of $1.4$ to $4.3\times T_{\rm ejection}$, i.e. $(1.1-3.5)\times 10^{47}\:{\rm
erg}$, simply by chance? This corresponds to a range of semi-major axes of 8.5 to 27 AU. If the distribution of $a$ follows $d F_b / d
{\rm log} a = $ constant [@2003gmbp.book.....H] from 0.1 to 6300 AU (see §\[sec:method\]), then the probability that the [$\rm \theta^1C$]{} binary falls in this range is 0.10. If we assume ejection occurred via Case 3, then $0.44<E_{\rm
ratio}[{\rm predicted}]<0.83$, corresponding to a range of semi-major axes of 16 to 30 AU and a probability of chance agreement of 0.057. The upper limit of allowed $a$ may be smaller than 6300 AU for conditions in the central regions of the ONC. For a stellar density of $\sim 10^4\:{\rm pc}^{-1}$, the average separation is 6000 AU. Most of these stars will have masses lower than [$\rm \theta^1C_2$]{}or BN. Hence it is not likely that these lower mass stars will disrupt the relatively more massive BN or [$\rm \theta^1C_2$]{} stars from being in a binary with [$\rm \theta^1C$]{}$_1$. If we adopt an upper limit smaller by a factor of 2, i.e. $3000$ AU, then the probabilities of the [$\rm \theta^1C$]{} binary falling in the above expected ranges by chance rises by just 7%.
\(6) [*Eccentricity ($e$([$\rm \theta^1C$]{})):*]{} Our numerical experiments (see Figure 2) show that a very broad range of eccentricities is expected for the [$\rm \theta^1C$]{} binary if it has ejected BN at the observed velocity (the average value for all the [[BN-Velocity]{}]{} outcomes is $e=0.60$ with a $1\sigma$ range from 0.34 to 0.86; for Case 3 this range is from 0.34 to 0.82). The observed value of $e=0.592\pm0.07$ is consistent with these expectations, especially being close to the peak of the distribution resulting from Case 3 (Figure \[fig:eratio\_e\]). However, to assess the probability of this agreement by chance we need to know the eccentricity distribution of ONC binaries, especially for massive stars. Unfortunately there are few observational constraints on this eccentricity distribution. If the binaries have existed long enough to suffer many interactions, then a thermal distribution, $dF_b/de = 2 e$, is expected, which is weighted towards high eccentricities. A binary drawn from a thermal distribution has a 0.62 chance to be in the range $0.34<e<0.86$ (0.56 to be in the range $0.34<e<0.82$ for Case 3). Of course, if the actual distribution of $e$ of ONC binaries is close to circular ($e\simeq
0$), then the probability of chance agreement for the eccentricity of [$\rm \theta^1C$]{} with the value expected from BN ejection would be very small. To be conservative, we adopt the probability of 0.62 implied by a thermal distribution of eccentricities.
\(7) [*Angle between the direction of BN’s motion and the angular momentum of [$\rm \theta^1C$]{} binary ($\alpha$):*]{} From our numerical experiments we find that for [[BN-Velocity]{}]{} events the angle $\alpha$ between $\hat{L}_{\rm{\theta^1C}}$ and $\hat{v}_{\rm{BN}}$ should satisfy $54^\circ < \alpha < 126^\circ$. This range contains about 70% of all [[BN-True]{}]{} events. For Case 3, the range is $58^\circ < \alpha < 122^\circ$. The observed value of $\alpha$ for the BN-[$\rm \theta^1C$]{} system is $\sim
90^\circ$ and thus contained within this range. If [$\rm \theta^1C$]{} and BN were unrelated, then the distribution of $\alpha$ should be $0.5\:{\rm
sin}\:\alpha$ over the range $0^\circ$ to $180^\circ$. Hence the probability for chance occurrence for $\alpha$ to be within the above range is 0.59 (0.53 for Case 3).
Combining the above individual probabilities and assuming, reasonably, that these properties are mutually independent, we find that the total probability of chance agreement of all 7 properties is small, $\epsilon = 0.023\times 0.27 \times 0.24 \times 0.16 \times 0.10
\times 0.62 \times 0.59 = 8.7\times 10^{-6}$. If we assume ejection happened via Case 3, which affects the last three probabilities, we obtain a total probability of chance agreement of $\epsilon =
4.0\times 10^{-6}$. The excellent agreement between the observed BN-[$\rm \theta^1C$]{}-system properties and the predicted final properties from our numerical simulations (§3) together with the low chance agreement probability ($\epsilon$) of all independent observed properties of the system strongly supports our proposed scenario that the observed system resulted from a strong binary-single ejection event involving [$\rm \theta^1C_1$]{}, [$\rm \theta^1C_2$]{}, and BN with a high probability of about $1-\epsilon
\approx 0.99999$.
From the present day velocities and projected distance between the [$\rm \theta^1C$]{} and BN this binary-single encounter must have happened about $4500$ years ago at the location shown in Fig. \[fig:onc\]. This scenario also explains the anomalously large proper motion of [$\rm \theta^1C$]{} [@1988AJ.....95.1744V], which will cause it to leave the central region of the cluster within $\sim 10^5$ years.
Based on a multi-frequency radial velocity analysis, have suggested [$\rm \theta^1C$]{} may actually harbor an additional star of $1.0\pm0.16\,{\rm{M_\odot}}$ in a close ($a=0.98\,{\rm{AU}}$; $P=61.5\,\rm{day}$), eccentric ($e=0.49$) orbit around [$\rm \theta^1C_1$]{}. If during the proposed scattering event that ejected BN, [$\rm \theta^1C_2$]{} or BN came close to this inner region then one would expect likely ejection of the solar mass star. To examine the likelihood of such close interactions, in Figure \[fig:closeapproach\] we show the cumulative histogram (weighted by cross-section) of closest approaches of either [$\rm \theta^1C_2$]{} or BN to [$\rm \theta^1C_1$]{} for all [[BN-True]{}]{} events. For example for Case 3 with Circular $e_i$, $\sim 60\%$ of the events happen with a closest approach that is $>5\,{\rm{AU}}$. At these distances we would expect the solar mass star to remain relatively undisturbed in its orbit. Case 1 and especially Case 2 involve somewhat closer approaches, although both of them still have at least a $20\%$ contribution to events with closest approach $>5\,{\rm{AU}}$. Future confirmation of the reality of the third star in the [$\rm \theta^1C$]{} system may help us place further constraints on how BN was ejected from [$\rm \theta^1C$]{}.
[cccc]{} Proper Motion in Dec. ($\mu_{\rm \delta,ONC}$) & $-2.3 \pm 0.9\:{\rm mas\:yr^{-1}}$ & $-1.8\pm 0.2 \:{\rm mas\:yr^{-1}}$ & 0.023\
Proper Motion in R.A. ($\mu_{\rm \alpha,ONC}{\rm cos}\delta$) & $+1.2\pm 0.8 \:{\rm mas\:yr^{-1}}$ & $+1.4\pm 0.2\:{\rm mas\:yr^{-1}}$ & 0.27\
Radial Velocity ($v_{\rm r,ONC}$) & $-2.6\pm 1.6\:{\rm km\:s^{-1}}$ & $-2.5\:{\rm km\:s^{-1}}$ & 0.24\
Mass of Secondary ($m_{\theta^1{\rm C}_2}$) & $7.3 - 38.2\,{\rm{M_\odot}}$ & $8.8\pm1.7\,{\rm{M_\odot}}$ & 0.16\
Eject. KE to Binary Total E ($E_{\rm ratio}$) & $0.23 - 0.72$ & $0.49\pm 0.22$ & 0.10 \[0.057\]\
Eccentricity ($e$) & $0.34 - 0.86$ & $0.592\pm0.07$ & 0.62\[0.56\]\
Angle between $\hat{L}_{\rm{\theta^1C}}$ & $\hat{v}_{\rm{BN}}$ ($\alpha$) & $54^\circ - 126^\circ$ & $\sim 90^\circ$ & 0.59\[0.53\]\
Combination of 7 Independent Properties & & & $8.7\times 10^{-6}$\[$4.0\times10^{-6}$\]\
\[tab:properties\]
Discussion and Summary {#sec:conclude}
======================
We present the following argument in favor of our proposed scenario that BN was ejected by a binary-single interaction involving [$\rm \theta^1C_1$]{}, [$\rm \theta^1C_2$]{}, and BN in the past. BN is a runaway star [@1995ApJ...455L.189P; @2004ApJ...607L..47T]. It had to be launched by an interaction with a multiple system that has a primary mass greater than BN’s mass along its past trajectory. The most massive binary in the ONC, [$\rm \theta^1C$]{}, is a system satisfying this condition, but there are a few other candidates including, potentially, the massive protostar source I. To test whether [$\rm \theta^1C$]{} binary ejected BN we consider 7 additional properties, namely recoil in 3 directions, sufficiently massive secondary, orbital binding energy, orbital eccentricity and angle between the orbital angular momentum vector and the direction of BN’s velocity. Aided in part by a large and well-sampled suite of numerical simulations we show that all of these 7 observed properties of the BN-[$\rm \theta^1C$]{} system agree well with the properties predicted if BN was ejected by [$\rm \theta^1C$]{}. There are two and only two possibilities: 1) [$\rm \theta^1C$]{} has all these properties [*by chance*]{}; 2) [$\rm \theta^1C$]{} has acquired these properties naturally as a result of BN’s ejection. We estimate the probability of chance agreement for all of the above properties to be low $\epsilon \lesssim 10^{-5}$. Hence, we conclude with about $(1-\epsilon)
\approx (1-10^{-5})$ probability that today’s BN-[$\rm \theta^1C$]{} system was created via a binary-single interaction involving [$\rm \theta^1C_1$]{}, [$\rm \theta^1C_2$]{}, and BN.
A summary of our numerical calculations is as follows. We performed $\sim 10^7$ numerical simulations that allowed us to properly sample the multidimensional parameter space effectively (§\[sec:method\]). We used two different $e_i$ distributions as limiting cases, and $200$ different $a_i$ values from the full range of possible $a_i$ in ONC from physical considerations for all three cases that can produce the observed BN-[$\rm \theta^1C$]{} system (§\[sec:method\]). We found that the predicted energies of the BN-[$\rm \theta^1C$]{} system agree well with the predictions from the scattering scenario (Figure \[fig:eratio\_e\]) for both assumed $e_i$ distributions and all three cases. Furthermore, the predicted distribution of the angle $\alpha$ between BN’s velocity vector $\vec{v}_{\rm{BN}}$ and [$\rm \theta^1C$]{}’s angular momentum vector $\vec{L}_{\theta^1\rm{C}}$ was consistent with the observed value of $\alpha$ (Figure \[fig:alpha\_sigma\]). Further calculations of cross-sections for the [[BN-Velocity]{}]{} and [[BN-True]{}]{} events constrained a range of initial binary semimajor axes for each Case (1, 2, 3) that would produce the observed BN-[$\rm \theta^1C$]{} system followed by a strong scattering (Figure \[fig:a\_sigma\]). For these ranges of $a_i$ ejection of BN in general is the dominant outcome (apart from weak fly-bys which has a formally infinite cross-section; Figure \[fig:branching\]). Our results indicate that all Cases can contribute to the production of the observed BN-[$\rm \theta^1C$]{} system. However, an interaction between an initial binary with members [$\rm \theta^1C_2$]{} and BN, and a single star [$\rm \theta^1C_1$]{}, our Case 3 (denoted by [$\rm [\theta^1C_2 , BN]\:\theta^1C_1 \longrightarrow [\theta^1C_1,\theta^1C_2]\: BN$]{}), is favored by about a factor of 2 (Figure \[fig:b\_sigma\]).
Ejection of BN from [$\rm \theta^1C$]{} has several important implications for our understanding of massive star formation in the KL nebula, where a core of gas appears to be collapsing to form at least one massive star, thought to be detected in the radio as source [*I*]{} (Figure 1).
If BN was ejected from [$\rm \theta^1C$]{}, then its passage near source [*I*]{} within $\sim0.5''$, i.e. a projected separation of $\sim 200$ AU, i.e. an expected physical separation [@2008ApJ...685..333G; @2011ApJ...728...15G] of $\sim 300$ AU in the KL nebula is coincidental. Our estimated ejection point (Figure 1) is $50.1''$ from source [*I*]{}’s present location, corresponding to 20,800 AU. The ONC-frame radial velocity is about half of the plane of sky velocity, implying BN also travelled 10,400 AU in the radial direction to reach source [*I*]{}, for a total distance of 23,200 AU. The probability to approach within 300 AU of source [*I*]{}, ignoring gravitational focussing, is thus $\pi
(300/23,200)^2 /4\pi = 4\times 10^{-5}$. Gravitational focussing by $\sim 20\,{\rm{M_\odot}}$ of total mass in and around source [*I*]{} [@2003ApJ...585..850M; @1992ApJ...393..225W] boosts the cross-section by $\sim$14%, so the probability of approach is $\sim
5\times 10^{-5}$. Thus the interaction of BN with KL is an improbable event. Of course, the chance of interaction of BN with [*any*]{} existing protostar, not necessarily source [*I*]{}, in the ONC is larger simply by a factor equal to the number of protostars in this volume around [$\rm \theta^1C$]{}. From X-ray observations [@2005ApJS..160..530G] there appear to be at least $\sim$10 such objects even in just the local vicinity of KL, so the total probability of an interaction between BN and a protostar can be at least an order of magnitude higher, but still leaving it as being highly unlikely. We note that the probability that [$\rm \theta^1C$]{} is masquerading as the system that ejected BN is even smaller than the probability of BN interacting with source [*I*]{}. Furthermore, the evidence for [$\rm \theta^1C$]{} to have ejected BN is based on 7 independent lines of evidence and so survived several tests for falsification (§\[sec:probability\]).
A close passage of BN with source [*I*]{} will have deflected BN’s motion by an angle\
$7.5^\circ (m_I/20\,{\rm{M_\odot}})(b/300{\rm
AU})^{-1}(v_{\rm BN}/30{\rm km\:s^{-1}})^{-2}$ towards source [ *I*]{}, where $b$ is the initial impact parameter and $v_{\rm BN}$ is the velocity of BN relative to source [*I*]{}. We expect that a reasonably accurate estimate of the original ONC-frame position angle (P.A.) of BN’s proper motion can be derived by considering the angle from the estimated position of the dynamical ejection from [$\rm \theta^1C$]{} (the cross in Figure 1) and BN’s current position, which is $-29^\circ.8$. The current observed P.A. of BN’s ONC-frame proper motion is variously estimated to be $-27^\circ.5\pm4^\circ$[@2008ApJ...685..333G] and $-18^\circ.8\pm4.6^\circ$[@2011ApJ...728...15G]. The average of these is $-23^\circ.1\pm3^\circ$, suggesting a projected deflection of $6^\circ.7\pm\sim 3^\circ$ towards source [*I*]{}. The total true deflection may be expected to be $\sim \sqrt{2}$ larger, i.e. $\sim
9^\circ.5\pm 3^\circ$, which is consistent with our previous estimate.
Close passage of BN near the accretion disk of source [*I*]{} about 500 years ago would induce tidal perturbations that enhance the effective viscosity of the disk and thus its accretion rate [@2004ApJ...607L..47T]. This can explain the enhanced, apparently explosive, outflow [@1993Natur.363...54A], which has observed timescales of $\sim 500-1000$ yr, if the inner region of the disk with an orbital time $\lesssim 500$ yr has been significantly perturbed. This corresponds to disk radii of $r\lesssim 180
(m_I/20\,{\rm{M_\odot}})^{1/3}$ AU, which is consistent with the estimate of closest approach based on deflection of BN’s trajectory.
The estimated current accretion rate to source [*I*]{} is $\sim
3\times 10^{-3}\,{\rm{M_\odot}}yr^{-1}$ to a $10\,{\rm{M_\odot}}$ protostar (and somewhat higher rates if the protostar is more massive) . This is about a factor of 10 higher than expected given the properties of the gas core [@2002Natur.416...59M; @2003ApJ...585..850M]. If constant over the last $1000$ yr, this would imply a total accreted mass of about $3\,{\rm{M_\odot}}$, which would be a significant fraction of the original accretion disk-mass around a $10-20\,{\rm{M_\odot}}$ protostar, since the disk mass is expected to be limited to $\sim 30\%-100\%$ of the stellar mass by gravitational torques [@2010ApJ...708.1585K]. The mass launched by a magneto-hydrodynamic outflow during this time is expected to be $\approx 30\%$ of this amount [@1994ApJ...429..808N], i.e. about $1\,{\rm{M_\odot}}$. This is consistent with the mass estimated [@1982ApJ...259L..97C] to be in the inner, “explosive” part of the outflow of about $3\,{\rm{M_\odot}}$.
Our results suggest that the formation of a massive star in the KL nebula, i.e. source [*I*]{}, has been affected by an external perturbation of a runaway B star, BN, ejected from a different region of the cluster, i.e. by [$\rm \theta^1C$]{}. For BN to be launched so close to a forming massive protostar does appear to be an intrinsically unlikely event, but multiple pieces of independent evidence strongly support this scenario. It also explains source [*I*]{}’s anomalously high accretion rate and the unusual, apparently “explosive” nature of the recent outflow from this source. It is possible that a significant fraction, up to $\sim 1/3$, of the accreted mass has been induced by this perturbation. In other respects, the Core Accretion model [@2002Natur.416...59M; @2003ApJ...585..850M] of massive star formation provides a reasonable description of the system, e.g., the presence of a massive core around the protostar and two wide-angle outflow cavities from which near-IR light emerges (Testi et al. 2010). The example of Orion BN-KL suggests that, occasionally, in crowded regions near the center of star clusters, the star formation model needs to be modified to account for tidal perturbations from external, passing stars.
Improved observational constraints on the properties of [$\rm \theta^1C$]{} especially its binary properties and ONC-frame proper motion, and the mass and current proper motion of BN will place even more stringent tests on the proposed ejection scenario and also help improve the dynamical mass constraints on source [*I*]{} from the deflection of BN.
We thank John Bally, Paola Caselli, Eric Ford, John Fregeau, Ciraco Goddi, Lincoln Greenhill, Stefan Kraus, Chris McKee, Francesco Palla, Hagai Perets, Dick Plambeck and Leonardo Testi for helpful discussions. We thank the anonymous referee for a detail review. JCT acknowledges support from NSF CAREER grant AST-0645412; NASA Astrophysics Theory and Fundamental Physics grant ATP09-0094; NASA Astrophysics Data Analysis Program ADAP10-0110 and a Faculty Enhancement Opportunity grant from the University of Florida. SC acknowledges support from the Theory Postdoctoral Fellowship from UF Department of Astronomy and College of Liberal Arts and Sciences.
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[^1]: Note that here and for the other [$\rm \theta^1C$]{} properties we have adopted $1\sigma$ errors when possible, but not all physical properties have well-defined uncertainties: e.g., the model dependent $m_{\rm BN}$ estimate given its observed luminosity.
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